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Proof step 1 from proof reorder-step-persistFO (conclusion) (back to tree)

⊦ reorder(l, fowl0, FO(t) ' + fowl1, rfo) ∧ isPersistFO(n, fowl0) ∧ fowl0[n] = FO(t) → reorder(l, fowl0 -p n, fowl1, rfo)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof reorder-step-persistFO (previous step) (back to tree)

fowl0[n] = FO(t), isPersistFO(n, fowl0), reorder(l, fowl0, FO(t) ' + fowl1, rfo), ¬ reorder(l, fowl0 -p n, fowl1, rfo) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: fowl0

Additional information

Used simplifier rules: reorder-base: ⊦ reorder(l, [], fowl0, rfo) ↔ fowl0 = [] ∧ (∀ t. propFOs(t, l, rfo(t)) = []); List('a) freely generated by [], +;; ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) a: ⊦ (x + y) + z = x + y + z; from specification list

Proof step 3 from proof reorder-step-persistFO (previous step) (back to tree)

(fow + fowl)[n] = FO(t), isPersistFO(n, fow + fowl), reorder(l, fow + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, (fow + fowl) -p n, fowl1, rfo), ∀ n, t, l, fowl1, rfo. ¬(fowl[n] = FO(t) ∧ isPersistFO(n, fowl) ∧ reorder(l, fowl, FO(t) ' + fowl1, rfo) ∧ ¬ reorder(l, fowl -p n, fowl1, rfo)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 4 from proof reorder-step-persistFO (previous step) (back to tree)

(fow ' + fowl)[n] = FO(t), isPersistFO(n, fow ' + fowl), reorder(l, fow ' + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, (fow ' + fowl) -p n, fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: n

Additional information

Used simplifier rules: Nat freely generated by 0, +1;;

Proof step 5 from proof reorder-step-persistFO (previous step) (back to tree)

(fow ' + fowl)[n] = FO(t), n = 0, isPersistFO(n, fow ' + fowl), reorder(l, fow ' + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, (fow ' + fowl) -p n, fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-rec-FO: ⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l))); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ FO(tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ (a ' + x) -p 0 = x; (s) ⊦ (a ' + x)[0] = a; (s)

Proof step 6 from proof reorder-step-persistFO (previous step) (back to tree)

FO(t) ' + fowl1 = fowl2 + FO(t) ' + fowl3, fowl2 | t = [], isPersistFO(0, FO(t) ' + fowl), reorder(l, fowl, fowl2 + fowl3, rfo), ¬ reorder(l, fowl, fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: fowl2

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s) ⊦ a ' ≠ []; (s) ⊦ a ' + x = a0 ' + y ↔ a = a0 ∧ x = y; (ws) ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ FO(tid).tid = tid; (s) ⊦ [] | t = []; (s) ⊦ # x = # z → (x + y = z + y0 ↔ x = z ∧ y = y0); (ws) ⊦ [] + x = x; (s)

Proof step 7 from proof reorder-step-persistFO (previous step) (back to tree)

(fow ' + fowl)[n] = FO(t), n = n1 +1, isPersistFO(n, fow ' + fowl), reorder(l, fow ' + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, (fow ' + fowl) -p n, fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ FO(tid).tid = tid; (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 = a ' + x; (s) ⊦ (a ' + x) -p (n + 1) = a + (x -p n); (s) ⊦ m +1 = m + 1; (s) ⊦ isPersistFO(n + 1, fow ' + fowl) ↔ isPersistFO(n, fowl) ∧ fow.FO? ∧ fowl[n].tid ≠ fow.tid; (s) ⊦ (a ' + x)[n + 1] = x[n]; (s)

Proof step 8 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), t ≠ fow.tid, fow.FO?, isPersistFO(n1, fowl), reorder(l, fow ' + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, fow ' + (fowl -p n1), fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: FOW, name:elim-FO, seq: ⊦ fow.FO? → (tid = fow.tid ↔ fow = FO(tid)), subst:[fow, tid] → [fow, t1]

Additional information

Used simplifier rules: ⊦ fow.FO? → (tid = fow.tid ↔ fow = FO(tid)); (elim) Used terms: fow, t1

Proof step 9 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fow = FO(t1), t ≠ t1, fow.FO?, isPersistFO(n1, fowl), reorder(l, fow ' + fowl, FO(t) ' + fowl1, rfo), ¬ reorder(l, fow ' + (fowl -p n1), fowl1, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: reorder-rec-FO: ⊦ reorder(l, FO(t) ' + fowl, fowl0, rfo) ↔ (∃ fowl1, fowl2. fowl0 = fowl1 + FO(t) ' + fowl2 ∧ fowl1 | t = [] ∧ reorder(l, fowl, fowl1 + fowl2, rfo)) ∨ fowl0 | t = [] ∧ FO(l) ∈ rfo(t) ∧ reorder(l, fowl, fowl0, rfo(t := rfo(t) -1l FO(l))); a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ FO(tid).FO?; (s) ⊦ [] + x = x; (s) ⊦ FO(tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s)

Proof step 10 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), t ≠ t1, isPersistFO(n1, fowl), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0), FO(t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl, fowl2 + fowl3, rfo) ∨ fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl, FO(t) ' + fowl1, rfo(t1 := rfo(t1) -1l FO(l))) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo), fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l)))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 5

Additional information

Proof step 11 from proof reorder-step-persistFO (previous step) (back to tree)

FO(t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl[n1] = FO(t), fowl2 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, fowl2 + fowl3, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo), fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l)))

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 ⊦ FO(tid) = FO(tid0) ↔ tid = tid0; (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws) ⊦ a ∈ a0 ' + x ↔ a = a0 ∨ a ∈ x; (s) ⊦ a ∈ x + y ↔ a ∈ x ∨ a ∈ y; (s) in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 12 from proof reorder-step-persistFO (previous step) (back to tree)

FO(t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl[n1] = FO(t), fowl2 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, fowl2 + fowl3, rfo), FO(t) ∈ fowl2, ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo), fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l)))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: list, name:firstoc-rew, seq: ⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ y), subst:[a, x] → [FO(t), fowl2]; ['a] → [FOW], <all paths>

Additional information

Used terms: FO(t), fowl2

Proof step 13 from proof reorder-step-persistFO (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ y, z. x = y + a ' + z ∧ ¬ a ∈ y)

A spec-lemma

Proof step 14 from proof reorder-step-persistFO (previous step) (back to tree)

FO(t) ' + fowl1 = fowl2 + FO(t1) ' + fowl3, fowl[n1] = FO(t), fowl2 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, fowl2 + fowl3, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0), ∃ y, z. fowl2 = y + FO(t) ' + z ∧ ¬ FO(t) ∈ y ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo), fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l)))

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) ⊦ FO(tid).tid = tid; (s) ⊦ fow ' | t = (t = fow.tid ⊃ fow ';[]); (s) ⊦ fowl + fowl0 | t = (fowl | t) + (fowl0 | t); (s) ⊦ ¬ a ∈ []; (s) ⊦ [] + x = x; (s) ⊦ [] | t = []; (s) ⊦ ¬ a ∈ y → (y + a0 ' + z = a ' + z0 ↔ y = [] ∧ a0 = a ∧ z = z0); (ws)

Proof step 15 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fowl4 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, FO(t) ' + fowl4 + fowl3, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl0. fowl4 + FO(t1) ' + fowl3 = fowl2 + FO(t1) ' + fowl0 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl0, rfo)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: n1, t, l, fowl4 + fowl3, rfo abort: F discard: F computed: T

Additional information

Used terms: n1, t, l, fowl4 + fowl3, rfo

Proof step 16 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fowl4 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, FO(t) ' + fowl4 + fowl3, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0), fowl[n1] ≠ FO(t) ∨ ¬ isPersistFO(n1, fowl) ∨ ¬ reorder(l, fowl, FO(t) ' + fowl4 + fowl3, rfo) ∨ reorder(l, fowl -p n1, fowl4 + fowl3, rfo) ⊦ ∃ fowl2, fowl0. fowl4 + FO(t1) ' + fowl3 = fowl2 + FO(t1) ' + fowl0 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl0, rfo)

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

Proof step 17 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fowl4 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, FO(t) ' + fowl4 + fowl3, rfo), reorder(l, fowl -p n1, fowl4 + fowl3, rfo), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl0. fowl4 + FO(t1) ' + fowl3 = fowl2 + FO(t1) ' + fowl0 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl0, rfo)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: fowl4, fowl3

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: fowl4, fowl3

Proof step 18 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fowl1 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, FO(t) ' + fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo), fowl1 | t1 = [] ∧ FO(l) ∈ rfo(t1) ∧ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l)))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 19 from proof reorder-step-persistFO (previous step) (back to tree)

fowl[n1] = FO(t), fowl1 | t1 = [], t ≠ t1, isPersistFO(n1, fowl), reorder(l, fowl, FO(t) ' + fowl1, rfo(t1 := rfo(t1) -1l FO(l))), FO(l) ∈ rfo(t1), ¬ reorder(l, fowl -p n1, fowl1, rfo(t1 := rfo(t1) -1l FO(l))), ∀ n0, t0, l0, fowl0, rfo0. fowl[n0] ≠ FO(t0) ∨ ¬ isPersistFO(n0, fowl) ∨ ¬ reorder(l0, fowl, FO(t0) ' + fowl0, rfo0) ∨ reorder(l0, fowl -p n0, fowl0, rfo0) ⊦ ∃ fowl2, fowl3. fowl1 = fowl2 + FO(t1) ' + fowl3 ∧ fowl2 | t1 = [] ∧ reorder(l, fowl -p n1, fowl2 + fowl3, rfo)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: n1, t, l, fowl1, rfo(t1 := rfo(t1) -1l FO(l))

Additional information

Used terms: n1, t, l, fowl1, rfo(t1 := rfo(t1) -1l FO(l))

Proof Tree for reorder-step-persistFO (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