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Proof step 1 from proof trace-append (conclusion) (back to tree)

⊦ trace(IOAut(start, step, ext))(⌜el⌝ + β) → (trace*(IOAut(start, step, ext))(el) ↔ true)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof trace-append (previous step) (back to tree)

¬(trace*(IOAut(start, step, ext))(el) ↔ true), trace(IOAut(start, step, ext))(⌜el⌝ + β) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 3 from proof trace-append (previous step) (back to tree)

trace(IOAut(start, step, ext))(⌜el⌝ + β), ¬ trace*(IOAut(start, step, ext))(el) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:trace, seq:⊦ trace(IOAut(start, step, ext))(β) ↔ (∃ σ, α. β = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α)), subst:[start, step, ext, β] → [start, step, ext, ⌜el⌝ + β]; ['e, 's, 'a] → ['e, 's, 'a], <all paths>

Additional information

Used terms: start, step, ext, ⌜el⌝ + β

Proof step 4 from proof trace-append (previous step) (back to tree)

⊦ trace(IOAut(start, step, ext))(β) ↔ (∃ σ, α. β = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α))

Lemma trace

Proof step 5 from proof trace-append (previous step) (back to tree)

¬ trace*(IOAut(start, step, ext))(el), ∃ σ, α. ⌜el⌝ + β = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: exec: ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); ⊦ IOAut(start, step, ext).start = start; (s)

Proof step 6 from proof trace-append (previous step) (back to tree)

⌜el⌝ + β = ext !(α), frag(IOAut(start, step, ext))(σ, α), start(σ[0]), ¬ trace*(IOAut(start, step, ext))(el) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fintrace, seq:⊦ trace*(IOAut(start, step, ext))(el) ↔ (∃ sl, al. el = ext !(al) ∧ exec*(IOAut(start, step, ext))(sl, al)), subst:[start, step, ext, el] → [start, step, ext, el]; ['e, 's, 'a] → ['e, 's, 'a], <all paths>

Additional information

Used terms: start, step, ext, el

Proof step 7 from proof trace-append (previous step) (back to tree)

⊦ trace*(IOAut(start, step, ext))(el) ↔ (∃ sl, al. el = ext !(al) ∧ exec*(IOAut(start, step, ext))(sl, al))

Lemma fintrace

Proof step 8 from proof trace-append (previous step) (back to tree)

⌜el⌝ + β = ext !(α), frag(IOAut(start, step, ext))(σ, α), start(σ[0]), ∀ sl, al. ¬(el = ext !(al) ∧ exec*(IOAut(start, step, ext))(sl, al)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: finexec: ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); ⊦ IOAut(start, step, ext).start = start; (s)

Proof step 9 from proof trace-append (previous step) (back to tree)

⌜el⌝ + β = ext !(α), frag(IOAut(start, step, ext))(σ, α), start(σ[0]), ∀ sl, al. el ≠ ext !(al) ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: remtau, name:remtau-split-weak, seq:⊦ ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β), subst:[ext, α, el, β] → [ext, α, el, β]; ['e, 'a] → ['e, 'a], <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream Used terms: ext, α, el, β

Proof step 10 from proof trace-append (previous step) (back to tree)

⊦ ext !(α) = ⌜el⌝ + β ↔ (∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β)

A spec-lemma

Proof step 11 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, α), start(σ[0]), ∀ sl, al. el ≠ ext !(al) ∨ ¬ frag*(IOAut(start, step, ext))(sl, al) ∨ ¬ start(sl[0]), ∃ al, α0. α = ⌜al⌝ + α0 ∧ ext !(⌜al⌝ ) = ⌜el⌝ ∧ ext !(α0) = β ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s)

Proof step 12 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, ⌜al⌝ + α0), start(σ[0]), ∀ sl, al0. ext !(al) ≠ ext !(al0) ∨ ¬ frag*(IOAut(start, step, ext))(sl, al0) ∨ ¬ start(sl[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: σ.take(# al + 1), al abort: F discard: T computed: F

Additional information

Used terms: σ.take(# al + 1), al

Proof step 13 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, ⌜al⌝ + α0), start(σ[0]), ext !(al) ≠ ext !(al) ∨ ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, ⌜al⌝ + α0), start(σ[0]), ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:frag-split, seq:⊦ ¬ # α < ⌜n⌝ ∧ ⌜n⌝ < # σ → ( frag(IOAut(start, step, ext))(σ, α) ↔ frag*(IOAut(start, step, ext))(σ.take(n + 1), α.take(n)) ∧ frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n))), subst:[step, α, start, ext, n, σ] → [step, ⌜al⌝ + α0, start, ext, # al, σ]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ (⌜x⌝ + σ).take(# x) = x; (s) ⊦ (⌜x⌝ + σ).drop(# x) = σ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ¬ n∞ + n∞1 < n∞1; (s) ⊦ 0 < m ↔ m ≠ 0; (s) ⊦ m + 1 ≠ 0; (s) ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s) a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic c: ⊦ m + n = n + m; from specification nat-basic ⊦ # 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 m < n, ¬ # σ < ⌜n⌝ ⊦ σ.take(n)[m] = σ[m]; (s) Used terms: ext, # al, ⌜al⌝ + α0, step, start, σ

Proof step 15 from proof trace-append (previous step) (back to tree)

⊦ ¬ # α < ⌜n⌝ ∧ ⌜n⌝ < # σ → ( frag(IOAut(start, step, ext))(σ, α) ↔ frag*(IOAut(start, step, ext))(σ.take(n + 1), α.take(n)) ∧ frag(IOAut(start, step, ext))(σ.drop(n), α.drop(n)))

Lemma frag-split

Proof step 16 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, ⌜al⌝ + α0), start(σ[0]), ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦ ¬ #(⌜al⌝ + α0) < ⌜# al⌝ ∧ ⌜# al⌝ < # σ

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: 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∞ + n∞1 < n∞1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s)

Proof step 17 from proof trace-append (previous step) (back to tree)

frag(IOAut(start, step, ext))(σ, ⌜al⌝ + α0), start(σ[0]), ¬ ⌜# al⌝ < # σ, ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:frag, seq:⊦ frag(IOAut(start, step, ext))(σ, α) ↔ # σ = # α + ⌜1⌝ ∧ (∀ n. ⌜n⌝ < # α → step(σ[n], α[n], σ[n +1])), subst:[σ, step, α, start, ext] → [σ, step, ⌜al⌝ + α0, start, ext]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: ext, ⌜al⌝ + α0, step, start, σ

Proof step 18 from proof trace-append (previous step) (back to tree)

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

Lemma frag

Proof step 19 from proof trace-append (previous step) (back to tree)

# σ = #(⌜al⌝ + α0) + ⌜1⌝ , start(σ[0]), ¬ ⌜# al⌝ < # σ, ∀ n. ⌜n⌝ < #(⌜al⌝ + α0) → step(σ[n], (⌜al⌝ + α0)[n], σ[n +1]), ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ # ⌜x⌝ = ⌜# x⌝ ; (s) ⊦ #(σ + σ1) = # σ + # σ1; (s) add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf

Proof step 20 from proof trace-append (previous step) (back to tree)

# σ = (⌜# al⌝ + # α0) + ⌜1⌝ , start(σ[0]), ¬ ⌜# al⌝ < # σ, ∀ n. ⌜n⌝ < ⌜# al⌝ + # α0 → step(σ[n], (⌜al⌝ + α0)[n], σ[n + 1]), ¬ frag*(IOAut(start, step, ext))(σ.take(# al + 1), al) ∨ ¬ start(σ.take(# al + 1)[0]) ⊦

User applied rule subst equation

And closed the goal!

Rule argument

pos: left 1, flags: discard

Additional information

Used simplifier rules: ⊦ ⌜n⌝ < ⌜1⌝ + n∞ ↔ ¬ n∞ < ⌜n⌝ ; (s) ⊦ ¬ n∞ + n∞1 < n∞1; (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

Proof Tree for trace-append (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