Proof step 1 from proof frag-rec (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

  ¬(frag(IOAut(start, step, ext))(⌜s '⌝ + σ, ⌜a '⌝ + α) ↔ step(s, a, hd(σ)) ∧ frag(IOAut(start, step, ext))(σ, α)) ⊦

Rule argument

Additional information

Proof step 2 from proof frag-rec (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 3:

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

• Proof step 4:

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

Rule argument

Additional information

Used terms: ext, ⌜a '⌝ + α, step, start, ⌜s '⌝ + σ

Proof step 3 from proof frag-rec (previous step) (back to tree)

Lemma frag

Proof step 4 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 5:

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

Rule argument

Additional information

Used simplifier rules: 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 ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) ⊦ m +1 = m + 1; (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⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ n < m → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ n∞1 = ⌜m - n⌝ + n∞2); (s) ⊦ ⌜(* n)⌝ + ⌜(* m)⌝ = ⌜n + m⌝ ; (s)

Proof step 5 from proof frag-rec (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 6:

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

• Proof step 7:

  ⊦ # σ = ⌜1⌝ + # α ∧ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + σ)[n], (⌜a '⌝ + α)[n], σ[n])) ↔ step(s, a, hd(σ)) ∧ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1]))

Rule argument

Additional information

Used terms: ext, α, step, start, σ

Proof step 6 from proof frag-rec (previous step) (back to tree)

Lemma frag

Proof step 7 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ⊦ # σ = ⌜1⌝ + # α ∧ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + σ)[n], (⌜a '⌝ + α)[n], σ[n])) ↔ step(s, a, hd(σ)) ∧ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n + 1]))

Rule argument

Additional information

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

Proof step 8 from proof frag-rec (previous step) (back to tree)

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

• Proof step 9:

  # σ = ⌜1⌝ + # α ⊦ # σ = ⌜1⌝ + # α ∧ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + σ)[n], (⌜a '⌝ + α)[n], σ[n])) ↔ step(s, a, hd(σ)) ∧ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n + 1]))

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used formulas: # σ = ⌜1⌝ + # α

Proof step 9 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 10:

  # σ = ⌜1⌝ + # α ⊦ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + σ)[n], (⌜a '⌝ + α)[n], σ[n])) ↔ step(s, a, hd(σ)) ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n + 1]))

Rule argument

Additional information

Used simplifier rules: add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf

Proof step 10 from proof frag-rec (previous step) (back to tree)

User applied rule apply elim lemma

And obtained the following new goals

• Proof step 11:

  # σ = ⌜1⌝ + # α, σ = s0 + σ0, σ ≠ [] ⊦ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + σ)[n], (⌜a '⌝ + α)[n], σ[n])) ↔ step(s, a, s0) ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n + 1]))

Rule argument

Additional information

Used simplifier rules: ⊦ # [] = ⌜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 ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: σ, s0, σ0

Proof step 11 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 12:

  # σ0 = # α ⊦ (∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n])) ↔ step(s, a, s0) ∧ (∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]))

Rule argument

Additional information

Used simplifier rules: 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 ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ ⌜a '⌝ + σ ≠ []; (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∞ + ⌜0⌝ = n∞; (s) ⊦ m ≤ n → (⌜(* n)⌝ + n∞1 = ⌜(* m)⌝ + n∞2 ↔ ⌜n - m⌝ + n∞1 = n∞2); (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 12 from proof frag-rec (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 13:

  # σ0 = # α, ∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]) ⊦ step(s, a, s0) ∧ (∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]))

• Proof step 17:

  # σ0 = # α, step(s, a, s0), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦ ∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n])

Rule argument

Additional information

Proof step 13 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 14:

  # σ0 = # α, ∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]) ⊦ step(s, a, s0) ∧ (⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]))

Rule argument

Additional information

Proof step 14 from proof frag-rec (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 15:

  # σ0 = # α, ¬ step(s, a, s0), ∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]) ⊦

• Proof step 16:

  # σ0 = # α, ⌜n⌝ < # α, ¬ step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]), ∀ n. ¬ # α < ⌜n⌝ → step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]) ⊦

Rule argument

Additional information

Proof step 15 from proof frag-rec (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ n∞ < ⌜0⌝ ; (s) ⊦ (⌜a '⌝ + σ)[0] = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: 0

Proof step 16 from proof frag-rec (previous step) (back to tree)

User applied rule all left

And closed the goal!

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜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 ⊦ (⌜a '⌝ + σ)[n + 1] = σ[n]; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: n + 1

Proof step 17 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 18:

  # σ0 = # α, step(s, a, s0), ¬ # α < ⌜n⌝ , ¬ step((⌜s '⌝ + ⌜s0 '⌝ + σ0)[n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦

Rule argument

Additional information

Proof step 18 from proof frag-rec (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 19:

  ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]

• Proof step 20:

  # σ0 = # α, step(s, a, s0), ¬ ⌜n⌝ < # ⌜s '⌝ , ¬ # α < ⌜n⌝ , ¬ step((⌜s0 '⌝ + σ0)[n - (# ⌜s '⌝ ).n], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ # a ' = 0 +1; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ (⌜a '⌝ + σ)[0] = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ⌜s '⌝ , ⌜s0 '⌝ + σ0, n

Proof step 19 from proof frag-rec (previous step) (back to tree)

A spec-lemma

Proof step 20 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  # σ0 = # α, n ≠ 0, step(s, a, s0), ¬ # α < ⌜n⌝ , ¬ step((⌜s0 '⌝ + σ0)[n - 1], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜n⌝ .n = n; (s) ⊦ # a ' = 0 +1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ m < 1 ↔ m = 0; (s) ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)

Proof step 21 from proof frag-rec (previous step) (back to tree)

Heuristic elimination applied rule apply elim lemma

And obtained the following new goals

• Proof step 22:

  # σ0 = # α, n = n0 + 1, n ≠ 0, step(s, a, s0), ¬ # α < ⌜n⌝ , ¬ step((⌜s0 '⌝ + σ0)[n0], (⌜a '⌝ + α)[n], (⌜s0 '⌝ + σ0)[n]), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦

Rule argument

Additional information

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

Proof step 22 from proof frag-rec (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 23:

  # σ0 = # α, ⌜n0⌝ < # α, step(s, a, s0), ¬ step((⌜s0 '⌝ + σ0)[n0], α[n0], σ0[n0]), ∀ n. ⌜n⌝ < # α → step((⌜s0 '⌝ + σ0)[n], α[n], σ0[n]) ⊦

Rule argument

Additional information

Used simplifier rules: 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 ⊦ (⌜a '⌝ + σ)[n + 1] = σ[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 ⊦ # σ < ⌜n⌝ + ⌜1⌝ ↔ ¬ ⌜n⌝ < # σ; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) ⊦ m + 1 ≠ 0; (s)

Proof step 23 from proof frag-rec (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: n0