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Proof step 1 from proof infappend-moves-gen (conclusion) (back to tree)

start(sf(0)), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ ))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∀ n. ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(⌜al⌝ ) = cext !(⌜caf(n) '⌝ ) ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

Additional information

Used formulas: ∀ n. ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(⌜al⌝ ) = cext !(⌜caf(n) '⌝ ) ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

Proof step 3 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(⌜al⌝ ) = cext !(⌜caf(n) '⌝ ) ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1) ⊦

User applied rule choice left

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 4 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(⌜alf(n0)⌝ ) = cext !(⌜caf(n0) '⌝ ) ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: infinite-append, name:infinite-append, seq:⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ.drop(ii(n)).take(# lf(n)) = lf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n)) ;∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ), subst:[lf] → [alf]; ['a] → ['a]

Additional information

Used terms: alf

Proof step 5 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) ∧ ¬ # σ < ⌜ii(n + 1)⌝ ∧ σ.drop(ii(n)).take(# lf(n)) = lf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → lf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

A spec-lemma

Proof step 6 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(⌜alf(n0)⌝ ) = cext !(⌜caf(n0) '⌝ ) ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∃ α, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(α) ⊃ ∃ n. # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 7 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: infinite-append, name:infinite-append-butlast, seq:∀ n. lf(n).last = lf(n + 1).head ∧ lf(n) ≠ [] ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# lf(n)) = lf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n) + 1⌝ ∧ (∀ m. n ≤ m → lf(m).tail = [] ∧ ii(m) = ii(n)) ;∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ), subst:[lf] → [slf]; ['a] → ['s]

Additional information

Used terms: slf

Proof step 8 from proof infappend-moves-gen (previous step) (back to tree)

∀ n. lf(n).last = lf(n + 1).head ∧ lf(n) ≠ [] ⊦ ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # lf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# lf(n)) = lf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n) + 1⌝ ∧ (∀ m. n ≤ m → lf(m).tail = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1))

A spec-lemma

Proof step 9 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n) ⊦ ∀ n. slf(n).last = slf(n + 1).head ∧ slf(n) ≠ []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 10 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ slf(n0).last = slf(n0 + 1).head ∧ slf(n0) ≠ []

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: n0 abort: F discard: F computed: T

Additional information

Used terms: n0

Proof step 11 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).last = sf(n0 + 1), slf(n0)[0] = sf(n0), ii(0) = 0, frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n) ⊦ slf(n0).last = slf(n0 + 1).head ∧ slf(n0) ≠ []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).last = sf(n0 + 1), slf(n0)[0] = sf(n0), ii(0) = 0, frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ sf(n0 + 1) = slf(n0 + 1).head ∧ slf(n0) ≠ []

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: get-zero-01, seq: ⊦ x ≠ [] → x[0] = x.head, subst:[x] → [slf(n0)]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); Used terms: slf(n0)

Proof step 13 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x ≠ [] → x[0] = x.head

A spec-lemma

Proof step 14 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).head = sf(n0), slf(n0).last = sf(n0 + 1), ii(0) = 0, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n) ⊦ sf(n0 + 1) = slf(n0 + 1).head ∧ slf(n0) ≠ []

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).head = sf(n0), slf(n0).last = sf(n0 + 1), ii(0) = 0, sf(n0 + 1) ≠ slf(n0 + 1).head, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [slf(n0), s, x0]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: slf(n0), s, x0

Proof step 16 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).last = sf(n0 + 1), ii(0) = 0, s = sf(n0), slf(n0) = s + x0, sf(n0 + 1) ≠ slf(n0 + 1).head, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a + x = a ' + x; (s)

Proof step 17 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).last = sf(n0 + 1), ii(0) = 0, slf(n0) = sf(n0) ' + x0, sf(n0 + 1) ≠ slf(n0 + 1).head, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), sf(n0) ∈ slf(n0), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: n0 + 1 abort: F discard: T computed: T

Additional information

Used terms: n0 + 1

Proof step 18 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0)) = cext !(caf(n0) '), ext !(alf(n0 + 1)) = cext !(caf(n0 + 1) '), slf(n0).last = sf(n0 + 1), slf(n0 + 1).last = sf((n0 + 1) + 1), slf(n0 + 1)[0] = sf(n0 + 1), ii(0) = 0, slf(n0) = sf(n0) ' + x0, sf(n0 + 1) ≠ slf(n0 + 1).head, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), frag*(IOAut(start, step, ext))(slf(n0 + 1), alf(n0 + 1)), start(sf(0)), sf(n0) ∈ slf(n0), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 19 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n0 + 1)) = cext !(caf(n0 + 1) '), ext !(alf(n0)) = cext !(caf(n0) '), slf(n0).last = sf(n0 + 1), slf(n0 + 1).last = sf(2 + n0), slf(n0 + 1)[0] = sf(n0 + 1), ii(0) = 0, slf(n0) = sf(n0) ' + x0, sf(n0 + 1) ≠ slf(n0 + 1).head, slf(n0) ≠ [], frag*(IOAut(start, step, ext))(slf(n0 + 1), alf(n0 + 1)), frag*(IOAut(start, step, ext))(slf(n0), alf(n0)), start(sf(0)), sf(n0) ∈ slf(n0), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: get-zero-01, seq: ⊦ x ≠ [] → x[0] = x.head, subst:[x] → [slf(n0 + 1)]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); Used terms: slf(n0 + 1)

Proof step 20 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x ≠ [] → x[0] = x.head

A spec-lemma

Proof step 21 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∃ σ, ii. ii(0) = 0 ∧ (∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (finp(σ) ⊃ ∃ n. # σ = ⌜ii(n) + 1⌝ ∧ (∀ m. n ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 22 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: trace, seq:⊦ trace(IOAut(start, step, ext))(β) ↔ (∃ σ, α. β = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α)), subst:[start, step, ext, β] → [start, step, ext, cext !(⌜caf⌝ )]; ['e, 's, 'a] → ['e, 's, 'a], <all paths>

Additional information

Used terms: start, step, ext, cext !(⌜caf⌝ )

Proof step 23 from proof infappend-moves-gen (previous step) (back to tree)

⊦ trace(IOAut(start, step, ext))(β) ↔ (∃ σ, α. β = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α))

Lemma trace

Proof step 24 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ σ, α. ¬(cext !(⌜caf⌝ ) = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: exec: ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); ⊦ IOAut(start, step, ext).start = start; (s)

Proof step 25 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ σ0, α0. cext !(⌜caf⌝ ) ≠ ext !(α0) ∨ ¬ frag(IOAut(start, step, ext))(σ0, α0) ∨ ¬ start(σ0[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: σ, α abort: F discard: T computed: T

Additional information

Used terms: σ, α

Proof step 26 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ii0 = ii

Additional information

Used formulas: ii0 = ii

Proof step 27 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0 = ii, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: # σ = # α + ⌜1⌝

Additional information

Used formulas: # σ = # α + ⌜1⌝

Proof step 29 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: cext !(⌜caf⌝ ) ≠ ext !(α)

Additional information

Used formulas: cext !(⌜caf⌝ ) ≠ ext !(α)

Proof step 30 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(α), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 31 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(α), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜α.take(ii(n))⌝ )

Additional information

Used formulas: ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜α.take(ii(n))⌝ )

Proof step 32 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(α), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜α.take(ii(n))⌝ ) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: α

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 33 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # ⌜al⌝ + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜al⌝ ), start(sf(0)), (finp(⌜al⌝ ) ⊃ # ⌜al⌝ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜⌜al⌝ .take(ii(n))⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # ⌜al⌝ < ⌜ii(n + 1)⌝ ∧ ⌜al⌝ .drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ = ⌜n0⌝ ↔ n = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ finp(⌜x⌝ ); (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 34 from proof infappend-moves-gen (previous step) (back to tree)

# σ = ⌜ii(n)⌝ + ⌜1⌝ , # al = ii(n), ii(0) = 0, cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ σ.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 35 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), # σ = ⌜ii(n)⌝ + ⌜1⌝ , ii(0) = 0, σ = ⌜sl⌝ , cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ σ.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ finp(⌜x⌝ ); (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s)

Proof step 36 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(n) + 1 = # sl, ii(0) = 0, ii(n0) = ii(n), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: n abort: F discard: F computed: F

Additional information

Used terms: n

Proof step 37 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n)) = ext !(al.take(ii(n))), # al = ii(n), ii(n) + 1 = # sl, ii(n0) = ii(n), ii(0) = 0, cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = n → x.take(n) = x; (s)

Proof step 38 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n)) = ext !(al), # al = ii(n), ii(n) + 1 = # sl, ii(0) = 0, ii(n0) = ii(n), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: ext !(⌜al⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s)

Proof step 39 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n)) = ext !(al), # al = ii(n), ii(n) + 1 = # sl, ii(n0) = ii(n), ii(0) = 0, ⌜el⌝ = ext !(⌜al⌝ ), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 40 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n)) = ext !(al), # al = ii(n), ii(n) + 1 = # sl, ii(n0) = ii(n), ii(0) = 0, cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [caf, n, x0, nf0]; ['a] → ['ca]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: caf, n, x0, nf0

Proof step 41 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), # al = ii(n), # x0 = n, ii(n) + 1 = # sl, caf = (⌜x0⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ii(n0) = ii(n), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ , start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ σ + σ0 = σ ↔ σ0 = [] ∨ # σ = ∞; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 42 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), # al = ii(# x0), # al + 1 = # sl, ii(0) = 0, ii(n0) = # al, cext !(⌜nf0⌝ ) ≠ [], start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-empty-equiv, seq: ⊦ ext !(α) = [] ↔ (∀ n. ⌜n⌝ < # α → ext(α[n]) = None), subst:[ext, α] → [cext, ⌜nf0⌝ ]; ['e, 'a] → ['e, 'ca], <all paths>

Additional information

Used terms: cext, ⌜nf0⌝

Proof step 43 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ext !(α) = [] ↔ (∀ n. ⌜n⌝ < # α → ext(α[n]) = None)

A spec-lemma

Proof step 44 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), # al = ii(# x0), # al + 1 = # sl, ii(n0) = # al, ii(0) = 0, start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al, ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∃ n. ¬(⌜n⌝ < # ⌜nf0⌝ → cext(⌜nf0⌝ [n]) = None) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜nf⌝ [n] = nf(n); (s)

Proof step 45 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), # al = ii(# x0), # al + 1 = # sl, ii(0) = 0, ii(n0) = # al, cext(nf0(n)) ≠ None, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: # x0 + n abort: F discard: T computed: F

Additional information

Used terms: # x0 + n

Proof step 46 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), ext !(alf(# x0 + n)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(# x0 + n) '), # al = ii(# x0), # al + 1 = # sl, slf(# x0 + n).last = sf((# x0 + n) + 1), slf(# x0 + n)[0] = sf(# x0 + n), ii(n0) = # al, ii(0) = 0, cext(nf0(n)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + n), alf(# x0 + n)), start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al, ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 16 terms: # x0 + n abort: F discard: F computed: F

Additional information

Used terms: # x0 + n

Proof step 47 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(# x0 + n)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(# x0 + n) '), cext !(x0) = ext !(al), cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(# x0 + n)) = ext !(al.take(ii(# x0 + n))), # al = ii(# x0), # al + 1 = # sl, slf(# x0 + n).last = sf((# x0 + n) + 1), slf(# x0 + n)[0] = sf(# x0 + n), ii(0) = 0, ii(n0) = # al, cext(nf0(n)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + n), alf(# x0 + n)), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext(a) ≠ None → ext !(a ') = ext(a).get '; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (⌜x⌝ + ⌜nf⌝ ).f(# x + n) = nf(n); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ (⌜x⌝ + ⌜nf⌝ ).f.take(# x + n) = x + nf.take(n); (s)

Proof step 48 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), ext !(alf(# x0 + n)) = cext(nf0(n)).get ', # al = ii(# x0), # al + 1 = # sl, ext !(al) + cext !(nf0.take(n)) = ext !(al.take(ii(# x0 + n))), slf(# x0 + n).last = sf((# x0 + n) + 1), slf(# x0 + n)[0] = sf(# x0 + n), ii(n0) = # al, ii(0) = 0, cext(nf0(n)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + n), alf(# x0 + n)), start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al, ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [nf0, n, x, nf1]; ['a] → ['ca]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: nf0, n, x, nf1

Proof step 49 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(# x0 + n)) = cext(nf0(n)).get ', cext !(x0) = ext !(al), # al = ii(# x0), # x = n, ext !(al) + cext !(x) = ext !(al.take(ii(# x0 + n))), # al + 1 = # sl, slf(# x0 + n).last = sf((# x0 + n) + 1), slf(# x0 + n)[0] = sf(# x0 + n), ii(0) = 0, ii(n0) = # al, nf0 = (⌜x⌝ + ⌜nf1⌝ ).f, cext(nf0(n)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + n), alf(# x0 + n)), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !((⌜x0⌝ + ⌜nf0⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 50 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al), ext !(alf(# x0 + # x)) = cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)).get ', # al = ii(# x0), # al + 1 = # sl, ext !(al) + cext !(x) = ext !(al.take(ii(# x0 + # x))), slf(# x0 + # x).last = sf((# x0 + # x) + 1), slf(# x0 + # x)[0] = sf(# x0 + # x), ii(n0) = # al, ii(0) = 0, cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + # x), alf(# x0 + # x)), start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al, ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜(⌜x⌝ + ⌜nf1⌝ ).f⌝ ).f(n) '), sf(n + 1)), ∀ n1. cext !((⌜x0⌝ + ⌜(⌜x⌝ + ⌜nf1⌝ ).f⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: option, name:elim-Some, seq: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)), subst:[opt, get] → [cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)), e]; ['a] → ['e]

Additional information

Used simplifier rules: ⊦ opt ≠ None → (get = opt.get ↔ opt = Some(get)); (elim) Used terms: cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)), e

Proof step 51 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(# x0 + # x)) = e ', cext !(x0) = ext !(al), # al = ii(# x0), ext !(al) + cext !(x) = ext !(al.take(ii(# x0 + # x))), # al + 1 = # sl, slf(# x0 + # x).last = sf((# x0 + # x) + 1), slf(# x0 + # x)[0] = sf(# x0 + # x), cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)) = Some(e), ii(0) = 0, ii(n0) = # al, cext((⌜x⌝ + ⌜nf1⌝ ).f(# x)) ≠ None, frag*(IOAut(start, step, ext))(slf(# x0 + # x), alf(# x0 + # x)), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !((⌜x0⌝ + ⌜(⌜x⌝ + ⌜nf1⌝ ).f⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜(⌜x⌝ + ⌜nf1⌝ ).f⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Some(a) ≠ None; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜x + nf⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ (⌜x⌝ + ⌜nf⌝ ).f = x + nf; (s) ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 52 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(# x0 + # x)) = e ', cext !(x0) = ext !(al), # al = ii(# x0), ext !(al) + cext !(x) = ext !(al.take(ii(# x0 + # x))), # al + 1 = # sl, slf(# x0 + # x).last = sf((# x0 + # x) + 1), slf(# x0 + # x)[0] = sf(# x0 + # x), cext(((x + nf1))(# x)) = Some(e), ii(0) = 0, ii(n0) = # al, frag*(IOAut(start, step, ext))(slf(# x0 + # x), alf(# x0 + # x)), start(sf(0)), ∀ n1. cext !((⌜x0⌝ + ⌜x⌝ + ⌜nf1⌝ ).f.take(n1)) = ext !(al.take(ii(n1))), ∀ n. move(IOAut(start, step, ext))(sf(n), (cext((⌜x0⌝ + ⌜x⌝ + ⌜nf1⌝ ).f(n)) = None ⊃ [];cext((⌜x0⌝ + ⌜x⌝ + ⌜nf1⌝ ).f(n)).get '), sf(n + 1)), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ # al < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # al < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. # x0 ≤ m → alf(m) = [] ∧ ii(m) = # al, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = # al ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 18 terms: # x0 + # x abort: F discard: T computed: F

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # x = n → x.take(n) = x; (s) ⊦ x = x + y ↔ y = []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ext !([]) = []; (s) ⊦ a ' ≠ []; (s) Used terms: # x0 + # x

Proof step 53 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # ⌜af⌝ + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), (finp(⌜af⌝ ) ⊃ # ⌜af⌝ = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜⌜af⌝ .take(ii(n))⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # ⌜af⌝ < ⌜ii(n + 1)⌝ ∧ ⌜af⌝ .drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ⌜nf⌝ .drop(n) = ⌜nf.drop(n)⌝ ; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ¬ finp(⌜nf⌝ ); (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ¬ ∞ < n∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s)

Proof step 54 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ )

Additional information

Used formulas: cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ )

Proof step 55 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ )

Additional information

Used simplifier rules: ⊦ σ1 ⊑ σ2 → (σ2 ⊑ σ1 ↔ σ1 = σ2); (s) Used formulas: ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ )

Proof step 56 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ ), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 2, left 4

Additional information

Proof step 57 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: ext !(⌜af⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 58 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-split-list-weak, seq: ⊦ ext !(α) = ⌜el⌝ ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(al) = el ∧ ext !(α0) = []), subst:[ext, α, el] → [ext, ⌜af⌝ , el]; ['e, 'a] → ['e, 'a], <all paths>

Additional information

Used terms: ext, ⌜af⌝ , el

Proof step 59 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ext !(α) = ⌜el⌝ ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(al) = el ∧ ext !(α0) = [])

A spec-lemma

Proof step 60 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∃ al, α0. ⌜af⌝ = ⌜al⌝ + α0 ∧ ext !(al) = el ∧ ext !(α0) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 61 from proof infappend-moves-gen (previous step) (back to tree)

ext !(α0) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + α0, start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α0

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ σ0 + ⌜x⌝ = ⌜nf0⌝ ↔ σ0 = ⌜nf0⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream

Proof step 62 from proof infappend-moves-gen (previous step) (back to tree)

ext !(α0) = [], ii(0) = 0, α0 = ⌜af0⌝ , ⌜af⌝ = ⌜al⌝ + α0, start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 63 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 3, flags:

Additional information

Proof step 64 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , start(sf(0)), ¬ ext !(⌜al⌝ + ⌜af0⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-append, seq: ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2), subst:[ext, al, α2] → [ext, al, ⌜af0⌝ ]; ['a, 'e] → ['a, 'e], <all paths>

Additional information

Used terms: ext, al, ⌜af0⌝

Proof step 65 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2)

A spec-lemma

Proof step 66 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , start(sf(0)), ¬ ⌜ext !(al)⌝ + ext !(⌜af0⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ σ + [] = σ; (s)

Proof step 67 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , start(sf(0)), ¬ ⌜ext !(al)⌝ ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: # al abort: F discard: F computed: F

Additional information

Used terms: # al

Proof step 68 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , start(sf(0)), ¬ ⌜ext !(al)⌝ ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∃ m. ii(m) ≤ # al ∧ # al < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 69 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , ii(m) < ii(m + 1), # al < ii(m + 1), start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: m + 1 abort: F discard: T computed: F

Additional information

Used terms: m + 1

Proof step 70 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], cext !(caf.take(m + 1)) = ext !(af.take(ii(m + 1))), ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , # al < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [caf, m + 1, x0, nf0]; ['a] → ['ca]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: caf, m + 1, x0, nf0

Proof step 71 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(af.take(ii(m + 1))), ext !(⌜af0⌝ ) = [], # x0 = m + 1, caf = (⌜x0⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , ii(m) < ii(m + 1), # al < ii(m + 1), start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 72 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], cext !(x0) = ext !(af.take(ii(m + 1))), # x0 = m + 1, ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , # al < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜cext !(x0)⌝ + cext !(⌜nf0⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, ii(m + 1), al0, af1]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, ii(m + 1), al0, af1

Proof step 73 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al0), ext !(⌜af0⌝ ) = [], # x0 = m + 1, # al0 = ii(m + 1), af = (⌜al0⌝ + ⌜af1⌝ ).f, ii(0) = 0, ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ , ii(m) < ii(m + 1), # al < ii(m + 1), start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜cext !(x0)⌝ + cext !(⌜nf0⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 74 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al0), ext !(⌜af0⌝ ) = [], # x0 = m + 1, # al0 = ii(m + 1), ⌜al0⌝ + ⌜af1⌝ = ⌜al⌝ + ⌜af0⌝ , ii(0) = 0, al0 ≠ [], ii(m) < # al0, # al < # al0, start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜ext !(al0)⌝ + cext !(⌜nf0⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜af0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ1, σ0] → [⌜al0⌝ + ⌜af1⌝ , ⌜al⌝ + ⌜af0⌝ ]; ['a] → ['a], <all paths>

Additional information

Used terms: ⌜al0⌝ + ⌜af1⌝ , ⌜al⌝ + ⌜af0⌝

Proof step 75 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 76 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], cext !(x0) = ext !(al0), #(⌜al⌝ + ⌜af0⌝ ) = #(⌜al0⌝ + ⌜af1⌝ ), # al0 = ii(m + 1), # x0 = m + 1, ii(0) = 0, al0 ≠ [], # al < # al0, ii(m) < # al0, start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜ext !(al0)⌝ + cext !(⌜nf0⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜af0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ¬ #(⌜al⌝ + ⌜af0⌝ ) < ⌜n⌝ → (⌜al⌝ + ⌜af0⌝ ).take(n) = (⌜al0⌝ + ⌜af1⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ ∞ < n∞; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + ∞ = ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 77 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al0), ext !(⌜af0⌝ ) = [], # x0 = m + 1, # al0 = ii(m + 1), ii(0) = 0, al0 ≠ [], ii(m) < # al0, # al < # al0, start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜ext !(al0)⌝ + cext !(⌜nf0⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜af0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. (⌜al⌝ + ⌜af0⌝ ).take(n) = (⌜al0⌝ + ⌜af1⌝ ).take(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: # al0 abort: F discard: F computed: F

Additional information

Used terms: # al0

Proof step 78 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af0⌝ ) = [], cext !(x0) = ext !(al0), # al0 = ii(m + 1), # x0 = m + 1, (⌜al⌝ + ⌜af0⌝ ).take(# al0) = (⌜al0⌝ + ⌜af1⌝ ).take(# al0), ii(0) = 0, al0 ≠ [], # al < # al0, ii(m) < # al0, start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜ext !(al0)⌝ + cext !(⌜nf0⌝ ), ∀ n. (⌜al⌝ + ⌜af0⌝ ).take(n) = (⌜al0⌝ + ⌜af1⌝ ).take(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜af0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ (⌜x⌝ + σ).take(# x) = x; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ # x < m → (⌜x⌝ + σ).take(m) = x + σ.take(m - # x); (s)

Proof step 79 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al0), ext !(⌜af0⌝ ) = [], # x0 = m + 1, # al0 = ii(m + 1), al + af0.take(# al0 - # al) = al0, ii(0) = 0, al0 ≠ [], ii(m) < # al0, # al < # al0, start(sf(0)), ii(m) ≤ # al, ¬ ⌜ext !(al)⌝ ⊑ ⌜ext !(al0)⌝ + cext !(⌜nf0⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜nf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜af0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. (⌜al⌝ + ⌜af0⌝ ).take(n) = (⌜al0⌝ + ⌜af1⌝ ).take(n) ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [# al0, # al, n2]

Additional information

Used simplifier rules: ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ nf.take(n) = [] ↔ n = 0; (s) ⊦ m ≤ m + n; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ # nf.take(n) = n; (s) ⊦ m < m + n ↔ n ≠ 0; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ σ1 ⊑ σ1 + σ2; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ m < n → m ≤ n; (s) n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: # al0, # al, n2

Proof step 80 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ ext !(⌜af⌝ ) ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 81 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ cext !(⌜caf⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: cext !(⌜caf⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 82 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 83 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef⌝ , ext !(⌜af⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , ext !(⌜af⌝ )

Proof step 84 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 85 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef⌝ = # ext !(⌜af⌝ ), ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ¬ # ⌜ef⌝ < ⌜n⌝ → ⌜ef⌝ .take(n) = ext !(⌜af⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 86 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ef.take(n) = ext !(⌜af⌝ ).take(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 13 terms: # el + 1 abort: F discard: T computed: F

Additional information

Used terms: # el + 1

Proof step 87 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = ext !(⌜af⌝ ).take(# el + 1), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [ext, ⌜af⌝ , # el + 1]; ['a, 'e] → ['a, 'e]

Additional information

Used terms: ext, ⌜af⌝ , # el + 1

Proof step 88 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 89 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = ext !(⌜af⌝ ).take(# el + 1), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ¬ # ext !(⌜af⌝ ) < ⌜# el + 1⌝ → (∃ m. ¬ # ⌜af⌝ < ⌜m⌝ ∧ ext !(⌜af⌝ ).take(# el + 1) = ext !(⌜af⌝ .take(m))), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ∞; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 90 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = ext !(af.take(m)), ext !(⌜af⌝ ).take(# el + 1) = ext !(af.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 91 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af⌝ ).take(# el + 1) = ext !(af.take(m)), ef.take(# el + 1) = ext !(af.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∃ m0. ii(m0) ≤ m ∧ m < ii(m0 + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 92 from proof infappend-moves-gen (previous step) (back to tree)

ext !(⌜af⌝ ).take(# el + 1) = ext !(af.take(m)), ef.take(# el + 1) = ext !(af.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), ii(m0) < ii(m0 + 1), m < ii(m0 + 1), start(sf(0)), ii(m0) ≤ m, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: m0 + 1 abort: F discard: T computed: F

Additional information

Used terms: m0 + 1

Proof step 93 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(af.take(ii(m0 + 1))), ef.take(# el + 1) = ext !(af.take(m)), ext !(⌜af⌝ ).take(# el + 1) = ext !(af.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), m < ii(m0 + 1), ii(m0) < ii(m0 + 1), start(sf(0)), ii(m0) ≤ m, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, m, al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, m, al, nf0

Proof step 94 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(af.take(ii(m0 + 1))), # al = m, ext !(⌜af⌝ ).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), af = (⌜al⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ⌜el⌝ = cext !(⌜caf⌝ ), ii(m0) < ii(m0 + 1), m < ii(m0 + 1), start(sf(0)), ii(m0) ≤ m, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 95 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !((⌜al⌝ + ⌜nf0⌝ ).f.take(ii(m0 + 1))), # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), # al < ii(m0 + 1), ii(m0) < ii(m0 + 1), start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: appinf-to-ge, seq: ⊦ # x ≤ n → (⌜x⌝ + ⌜nf⌝ ).f.take(n) = x + nf.take(n - # x), subst:[x, nf, n] → [al, nf0, ii(m0 + 1)]; ['a] → ['a], <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: al, nf0, ii(m0 + 1)

Proof step 96 from proof infappend-moves-gen (previous step) (back to tree)

⊦ # x ≤ n → (⌜x⌝ + ⌜nf⌝ ).f.take(n) = x + nf.take(n - # x)

A spec-lemma

Proof step 97 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(al + nf0.take(ii(m0 + 1) - # al)), # ext !(⌜nf0⌝ ) = ∞, (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), ii(m0) < ii(m0 + 1), # al < ii(m0 + 1), start(sf(0)), ii(m0) ≤ # al, # al ≤ ii(m0 + 1), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s)

Proof step 98 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(al) + ext !(nf0.take(ii(m0 + 1) - # al)), # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), # al < ii(m0 + 1), ii(m0) < ii(m0 + 1), start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [ii(m0 + 1), # al, n2]

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: ii(m0 + 1), # al, n2

Proof step 99 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(al) + ext !(nf0.take(n2)), # ext !(⌜nf0⌝ ) = ∞, (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), ii(0) = 0, ii(m0 + 1) = n2 + # al, ⌜el⌝ = cext !(⌜caf⌝ ), ii(m0) < ii(m0 + 1), # al < ii(m0 + 1), start(sf(0)), ii(m0) ≤ # al, # al ≤ ii(m0 + 1), ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < m + n ↔ n ≠ 0; (s) ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → m ≤ n; (s)

Proof step 100 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m0 + 1)) = ext !(al) + ext !(nf0.take(n2)), # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(m0 + 1) = n2 + # al, ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [caf, m0 + 1, x0, caf0]; ['a] → ['ca]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: caf, m0 + 1, x0, caf0

Proof step 101 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al) + ext !(nf0.take(n2)), # ext !(⌜nf0⌝ ) = ∞, # x0 = m0 + 1, (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), caf = (⌜x0⌝ + ⌜caf0⌝ ).f, ii(0) = 0, ii(m0 + 1) = n2 + # al, ⌜el⌝ = cext !(⌜caf⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 102 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al) + ext !(nf0.take(n2)), # x0 = m0 + 1, # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(m0 + 1) = n2 + # al, ii(0) = 0, ⌜el⌝ = ⌜cext !(x0)⌝ + cext !(⌜caf0⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ0, σ1] → [⌜el⌝ , ⌜cext !(x0)⌝ + cext !(⌜caf0⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜el⌝ , ⌜cext !(x0)⌝ + cext !(⌜caf0⌝ )

Proof step 103 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

A spec-lemma

Proof step 104 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al) + ext !(nf0.take(n2)), # ⌜el⌝ = #(⌜cext !(x0)⌝ + cext !(⌜caf0⌝ )), # ext !(⌜nf0⌝ ) = ∞, # x0 = m0 + 1, (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), ii(0) = 0, ii(m0 + 1) = n2 + # al, n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ⌜n⌝ < # ⌜el⌝ → ⌜el⌝ [n] = (⌜cext !(x0)⌝ + cext !(⌜caf0⌝ ))[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 105 from proof infappend-moves-gen (previous step) (back to tree)

cext !(x0) = ext !(al) + ext !(nf0.take(n2)), # x0 = m0 + 1, # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(m0 + 1) = n2 + # al, ii(0) = 0, ⌜# el⌝ = ⌜# cext !(x0)⌝ + # cext !(⌜caf0⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. n < # el → el[n] = (⌜cext !(x0)⌝ + cext !(⌜caf0⌝ ))[n] ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 106 from proof infappend-moves-gen (previous step) (back to tree)

# ext !(⌜nf0⌝ ) = ∞, # x0 = m0 + 1, (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), ii(0) = 0, ii(m0 + 1) = n2 + # al, ⌜# el⌝ = ⌜#(ext !(al) + ext !(nf0.take(n2)))⌝ + # cext !(⌜caf0⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. n < # el → el[n] = (⌜ext !(al) + ext !(nf0.take(n2))⌝ + cext !(⌜caf0⌝ ))[n], ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ #(x + y) = # x + # y; (s)

Proof step 107 from proof infappend-moves-gen (previous step) (back to tree)

# x0 = m0 + 1, # ext !(⌜nf0⌝ ) = ∞, ef.take(# el + 1) = ext !(al), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(# el + 1) = ext !(al), ii(m0 + 1) = n2 + # al, ii(0) = 0, ⌜# el⌝ = (⌜# ext !(al)⌝ + ⌜# ext !(nf0.take(n2))⌝ ) + # cext !(⌜caf0⌝ ), n2 ≠ 0, ii(m0) < n2 + # al, start(sf(0)), ii(m0) ≤ # al, ¬ ⌜ef⌝ ⊑ ⌜el⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜x0⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. n < # el → el[n] = ((⌜ext !(al)⌝ + ⌜ext !(nf0.take(n2))⌝ ) + cext !(⌜caf0⌝ ))[n] ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 4, flags: rotate, discard

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞ = ⌜n⌝ ↔ n∞ = ⌜0⌝ ; (s) ⊦ ⌜n⌝ + n∞ = ⌜0⌝ ↔ n∞ = ⌜0⌝ ∧ n = 0; (ws) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 108 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ cext !(⌜caf⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 109 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ⌜ef0⌝ , ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜ef⌝ , ⌜ef0⌝ ]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , ⌜ef0⌝

Proof step 110 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 111 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ β. ⌜ef⌝ + β ≠ ⌜ef0⌝ ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ = ⌜nf0⌝ ↔ nf = nf0; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜nf⌝ + σ = ⌜nf⌝ ; (s)

Proof step 112 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∀ n. ef.take(n) = ef0.take(n)

Additional information

Used formulas: ∀ n. ef.take(n) = ef0.take(n)

Proof step 113 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ef.take(n) = ef0.take(n) ⊦

User applied rule expand left

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 114 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n. ef.take(n) = ef0.take(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: n + 1 abort: F discard: T computed: F

Additional information

Used terms: n + 1

Proof step 115 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n + 1) = ef0.take(n + 1), ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-unwind-end-inf, seq: ⊦ nf.take(m + 1) = nf.take(m) + nf(m) ', subst:[nf, m] → [ef, n]; ['a] → ['e], <all paths>

Additional information

Used terms: ef, n

Proof step 116 from proof infappend-moves-gen (previous step) (back to tree)

⊦ nf.take(m + 1) = nf.take(m) + nf(m) '

A spec-lemma

Proof step 117 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) + ef(n) ' = ef0.take(n + 1), ii(0) = 0, ⌜ef0⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list

Proof step 118 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) + ef(n) ' = ef0.take(n + 1), ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ef(n) ∈ ef0.take(n + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-unwind-end-inf, seq: ⊦ nf.take(m + 1) = nf.take(m) + nf(m) ', subst:[nf, m] → [ef0, n]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ⊦ x + a ' = y + a0 ' ↔ x = y ∧ a = a0; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: ef0, n

Proof step 119 from proof infappend-moves-gen (previous step) (back to tree)

⊦ nf.take(m + 1) = nf.take(m) + nf(m) '

A spec-lemma

Proof step 120 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ⌜ef0⌝ = cext !(⌜caf⌝ ), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦ ∀ n. ef.take(n) = ef0.take(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 121 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef0⌝ , cext !(⌜caf⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef0⌝ , cext !(⌜caf⌝ )

Proof step 122 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 123 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef0⌝ = # cext !(⌜caf⌝ ), ii(0) = 0, ⌜ef⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ¬ # ⌜ef0⌝ < ⌜n⌝ → ⌜ef0⌝ .take(n) = cext !(⌜caf⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 124 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜ef⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n0. ef0.take(n0) = cext !(⌜caf⌝ ).take(n0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef⌝ , ext !(⌜af⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , ext !(⌜af⌝ )

Proof step 125 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 126 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef⌝ = # ext !(⌜af⌝ ), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef0.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ¬ # ⌜ef⌝ < ⌜n⌝ → ⌜ef⌝ .take(n) = ext !(⌜af⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 127 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n0. ef0.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n0. ef.take(n0) = ext !(⌜af⌝ ).take(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: n abort: F discard: T computed: T

Additional information

Used terms: n

Proof step 128 from proof infappend-moves-gen (previous step) (back to tree)

ef0.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef.take(n0) = ext !(⌜af⌝ ).take(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: n abort: F discard: T computed: T

Additional information

Used terms: n

Proof step 129 from proof infappend-moves-gen (previous step) (back to tree)

ef0.take(n) = cext !(⌜caf⌝ ).take(n), ef.take(n) = ext !(⌜af⌝ ).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 2, flags: discard

Additional information

Proof step 130 from proof infappend-moves-gen (previous step) (back to tree)

ef0.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ext !(⌜af⌝ ).take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 1, flags: discard

Additional information

Proof step 131 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(⌜caf⌝ ).take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [cext, ⌜caf⌝ , n]; ['a, 'e] → ['ca, 'e]

Additional information

Used terms: cext, ⌜caf⌝ , n

Proof step 132 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 133 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(⌜caf⌝ ).take(n), ef ≠ ef0, start(sf(0)), ¬ # cext !(⌜caf⌝ ) < ⌜n⌝ → (∃ m. ¬ # ⌜caf⌝ < ⌜m⌝ ∧ cext !(⌜caf⌝ ).take(n) = cext !(⌜caf⌝ .take(m))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ ∞ < n∞; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 134 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ).take(n) = cext !(caf.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(caf.take(m)), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: m abort: F discard: T computed: F

Additional information

Used terms: m

Proof step 135 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(af.take(ii(m))), cext !(⌜caf⌝ ).take(n) = cext !(caf.take(m)), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(caf.take(m)), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 136 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(af.take(ii(m))), cext !(⌜caf⌝ ).take(n) = cext !(caf.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(caf.take(m)), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, ii(m), al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, ii(m), al, nf0

Proof step 137 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), # al = ii(m), cext !(⌜caf⌝ ).take(n) = cext !(caf.take(m)), af = (⌜al⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ext !(⌜af⌝ ).take(n) ≠ cext !(caf.take(m)), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 138 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), # al = ii(m), # ext !(⌜nf0⌝ ) = ∞, cext !(⌜caf⌝ ).take(n) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), (⌜ext !(al)⌝ + ext !(⌜nf0⌝ )).take(n) ≠ ext !(al), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name:extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[y, x] → [cext !(⌜caf⌝ ).take(n), ext !(al)]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ ¬ ∞ < n∞; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) ⊦ (⌜x⌝ + σ).take(# x) = x; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: cext !(⌜caf⌝ ).take(n), ext !(al)

Proof step 139 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 140 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: cext !(⌜caf⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 141 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 142 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜el⌝ = cext !(⌜caf⌝ ), ⌜el⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜el⌝ ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name: remtau-split-list-inf, seq:⊦ el ≠ [] → ( ext !(⌜af⌝ ) = ⌜el⌝ ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(el.last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = [])), subst:[el, ext, af] → [el, cext, caf]; ['e, 'a] → ['e, 'ca], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜[]⌝ = []; (s) ⊦ [] ⊑ σ; (s) Used terms: el, cext, caf

Proof step 143 from proof infappend-moves-gen (previous step) (back to tree)

⊦ el ≠ [] → (ext !(⌜af⌝ ) = ⌜el⌝ ↔ (∃ al, af0. ⌜af⌝ = ⌜al⌝ + ⌜af0⌝ ∧ al ≠ [] ∧ ext !(al) = el ∧ Some(el.last) = ext(al.last) ∧ ext !(⌜af0⌝ ) = []))

A spec-lemma

Proof step 144 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, el ≠ [], ⌜el⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜el⌝ ⊑ ext !(⌜af⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∃ _al, caf0. ⌜caf⌝ = ⌜_al⌝ + ⌜caf0⌝ ∧ _al ≠ [] ∧ cext !(_al) = el ∧ Some(el.last) = cext(_al.last) ∧ cext !(⌜caf0⌝ ) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 145 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf0⌝ ) = [], Some(cext !(_al).last) = cext(_al.last), ii(0) = 0, ⌜caf⌝ = ⌜_al⌝ + ⌜caf0⌝ , cext !(_al) ≠ [], _al ≠ [], ⌜cext !(_al)⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜cext !(_al)⌝ ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: # _al abort: F discard: T computed: F

Additional information

Used terms: # _al

Proof step 146 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf0⌝ ) = [], cext !(caf.take(# _al)) = ext !(af.take(ii(# _al))), Some(cext !(_al).last) = cext(_al.last), ii(0) = 0, ⌜caf⌝ = ⌜_al⌝ + ⌜caf0⌝ , cext !(_al) ≠ [], _al ≠ [], ⌜cext !(_al)⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜cext !(_al)⌝ ⊑ ext !(⌜af⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [caf, # _al, _al0, nf0]; ['a] → ['ca]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: caf, # _al, _al0, nf0

Proof step 147 from proof infappend-moves-gen (previous step) (back to tree)

cext !(_al0) = ext !(af.take(ii(# _al))), cext !(⌜caf0⌝ ) = [], # _al0 = # _al, Some(cext !(_al).last) = cext(_al.last), caf = (⌜_al0⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ⌜caf⌝ = ⌜_al⌝ + ⌜caf0⌝ , cext !(_al) ≠ [], _al ≠ [], ⌜cext !(_al)⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜cext !(_al)⌝ ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜nf⌝ = ⌜nf0⌝ ↔ nf = nf0; (s) ⊦ # x = # y → (⌜x⌝ + σ = ⌜y⌝ + σ0 ↔ x = y ∧ σ = σ0); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 148 from proof infappend-moves-gen (previous step) (back to tree)

cext !(_al) = ext !(af.take(ii(# _al))), cext !(⌜caf0⌝ ) = [], Some(cext !(_al).last) = cext(_al.last), ii(0) = 0, cext !(_al) ≠ [], _al ≠ [], ⌜cext !(_al)⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜cext !(_al)⌝ ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !((⌜_al⌝ + ⌜caf0⌝ ).f(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !((⌜_al⌝ + ⌜caf0⌝ ).f(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, ii(# _al), al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ σ + σ0 = σ ↔ σ0 = [] ∨ # σ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ σ1 ⊑ σ1 + σ2; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, ii(# _al), al, nf0

Proof step 149 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ cext !(⌜caf⌝ ) ⊑ ext !(⌜af⌝ ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 150 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ¬ ⌜ef⌝ ⊑ ext !(⌜af⌝ ), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: prefix-def, seq: ⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0), subst:[σ, σ0] → [⌜ef⌝ , ext !(⌜af⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , ext !(⌜af⌝ )

Proof step 151 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ ⊑ σ0 ↔ (∃ σ1. σ + σ1 = σ0)

A spec-lemma

Proof step 152 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ β. ⌜ef⌝ + β ≠ ext !(⌜af⌝ ) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ⌜nf⌝ + σ = ⌜nf⌝ ; (s)

Proof step 153 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: ext !(⌜af⌝ )

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;;

Proof step 154 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), ⌜ef⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s)

Proof step 155 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef⌝ , cext !(⌜caf⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , cext !(⌜caf⌝ )

Proof step 156 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 157 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef⌝ = # cext !(⌜caf⌝ ), ii(0) = 0, ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ¬ # ⌜ef⌝ < ⌜n⌝ → ⌜ef⌝ .take(n) = cext !(⌜caf⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 158 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ef.take(n) = cext !(⌜caf⌝ ).take(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: # el + 1 abort: F discard: T computed: F

Additional information

Used terms: # el + 1

Proof step 159 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = cext !(⌜caf⌝ ).take(# el + 1), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [cext, ⌜caf⌝ , # el + 1]; ['a, 'e] → ['ca, 'e]

Additional information

Used terms: cext, ⌜caf⌝ , # el + 1

Proof step 160 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 161 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = cext !(⌜caf⌝ ).take(# el + 1), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ¬ # cext !(⌜caf⌝ ) < ⌜# el + 1⌝ → (∃ m. ¬ # ⌜caf⌝ < ⌜m⌝ ∧ cext !(⌜caf⌝ ).take(# el + 1) = cext !(⌜caf⌝ .take(m))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ∞; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 162 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(# el + 1) = cext !(caf.take(m)), cext !(⌜caf⌝ ).take(# el + 1) = cext !(caf.take(m)), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 163 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(af.take(ii(m))), cext !(⌜caf⌝ ).take(# el + 1) = cext !(caf.take(m)), ef.take(# el + 1) = cext !(caf.take(m)), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, ii(m), al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, ii(m), al, nf0

Proof step 164 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), # al = ii(m), ef.take(# el + 1) = cext !(caf.take(m)), cext !(⌜caf⌝ ).take(# el + 1) = cext !(caf.take(m)), af = (⌜al⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s)

Proof step 165 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), # al = ii(m), cext !(⌜caf⌝ ).take(# el + 1) = ext !(al), ef.take(# el + 1) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜el⌝ = ⌜ext !(al)⌝ + ext !(⌜nf0⌝ ), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !((⌜al⌝ + ⌜nf0⌝ ).f.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:extensionality, seq: ⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n]), subst:[σ1, σ0] → [⌜el⌝ , ⌜ext !(al)⌝ + ext !(⌜nf0⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜el⌝ , ⌜ext !(al)⌝ + ext !(⌜nf0⌝ )

Proof step 166 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ⌜n⌝ < # σ0 → σ0[n] = σ1[n])

A spec-lemma

Proof step 167 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), #(⌜ext !(al)⌝ + ext !(⌜nf0⌝ )) = # ⌜el⌝ , # al = ii(m), ef.take(# el + 1) = ext !(al), cext !(⌜caf⌝ ).take(# el + 1) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !((⌜al⌝ + ⌜nf0⌝ ).f.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ⌜n⌝ < #(⌜ext !(al)⌝ + ext !(⌜nf0⌝ )) → (⌜ext !(al)⌝ + ext !(⌜nf0⌝ ))[n] = ⌜el⌝ [n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 168 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(al), # al = ii(m), ⌜# ext !(al)⌝ + # ext !(⌜nf0⌝ ) = ⌜# el⌝ , ef.take(# el + 1) = ext !(al), cext !(⌜caf⌝ ).take(# el + 1) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !((⌜al⌝ + ⌜nf0⌝ ).f.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # el → (⌜ext !(al)⌝ + ext !(⌜nf0⌝ ))[n] = el[n] ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 4, flags: rotate, discard

Additional information

Used simplifier rules: ⊦ # nf.take(n) = n; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ ⌜n⌝ + n∞ = ⌜n⌝ ↔ n∞ = ⌜0⌝ ; (s) ⊦ ⌜n⌝ + n∞ = ⌜0⌝ ↔ n∞ = ⌜0⌝ ∧ n = 0; (ws) ⊦ # σ = ⌜0⌝ ↔ σ = []; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 169 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ⌜ef⌝ ≠ ext !(⌜af⌝ ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ = ⌜nf0⌝ ↔ nf = nf0; (s)

Proof step 170 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∀ n. ef.take(n) = ef0.take(n)

Additional information

Used formulas: ∀ n. ef.take(n) = ef0.take(n)

Proof step 171 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = ext !(⌜af⌝ ), ⌜ef⌝ = cext !(⌜caf⌝ ), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ef.take(n) = ef0.take(n) ⊦

User applied rule expand left

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 172 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n. ef.take(n) = ef0.take(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: n + 1 abort: F discard: T computed: F

Additional information

Used terms: n + 1

Proof step 173 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n + 1) = ef0.take(n + 1), ii(0) = 0, ⌜ef0⌝ = ext !(⌜af⌝ ), ⌜ef⌝ = cext !(⌜caf⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-unwind-end-inf, seq: ⊦ nf.take(m + 1) = nf.take(m) + nf(m) ', subst:[nf, m] → [ef0, n]; ['a] → ['e], <all paths>

Additional information

Used terms: ef0, n

Proof step 174 from proof infappend-moves-gen (previous step) (back to tree)

⊦ nf.take(m + 1) = nf.take(m) + nf(m) '

A spec-lemma

Proof step 175 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n + 1) = ef0.take(n) + ef0(n) ', ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list

Proof step 176 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n + 1) = ef0.take(n) + ef0(n) ', ii(0) = 0, ⌜ef0⌝ = ext !(⌜af⌝ ), ⌜ef⌝ = cext !(⌜caf⌝ ), ef(n) ≠ ef0(n), start(sf(0)), ef0(n) ∈ ef.take(n + 1), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-unwind-end-inf, seq: ⊦ nf.take(m + 1) = nf.take(m) + nf(m) ', subst:[nf, m] → [ef, n]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ⊦ x + a ' = y + a0 ' ↔ x = y ∧ a = a0; (ws) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: ef, n

Proof step 177 from proof infappend-moves-gen (previous step) (back to tree)

⊦ nf.take(m + 1) = nf.take(m) + nf(m) '

A spec-lemma

Proof step 178 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef0⌝ = ext !(⌜af⌝ ), ⌜ef⌝ = cext !(⌜caf⌝ ), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦ ∀ n. ef.take(n) = ef0.take(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 179 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ⌜ef⌝ = cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef⌝ , cext !(⌜caf⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef⌝ , cext !(⌜caf⌝ )

Proof step 180 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 181 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef⌝ = # cext !(⌜caf⌝ ), ii(0) = 0, ⌜ef0⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ¬ # ⌜ef⌝ < ⌜n⌝ → ⌜ef⌝ .take(n) = cext !(⌜caf⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 182 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: n abort: F discard: F computed: T

Additional information

Used terms: n

Proof step 183 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ⌜ef0⌝ = ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: extensionality-to, seq:⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n)), subst:[σ0, σ1] → [⌜ef0⌝ , ext !(⌜af⌝ )]; ['a] → ['e], <all paths>

Additional information

Used terms: ⌜ef0⌝ , ext !(⌜af⌝ )

Proof step 184 from proof infappend-moves-gen (previous step) (back to tree)

⊦ σ0 = σ1 ↔ # σ0 = # σ1 ∧ (∀ n. ¬ # σ0 < ⌜n⌝ → σ0.take(n) = σ1.take(n))

A spec-lemma

Proof step 185 from proof infappend-moves-gen (previous step) (back to tree)

# ⌜ef0⌝ = # ext !(⌜af⌝ ), ef.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ¬ # ⌜ef0⌝ < ⌜n⌝ → ⌜ef0⌝ .take(n) = ext !(⌜af⌝ ).take(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 186 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau_of_to-to_of_rmtau, seq:⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m))), subst:[ext, α, n] → [cext, ⌜caf⌝ , n]; ['a, 'e] → ['ca, 'e]

Additional information

Used terms: cext, ⌜caf⌝ , n

Proof step 187 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # ext !(α) < ⌜n⌝ → (∃ m. ¬ # α < ⌜m⌝ ∧ ext !(α).take(n) = ext !(α.take(m)))

A spec-lemma

Proof step 188 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ¬ # cext !(⌜caf⌝ ) < ⌜n⌝ → (∃ m. ¬ # ⌜caf⌝ < ⌜m⌝ ∧ cext !(⌜caf⌝ ).take(n) = cext !(⌜caf⌝ .take(m))), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ ∞ < n∞; (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 189 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) = cext !(⌜caf⌝ ).take(n), ef.take(n) = cext !(caf.take(m)), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n1. cext !(caf.take(n1)) = ext !(af.take(ii(n1))), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: m abort: F discard: T computed: F

Additional information

Used terms: m

Proof step 190 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(m)) = ext !(af.take(ii(m))), ef.take(n) = cext !(caf.take(m)), ef.take(n) = cext !(⌜caf⌝ ).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 191 from proof infappend-moves-gen (previous step) (back to tree)

ef.take(n) = cext !(⌜caf⌝ ).take(n), ef.take(n) = cext !(caf.take(m)), ef.take(n) = ext !(af.take(ii(m))), ii(0) = 0, ∞ = # ext !(⌜af⌝ ), ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-fromto-inf, seq: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n, subst:[nf, n, x0, nf0] → [af, ii(m), al, nf0]; ['a] → ['a]

Additional information

Used simplifier rules: ⊦ x0 = nf.take(n) ∧ nf0 = nf.drop(n) ↔ nf = (⌜x0⌝ + ⌜nf0⌝ ).f ∧ # x0 = n; (elim) Used terms: af, ii(m), al, nf0

Proof step 192 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(m), ef.take(n) = ext !(al), ef.take(n) = cext !(caf.take(m)), ef.take(n) = cext !(⌜caf⌝ ).take(n), af = (⌜al⌝ + ⌜nf0⌝ ).f, ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ∞ = # ext !(⌜af⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ af.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n0. ef0.take(n0) = ext !(⌜af⌝ ).take(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ext !(⌜al⌝ + α2) = ⌜ext !(al)⌝ + ext !(α2); (s) ⊦ ⌜(⌜x⌝ + ⌜nf⌝ ).f⌝ = ⌜x⌝ + ⌜nf⌝ ; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 193 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(m), # ext !(⌜nf0⌝ ) = ∞, ef.take(n) = cext !(⌜caf⌝ ).take(n), ef.take(n) = cext !(caf.take(m)), ef.take(n) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n0. ef0.take(n0) = (⌜ef.take(n)⌝ + ext !(⌜nf0⌝ )).take(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: n abort: F discard: F computed: T

Additional information

Used terms: n

Proof step 194 from proof infappend-moves-gen (previous step) (back to tree)

# ext !(⌜nf0⌝ ) = ∞, # al = ii(m), ef.take(n) = ext !(al), ef.take(n) = cext !(caf.take(m)), ef.take(n) = cext !(⌜caf⌝ ).take(n), ef0.take(n) = (⌜ef.take(n)⌝ + ext !(⌜nf0⌝ )).take(n), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ ef0.take(n), ef ≠ ef0, start(sf(0)), ∀ n0. ef0.take(n0) = (⌜ef.take(n)⌝ + ext !(⌜nf0⌝ )).take(n0), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 6, flags: discard

Additional information

Proof step 195 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(m), # ext !(⌜nf0⌝ ) = ∞, ef.take(n) = cext !(⌜caf⌝ ).take(n), ef.take(n) = cext !(caf.take(m)), ef.take(n) = ext !(al), ii(0) = 0, ∞ = # cext !(⌜caf⌝ ), ef.take(n) ≠ (⌜ef.take(n)⌝ + ext !(⌜nf0⌝ )).take(n), ef ≠ ef0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ (⌜al⌝ + ⌜nf0⌝ ).f.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. ef.take(n0) = cext !(⌜caf⌝ ).take(n0), ∀ n0. ef0.take(n0) = (⌜ef.take(n)⌝ + ext !(⌜nf0⌝ )).take(n0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-le-to, seq: ⊦ ¬ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.take(n), subst:[n, σ, σ0] → [n, ⌜ef.take(n)⌝ , ext !(⌜nf0⌝ )]; ['a] → ['e], <all paths>

Additional information

Used simplifier rules: ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ nf.take(n).take(n) = nf.take(n); (s) ⊦ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.l + σ0.take(n - # σ.l); (s) ⊦ ⌜x⌝ .l = x; (s) ⊦ # nf.take(n) = n; (s) ⊦ m - m = 0; (s) ⊦ σ.take(0) = []; (s) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: n, ⌜ef.take(n)⌝ , ext !(⌜nf0⌝ )

Proof step 196 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # σ < ⌜n⌝ → (σ + σ0).take(n) = σ.take(n)

A spec-lemma

Proof step 197 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(α), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦ ∀ n. cext !(⌜caf.take(n)⌝ ) = ext !(⌜α.take(ii(n))⌝ )

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 198 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(⌜caf⌝ ) ≠ ext !(α), cext !(caf.take(n1)) ≠ ext !(α.take(ii(n1))), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 3

Additional information

Proof step 199 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(caf.take(n1)) ≠ ext !(α.take(ii(n1))), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: n1

Additional information

Used simplifier rules: Nat freely generated by 0, +1;; ⊦ nf.take(0) = []; (s) ⊦ σ.take(0) = []; (s) ⊦ ext !([]) = []; (s)

Proof step 200 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(caf.take(n2 +1)) ≠ ext !(α.take(ii(n2 +1))), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ σ, α, ii, cext, caf, ext, start, sf, n0, slf, n, alf, step. ¬ ( # σ = # α + ⌜1⌝ ∧ ii(0) = 0 ∧ cext !(caf.take(n2)) ≠ ext !(α.take(ii(n2))) ∧ start(sf(0)) ∧ (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∧ (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∧ (∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1)) ∧ (∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1))) ∧ (∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 201 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ σ0, α0, ii0, cext0, caf0, ext0, start0, sf0, n1, slf0, n3, alf0, step0. # σ0 ≠ # α0 + ⌜1⌝ ∨ ii0(0) ≠ 0 ∨ cext0 !(caf0.take(n2)) = ext0 !(α0.take(ii0(n2))) ∨ ¬ start0(sf0(0)) ∨ ¬ (finp(σ0) ⊃ # σ0 = ⌜ii0(n1)⌝ + ⌜1⌝ ∧ (∀ m. n1 ≤ m → slf0(m).tail = [] ∧ ii0(m) = ii0(n1));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)) ∨ ¬ (finp(α0) ⊃ # α0 = ⌜ii0(n3)⌝ ∧ (∀ m. n3 ≤ m → alf0(m) = [] ∧ ii0(m) = ii0(n3));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)) ∨ ¬ (∀ n0. frag*(IOAut(start0, step0, ext0))(slf0(n0), alf0(n0)) ∧ ext0 !(alf0(n0)) = cext0 !(caf0(n0) ') ∧ slf0(n0)[0] = sf0(n0) ∧ slf0(n0).last = sf0(n0 + 1)) ∨ ¬(∀ n. move(IOAut(start0, step0, ext0))(sf0(n), cext0 !(caf0(n) '), sf0(n + 1))) ∨ ¬(∀ n. ii0(n + 1) = ii0(n) + # alf0(n) ∧ ¬ # α0 < ⌜ii0(n + 1)⌝ ∧ α0.drop(ii0(n)).take(# alf0(n)) = alf0(n)) ∨ ¬(∀ m, n. m ≤ n → ii0(m) ≤ ii0(n)) ∨ ¬(∀ n. ii0(n + 1) = ii0(n) + # slf0(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ0 ∧ σ0.drop(ii0(n)).take(# slf0(n)) = slf0(n)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: σ, α, ii, cext, caf, ext, start, sf, n0, slf, n, alf, step abort: F discard: T computed: T

Additional information

Used terms: σ, α, ii, cext, caf, ext, start, sf, n0, slf, n, alf, step

Proof step 202 from proof infappend-moves-gen (previous step) (back to tree)

# σ = # α + ⌜1⌝ , ii(0) = 0, cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), # σ ≠ # α + ⌜1⌝ ∨ ii(0) ≠ 0 ∨ cext !(caf.take(n2)) = ext !(α.take(ii(n2))) ∨ ¬ start(sf(0)) ∨ ¬ (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∨ ¬ (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∨ ¬(∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1)) ∨ ¬(∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1))) ∨ ¬(∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)) ∨ ¬(∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∨ ¬(∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 203 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , ii(0) = 0, cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: n2 abort: F discard: F computed: F

Additional information

Used terms: n2

Proof step 204 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , α.drop(ii(n2)).take(# alf(n2)) = alf(n2), ii(0) = 0, ii(n2 + 1) = ii(n2) + # alf(n2), cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: n2 abort: F discard: F computed: F

Additional information

Used terms: n2

Proof step 205 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), ii(n2 + 1) = ii(n2) + # alf(n2), ii(0) = 0, ii(n2 + 1) = ii(n2) + # slf(n2) - 1, cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), ⌜ii(n2 + 1)⌝ < # σ, start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: n2 abort: F discard: F computed: F

Additional information

Used terms: n2

Proof step 206 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), ext !(alf(n2)) = cext !(caf(n2) '), # σ = # α + ⌜1⌝ , slf(n2).last = sf(n2 + 1), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), slf(n2)[0] = sf(n2), ii(n2 + 1) = ii(n2) + # slf(n2) - 1, ii(0) = 0, ii(n2 + 1) = ii(n2) + # alf(n2), cext !(caf.take(n2 + 1)) ≠ ext !(α.take(ii(n2 + 1))), ⌜ii(n2 + 1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: to-unwind-end-inf, seq: ⊦ nf.take(m + 1) = nf.take(m) + nf(m) ', subst:[nf, m] → [caf, n2]; ['a] → ['ca], <all paths>

Additional information

Used terms: caf, n2

Proof step 207 from proof infappend-moves-gen (previous step) (back to tree)

⊦ nf.take(m + 1) = nf.take(m) + nf(m) '

A spec-lemma

Proof step 208 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , slf(n2).last = sf(n2 + 1), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), slf(n2)[0] = sf(n2), ii(n2 + 1) = ii(n2) + # alf(n2), ii(0) = 0, ii(n2 + 1) = ii(n2) + # slf(n2) - 1, cext !(caf.take(n2) + caf(n2) ') ≠ ext !(α.take(ii(n2 + 1))), ⌜ii(n2 + 1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s)

Proof step 209 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), ext !(alf(n2)) = cext !(caf(n2) '), # σ = # α + ⌜1⌝ , slf(n2).last = sf(n2 + 1), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), slf(n2)[0] = sf(n2), ii(n2 + 1) = ii(n2) + # slf(n2) - 1, ii(0) = 0, ii(n2 + 1) = ii(n2) + # alf(n2), cext !(caf.take(n2)) + ext !(alf(n2)) ≠ ext !(α.take(ii(n2 + 1))), ⌜ii(n2 + 1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule subst equation

And obtained the following new goals

Rule argument

pos: left 10, flags: discard

Additional information

Proof step 210 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , ii(n2) + # alf(n2) = ii(n2) + # slf(n2) - 1, slf(n2).last = sf(n2 + 1), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), slf(n2)[0] = sf(n2), ii(0) = 0, cext !(caf.take(n2)) + ext !(alf(n2)) ≠ ext !(α.take(ii(n2) + # alf(n2))), ⌜ii(n2) + # alf(n2)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2) + # alf(n2)⌝ , (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)

Proof step 211 from proof infappend-moves-gen (previous step) (back to tree)

cext !(caf.take(n2)) = ext !(α.take(ii(n2))), ext !(alf(n2)) = cext !(caf(n2) '), # σ = # α + ⌜1⌝ , ii(n2) + # alf(n2) = ii(n2) + # slf(n2) - 1, slf(n2).last = sf(n2 + 1), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), slf(n2)[0] = sf(n2), ii(0) = 0, cext !(caf.take(n2)) + ext !(alf(n2)) ≠ ext !(α.take(ii(n2) + # alf(n2))), ⌜ii(n2)⌝ + ⌜# alf(n2)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2)⌝ + ⌜# alf(n2)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: from-append-tofrom, seq: ⊦ ¬ # σ < ⌜n + n1⌝ → σ.take(n) + σ.drop(n).take(n1) = σ.take(n + n1), subst:[σ, n, n1] → [α, ii(n2), # alf(n2)]; ['a] → ['a], <rotated> <all paths>

Additional information

Used simplifier rules: ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: α, ii(n2), # alf(n2)

Proof step 212 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # σ < ⌜n + n1⌝ → σ.take(n) + σ.drop(n).take(n1) = σ.take(n + n1)

A spec-lemma

Proof step 213 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), cext !(caf.take(n2)) = ext !(α.take(ii(n2))), # σ = # α + ⌜1⌝ , ii(n2) + # alf(n2) = ii(n2) + # slf(n2) - 1, slf(n2).last = sf(n2 + 1), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), slf(n2)[0] = sf(n2), ii(0) = 0, cext !(caf.take(n2)) + ext !(alf(n2)) ≠ ext !(α.take(ii(n2)) + α.drop(ii(n2)).take(# alf(n2))), ⌜ii(n2)⌝ + ⌜# alf(n2)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2) + # alf(n2)⌝ , ¬ # α < ⌜ii(n2)⌝ + ⌜# alf(n2)⌝ , (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:swap-from-to, seq: ⊦ ¬ # σ < ⌜m⌝ + ⌜n⌝ → σ.drop(m).take(n) = σ.take(m + n).drop(m), subst:[σ, m, n] → [α, ii(n2), # alf(n2)]; ['a] → ['a], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: α, ii(n2), # alf(n2)

Proof step 214 from proof infappend-moves-gen (previous step) (back to tree)

⊦ ¬ # σ < ⌜m⌝ + ⌜n⌝ → σ.drop(m).take(n) = σ.take(m + n).drop(m)

A spec-lemma

Proof step 215 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 216 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 12

Additional information

Proof step 217 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), ¬ frag(IOAut(start, step, ext))(σ, α), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [σ, step, α, start, ext]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: ext, α, start, σ, step

Proof step 218 from proof infappend-moves-gen (previous step) (back to tree)

⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1]))

Lemma frag

Proof step 219 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), # σ ≠ # α + ⌜1⌝ ∨ (∃ n. ¬(⌜n⌝ < # α → step(σ[n], α[n], σ[n +1]))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 220 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, ⌜n1⌝ < # α, start(sf(0)), ¬ step(σ[n1], α[n1], σ[n1 + 1]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

Proof step 221 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , # α = ⌜ii(n)⌝ , ii(0) = 0, ⌜n1⌝ < # α, finp(α), start(sf(0)), ¬ step(σ[n1], α[n1], σ[n1 + 1]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ ⌜n⌝ = ⌜n0⌝ ↔ n = n0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as)

Proof step 222 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), # σ = ⌜ii(n)⌝ + ⌜1⌝ , ii(0) = 0, n1 < ii(n), start(sf(0)), ¬ step(σ[n1], al[n1], σ[n1 + 1]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 8

Additional information

Used simplifier rules: ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜n⌝ ≠ ∞; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 223 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # σ = ⌜ii(n)⌝ + ⌜1⌝ , # al = ii(n), # σ = ⌜ii(n0)⌝ + ⌜1⌝ , ii(0) = 0, n1 < ii(n), finp(σ), start(sf(0)), ¬ step(σ[n1], al[n1], σ[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as)

Proof step 224 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, ii(0) = 0, ii(n0) = ii(n), n1 < ii(n), start(sf(0)), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Additional information

Used formulas: ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Proof step 225 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, ii(n0) = ii(n), ii(0) = 0, n1 < ii(n), start(sf(0)), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 226 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, ii(0) = 0, ii(n0) = ii(n), n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 227 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, al.drop(ii(m)).take(# alf(m)) = alf(m), ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, m ≤ n, ¬ ii(n) < ii(m + 1), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ m < m + n ↔ n ≠ 0; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 228 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, al.drop(ii(m)).take(# alf(m)) = alf(m), ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), start(sf(0)), ii(m + 1) ≤ ii(n), ii(m) ≤ n1, m ≤ n, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 229 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, al.drop(ii(m)).take(# alf(m)) = alf(m), sl.drop(ii(m)).take(# slf(m)) = slf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ ii(n) < ii(m + 1), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m ≤ n → (n < m ↔ false); (s)

Proof step 230 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, al.drop(ii(m)).take(# alf(m)) = alf(m), sl.drop(ii(m)).take(# slf(m)) = slf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 22 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 231 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), al.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), frag*(IOAut(start, step, ext))(slf(m), alf(m)), start(sf(0)), ii(m + 1) ≤ ii(n), ii(m) ≤ n1, m ≤ n, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(start, step, ext), slf(m), alf(m)]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: IOAut(start, step, ext), slf(m), alf(m)

Proof step 232 from proof infappend-moves-gen (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

Lemma fragstar

Proof step 233 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m)) = alf(m), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n. n < # alf(m) → IOAut(start, step, ext).step(slf(m)[n], alf(m)[n], slf(m)[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s)

Proof step 234 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), al.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), start(sf(0)), ii(m + 1) ≤ ii(n), ii(m) ≤ n1, m ≤ n, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [al.drop(ii(m)).take(# alf(m)), alf(m)]; ['a] → ['a], <all paths>

Additional information

Used terms: al.drop(ii(m)).take(# alf(m)), alf(m)

Proof step 235 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 236 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al.drop(ii(m)).take(# alf(m)) = # alf(m), # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n. n < # al.drop(ii(m)).take(# alf(m)) → al.drop(ii(m)).take(# alf(m))[n] = alf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x.take(n) = n; (s)

Proof step 237 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), start(sf(0)), ii(m + 1) ≤ ii(n), ii(m) ≤ n1, m ≤ n, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 31 terms: n1 - ii(m) abort: F discard: F computed: F

Additional information

Used terms: n1 - ii(m)

Proof step 238 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ step(sl[n1], al[n1], sl[n1 + 1]), n1 - ii(m) < # alf(m) → al.drop(ii(m)).take(# alf(m))[n1 - ii(m)] = alf(m)[n1 - ii(m)], ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 239 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), al.drop(ii(m)).take(# alf(m))[n1 - ii(m)] = alf(m)[n1 - ii(m)], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), n1 < ii(n), ii(m) < ii(m + 1), ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m) ≤ n1, ii(m + 1) ≤ ii(n), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n1, ii(m), n3]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n1, ii(m), n3

Proof step 240 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), n1 = n3 + ii(m), alf(m) ≠ [], ii(m) < ii(n), ii(m) < ii(m + 1), n1 < ii(n), n1 < ii(m + 1), start(sf(0)), ii(m + 1) ≤ ii(n), ii(m) ≤ n1, m ≤ n, ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m + n < m + n0 ↔ n < n0; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m ≤ m + n; (s)

Proof step 241 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [sl.drop(ii(m)).take(# slf(m)), slf(m)]; ['a] → ['s], <all paths>

Additional information

Used terms: sl.drop(ii(m)).take(# slf(m)), slf(m)

Proof step 242 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 243 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # sl.drop(ii(m)).take(# slf(m)) = # slf(m), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n. n < # sl.drop(ii(m)).take(# slf(m)) → sl.drop(ii(m)).take(# slf(m))[n] = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x.take(n) = n; (s)

Proof step 244 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), start(sf(0)), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 31 terms: n3 abort: F discard: F computed: F

Additional information

Used terms: n3

Proof step 245 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), n3 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1], ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2] ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n3 < # slf(m)

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used formulas: n3 < # slf(m)

Proof step 246 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), start(sf(0)), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), n3 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n → ¬ n < m; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ # x = 0 ↔ x = []; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 247 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(n) + 1 = # sl, slf(m).head = sf(m), slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], slf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), start(sf(0)), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1] ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [ii(m) + # slf(m), m0]

Additional information

Used simplifier rules: ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: ii(m) + # slf(m), m0

Proof step 248 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, ii(m) + # slf(m) = m0 + 1, slf(m).head = sf(m), slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(m + 1) = m0, ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), ii(m) + # slf(m) ≠ 0, alf(m) ≠ [], slf(m) ≠ [], n3 < # slf(m), n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1], ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 249 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).head = sf(m), slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], slf(m) ≠ [], n3 < # slf(m), n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1], ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2] ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [slf(m), s, sl0]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: slf(m), s, sl0

Proof step 250 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(m) + # slf(m) = ii(m + 1) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), s = sf(m), slf(m) = s + sl0, alf(m) ≠ [], slf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), start(sf(0)), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], slf(m)[n2 + 1]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) ⊦ a + x = a ' + x; (s)

Proof step 251 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), sl.drop(ii(m)).take(# slf(m))[n3] = slf(m)[n3], al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), slf(m) = sf(m) ' + sl0, alf(m) ≠ [], slf(m) ≠ [], n3 < # slf(m), n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), sf(m) ∈ slf(m), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1], ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], sl0[n2]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-finto, seq:¬ # x < n, m < n ⊦ x.take(n)[m] = x[m], subst:[x, n, m] → [sl.drop(ii(m)), # slf(m), n3]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: ¬ # x < n ⊦ # x.drop(n) = # x - n; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic ⊦ m < n → n ≠ 0; (s) ⊦ m +1 = m + 1; (s) ⊦ m + m0 < n → ¬ n + n0 < m; (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m + 1 ≠ 0; (s) ⊦ m ≤ m0 ∧ n ≤ n0 → ¬ m0 + n0 < m + n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: sl.drop(ii(m)), # slf(m), n3

Proof step 252 from proof infappend-moves-gen (previous step) (back to tree)

¬ # x < n, m < n ⊦ x.take(n)[m] = x[m]

A spec-lemma

Proof step 253 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(m) + # slf(m) = ii(m + 1) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al.drop(ii(m)).take(# alf(m))[n3] = alf(m)[n3], sl.drop(ii(m))[n3] = slf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), slf(m) = sf(m) ' + sl0, alf(m) ≠ [], slf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), start(sf(0)), sf(m) ∈ slf(m), m ≤ n, ii(m + 1) ≤ ii(n), ¬ # sl.drop(ii(m)) < # slf(m), ¬ step(sl[n3 + ii(m)], al[n3 + ii(m)], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], sl0[n2]), ∀ n2. n2 < # alf(m) → al.drop(ii(m)).take(# alf(m))[n2] = alf(m)[n2], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ n1. n1 < # slf(m) → sl.drop(ii(m)).take(# slf(m))[n1] = slf(m)[n1] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≤ n ↔ m < n; (s) ⊦ m < n → ¬ n < m; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ¬ # x < n, m < n ⊦ x.take(n)[m] = x[m]; (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (s) trans-Nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m < n → m ≤ n; (s) ¬ # x < n ⊦ # x.drop(n) = # x - n; (s) ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ # x = 0 ↔ x = []; (s) m + m0 < # x ⊦ x.drop(m)[m0] = x[m + m0]; (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat ⊦ ¬ m + n < m; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ m + m0 < n → ¬ n + n0 < m; (s)

Proof step 254 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(m) + # slf(m) = ii(m + 1) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al[ii(m) + n3] = alf(m)[n3], sl[ii(m) + n3] = slf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), slf(m) = sf(m) ' + sl0, alf(m) ≠ [], sl ≠ [], slf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), ii(m) < # sl, ii(m + 1) < # sl, start(sf(0)), sf(m) ∈ slf(m), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(slf(m)[n3], alf(m)[n3], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m))[n2] = alf(m)[n2], ∀ n1. n1 < # slf(m) → sl[ii(m) + n1] = slf(m)[n1], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], sl0[n2]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 31 terms: n3 abort: F discard: F computed: T

Additional information

Used terms: n3

Proof step 255 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , ext !(alf(m)) = cext !(caf(m) '), # al = ii(n), # slf(m) = # alf(m) + 1, ii(n) + 1 = # sl, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), sl[ii(m) + n3] = slf(m)[n3], al[ii(m) + n3] = alf(m)[n3], ii(n0) = ii(n), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), slf(m) = sf(m) ' + sl0, alf(m) ≠ [], sl ≠ [], slf(m) ≠ [], ii(m + 1) < # sl, ii(m) < # sl, n3 < # slf(m), n3 + ii(m) < ii(n), ii(m) < ii(n), ii(m) < ii(m + 1), n3 < # alf(m), start(sf(0)), sf(m) ∈ slf(m), ii(m + 1) ≤ ii(n), m ≤ n, ¬ step(slf(m)[n3], alf(m)[n3], sl[(n3 + ii(m)) + 1]), n3 < # alf(m) → step(slf(m)[n3], alf(m)[n3], sl0[n3]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], sl0[n2]), ∀ n1. n1 < # slf(m) → sl[ii(m) + n1] = slf(m)[n1], ∀ n2. n2 < # alf(m) → al.drop(ii(m))[n2] = alf(m)[n2] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 256 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # slf(m) = # alf(m) + 1, # al = ii(n), ii(m) + # slf(m) = ii(m + 1) + 1, ii(n) + 1 = # sl, slf(m).last = sf(m + 1), al[ii(m) + n3] = alf(m)[n3], sl[ii(m) + n3] = slf(m)[n3], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(n0) = ii(n), slf(m) = sf(m) ' + sl0, alf(m) ≠ [], sl ≠ [], slf(m) ≠ [], n3 < # alf(m), ii(m) < ii(m + 1), ii(m) < ii(n), n3 + ii(m) < ii(n), n3 < # slf(m), ii(m) < # sl, ii(m + 1) < # sl, start(sf(0)), step(slf(m)[n3], alf(m)[n3], sl0[n3]), sf(m) ∈ slf(m), m ≤ n, ii(m + 1) ≤ ii(n), ¬ step(slf(m)[n3], alf(m)[n3], sl[(n3 + ii(m)) + 1]), ∀ n2. n2 < # alf(m) → al.drop(ii(m))[n2] = alf(m)[n2], ∀ n1. n1 < # slf(m) → sl[ii(m) + n1] = slf(m)[n1], ∀ n2. n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], sl0[n2]), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 31 terms: n3 + 1 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n3 + 1

Proof step 257 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ⌜ext !(al)⌝ , # al = ii(n), ii(n) + 1 = # sl, ii(n0) = ii(n), ii(0) = 0, n1 < ii(n), start(sf(0)), ¬ step(sl[n1], al[n1], sl[n1 + 1]), ∀ n2. ii(n2 + 1) = ii(n2) + # slf(n2) - 1 ∧ ¬ ii(n) < ii(n2 + 1) ∧ sl.drop(ii(n2)).take(# slf(n2)) = slf(n2), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n), ∀ n2. ii(n2 + 1) = ii(n2) + # alf(n2) ∧ ¬ ii(n) < ii(n2 + 1) ∧ al.drop(ii(n2)).take(# alf(n2)) = alf(n2), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 1, left 2, left 3, left 4, left 7, left 8, left 9, left 10, left 11, left 12, left 13, left 14

Additional information

Proof step 258 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

User applied rule induction

And obtained the following new goals

Rule argument

indvar: n wfpred: <

Additional information

Used simplifier rules: <; (wellfounded) Used terms: n

Proof step 259 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < n → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: n - 1 abort: F discard: F computed: F

Additional information

Used terms: n - 1

Proof step 260 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < n → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1), n - 1 ≤ n ∧ ii(n - 1) ≤ n1 ∧ n1 < ii((n - 1) + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m ≤ m + n; (s) n0 ≤ n ⊦ n - n0 ≤ m ↔ n ≤ m + n0; (s) ⊦ m < n → n ≠ 0; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 261 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < n → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1), ii(n - 1) ≤ n1 ∧ n1 < ii((n - 1) + 1)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [n, m0]

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m0

Proof step 262 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n = m0 + 1, n ≠ 0, n1 < ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < n → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ n ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1), ii(m0) ≤ n1 ∧ n1 < ii(m0 + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 263 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(m0), n1 < ii(m0 + 1), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < m0 + 1 → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ m0 + 1 ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

User applied rule apply induction

And obtained the following new goals

Rule argument

subst: [n1, ii, n2] → [m0, ii, n1]

Additional information

Used simplifier rules: ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ n < n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m0, ii, n1

Proof step 264 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, n1 < ii(m0 + 1), n1 < ii(m0), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∃ m. m ≤ m0 ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1), ∀ n1, ii, n2. n1 < m0 + 1 → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ m0 + 1 ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 265 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii(m) < ii(m0 + 1), ii(m) < ii(m0), ii(m) < ii(m + 1), n1 < ii(m0 + 1), n1 < ii(m0), n1 < ii(m + 1), ii(m) ≤ n1, m ≤ m0, ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n1, ii, n2. n1 < m0 + 1 → ii(0) = 0 ∧ n2 < ii(n1) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) → (∃ m. m ≤ n1 ∧ ii(m) ≤ n2 ∧ n2 < ii(m + 1)) ⊦ ∃ m. m ≤ m0 + 1 ∧ ii(m) ≤ n1 ∧ n1 < ii(m + 1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: m

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ m < n → n ≠ 0; (s) ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: m

Proof step 266 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, ⌜n1⌝ < # α, start(sf(0)), ¬ finp(α), ¬ step(σ[n1], α[n1], σ[n1 + 1]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ⌜nf⌝ .drop(n) = ⌜nf.drop(n)⌝ ; (s) ⊦ ⌜nf⌝ [n] = nf(n); (s) ⊦ # ⌜nf⌝ = ∞; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ ¬ ∞ < n∞; (s) ⊦ ⌜m⌝ + ∞ = ∞; (s)

Proof step 267 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ii(0) = 0, start(sf(0)), ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: n1 abort: F discard: F computed: F

Additional information

Used terms: n1

Proof step 268 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ii(0) = 0, start(sf(0)), ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∃ m. ii(m) ≤ n1 ∧ n1 < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 269 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ii(0) = 0, ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 270 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(0) = 0, n1 < ii(m + 1), ii(m) < ii(m + 1), frag*(IOAut(start, step, ext))(slf(m), alf(m)), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(start, step, ext), slf(m), alf(m)]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: IOAut(start, step, ext), slf(m), alf(m)

Proof step 271 from proof infappend-moves-gen (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

Lemma fragstar

Proof step 272 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(0) = 0, ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → IOAut(start, step, ext).step(slf(m)[n], alf(m)[n], slf(m)[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s)

Proof step 273 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(0) = 0, n1 < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 16 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 274 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ m < m + n ↔ n ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic

Proof step 275 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 16 terms: m abort: F discard: F computed: F

Additional information

Used terms: m

Proof step 276 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), sf0.drop(ii(m)).take(# slf(m)) = slf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [sf0.drop(ii(m)).take(# slf(m)), slf(m)]; ['a] → ['s], <all paths>

Additional information

Used terms: sf0.drop(ii(m)).take(# slf(m)), slf(m)

Proof step 277 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 278 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # sf0.drop(ii(m)).take(# slf(m)) = # slf(m), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # sf0.drop(ii(m)).take(# slf(m)) → sf0.drop(ii(m)).take(# slf(m))[n] = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ nf.drop(n)(m) = nf(m + n); (s) m < n ⊦ nf.take(n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ # nf.take(n) = n; (s)

Proof step 279 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 16 terms: n1 - ii(m) abort: F discard: F computed: F

Additional information

Used terms: n1 - ii(m)

Proof step 280 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], n1 < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), n1 - ii(m) < # slf(m) → sf0((n1 - ii(m)) + ii(m)) = slf(m)[n1 - ii(m)], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ m ≤ n → (n < m ↔ false); (s) ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 281 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], ii(m) < ii(m + 1), n1 < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), n1 < # slf(m) + ii(m) → sf0((n1 - ii(m)) + ii(m)) = slf(m)[n1 - ii(m)], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-sub, seq:n ≤ m ⊦ n0 = m - n ↔ m = n0 + n, subst:[m, n, n0] → [n1, ii(m), n2]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n1, ii(m), n2

Proof step 282 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, n1 = n2 + ii(m), alf(m) ≠ [], n1 < ii(m + 1), ii(m) < ii(m + 1), start(sf(0)), ii(m) ≤ n1, ¬ step(sf0(n1), af(n1), sf0(n1 + 1)), n1 < # slf(m) + ii(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m + n < m + n0 ↔ n < n0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m ≤ m + n; (s)

Proof step 283 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), af.drop(ii(m)).take(# alf(m)) = alf(m), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), ¬ step(sf0(n2 + ii(m)), af(n2 + ii(m)), sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [af.drop(ii(m)).take(# alf(m)), alf(m)]; ['a] → ['a], <all paths>

Additional information

Used terms: af.drop(ii(m)).take(# alf(m)), alf(m)

Proof step 284 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 285 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # af.drop(ii(m)).take(# alf(m)) = # alf(m), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), start(sf(0)), ¬ step(sf0(n2 + ii(m)), af(n2 + ii(m)), sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # af.drop(ii(m)).take(# alf(m)) → af.drop(ii(m)).take(# alf(m))[n] = alf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ nf.drop(n)(m) = nf(m + n); (s) m < n ⊦ nf.take(n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ # nf.take(n) = n; (s)

Proof step 286 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), ¬ step(sf0(n2 + ii(m)), af(n2 + ii(m)), sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n] ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 23 terms: n2 abort: F discard: F computed: T

Additional information

Used terms: n2

Proof step 287 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), start(sf(0)), ¬ step(sf0(n2 + ii(m)), af(n2 + ii(m)), sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], n2 < # alf(m) → af(n2 + ii(m)) = alf(m)[n2], ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 288 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # slf(m) - 1, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), alf(m) ≠ [], n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), ¬ step(sf0(n2 + ii(m)), alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n] ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n2 < # slf(m)

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ m ≤ n → (n < m ↔ false); (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic ⊦ ¬ m + n < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used formulas: n2 < # slf(m)

Proof step 289 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).last = sf(m + 1), slf(m)[0] = sf(m), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, alf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), ¬ step(sf0(n2 + ii(m)), alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 < # slf(m) → sf0(n2 + ii(m)) = slf(m)[n2], ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n → ¬ n < m; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ # x = 0 ↔ x = []; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 290 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, slf(m).head = sf(m), slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, ii(m + 1) = ii(m) + # slf(m) - 1, sf0(n2 + ii(m)) = slf(m)[n2], alf(m) ≠ [], slf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [ii(m) + # slf(m), m0]

Additional information

Used simplifier rules: ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) ⊦ # x = 0 ↔ x = []; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: ii(m) + # slf(m), m0

Proof step 291 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = m0 + 1, slf(m).head = sf(m), slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = m0, ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], ii(m) + # slf(m) ≠ 0, alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + 1 ≠ 0; (s)

Proof step 292 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).head = sf(m), slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n] ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [slf(m), s, x0]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: slf(m), s, x0

Proof step 293 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, s = sf(m), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = s + x0, alf(m) ≠ [], slf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], slf(m)[n + 1]), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) ⊦ a + x = a ' + x; (s)

Proof step 294 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 27 terms: n2 abort: F discard: F computed: T

Additional information

Used terms: n2

Proof step 295 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 < # alf(m) → step(slf(m)[n2], alf(m)[n2], x0[n2]), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 296 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 27 terms: n2 + 1 abort: F discard: F computed: F

Additional information

Used terms: n2 + 1

Proof step 297 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 + 1 < # alf(m) → af((n2 + 1) + ii(m)) = alf(m)[n2 + 1], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: n2 + 1 < # alf(m)

Additional information

Used formulas: n2 + 1 < # alf(m)

Proof step 298 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), n2 + 1 < # alf(m), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 + 1 < # alf(m) → af((n2 + 1) + ii(m)) = alf(m)[n2 + 1], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 299 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], af((n2 + 1) + ii(m)) = alf(m)[n2 + 1], ii(m + 1) = ii(m) + # alf(m), ii(0) = 0, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], n2 + 1 < # alf(m), ii(m) < ii(m + 1), n2 < # alf(m), n2 < # slf(m), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n] ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 30 terms: n2 + 1 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n2 + 1

Proof step 300 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m) = # alf(m) + 1, ii(m) + # slf(m) = ii(m + 1) + 1, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + # alf(m), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], n2 < # slf(m), n2 < # alf(m), ii(m) < ii(m + 1), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ n2 + 1 < # alf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), n2 + 1 < # alf(m) → af((n2 + 1) + ii(m)) = alf(m)[n2 + 1], ∀ n. n < # slf(m) → sf0(n + ii(m)) = slf(m)[n], ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < # alf(m) → af(n + ii(m)) = alf(m)[n], ∀ n. n < # alf(m) → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m < m + n ↔ n ≠ 0; (s) n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ ¬ m + n < m; (s) ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) ⊦ m + 1 ≠ 0; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 301 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, slf(m).last = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 20 terms: m + 1 abort: F discard: T computed: F

Additional information

Used terms: m + 1

Proof step 302 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m).last = sf(m + 1), slf(m + 1).last = sf((m + 1) + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 303 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 22 terms: m + 1 abort: F discard: T computed: F

Additional information

Used terms: m + 1

Proof step 304 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m + 1)).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii((m + 1) + 1) = ii(m + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 305 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ sf0.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 23 terms: m + 1 abort: F discard: T computed: F

Additional information

Used terms: m + 1

Proof step 306 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), sf0.drop(ii(m + 1)).take(# slf(m + 1)) = slf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii((m + 1) + 1) = ii(m + 1) + # slf(m + 1) - 1, sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m + n = m + n0 ↔ n = n0; (s) eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic ⊦ n + 0 = n; (s) ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)

Proof step 307 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m + 1) = 1 + # alf(m + 1), # alf(m) = n2 + 1, # slf(m) = 2 + n2, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1)) = slf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = ii(m) + n2 + # slf(m + 1), ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1)), slf(m + 1)]; ['a] → ['s], <all paths>

Additional information

Used terms: sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1)), slf(m + 1)

Proof step 308 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 309 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1)) = # slf(m + 1), # slf(m) = 2 + n2, # alf(m) = n2 + 1, # slf(m + 1) = 1 + # alf(m + 1), slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii(2 + m) = ii(m) + n2 + # slf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. n < # sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1)) → sf0.drop(ii(m) + n2 + 1).take(# slf(m + 1))[n] = slf(m + 1)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ nf.drop(n)(m) = nf(m + n); (s) m < n ⊦ nf.take(n)[m] = nf(m); (s) ⊦ m < n → (n ≤ m ↔ false); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss) ⊦ # nf.take(n) = n; (s)

Proof step 310 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # slf(m + 1) = 1 + # alf(m + 1), # alf(m) = n2 + 1, # slf(m) = 2 + n2, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = ii(m) + n2 + # slf(m + 1), ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], frag*(IOAut(start, step, ext))(slf(m + 1), alf(m + 1)), start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(start, step, ext), slf(m + 1), alf(m + 1)]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: IOAut(start, step, ext), slf(m + 1), alf(m + 1)

Proof step 311 from proof infappend-moves-gen (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

Lemma fragstar

Proof step 312 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m + 1) = # alf(m + 1) + 1, # slf(m) = 2 + n2, # alf(m) = n2 + 1, # slf(m + 1) = 1 + # alf(m + 1), slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii(2 + m) = ii(m) + n2 + # slf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. n < # alf(m + 1) → IOAut(start, step, ext).step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s)

Proof step 313 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m + 1) = # alf(m + 1) + 1, # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii(2 + m) = ii(m) + n2 + # slf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 24 terms: 0 abort: F discard: F computed: F

Additional information

Used terms: 0

Proof step 314 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, # slf(m + 1) = # alf(m + 1) + 1, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = ii(m) + n2 + # slf(m + 1), ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), 0 < # alf(m + 1) → step(slf(m + 1)[0], alf(m + 1)[0], slf(m + 1)[0 + 1]), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 315 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m + 1) = # alf(m + 1) + 1, # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii(2 + m) = ii(m) + n2 + # slf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, slf(m + 1)[1]), ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 26 terms: 0 abort: F discard: F computed: F

Additional information

Used terms: 0

Proof step 316 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, # slf(m + 1) = # alf(m + 1) + 1, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = ii(m) + n2 + # slf(m + 1), ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, slf(m + 1)[1]), 0 < # slf(m + 1) → sf0(0 + ii(m) + n2 + 1) = slf(m + 1)[0], ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x ≠ [] → x[0] = x.head; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 317 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # slf(m + 1) = # alf(m + 1) + 1, # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, ii(2 + m) = ii(m) + n2 + # slf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, slf(m + 1)[1]), slf(m + 1) ≠ [] → sf0(1 + ii(m) + n2) = slf(m + 1).head, ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-data, name:elim-add, seq: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail), subst:[x, head, tail] → [slf(m + 1), s, x1]; ['a] → ['s]

Additional information

Used simplifier rules: eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic ⊦ # [] = 0; (s) ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: slf(m + 1), s, x1

Proof step 318 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, # slf(m + 1) = # alf(m + 1) + 1, slf(m).last = sf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# alf(m + 1)) = alf(m + 1), slf(m + 1)[0] = sf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = ii(m) + n2 + # slf(m + 1), ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # alf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = sf(m) ' + x0, slf(m + 1) = s + x1, alf(m) ≠ [], slf(m + 1) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf0((n2 + ii(m)) + 1)), slf(m + 1) ≠ [] → sf0(1 + ii(m) + n2) = s, alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, slf(m + 1)[1]), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # slf(m + 1) → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # alf(m + 1) → step(slf(m + 1)[n], alf(m + 1)[n], slf(m + 1)[n + 1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) n ≠ 0 ⊦ (a ' + x)[(* n)] = x[n -1]; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ m + n = m + n0 ↔ n = n0; (s) ⊦ (a ' + x).head = a; (s) eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ a + x = a ' + x; (s)

Proof step 319 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # x1 = # alf(m + 1), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), slf(m).last = sf(m + 1), af.drop(ii(m) + n2 + 1).take(# x1) = alf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # x1, ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, sf0(1 + ii(m) + n2) = sf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m + 1) = sf(m + 1) ' + x1, slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m + 1) ∈ slf(m + 1), sf(m) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf(m + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, x1[0]), ∀ n. ¬ # x1 < n → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # x1 → step(slf(m + 1)[n], alf(m + 1)[n], x1[n]), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-last, name:last-elim, seq:x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a ', subst:[x, a, y] → [slf(m), s, x2]; ['a] → ['s]

Additional information

Used simplifier rules: x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: slf(m), s, x2

Proof step 320 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, # x1 = # alf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# x1) = alf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # x1, s = sf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], sf0(1 + ii(m) + n2) = sf(m + 1), slf(m) = sf(m) ' + x0, slf(m + 1) = sf(m + 1) ' + x1, slf(m) = x2 + s ', alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m) ∈ slf(m), sf(m + 1) ∈ slf(m + 1), ¬ step(slf(m)[n2], alf(m)[n2], sf(m + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, x1[0]), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # x1 → step(slf(m + 1)[n], alf(m + 1)[n], x1[n]), ∀ n. ¬ # x1 < n → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list

Proof step 321 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # x1 = # alf(m + 1), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# x1) = alf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # x1, ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, sf0(1 + ii(m) + n2) = sf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = x2 + sf(m + 1) ', slf(m + 1) = sf(m + 1) ' + x1, slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m + 1) ∈ slf(m + 1), sf(m) ∈ slf(m), sf(m + 1) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf(m + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, x1[0]), ∀ n. ¬ # x1 < n → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # x1 → step(slf(m + 1)[n], alf(m + 1)[n], x1[n]), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 29 terms: n2 abort: F discard: F computed: T

Additional information

Used terms: n2

Proof step 322 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(m + 1)) = cext !(caf(m + 1) '), ext !(alf(m)) = cext !(caf(m) '), cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), # alf(m) = n2 + 1, # slf(m) = 2 + n2, # x1 = # alf(m + 1), slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# x1) = alf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(0) = 0, ii(m + 1) = ii(m) + n2 + 1, ii(2 + m) = (ii(m) + n2 + 1) + # x1, sf0(n2 + ii(m)) = slf(m)[n2], sf0(1 + ii(m) + n2) = sf(m + 1), slf(m) = sf(m) ' + x0, slf(m + 1) = sf(m + 1) ' + x1, slf(m) = x2 + sf(m + 1) ', alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m + 1) ∈ slf(m), sf(m) ∈ slf(m), sf(m + 1) ∈ slf(m + 1), ¬ step(slf(m)[n2], alf(m)[n2], sf(m + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, x1[0]), ¬ n2 < n2 → step(slf(m)[n2], alf(m)[n2], x0[n2]), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. n < # x1 → step(slf(m + 1)[n], alf(m + 1)[n], x1[n]), ∀ n. ¬ # x1 < n → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n < n; (s)

Proof step 323 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(⌜af⌝ ), ext !(alf(m)) = cext !(caf(m) '), ext !(alf(m + 1)) = cext !(caf(m + 1) '), # x1 = # alf(m + 1), # slf(m) = 2 + n2, # alf(m) = n2 + 1, slf(m + 1).last = sf(2 + m), af.drop(ii(m) + n2 + 1).take(# x1) = alf(m + 1), af(n2 + ii(m)) = alf(m)[n2], ii(2 + m) = (ii(m) + n2 + 1) + # x1, ii(m + 1) = ii(m) + n2 + 1, ii(0) = 0, sf0(1 + ii(m) + n2) = sf(m + 1), sf0(n2 + ii(m)) = slf(m)[n2], slf(m) = x2 + sf(m + 1) ', slf(m + 1) = sf(m + 1) ' + x1, slf(m) = sf(m) ' + x0, alf(m) ≠ [], slf(m) ≠ [], start(sf(0)), step(slf(m)[n2], alf(m)[n2], x0[n2]), sf(m + 1) ∈ slf(m + 1), sf(m) ∈ slf(m), sf(m + 1) ∈ slf(m), ¬ step(slf(m)[n2], alf(m)[n2], sf(m + 1)), alf(m + 1) ≠ [] → step(sf(m + 1), alf(m + 1).head, x1[0]), ∀ n. ¬ # x1 < n → sf0(n + ii(m) + n2 + 1) = slf(m + 1)[n], ∀ n. n < # x1 → step(slf(m + 1)[n], alf(m + 1)[n], x1[n]), ∀ n. ¬ n2 < n → step(slf(m)[n], alf(m)[n], x0[n]), ∀ n. ¬ n2 < n → af(n + ii(m)) = alf(m)[n], ∀ n. n < 2 + n2 → sf0(n + ii(m)) = slf(m)[n], ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 31 terms: n2 + 1 abort: F discard: F computed: F

Additional information

Used simplifier rules: ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ x = a ' + x0 → x[m + 1] = x0[m]; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list in-15: ⊦ x + a ' = y → a ∈ y; (forward) from specification list a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: n2 + 1

Proof step 324 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , ii(0) = 0, start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: 0 abort: F discard: F computed: F

Additional information

Used terms: 0

Proof step 325 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , σ.drop(ii(0)).take(# slf(0)) = slf(0), ii(0) = 0, ii(0 + 1) = ii(0) + # slf(0) - 1, ⌜ii(0 + 1)⌝ < # σ, start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ σ.drop(0) = σ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ n + 0 = n; (s)

Proof step 326 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , σ.take(# slf(0)) = slf(0), ii(0) = 0, ii(1) = # slf(0) - 1, ⌜ii(1)⌝ < # σ, start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: 0 abort: F discard: F computed: F

Additional information

Used terms: 0

Proof step 327 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ = # α + ⌜1⌝ , slf(0).last = sf(0 + 1), σ.take(# slf(0)) = slf(0), slf(0)[0] = sf(0), ii(1) = # slf(0) - 1, ii(0) = 0, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic

Proof step 328 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(0)) = cext !(caf(0) '), cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , slf(0).last = sf(1), σ.take(# slf(0)) = slf(0), slf(0)[0] = sf(0), ii(0) = 0, ii(1) = # slf(0) - 1, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-getput, name: extension, seq: ⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n]), subst:[x, y] → [σ.take(# slf(0)), slf(0)]; ['a] → ['s], <all paths>

Additional information

Used terms: σ.take(# slf(0)), slf(0)

Proof step 329 from proof infappend-moves-gen (previous step) (back to tree)

⊦ x = y ↔ # x = # y ∧ (∀ n. n < # x → x[n] = y[n])

A spec-lemma

Proof step 330 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ.take(# slf(0)) = # slf(0), # σ = # α + ⌜1⌝ , slf(0).last = sf(1), slf(0)[0] = sf(0), ii(1) = # slf(0) - 1, ii(0) = 0, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. n < # σ.take(# slf(0)) → σ.take(# slf(0))[n] = slf(0)[n] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 331 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(0)) = cext !(caf(0) '), cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , # σ.take(# slf(0)) = # slf(0), slf(0).last = sf(1), slf(0)[0] = sf(0), ii(0) = 0, ii(1) = # slf(0) - 1, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. n < # slf(0) → σ.take(# slf(0))[n] = slf(0)[n], ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 15 terms: 0 abort: F discard: T computed: F

Additional information

Used terms: 0

Proof step 332 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ.take(# slf(0)) = # slf(0), # σ = # α + ⌜1⌝ , slf(0).last = sf(1), slf(0)[0] = sf(0), ii(1) = # slf(0) - 1, ii(0) = 0, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), 0 < # slf(0) → σ.take(# slf(0))[0] = slf(0)[0], (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 333 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(0)) = cext !(caf(0) '), cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , # σ.take(# slf(0)) = # slf(0), slf(0).last = sf(1), slf(0)[0] = sf(0), ii(0) = 0, ii(1) = # slf(0) - 1, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), slf(0) ≠ [] → σ.take(# slf(0))[0] = slf(0).head, (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 13

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); ⊦ ¬ n∞ < ⌜0⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # [] = 0; (s)

Proof step 334 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ.take(# slf(0)) = # slf(0), # σ = # α + ⌜1⌝ , slf(0).last = sf(1), slf(0)[0] = sf(0), σ.take(# slf(0))[0] = slf(0).head, ii(1) = # slf(0) - 1, ii(0) = 0, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name: get-to, seq:m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m], subst:[σ, n, m] → [σ, # slf(0), 0]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ x ≠ [] → x[0] = x.head; (s) Used terms: σ, # slf(0), 0

Proof step 335 from proof infappend-moves-gen (previous step) (back to tree)

m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]

A spec-lemma

Proof step 336 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(0)) = cext !(caf(0) '), cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , # σ.take(# slf(0)) = # slf(0), slf(0).last = sf(1), σ.take(# slf(0))[0] = slf(0).head, slf(0)[0] = sf(0), ii(0) = 0, ii(1) = # slf(0) - 1, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ 0 < # slf(0) ∧ ¬ # σ < ⌜# slf(0)⌝

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 337 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ.take(# slf(0)) = # slf(0), # σ = # α + ⌜1⌝ , slf(0).last = sf(1), slf(0)[0] = sf(0), σ.take(# slf(0))[0] = slf(0).head, ii(1) = # slf(0) - 1, ii(0) = 0, ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦ slf(0) ≠ [] ∧ ¬ # σ < ⌜# slf(0)⌝

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); ⊦ ¬ n∞ < ⌜0⌝ ; (s) ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ σ.take(0) = []; (s) ⊦ # [] = 0; (s)

Proof step 338 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(0)) = cext !(caf(0) '), cext !(⌜caf⌝ ) = ext !(α), # σ = # α + ⌜1⌝ , # σ.take(# slf(0)) = # slf(0), slf(0).last = sf(1), σ.take(# slf(0))[0] = slf(0).head, slf(0)[0] = sf(0), ii(0) = 0, ii(1) = # slf(0) - 1, # σ < ⌜# slf(0)⌝ , ⌜ii(1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ ¬ n < n; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ m ≤ n ↔ ¬ n < m; (ss) ⊦ x ≠ [] → x[0] = x.head; (s) lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf

Proof step 339 from proof infappend-moves-gen (previous step) (back to tree)

cext !(⌜caf⌝ ) = ext !(α), ext !(alf(0)) = cext !(caf(0) '), # σ.take(# slf(0)) = # slf(0), # σ = # α + ⌜1⌝ , slf(0).head = sf(0), slf(0).last = sf(1), σ.take(# slf(0))[0] = sf(0), ii(1) = # slf(0) - 1, ii(0) = 0, slf(0) ≠ [], ii(1) < # slf(0), ⌜ii(1)⌝ < # σ, # σ < ⌜# slf(0)⌝ , frag*(IOAut(start, step, ext))(slf(0), alf(0)), start(sf(0)), ¬ start(σ[0]), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

Heuristic elimination applied rule apply elim lemma

And closed the goal!

Rule argument

spec: nat, name:elim-pred-c, seq:n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1, subst:[n, m] → [# slf(0), m0]

Additional information

Used simplifier rules: ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ m + 1 ≠ 0; (s) m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf ¬ # σ < ⌜n⌝ ⊦ # σ.take(n) = n; (s) ⊦ # x = 0 ↔ x = []; (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ls-trans: ⊦ n∞ < n∞1 ∧ n∞1 < n∞2 → n∞ < n∞2; (forward) from specification nat-inf f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic ⊦ ¬ n∞ < ⌜0⌝ ; (s) n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # slf(0), m0

Proof step 340 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, # σ ≠ # α + ⌜1⌝ , start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 4

Additional information

Proof step 341 from proof infappend-moves-gen (previous step) (back to tree)

# α = ⌜ii(n)⌝ , ii(0) = 0, # σ ≠ # α + ⌜1⌝ , finp(α), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: frag-finite-al: ⊦ frag(ioa)(σ, ⌜al⌝ ) ↔ frag*(ioa)(σ.l, al) ∧ finp(σ); ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ ⌜n⌝ = ⌜n0⌝ ↔ n = n0; (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as)

Proof step 342 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(0) = 0, # σ ≠ ⌜ii(n)⌝ + ⌜1⌝ , start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(σ.l, al) ∨ ¬ finp(σ) ∨ ¬ start(σ[0]) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

Proof step 343 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), # σ = ⌜ii(n0)⌝ + ⌜1⌝ , ii(0) = 0, # σ ≠ ⌜ii(n)⌝ + ⌜1⌝ , finp(σ), start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(σ.l, al) ∨ ¬ finp(σ) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ [n] = x[n]; (s) ⊦ finp(⌜x⌝ ); (s) ⊦ ⌜x⌝ .l = x; (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ n∞ = ⌜m⌝ + ⌜n0⌝ → (⌜n⌝ < n∞ ↔ n < m + n0); (s) ⊦ finp(σ) ↔ (∃ x. ⌜x⌝ = σ); (as)

Proof step 344 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(n0) + 1 = # sl, ii(0) = 0, ii(n) + 1 ≠ # sl, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n0) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 11 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 345 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(n0) + 1 = # sl, ii(0) = 0, ii(n) + 1 ≠ # sl, start(sf(0)), n0 ≤ n → slf(n).tail = [] ∧ ii(n) = ii(n0), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n0) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 6

Additional information

Proof step 346 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(n0) + 1 = # sl, ii(0) = 0, ii(n) + 1 ≠ # sl, start(sf(0)), ¬ n0 ≤ n, ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n0) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 347 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(n0) + 1 = # sl, ii(0) = 0, ii(n) + 1 ≠ # sl, n < n0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n1. ii(n1 + 1) = ii(n1) + # slf(n1) - 1 ∧ ¬ ii(n0) < ii(n1 + 1) ∧ sl.drop(ii(n1)).take(# slf(n1)) = slf(n1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: n0 abort: F discard: T computed: F

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: n0

Proof step 348 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(0) = 0, # σ ≠ ⌜ii(n)⌝ + ⌜1⌝ , start(sf(0)), ¬ finp(σ), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), cext !(⌜caf⌝ ) ≠ ⌜ext !(al)⌝ ∨ ¬ frag*(IOAut(start, step, ext))(σ.l, al) ∨ ¬ finp(σ) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ finp(⌜nf⌝ ); (s) ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ⌜nf⌝ .drop(n) = ⌜nf.drop(n)⌝ ; (s) ⊦ ⌜m⌝ < ∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ ≠ ∞; (s) ⊦ ⌜m⌝ + n∞ = ∞ ↔ n∞ = ∞; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss)

Proof step 349 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(0) = 0, start(sf(0)), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n0. ii(n0 + 1) = ii(n0) + # slf(n0) - 1 ∧ sf0.drop(ii(n0)).take(# slf(n0)) = slf(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 9 terms: # al abort: F discard: T computed: F

Additional information

Used terms: # al

Proof step 350 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(0) = 0, start(sf(0)), ∀ n0. ii(n0 + 1) = ii(n0) + # slf(n0) - 1 ∧ sf0.drop(ii(n0)).take(# slf(n0)) = slf(n0), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1), ∃ m. ii(m) ≤ # al ∧ # al < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 351 from proof infappend-moves-gen (previous step) (back to tree)

# al = ii(n), ii(0) = 0, ii(m) < ii(m + 1), ii(n) < ii(m + 1), start(sf(0)), ii(m) ≤ ii(n), ∀ n0. ii(n0 + 1) = ii(n0) + # slf(n0) - 1 ∧ sf0.drop(ii(n0)).take(# slf(n0)) = slf(n0), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n), ∀ n1. ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ ii(n) < ii(n1 + 1) ∧ al.drop(ii(n1)).take(# alf(n1)) = alf(n1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: m

Additional information

Used terms: m

Proof step 352 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, # σ ≠ # α + ⌜1⌝ , start(sf(0)), ¬ finp(α), (finp(σ) ⊃ # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: frag-inf-al: ⊦ frag(ioa)(σ, ⌜af⌝ ) ↔ fragω(ioa)(σ.f, af) ∧ ¬ finp(σ); ⊦ ⌜nf⌝ .take(n) = nf.take(n); (s) ⊦ ⌜nf⌝ .drop(n) = ⌜nf.drop(n)⌝ ; (s) ⊦ ¬ ∞ < n∞; (s) ⊦ # ⌜nf⌝ = ∞; (s) add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜m⌝ + ∞ = ∞; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss)

Proof step 353 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, # σ ≠ ∞, start(sf(0)), finp(σ) → # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ) ∨ ¬ fragω(IOAut(start, step, ext))(σ.f, af) ∨ finp(σ) ∨ ¬ start(σ[0]) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: σ

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ # ⌜nf⌝ = ∞; (s)

Proof step 354 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, σ = ⌜sl⌝ , # σ ≠ ∞, start(sf(0)), finp(σ) → # σ = ⌜ii(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0)), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ⌜ii(n + 1)⌝ < # σ ∧ σ.drop(ii(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), cext !(⌜caf⌝ ) ≠ ext !(⌜af⌝ ) ∨ ¬ fragω(IOAut(start, step, ext))(σ.f, af) ∨ finp(σ) ∨ ¬ start(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf ⊦ ⌜n⌝ + ⌜m⌝ = ⌜n0⌝ ↔ n + m = n0; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ finp(⌜x⌝ ); (s) ⊦ ⌜x⌝ .take(n) = x.take(n); (s) ⊦ ⌜x⌝ .drop(n) = ⌜x.drop(n)⌝ ; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ ⌜n⌝ ≠ ∞; (s)

Proof step 355 from proof infappend-moves-gen (previous step) (back to tree)

ii(n0) + 1 = # sl, ii(0) = 0, start(sf(0)), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ii(n + 1) < # sl ∧ sl.drop(ii(n)).take(# slf(n)) = slf(n), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: # sl abort: F discard: T computed: F

Additional information

Used terms: # sl

Proof step 356 from proof infappend-moves-gen (previous step) (back to tree)

ii(n0) + 1 = # sl, ii(0) = 0, start(sf(0)), ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ii(n + 1) < # sl ∧ sl.drop(ii(n)).take(# slf(n)) = slf(n), ∃ m. ii(m) ≤ # sl ∧ # sl < ii(m + 1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 357 from proof infappend-moves-gen (previous step) (back to tree)

ii(n0) + 1 = # sl, ii(0) = 0, ii(m) < ii(m + 1), # sl < ii(m + 1), start(sf(0)), ii(m) ≤ # sl, ∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii(m) = ii(n0), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ af.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. ii(n + 1) = ii(n) + # slf(n) - 1 ∧ ii(n + 1) < # sl ∧ sl.drop(ii(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: m abort: F discard: T computed: F

Additional information

Used simplifier rules: ⊦ m < n → ¬ n < m; (s) Used terms: m

Proof step 358 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0 ≠ ii, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), cext !(⌜caf⌝ ) ≠ ext !(α) ∨ ¬ frag(IOAut(start, step, ext))(σ, α) ∨ ¬ start(σ[0]) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 13

Additional information

Proof step 359 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0 ≠ ii, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦

User applied rule expand left

And obtained the following new goals

Rule argument

pos: left 3

Additional information

Proof step 360 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n1) ≠ ii(n1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: n1

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 361 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 +1) ≠ ii(n2 +1), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ ii, ii0, start, sf, α, n, alf, σ, n0, slf, step, ext, cext, caf. ¬ ( ii(0) = 0 ∧ ii0(0) = 0 ∧ ii0(n2) ≠ ii(n2) ∧ start(sf(0)) ∧ (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∧ (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)) ∧ (∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n)) ∧ (∀ m, n. m ≤ n → ii0(m) ≤ ii0(n)) ∧ (∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)) ∧ (∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∧ (∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1))) ∧ (∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)

Proof step 362 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ ii1, ii2, start0, sf0, α0, n1, alf0, σ0, n3, slf0, step0, ext0, cext0, caf0. ii1(0) ≠ 0 ∨ ii2(0) ≠ 0 ∨ ii2(n2) = ii1(n2) ∨ ¬ start0(sf0(0)) ∨ ¬ (finp(α0) ⊃ # α0 = ⌜ii1(n1)⌝ ∧ (∀ m. n1 ≤ m → alf0(m) = [] ∧ ii1(m) = ii1(n1));∀ n. ∃ m. ii1(m) ≤ n ∧ n < ii1(m + 1)) ∨ ¬ (finp(σ0) ⊃ # σ0 = ⌜ii2(n3)⌝ + ⌜1⌝ ∧ (∀ m. n3 ≤ m → slf0(m).tail = [] ∧ ii2(m) = ii2(n3));∀ n. ∃ m. ii2(m) ≤ n ∧ n < ii2(m + 1)) ∨ ¬(∀ n. ii2(n + 1) = ii2(n) + # slf0(n) - 1 ∧ ⌜ii2(n + 1)⌝ < # σ0 ∧ σ0.drop(ii2(n)).take(# slf0(n)) = slf0(n)) ∨ ¬(∀ m, n. m ≤ n → ii2(m) ≤ ii2(n)) ∨ ¬(∀ n. ii1(n + 1) = ii1(n) + # alf0(n) ∧ ¬ # α0 < ⌜ii1(n + 1)⌝ ∧ α0.drop(ii1(n)).take(# alf0(n)) = alf0(n)) ∨ ¬(∀ m, n. m ≤ n → ii1(m) ≤ ii1(n)) ∨ ¬(∀ n. move(IOAut(start0, step0, ext0))(sf0(n), cext0 !(caf0(n) '), sf0(n + 1))) ∨ ¬ (∀ n0. frag*(IOAut(start0, step0, ext0))(slf0(n0), alf0(n0)) ∧ ext0 !(alf0(n0)) = cext0 !(caf0(n0) ') ∧ slf0(n0)[0] = sf0(n0) ∧ slf0(n0).last = sf0(n0 + 1)) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 13 terms: ii, ii0, start, sf, α, n, alf, σ, n0, slf, step, ext, cext, caf abort: F discard: T computed: T

Additional information

Used terms: ii, ii0, start, sf, α, n, alf, σ, n0, slf, step, ext, cext, caf

Proof step 363 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ii(0) ≠ 0 ∨ ii0(0) ≠ 0 ∨ ii0(n2) = ii(n2) ∨ ¬ start(sf(0)) ∨ ¬ (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)) ∨ ¬ (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)) ∨ ¬(∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n)) ∨ ¬(∀ m, n. m ≤ n → ii0(m) ≤ ii0(n)) ∨ ¬(∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)) ∨ ¬(∀ m, n. m ≤ n → ii(m) ≤ ii(n)) ∨ ¬(∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1))) ∨ ¬(∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1)) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 13

Additional information

Proof step 364 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2) = ii(n2), ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: n2 abort: F discard: T computed: T

Additional information

Used terms: n2

Proof step 365 from proof infappend-moves-gen (previous step) (back to tree)

σ.drop(ii0(n2)).take(# slf(n2)) = slf(n2), ii(0) = 0, ii0(n2) = ii(n2), ii0(0) = 0, ii0(n2 + 1) = ii0(n2) + # slf(n2) - 1, ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 366 from proof infappend-moves-gen (previous step) (back to tree)

σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), ii(0) = 0, ii0(n2 + 1) = ii(n2) + # slf(n2) - 1, ii0(0) = 0, ii0(n2) = ii(n2), ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 12 terms: n2 abort: F discard: T computed: T

Additional information

Used terms: n2

Proof step 367 from proof infappend-moves-gen (previous step) (back to tree)

σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), ii(0) = 0, ii(n2 + 1) = ii(n2) + # alf(n2), ii0(n2) = ii(n2), ii0(0) = 0, ii0(n2 + 1) = ii(n2) + # slf(n2) - 1, ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 14 terms: n2 abort: F discard: T computed: F

Additional information

Used terms: n2

Proof step 368 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), slf(n2).last = sf(n2 + 1), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), slf(n2)[0] = sf(n2), ii(n2 + 1) = ii(n2) + # alf(n2), ii(0) = 0, ii0(n2 + 1) = ii(n2) + # slf(n2) - 1, ii0(0) = 0, ii0(n2) = ii(n2), ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, frag*(IOAut(start, step, ext))(slf(n2), alf(n2)), start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fragstar, seq:⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1])), subst:[ioa, sl, al] → [IOAut(start, step, ext), slf(n2), alf(n2)]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: IOAut(start, step, ext), slf(n2), alf(n2)

Proof step 369 from proof infappend-moves-gen (previous step) (back to tree)

⊦ frag*(ioa)(sl, al) ↔ # sl = # al + 1 ∧ (∀ n. n < # al → ioa.step(sl[n], al[n], sl[n +1]))

Lemma fragstar

Proof step 370 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), # slf(n2) = # alf(n2) + 1, slf(n2).last = sf(n2 + 1), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), slf(n2)[0] = sf(n2), ii(0) = 0, ii(n2 + 1) = ii(n2) + # alf(n2), ii0(n2) = ii(n2), ii0(0) = 0, ii0(n2 + 1) = ii(n2) + # slf(n2) - 1, ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. n < # alf(n2) → IOAut(start, step, ext).step(slf(n2)[n], alf(n2)[n], slf(n2)[n +1]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).step = step; (s) ⊦ m +1 = m + 1; (s)

Proof step 371 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n2)) = cext !(caf(n2) '), # slf(n2) = # alf(n2) + 1, slf(n2).last = sf(n2 + 1), α.drop(ii(n2)).take(# alf(n2)) = alf(n2), σ.drop(ii(n2)).take(# slf(n2)) = slf(n2), slf(n2)[0] = sf(n2), ii(n2 + 1) = ii(n2) + # alf(n2), ii(0) = 0, ii0(n2 + 1) = ii(n2) + # slf(n2) - 1, ii0(0) = 0, ii0(n2) = ii(n2), ii0(n2 + 1) ≠ ii(n2 + 1), ⌜ii0(n2 + 1)⌝ < # σ, start(sf(0)), ¬ # α < ⌜ii(n2 + 1)⌝ , (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n1. n1 < # alf(n2) → step(slf(n2)[n1], alf(n2)[n1], slf(n2)[n1 + 1]) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 2, flags: discard

Additional information

Used simplifier rules: ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s) ⊦ n + 0 = n; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic

Proof step 372 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 373 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦ ii0(n1 + 1) = ii0(n1) + # slf(n1) - 1 ∧ ⌜ii0(n1 + 1)⌝ < # σ ∧ σ.drop(ii0(n1)).take(# slf(n1)) = slf(n1)

User applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 374 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 375 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), ii0(n1) < ii0(m), start(sf(0)), m ≤ n1, (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 10 terms: m, n1

Additional information

Used simplifier rules: ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m, n1

Proof step 376 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 377 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦ ii(n1 + 1) = ii(n1) + # alf(n1) ∧ ¬ # α < ⌜ii(n1 + 1)⌝ ∧ α.drop(ii(n1)).take(# alf(n1)) = alf(n1)

User applied rule all left

And closed the goal!

Rule argument

pos: left 10 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 378 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ m, n. m ≤ n → ii(m) ≤ ii(n)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m ≤ n ↔ ¬ n < m; (ss)

Proof step 379 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), ii(n1) < ii(m), start(sf(0)), m ≤ n1, (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: m, n1

Additional information

Used simplifier rules: ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m, n1

Proof step 380 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 381 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), ¬ move(IOAut(start, step, ext))(sf(n1), cext !(caf(n1) '), sf(n1 + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 8 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 382 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n1)) = cext !(caf(n1) '), slf(n1).last = sf(n1 + 1), slf(n1)[0] = sf(n1), ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), frag*(IOAut(start, step, ext))(slf(n1), alf(n1)), start(sf(0)), ¬ move(IOAut(start, step, ext))(sf(n1), cext !(caf(n1) '), sf(n1 + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 383 from proof infappend-moves-gen (previous step) (back to tree)

ext !(alf(n1)) = cext !(caf(n1) '), slf(n1).last = sf(n1 + 1), slf(n1)[0] = sf(n1), ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), frag*(IOAut(start, step, ext))(slf(n1), alf(n1)), start(sf(0)), ¬ move(IOAut(start, step, ext))(sf(n1), ext !(alf(n1)), sf(n1 + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 12 terms: n1

Additional information

Used terms: n1

Proof step 384 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1) ⊦ ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 385 from proof infappend-moves-gen (previous step) (back to tree)

ii(0) = 0, ii0(0) = 0, ii0(n2 + 1) ≠ ii(n2 + 1), start(sf(0)), (finp(σ) ⊃ # σ = ⌜ii0(n0)⌝ + ⌜1⌝ ∧ (∀ m. n0 ≤ m → slf(m).tail = [] ∧ ii0(m) = ii0(n0));∀ n. ∃ m. ii0(m) ≤ n ∧ n < ii0(m + 1)), (finp(α) ⊃ # α = ⌜ii(n)⌝ ∧ (∀ m. n ≤ m → alf(m) = [] ∧ ii(m) = ii(n));∀ n. ∃ m. ii(m) ≤ n ∧ n < ii(m + 1)), ∀ n0. frag*(IOAut(start, step, ext))(slf(n0), alf(n0)) ∧ ext !(alf(n0)) = cext !(caf(n0) ') ∧ slf(n0)[0] = sf(n0) ∧ slf(n0).last = sf(n0 + 1), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), ∀ m, n. m ≤ n → ii(m) ≤ ii(n), ∀ n. ii(n + 1) = ii(n) + # alf(n) ∧ ¬ # α < ⌜ii(n + 1)⌝ ∧ α.drop(ii(n)).take(# alf(n)) = alf(n), ∀ m, n. m ≤ n → ii0(m) ≤ ii0(n), ∀ n. ii0(n + 1) = ii0(n) + # slf(n) - 1 ∧ ⌜ii0(n + 1)⌝ < # σ ∧ σ.drop(ii0(n)).take(# slf(n)) = slf(n) ⊦ frag*(IOAut(start, step, ext))(slf(n1), alf(n1)) ∧ ext !(alf(n1)) = cext !(caf(n1) ') ∧ slf(n1)[0] = sf(n1) ∧ slf(n1).last = sf(n1 + 1)

User applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: n1 abort: F discard: T computed: T

Additional information

Used terms: n1

Proof step 386 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦ ∀ n. ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(⌜al⌝ ) = cext !(⌜caf(n) '⌝ ) ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 387 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∀ n. move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)) ⊦ ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(al) = cext !(caf(n) ') ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: n abort: F discard: T computed: F

Additional information

Used terms: n

Proof step 388 from proof infappend-moves-gen (previous step) (back to tree)

move(IOAut(start, step, ext))(sf(n), cext !(caf(n) '), sf(n + 1)), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )) ⊦ ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(al) = cext !(caf(n) ') ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(start, step, ext), sf(n), cext !(caf(n) '), sf(n + 1)]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), sf(n), cext !(caf(n) '), sf(n + 1)

Proof step 389 from proof infappend-moves-gen (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

Lemma move

Proof step 390 from proof infappend-moves-gen (previous step) (back to tree)

start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )), ∃ al, sl. frag*(IOAut(start, step, ext))(sl, al) ∧ sf(n) = sl[0] ∧ sf(n + 1) = sl.last ∧ IOAut(start, step, ext).ext !(al) = cext !(caf(n) ') ⊦ ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(al) = cext !(caf(n) ') ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 391 from proof infappend-moves-gen (previous step) (back to tree)

ext !(al) = cext !(caf(n) '), sf(n) = sl[0], sf(n + 1) = sl.last, frag*(IOAut(start, step, ext))(sl, al), start(sf(0)), ¬ trace(IOAut(start, step, ext))(cext !(⌜caf⌝ )) ⊦ ∃ sl, al. frag*(IOAut(start, step, ext))(sl, al) ∧ ext !(al) = cext !(caf(n) ') ∧ sl[0] = sf(n) ∧ sl.last = sf(n + 1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: sl, al

Additional information

Used terms: sl, al

Proof Tree for infappend-moves-gen (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree