Proof step 1 from proof fintrace (conclusion) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 2:

  ⊦ trace(ioa)(⌜el⌝ ) ↔ trace*(ioa)(el)

• Proof step 3:

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

Rule argument

Additional information

Used terms: IOAut(start, step, ext), el

Proof step 2 from proof fintrace (previous step) (back to tree)

Lemma tracestar-def

Proof step 3 from proof fintrace (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 4:

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

• Proof step 5:

  ⊦ (∃ σ, α. ⌜el⌝ = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α)) ↔ (∃ sl, al. el = ext !(al) ∧ exec*(IOAut(start, step, ext))(sl, al))

Rule argument

Additional information

Used terms: start, step, ext, ⌜el⌝

Proof step 4 from proof fintrace (previous step) (back to tree)

Lemma trace

Proof step 5 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

  ⊦ (∃ σ, α. ⌜el⌝ = 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]))

Rule argument

Additional information

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

Proof step 6 from proof fintrace (previous step) (back to tree)

User applied rule case distinction

And obtained the following new goals

• Proof step 7:

  ∃ σ, α. ⌜el⌝ = 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])

• Proof step 20:

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

Rule argument

Additional information

Proof step 7 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 8:

  ⌜el⌝ = 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])

Rule argument

Additional information

Proof step 8 from proof fintrace (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 9:

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

• Proof step 10:

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

Rule argument

Additional information

Used terms: el, ext, α

Proof step 9 from proof fintrace (previous step) (back to tree)

A spec-lemma

Proof step 10 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 11:

  ext !(α0) = [], 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])

Rule argument

Additional information

Proof step 11 from proof fintrace (previous step) (back to tree)

User applied rule exists right

And obtained the following new goals

• Proof step 12:

  ext !(α0) = [], 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])

Rule argument

Additional information

Used terms: σ.take(# al + 1), al

Proof step 12 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 13:

  ext !(α0) = [], 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])

Rule argument

Additional information

Proof step 13 from proof fintrace (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 14:

  ⊦ ¬ # 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)))

• Proof step 15:

  ext !(α0) = [], 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

Rule argument

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 14 from proof fintrace (previous step) (back to tree)

Lemma frag-split

Proof step 15 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 16:

  ext !(α0) = [], 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])

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 16 from proof fintrace (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 17:

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

• Proof step 18:

  ext !(α0) = [], # 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])

Rule argument

Additional information

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

Proof step 17 from proof fintrace (previous step) (back to tree)

Lemma frag

Proof step 18 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 19:

  ext !(α0) = [], # 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])

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 19 from proof fintrace (previous step) (back to tree)

User applied rule subst equation

And closed the goal!

Rule argument

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 step 20 from proof fintrace (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 21:

  frag*(IOAut(start, step, ext))(sl, al), start(sl[0]) ⊦ ∃ σ, α. ⌜ext !(al)⌝ = ext !(α) ∧ frag(IOAut(start, step, ext))(σ, α) ∧ start(σ[0])

Rule argument

Additional information

Proof step 21 from proof fintrace (previous step) (back to tree)

User applied rule exists right

And closed the goal!

Rule argument

Additional information

Used simplifier rules: fragstar-def: ⊦ frag(ioa)(⌜sl⌝ , ⌜al⌝ ) ↔ frag*(ioa)(sl, al); ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; (s) ⊦ ⌜x⌝ [n] = x[n]; (s) Used terms: ⌜sl⌝ , ⌜al⌝