Proof step 1 from proof one-pending-no-fo-no-pending (conclusion) (back to tree)

⊦ onePendingW(t, l, p) ∧ FOFollowsW(t, p(l)) ∧ p(l) = FO(t) ' + fowl ∧ ¬ FO(t) ∈ fowl → noPendingW(t, p)

System applied rule normalize

And obtained the following new goals

Proof step 2:
p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), onePendingW(t, l, p), ¬ noPendingW(t, p), ¬ FO(t) ∈ fowl ⊦

Rule argument

Additional information

Proof step 2 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), onePendingW(t, l, p), ¬ noPendingW(t, p), ¬ FO(t) ∈ fowl ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
p(l) = FO(t) ' + fowl, p(l0).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

Rule argument

Additional information

Used simplifier rules: onePendingW: ⊦ onePendingW(t, l, p) ↔ (∀ l0. l0 ≠ l → p(l0) |W | t = []); noPendingW: ⊦ noPendingW(t, p) ↔ (∀ l. p(l) |W | t = []); filterwrites-def: ⊦ fowl |W = fowl.filter(λ fow. fow.W?); filtertid-def: ⊦ fowl | t = fowl.filter(λ fow. fow.tid = t); a: ⊦ (x + y) + z = x + y + z; from specification list in-14: ⊦ a ' + x = y → a ∈ y; (forward) from specification list

Proof step 3 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, p(l0).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

User applied rule all left

And obtained the following new goals

Proof step 4:
p(l) = FO(t) ' + fowl, p(l0).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [], ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

Rule argument

pos: left 6 terms: l0 abort: F discard: F computed: T

Additional information

Used terms: l0

Proof step 4 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, p(l0).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [], ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
p(l) = FO(t) ' + fowl, p(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

Rule argument

Additional information

Proof step 5 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, p(l).filter(.W?).filter(λ fow. fow.tid = t) ≠ [], FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ x = [] ↔ ¬(∃ a. a ∈ x)
Proof step 7:
p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [], ∃ a. a ∈ p(l).filter(.W?).filter(λ fow. fow.tid = t) ⊦

Rule argument

spec: list, name:some-element, seq: ⊦ x = [] ↔ ¬(∃ a. a ∈ x), subst:[x] → [p(l).filter(.W?).filter(λ fow. fow.tid = t)]; ['a] → [FOW], <all paths>

Additional information

Used terms: p(l).filter(.W?).filter(λ fow. fow.tid = t)

Proof step 6 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

⊦ x = [] ↔ ¬(∃ a. a ∈ x)

A spec-lemma

Proof step 7 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow. fow.tid = t) = [], ∃ a. a ∈ p(l).filter(.W?).filter(λ fow. fow.tid = t) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
p(l) = FO(fow.tid) ' + fowl, fow.W?, FOFollowsW(fow.tid, p(l)), fow ∈ p(l), FO(fow.tid) ∈ p(l), ¬ FO(fow.tid) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [] ⊦

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.filter(p) ↔ a ∈ x ∧ p(a); (s)

Proof step 8 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(fow.tid) ' + fowl, fow.W?, FOFollowsW(fow.tid, p(l)), fow ∈ p(l), FO(fow.tid) ∈ p(l), ¬ FO(fow.tid) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = fow.tid) = [] ⊦

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

Proof step 9:
fow = W(v, t), p(l) = FO(t) ' + fowl, fow.W?, FOFollowsW(t, p(l)), FO(t) ∈ p(l), fow ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

spec: FOW, name:elim-W, seq: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)), subst:[fow, val, tid] → [fow, v, t]

Additional information

Used simplifier rules: ⊦ fow.W? → (val = fow.val ∧ tid = fow.tid ↔ fow = W(val, tid)); (elim) Used terms: fow, v, t

Proof step 9 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

fow = W(v, t), p(l) = FO(t) ' + fowl, fow.W?, FOFollowsW(t, p(l)), FO(t) ∈ p(l), fow ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), W(v, t) ∈ p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

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 ⊦ W(v, tid).W?; (s)

Proof step 10 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), W(v, t) ∈ p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a)
Proof step 12:
p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∃ n. n < # p(l) ∧ p(l)[n] = W(v, t) ⊦

Rule argument

spec: list-del, name:in-get, seq: ⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a), subst:[a, x] → [W(v, t), p(l)]; ['a] → [FOW], <all paths>

Additional information

Used terms: W(v, t), p(l)

Proof step 11 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a)

A spec-lemma

Proof step 12 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l) = FO(t) ' + fowl, FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∃ n. n < # p(l) ∧ p(l)[n] = W(v, t) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

Additional information

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

Proof step 13 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FOFollowsW(t, p(l)), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ FOFollowsW(t, fowl) ↔ (∀ n. n < # fowl ∧ fowl[n].W? ∧ fowl[n].tid = t → (∃ n0. n < n0 ∧ n0 < # fowl ∧ fowl[n0] = FO(t)))
Proof step 15:
p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n. n < # p(l) ∧ p(l)[n].W? ∧ p(l)[n].tid = t → (∃ n0. n < n0 ∧ n0 < # p(l) ∧ p(l)[n0] = FO(t)) ⊦

Rule argument

spec: <toplevel>, name:FOFollowsW-def, seq:⊦ FOFollowsW(t, fowl) ↔ (∀ n. n < # fowl ∧ fowl[n].W? ∧ fowl[n].tid = t → (∃ n0. n < n0 ∧ n0 < # fowl ∧ fowl[n0] = FO(t))), subst:[t, fowl] → [t, p(l)], <all paths>

Additional information

Used terms: t, p(l)

Proof step 14 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

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

Lemma FOFollowsW-def

Proof step 15 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n. n < # p(l) ∧ p(l)[n].W? ∧ p(l)[n].tid = t → (∃ n0. n < n0 ∧ n0 < # p(l) ∧ p(l)[n0] = FO(t)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 16:
p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. n0 < # p(l) ∧ p(l)[n0].W? ∧ p(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p(l) ∧ p(l)[n1] = FO(p(l)[n0].tid)) ⊦

Rule argument

Additional information

Proof step 16 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. n0 < # p(l) ∧ p(l)[n0].W? ∧ p(l)[n0].tid = t → (∃ n1. n0 < n1 ∧ n1 < # p(l) ∧ p(l)[n1] = FO(p(l)[n0].tid)) ⊦

User applied rule all left

And obtained the following new goals

Proof step 17:
p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, n < # p(l) ∧ p(l)[n].W? ∧ p(l)[n].tid = t → (∃ n1. n < n1 ∧ n1 < # p(l) ∧ p(l)[n1] = FO(p(l)[n].tid)), ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

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

Additional information

Used terms: n

Proof step 17 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, n < # p(l) ∧ p(l)[n].W? ∧ p(l)[n].tid = t → (∃ n1. n < n1 ∧ n1 < # p(l) ∧ p(l)[n1] = FO(p(l)[n].tid)), ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
p(l)[n1] = FO(t), p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n1 < # p(l), n < n1, n < # p(l), n + 1 < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = 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 ⊦ W(v, tid).tid = tid; (s) ⊦ W(v, tid).W?; (s)

Proof step 18 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

p(l)[n1] = FO(t), p(l)[n] = W(v, t), p(l) = FO(t) ' + fowl, p(l) ≠ [], n1 < # p(l), n < n1, n < # p(l), n + 1 < # p(l), FO(t) ∈ p(l), ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

User applied rule subst equation

And obtained the following new goals

Proof step 19:
(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), FO(t) ' + fowl ≠ [], n + 1 < #(FO(t) ' + fowl), n < #(FO(t) ' + fowl), n < n1, n1 < #(FO(t) ' + fowl), FO(t) ∈ FO(t) ' + fowl, ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

pos: left 3, flags: discard

Additional information

Proof step 19 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), FO(t) ' + fowl ≠ [], n + 1 < #(FO(t) ' + fowl), n < #(FO(t) ' + fowl), n < n1, n1 < #(FO(t) ' + fowl), FO(t) ∈ FO(t) ' + fowl, ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), fowl ≠ [], n < # fowl, n < n1, n1 ≤ # fowl, ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # x = 0 ↔ x = []; (s) fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic ⊦ m < n → ¬ n < m; (s) ⊦ 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 + n < m + n0 ↔ n < n0; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ m +1 = m + 1; (s) ⊦ #(a ' + x) = (# x)+1; (s) ⊦ m < n + 1 ↔ ¬ n < m; (s) ⊦ ¬ a ∈ x → (a ∈ a0 ' + x ↔ a = a0); (s) ⊦ a ' ≠ []; (s) ⊦ x + y = [] ↔ x = [] ∧ y = []; (s)

Proof step 20 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), fowl ≠ [], n < # fowl, n < n1, n1 ≤ # fowl, ¬ FO(t) ∈ fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a)
Proof step 22:
(FO(t) ' + fowl)[n1] = FO(t), (FO(t) ' + fowl)[n] = W(v, t), fowl ≠ [], n < n1, n < # fowl, n1 ≤ # fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n. ¬(n < # fowl ∧ fowl[n] = FO(t)) ⊦

Rule argument

spec: list-del, name:in-get, seq: ⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a), subst:[a, x] → [FO(t), fowl]; ['a] → [FOW], <all paths>

Additional information

Used terms: FO(t), fowl

Proof step 21 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

⊦ a ∈ x ↔ (∃ n. n < # x ∧ x[n] = a)

A spec-lemma

Proof step 22 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n1] = FO(t), (FO(t) ' + fowl)[n] = W(v, t), fowl ≠ [], n < n1, n < # fowl, n1 ≤ # fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n. ¬(n < # fowl ∧ fowl[n] = FO(t)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), fowl ≠ [], n < # fowl, n < n1, n1 ≤ # fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t) ⊦

Rule argument

Additional information

Proof step 23 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n] = W(v, t), (FO(t) ' + fowl)[n1] = FO(t), fowl ≠ [], n < # fowl, n < n1, n1 ≤ # fowl, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 24:
(FO(t) ' + fowl)[n1] = FO(t), (FO(t) ' + fowl)[n] = W(v, t), n1 = n2 +1, fowl ≠ [], n < n1, n < # fowl, n1 ≤ # fowl, ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t), ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Rule argument

var: n1

Additional information

Used simplifier rules: Nat freely generated by 0, +1;; ⊦ (a ' + x)[0] = a; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ FO(tid) = FO(tid0) ↔ tid = tid0; (s) ⊦ 0 ≤ m; (s) ⊦ ¬ n < 0; (s)

Proof step 24 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n1] = FO(t), (FO(t) ' + fowl)[n] = W(v, t), n1 = n2 +1, fowl ≠ [], n < n1, n < # fowl, n1 ≤ # fowl, ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t), ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [] ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
(FO(t) ' + fowl)[n] = W(v, t), fowl[n2] = FO(t), fowl ≠ [], n2 < # fowl, n < # fowl, n ≤ n2, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ 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 < n + 1 ↔ ¬ n < m; (s) ⊦ m +1 = m + 1; (s) ⊦ m + 1 ≤ n ↔ m < n; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[n + 1] = x[n]; (s)

Proof step 25 from proof one-pending-no-fo-no-pending (previous step) (back to tree)

(FO(t) ' + fowl)[n] = W(v, t), fowl[n2] = FO(t), fowl ≠ [], n2 < # fowl, n < # fowl, n ≤ n2, ∀ l0. l0 ≠ l → p(l0).filter(.W?).filter(λ fow0. fow0.tid = t) = [], ∀ n0. ¬ n0 < # fowl ∨ fowl[n0] ≠ FO(t) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: n2

Additional information

Used terms: n2

Proof Tree for one-pending-no-fo-no-pending (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.