Proof step 1 from proof relyp-fo-follows-w-stable (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  FOFollowsW(t, p0(l)), relyp(t, p0, p1), ¬ FOFollowsW(t, p1(l)) ⊦

Rule argument

Additional information

Proof step 2 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 3:

  ⊦ relyp(t, p, p0) ↔ (∀ l. relyp(t, p(l), p0(l)))

• Proof step 4:

  FOFollowsW(t, p0(l)), ¬ FOFollowsW(t, p1(l)), ∀ l. relyp(t, p0(l), p1(l)) ⊦

Rule argument

Additional information

Used terms: t, p0, p1

Proof step 3 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Lemma allrelyp-def

Proof step 4 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 5:

  FOFollowsW(t, p0(l)), relyp(t, p0(l), p1(l)), ¬ FOFollowsW(t, p1(l)) ⊦

Rule argument

Additional information

Used terms: l

Proof step 5 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 6:

  ⊦ relyp(t, fowl, fowl0) ↔ (∃ fowl1, fowl2. fowl ⊒ fowl1 ∧ fowl0 = fowl1 + fowl2 ∧ fowl2 | t = [])

• Proof step 7:

  FOFollowsW(t, p0(l)), ¬ FOFollowsW(t, p1(l)), ∃ fowl1, fowl2. p0(l) ⊒ fowl1 ∧ p1(l) = fowl1 + fowl2 ∧ fowl2 | t = [] ⊦

Rule argument

Additional information

Used terms: t, p0(l), p1(l)

Proof step 6 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Lemma relyp-def

Proof step 7 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, FOFollowsW(t, p0(l)), p0(l) ⊒ fowl1, ¬ FOFollowsW(t, fowl1 + fowl2) ⊦

Rule argument

Additional information

Used simplifier rules: filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t);

Proof step 8 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

  ⊦ FOFollowsW(t, fowl) ↔ (∀ n. n < # fowl ∧ fowl[n].W? ∧ fowl[n].tid = t → (∃ n0. n < n0 ∧ n0 < # fowl ∧ fowl[n0] = FO(t)))

• Proof step 10:

  fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, p0(l) ⊒ fowl1, ¬ FOFollowsW(t, fowl1 + fowl2), ∀ n. n < # p0(l) ∧ p0(l)[n].W? ∧ p0(l)[n].tid = t → (∃ n0. n < n0 ∧ n0 < # p0(l) ∧ p0(l)[n0] = FO(t)) ⊦

Rule argument

Additional information

Used terms: t, p0(l)

Proof step 9 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Lemma FOFollowsW-def

Proof step 10 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, p0(l) ⊒ fowl1, ¬ FOFollowsW(t, fowl1 + fowl2), ∀ n. n < # p0(l) ∧ p0(l)[n].W? ∧ p0(l)[n].tid = t → (∃ n0. n < n0 ∧ n0 < # p0(l) ∧ p0(l)[n0] = FO(p0(l)[n].tid)) ⊦

Rule argument

Additional information

Proof step 11 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

  ⊦ FOFollowsW(t, fowl) ↔ (∀ n. n < # fowl ∧ fowl[n].W? ∧ fowl[n].tid = t → (∃ n0. n < n0 ∧ n0 < # fowl ∧ fowl[n0] = FO(t)))

• Proof step 13:

  fowl2.filter(λ fow. fow.tid = t) = [], p1(l) = fowl1 + fowl2, p0(l) ⊒ fowl1, ∀ n. n < # p0(l) ∧ p0(l)[n].W? ∧ p0(l)[n].tid = t → (∃ n0. n < n0 ∧ n0 < # p0(l) ∧ p0(l)[n0] = FO(p0(l)[n].tid)), ∃ n. ¬ ( n < #(fowl1 + fowl2) ∧ (fowl1 + fowl2)[n].W? ∧ (fowl1 + fowl2)[n].tid = t → (∃ n0. n < n0 ∧ n0 < #(fowl1 + fowl2) ∧ (fowl1 + fowl2)[n0] = FO(t))) ⊦

Rule argument

Additional information

Used terms: t, fowl1 + fowl2

Proof step 12 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Lemma FOFollowsW-def

Proof step 13 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  fowl2.filter(λ fow. fow.tid = (fowl1 + fowl2)[n].tid) = [], p1(l) = fowl1 + fowl2, (fowl1 + fowl2)[n].W?, n < # fowl1 + # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = (fowl1 + fowl2)[n].tid → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO((fowl1 + fowl2)[n].tid)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s) 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 14 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 15:

  fowl2.filter(λ fow. fow.tid = t) = [], (fowl1 + fowl2)[n] = W(v, t), p1(l) = fowl1 + fowl2, (fowl1 + fowl2)[n].W?, n < # fowl1 + # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)); (elim) Used terms: (fowl1 + fowl2)[n], v, t

Proof step 15 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  fowl2.filter(λ fow. fow.tid = t) = [], (fowl1 + fowl2)[n] = W(v, t), p1(l) = fowl1 + fowl2, n < # fowl1 + # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ W(v, tid).W?; (s)

Proof step 16 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

  ⊦ n < # x → (x + y)[n] = x[n]

• Proof step 18:

  fowl2.filter(λ fow. fow.tid = t) = [], (fowl1 + fowl2)[n] = W(v, t), p1(l) = fowl1 + fowl2, n < # fowl1 + # fowl2, p0(l) ⊒ fowl1, ¬ n < # fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

• Proof step 30:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, n < # fowl1 + # fowl2, n < # fowl1, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: n, fowl1, fowl2

Proof step 17 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 18 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl2[n - # fowl1] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n < # fowl1 + # fowl2, # fowl1 ≤ n, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

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-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ ¬ n < # x → (x + y)[n] = y[n - # x]; (s)

Proof step 19 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 20:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl2[n2] = W(v, t), n = n2 + # fowl1, p1(l) = fowl1 + fowl2, fowl2 ≠ [], n < # fowl1 + # fowl2, # fowl1 ≤ n, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, # fowl1, n2

Proof step 20 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (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 < m + n0 ↔ n < n0; (s) ⊦ m ≤ m + n; (s)

Proof step 21 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 22:

  ⊦ x = [] ↔ ¬(∃ a. a ∈ x)

• Proof step 23:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ∀ a. ¬ a ∈ fowl2.filter(λ fow. fow.tid = t) ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: fowl2.filter(λ fow. fow.tid = t)

Proof step 22 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 23 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 24:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ∀ a. ¬ a ∈ fowl2 ∨ a.tid ≠ t ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ a ∈ x.filter(p) ↔ a ∈ x ∧ p(a); (s)

Proof step 24 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 25:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ¬ W(v, t) ∈ fowl2 ∨ W(v, t).tid ≠ t ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: W(v, t)

Proof step 25 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 26:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ¬ W(v, t) ∈ fowl2, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ W(v, tid).tid = tid; (s)

Proof step 26 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 27:

  ⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a)

• Proof step 28:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ∀ n. ¬(n < # fowl2 ∧ fowl2[n] = W(v, t)) ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: W(v, t), fowl2

Proof step 27 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 28 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 29:

  fowl2[n2] = W(v, t), p1(l) = fowl1 + fowl2, fowl2 ≠ [], n2 < # fowl2, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ∀ n. ¬ n < # fowl2 ∨ fowl2[n] ≠ W(v, t) ⊦ ∃ n0. n2 + # fowl1 < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Proof step 29 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n2

Proof step 30 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 31:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1 + # fowl2, n < # fowl1, p0(l) ⊒ fowl1, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic

Proof step 31 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 32:

  ⊦ x ⊒ y ↔ (∃ z. z + y = x)

• Proof step 33:

  fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1, n < # fowl1 + # fowl2, ∀ n0. n0 < # p0(l) ∧ p0(l)[n0].W? ∧ p0(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p0(l) ∧ p0(l)[n1] = FO(p0(l)[n0].tid)), ∃ z. z + fowl1 = p0(l) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: p0(l), fowl1

Proof step 32 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 33 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 34:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1, n < # fowl1 + # fowl2, ∀ n0. n0 < # fowl + # fowl1 ∧ (fowl + fowl1)[n0].W? ∧ (fowl + fowl1)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # fowl + # fowl1 ∧ (fowl + fowl1)[n1] = FO((fowl + fowl1)[n0].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ #(x + y) = # x + # y; (s)

Proof step 34 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule all left

And obtained the following new goals

• Proof step 35:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1 + # fowl2, n < # fowl1, # fowl + n < # fowl + # fowl1 ∧ (fowl + fowl1)[# fowl + n].W? ∧ (fowl + fowl1)[# fowl + n].tid = t → (∃ n1. # fowl + n < n1 ∧ n1 < # fowl + # fowl1 ∧ (fowl + fowl1)[n1] = FO((fowl + fowl1)[# fowl + n].tid)) ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used terms: # fowl + n

Proof step 35 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 36:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], (fowl + fowl1)[n1] = FO(t), fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n1 < # fowl + # fowl1, # fowl + n < n1, n < # fowl1 + # fowl2, n + 1 < # fowl1, n < # fowl1 ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

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) ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (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 ⊦ W(v, tid).tid = tid; (s) ⊦ W(v, tid).W?; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (x + y)[n + # x] = y[n]; (s)

Proof step 36 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 37:

  ⊦ ¬ n < # x → (x + y)[n] = y[n - # x]

• Proof step 38:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), fowl1[n1 - # fowl] = FO(t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1, n + 1 < # fowl1, n < # fowl1 + # fowl2, # fowl + n < n1, n1 < # fowl + # fowl1, ¬ n1 < # fowl ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ n < # x → (x + y)[n] = x[n]; (s) a: ⊦ (x + y) + z = x + y + z; from specification list f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic 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 ⊦ # x = 0 ↔ x = []; (s) ⊦ m +1 = m + 1; (s) ⊦ m + m0 < n → ¬ n + n0 < m; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic Used terms: n1, fowl, fowl1

Proof step 37 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

A spec-lemma

Proof step 38 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 39:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n] = W(v, t), fowl1[n1 - # fowl] = FO(t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1, n + 1 < # fowl1, n < # fowl1 + # fowl2, # fowl + n < n1, n1 < # fowl + # fowl1, # fowl ≤ n1 ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

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-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat ⊦ m < n ↔ ¬ n ≤ m; (ss)

Proof step 39 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 40:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n2] = FO(t), fowl1[n] = W(v, t), n1 = n2 + # fowl, p1(l) = fowl1 + fowl2, fowl1 ≠ [], n1 < # fowl + # fowl1, # fowl + n < n1, n < # fowl1 + # fowl2, n + 1 < # fowl1, n < # fowl1, # fowl ≤ n1 ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n1, # fowl, n2

Proof step 40 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 41:

  fowl + fowl1 = p0(l), fowl2.filter(λ fow. fow.tid = t) = [], fowl1[n2] = FO(t), fowl1[n] = W(v, t), p1(l) = fowl1 + fowl2, fowl1 ≠ [], n < # fowl1, n2 < # fowl1, n < n2, n < # fowl1 + # fowl2, n + 1 < # fowl1 ⊦ ∃ n0. n < n0 ∧ n0 < # fowl1 + # fowl2 ∧ (fowl1 + fowl2)[n0] = FO(t)

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) ⊦ m < n → n ≠ 0; (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 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) ⊦ m ≤ m + n; (s)

Proof step 41 from proof relyp-fo-follows-w-stable (previous step) (back to tree)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ n < # x → (x + y)[n] = x[n]; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m < n ↔ ¬ n ≤ m; (ss) fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat ⊦ ¬ 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: n2