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Proof step 1 from proof move-trans (conclusion) (back to tree)

⊦ move(IOAut(start, step, ext))(s, el1, s1) ∧ move(IOAut(start, step, ext))(s1, el2, s2) → move(IOAut(start, step, ext))(s, el1 + el2, s2)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof move-trans (previous step) (back to tree)

move(IOAut(start, step, ext))(s, el1, s1), move(IOAut(start, step, ext))(s1, el2, s2), ¬ move(IOAut(start, step, ext))(s, el1 + el2, s2) ⊦

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), s, el1, s1]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s, el1, s1

Proof step 3 from proof move-trans (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 4 from proof move-trans (previous step) (back to tree)

move(IOAut(start, step, ext))(s1, el2, s2), ¬ move(IOAut(start, step, ext))(s, el1 + el2, s2), ∃ al, sl. frag*(IOAut(start, step, ext))(sl, al) ∧ s = sl[0] ∧ s1 = sl.last ∧ IOAut(start, step, ext).ext !(al) = el1 ⊦

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 5 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl, al), move(IOAut(start, step, ext))(sl.last, el2, s2), ¬ move(IOAut(start, step, ext))(sl[0], ext !(al) + el2, s2) ⊦

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), sl.last, el2, s2]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), sl.last, el2, s2

Proof step 6 from proof move-trans (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 7 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl, al), ¬ move(IOAut(start, step, ext))(sl[0], ext !(al) + el2, s2), ∃ al, sl0. frag*(IOAut(start, step, ext))(sl0, al) ∧ sl.last = sl0[0] ∧ s2 = sl0.last ∧ IOAut(start, step, ext).ext !(al) = el2 ⊦

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 8 from proof move-trans (previous step) (back to tree)

sl.last = sl0[0], frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl, al), ¬ move(IOAut(start, step, ext))(sl[0], ext !(al) + ext !(al0), sl0.last) ⊦

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), sl[0], ext !(al) + ext !(al0), sl0.last]; ['s, 'e, 'a] → ['s, 'e, 'a], <all paths>

Additional information

Used terms: IOAut(start, step, ext), sl[0], ext !(al) + ext !(al0), sl0.last

Proof step 9 from proof move-trans (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 10 from proof move-trans (previous step) (back to tree)

sl.last = sl0[0], frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl0, al0), ∀ al1, sl1. ¬(frag*(IOAut(start, step, ext))(sl1, al1) ∧ sl[0] = sl1[0] ∧ sl0.last = sl1.last ∧ IOAut(start, step, ext).ext !(al1) = ext !(al) + ext !(al0)) ⊦

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 11 from proof move-trans (previous step) (back to tree)

sl.last = sl0[0], frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl, al), ∀ al1, sl1. ¬ frag*(IOAut(start, step, ext))(sl1, al1) ∨ sl[0] ≠ sl1[0] ∨ sl0.last ≠ sl1.last ∨ ext !(al1) ≠ ext !(al) + ext !(al0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: al + al0, sl.butlast + sl0 abort: F discard: T computed: F

Additional information

Used terms: al + al0, sl.butlast + sl0

Proof step 12 from proof move-trans (previous step) (back to tree)

sl.last = sl0[0], frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))(sl.butlast + sl0, al + al0) ∨ sl[0] ≠ (sl.butlast + sl0)[0] ∨ sl0.last ≠ (sl.butlast + sl0).last ∨ ext !(al + al0) ≠ ext !(al) + ext !(al0) ⊦

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 ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ ext !(al + al0) = ext !(al) + ext !(al0); (s)

Proof step 13 from proof move-trans (previous step) (back to tree)

sl.last = sl0[0], frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl, al), ¬ frag*(IOAut(start, step, ext))(sl.butlast + sl0, al + al0) ∨ sl[0] ≠ (sl.butlast + sl0)[0] ∨ sl0.last ≠ (sl.butlast + sl0).last ⊦

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] → [sl, s, sl1]; ['a] → ['s]

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); x ≠ [] ⊦ a = x.last ∧ y = x.butlast ↔ x = y + a '; (elim) Used terms: sl, s, sl1

Proof step 14 from proof move-trans (previous step) (back to tree)

s = sl0[0], sl = sl1 + s ', sl ≠ [], frag*(IOAut(start, step, ext))(sl, al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ sl[0] ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

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 ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 15 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl1 + sl0[0] ', al), ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ (sl1 + sl0[0] ')[0] ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:get-append-less, seq: ⊦ n < # x → (x + y)[n] = x[n], subst:[n, x, y] → [0, sl1, sl0[0] ']; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: 0, sl1, sl0[0] '

Proof step 16 from proof move-trans (previous step) (back to tree)

⊦ n < # x → (x + y)[n] = x[n]

A spec-lemma

Proof step 17 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl1 + sl0[0] ', al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ 0 < # sl1, ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ (sl1 + sl0[0] ')[0] ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

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 ⊦ [] + x = x; (s) ⊦ a '[0] = a; (s) ⊦ # x = 0 ↔ x = []; (s) ⊦ 0 < m ↔ m ≠ 0; (s)

Proof step 18 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl0[0] ', al), ¬ frag*(IOAut(start, step, ext))(sl0, al + al0) ⊦

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), sl0[0] ', al]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ # a ' = 0 +1; (s) ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 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 ⊦ m +1 = m + 1; (s) ⊦ IOAut(start, step, ext).step = step; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ # [] = 0; (s) ⊦ ¬ n < 0; (s) Used terms: IOAut(start, step, ext), sl0[0] ', al

Proof step 19 from proof move-trans (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 20 from proof move-trans (previous step) (back to tree)

0 < # sl1, frag*(IOAut(start, step, ext))(sl1 + sl0[0] ', al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ sl1[0] ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

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 21 from proof move-trans (previous step) (back to tree)

sl1 ≠ [], frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(sl1 + sl0[0] ', al), ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ sl1.head ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

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] → [sl1, s, sl]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: sl1, s, sl

Proof step 22 from proof move-trans (previous step) (back to tree)

sl1 = s + sl, sl1 ≠ [], frag*(IOAut(start, step, ext))(sl1 + sl0[0] ', al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))(sl1 + sl0, al + al0) ∨ s ≠ (sl1 + sl0)[0] ∨ sl0.last ≠ (sl1 + sl0).last ⊦

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 ⊦ a + x = a ' + x; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 23 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))((s ' + sl) + sl0[0] ', al), ¬ frag*(IOAut(start, step, ext))((s ' + sl) + sl0, al + al0) ∨ sl0.last ≠ ((s ' + sl) + sl0).last ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-last, name:lapp, seq: ⊦ y ≠ [] → (x + y).last = y.last, subst:[x, y] → [s ', sl + sl0]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: s ', sl + sl0

Proof step 24 from proof move-trans (previous step) (back to tree)

⊦ y ≠ [] → (x + y).last = y.last

A spec-lemma

Proof step 25 from proof move-trans (previous step) (back to tree)

sl + sl0 ≠ [], frag*(IOAut(start, step, ext))((s ' + sl) + sl0[0] ', al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))((s ' + sl) + sl0, al + al0) ∨ sl0.last ≠ (sl + sl0).last ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: not-finfrag-empty: ⊦ ¬ frag*(IOAut(start, step, ext))([], al); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ y ≠ [] → (x + y).last = y.last; (s) ⊦ x ≠ [] → x[0] = x.head; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 26 from proof move-trans (previous step) (back to tree)

sl0 ≠ [], frag*(IOAut(start, step, ext))(sl0, al0), frag*(IOAut(start, step, ext))(s ' + sl + sl0.head ', al), ¬ frag*(IOAut(start, step, ext))((s ' + sl) + sl0, al + al0) ⊦

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] → [sl0, s0, sl1]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ x ≠ [] → (head = x.head ∧ tail = x.tail ↔ x = head + tail); (elim) Used terms: sl0, s0, sl1

Proof step 27 from proof move-trans (previous step) (back to tree)

sl0 = s0 + sl1, sl0 ≠ [], frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(sl0, al0), ¬ frag*(IOAut(start, step, ext))((s ' + sl) + sl0, al + al0) ⊦

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 ⊦ a + x = a ' + x; (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 28 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 ' + sl1, al + al0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:finfrag-split, seq:⊦ ¬ # al < n ∧ n < # sl → ( frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al.take(n)) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al.drop(n))), subst:[n, al, step, start, ext, sl] → [# al, al + al0, step, start, ext, s ' + sl + s0 ' + sl1]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: ext, # al, al + al0, step, start, s ' + sl + s0 ' + sl1

Proof step 29 from proof move-trans (previous step) (back to tree)

⊦ ¬ # al < n ∧ n < # sl → ( frag*(IOAut(start, step, ext))(sl, al) ↔ frag*(IOAut(start, step, ext))(sl.take(n + 1), al.take(n)) ∧ frag*(IOAut(start, step, ext))(sl.drop(n), al.drop(n)))

Lemma finfrag-split

Proof step 30 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 ' + sl1, al + al0) ⊦ ¬ #(al + al0) < # al ∧ # al < #(s ' + sl + s0 ' + sl1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ ¬ m + n < m; (s)

Proof step 31 from proof move-trans (previous step) (back to tree)

frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), 2 + # sl + # sl1 ≤ # al, ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 ' + sl1, al + al0) ⊦

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), s ' + sl + s0 ', al]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: IOAut(start, step, ext), s ' + sl + s0 ', al

Proof step 32 from proof move-trans (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 33 from proof move-trans (previous step) (back to tree)

#(s ' + sl + s0 ') = # al + 1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), 2 + # sl + # sl1 ≤ # al, ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 ' + sl1, al + al0), ∀ n. n < # al → IOAut(start, step, ext).step((s ' + sl + s0 ')[n], al[n], (s ' + sl + s0 ')[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) a: ⊦ (x + y) + z = x + y + z; from specification list c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ (a ' + x)[n + 1] = x[n]; (s) ⊦ m +1 = m + 1; (s) n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) ⊦ # a ' = 0 +1; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 34 from proof move-trans (previous step) (back to tree)

# al = # sl + 1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), 2 + # sl + # sl1 ≤ # al, ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 ' + sl1, al + al0), ∀ n. n < # al → step((s ' + sl + s0 ')[n], al[n], (sl + s0 ')[n]) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 1, flags: discard

Additional information

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

Proof step 35 from proof move-trans (previous step) (back to tree)

# al < #(s ' + sl + s0 ' + sl1), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), ¬ #(al + al0) < # al, ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).take(# al + 1), (al + al0).take(# al)) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), (al + al0).drop(# al)) ⊦

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 < n0 ⊦ (* n) ≠ m0 + (* n0); (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + y).drop(# x) = y; (s) ⊦ (x + y).take(# x) = x; (s) ⊦ a + x = a ' + x; (s) ⊦ (a ' + x).take(n + 1) = a + x.take(n); (s) ⊦ ¬ m + n < m; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 36 from proof move-trans (previous step) (back to tree)

# al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), ¬ frag*(IOAut(start, step, ext))(s ' + (sl + s0 ' + sl1).take(# al), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-to-gr, seq: ⊦ # x < n → (x + y).take(n) = x + y.take(n - # x), subst:[n, x, y] → [# al, sl, s0 ' + sl1]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: # al, sl, s0 ' + sl1

Proof step 37 from proof move-trans (previous step) (back to tree)

⊦ # x < n → (x + y).take(n) = x + y.take(n - # x)

A spec-lemma

Proof step 38 from proof move-trans (previous step) (back to tree)

# al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ # sl < # al, ¬ frag*(IOAut(start, step, ext))(s ' + (sl + s0 ' + sl1).take(# al), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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 ⊦ n ≤ # x → (x + y).take(n) = x.take(n); (s) ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 39 from proof move-trans (previous step) (back to tree)

# al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), # al ≤ # sl, ¬ frag*(IOAut(start, step, ext))(s ' + sl.take(# al), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: list-del, name:elim-firstn-restn, seq: ⊦ n ≤ # x → (y = x.take(n) ∧ z = x.drop(n) ↔ x = y + z ∧ # y = n), subst:[x, n, y, z] → [sl, # al, sl0, sl2]; ['a] → ['s]

Additional information

Used simplifier rules: ⊦ n ≤ # x → (y = x.take(n) ∧ z = x.drop(n) ↔ x = y + z ∧ # y = n); (elim) Used terms: sl, # al, sl0, sl2

Proof step 40 from proof move-trans (previous step) (back to tree)

# sl0 = # al, sl = sl0 + sl2, # al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), # al ≤ # sl, ¬ frag*(IOAut(start, step, ext))(s ' + sl0, al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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 < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ m < m + n ↔ n ≠ 0; (s) ⊦ m ≤ m + n; (s) lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 41 from proof move-trans (previous step) (back to tree)

# sl0 = # al, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + (sl0 + sl2) + s0 ', al), ¬ frag*(IOAut(start, step, ext))(s ' + sl0, al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + (sl0 + sl2) + s0 ' + sl1).drop(# al), al0) ⊦

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), s ' + (sl0 + sl2) + s0 ', al]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ # a ' = 0 +1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ m = m0 + 1 + n → m ≠ n; (s) Used terms: IOAut(start, step, ext), s ' + (sl0 + sl2) + s0 ', al

Proof step 42 from proof move-trans (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 43 from proof move-trans (previous step) (back to tree)

# al < 2 + # sl + # sl1, # sl < # al, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(# al - # sl), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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 < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ m < m + n ↔ n ≠ 0; (s) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m +1 = m + 1; (s) ⊦ # x = 0 ↔ x = []; (s) trans-nat: ⊦ n < n0 ∧ n0 < n1 → n < n1; (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

Proof step 44 from proof move-trans (previous step) (back to tree)

al ≠ [], # al < 2 + # sl + # sl1, # sl < # al, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(# al - # sl), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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] → [# al, # sl, n0]

Additional information

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

Proof step 45 from proof move-trans (previous step) (back to tree)

# al = n0 + # sl, al ≠ [], # sl < # al, # al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), # sl ≤ # al, ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(n0), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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 < n0 ⊦ (* n) ≠ m0 + (* n0); (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m < m + n ↔ n ≠ 0; (s) 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 < n → m ≤ n; (s)

Proof step 46 from proof move-trans (previous step) (back to tree)

# al = n0 + # sl, al ≠ [], n0 ≠ 0, n0 < 2 + # sl1, # al < 2 + # sl + # sl1, # sl < # al, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(n0), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-to-le, seq: ⊦ n ≤ # x → (x + y).take(n) = x.take(n), subst:[n, x, y] → [n0, s0 ', sl1]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: n0, s0 ', sl1

Proof step 47 from proof move-trans (previous step) (back to tree)

⊦ n ≤ # x → (x + y).take(n) = x.take(n)

A spec-lemma

Proof step 48 from proof move-trans (previous step) (back to tree)

# al = n0 + # sl, al ≠ [], n0 ≠ 0, # sl < # al, # al < 2 + # sl + # sl1, n0 < 2 + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), ¬ n0 ≤ # s0 ', ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(n0), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s) ⊦ # a ' = 0 +1; (s)

Proof step 49 from proof move-trans (previous step) (back to tree)

# al = n0 + # sl, al ≠ [], n0 ≠ 0, n0 ≠ 1, n0 < 2 + # sl1, # al < 2 + # sl + # sl1, # sl < # al, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))(s ' + sl + (s0 ' + sl1).take(n0), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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), s ' + sl + s0 ', al]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ #(x + y) = # x + # y; (s) ⊦ # a ' = 0 +1; (s) n < n0 ⊦ m + (* n) = m0 + (* n0) ↔ m = m0 + (n0 - n); (s) ⊦ m +1 = m + 1; (s) ⊦ (a ' + x)[n + 1] = x[n]; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ IOAut(start, step, ext).step = step; (s) eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic 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 Used terms: IOAut(start, step, ext), s ' + sl + s0 ', al

Proof step 50 from proof move-trans (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 51 from proof move-trans (previous step) (back to tree)

# al = n0 + # sl, al ≠ [], n0 ≠ 0, # sl < # al, # al < 2 + # sl + # sl1, n0 < 2 + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), n0 ≤ # s0 ', ¬ frag*(IOAut(start, step, ext))(s ' + sl + s0 '.take(n0), al) ∨ ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

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) n < n0 ⊦ (* n) ≠ m0 + (* n0); (s) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) ⊦ m + n < m + n0 ↔ n < n0; (s) ⊦ m ≤ m; (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 eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.take(1) = a '; (s) ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s) ⊦ # a ' = 0 +1; (s)

Proof step 52 from proof move-trans (previous step) (back to tree)

# al = 1 + # sl, al ≠ [], # al < 2 + # sl + # sl1, # sl < # al, frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), ¬ frag*(IOAut(start, step, ext))((s ' + sl + s0 ' + sl1).drop(# al), al0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, name:append-from-le, seq: ⊦ n ≤ # x → (x + y).drop(n) = x.drop(n) + y, subst:[n, x, y] → [# al, s ' + sl, s0 ' + sl1]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: # al, s ' + sl, s0 ' + sl1

Proof step 53 from proof move-trans (previous step) (back to tree)

⊦ n ≤ # x → (x + y).drop(n) = x.drop(n) + y

A spec-lemma

Proof step 54 from proof move-trans (previous step) (back to tree)

# al = 1 + # sl, al ≠ [], # sl < # al, # al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), # al ≤ #(s ' + sl), ¬ frag*(IOAut(start, step, ext))((s ' + sl).drop(# al) + s0 ' + sl1, al0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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 c: ⊦ m + n = n + m; from specification nat-basic a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic ⊦ m < n → (n ≤ m + 1 ↔ n = m + 1); (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s)

Proof step 55 from proof move-trans (previous step) (back to tree)

# al = # sl + 1, al ≠ [], # sl < # al, # al < 2 + # sl + # sl1, frag*(IOAut(start, step, ext))(s0 ' + sl1, al0), frag*(IOAut(start, step, ext))(s ' + sl + s0 ', al), ¬ frag*(IOAut(start, step, ext))((s ' + sl).drop(# al) + s0 ' + sl1, al0) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list-del, name:rest-empty, seq: ⊦ # x = n → x.drop(n) = [], subst:[x, n] → [s ' + sl, # al]; ['a] → ['s], <all paths>

Additional information

Used simplifier rules: ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m +1 = m + 1; (s) ⊦ [] + x = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: s ' + sl, # al

Proof step 56 from proof move-trans (previous step) (back to tree)

⊦ # x = n → x.drop(n) = []

A spec-lemma

Proof Tree for move-trans (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