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

⊦ traceω(IOAut(start, step, ext))(ef) ↔ (∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ execω(IOAut(start, step, ext))(sf, af))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: infexec: ⊦ execω(ioa)(sf, af) ↔ fragω(ioa)(sf, af) ∧ ioa.start(sf(0)); ⊦ IOAut(start, step, ext).start = start; (s)

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

⊦ traceω(IOAut(start, step, ext))(ef) ↔ (∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0)))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:traceomega, seq: ⊦ trace(ioa)(⌜ef⌝ ) ↔ traceω(ioa)(ef), subst:[ioa, ef] → [IOAut(start, step, ext), ef]; ['e, 's, 'a] → ['e, 's, 'a], <rotated> <all paths>

Additional information

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

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

⊦ trace(ioa)(⌜ef⌝ ) ↔ traceω(ioa)(ef)

Lemma traceomega

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

⊦ trace(IOAut(start, step, ext))(⌜ef⌝ ) ↔ (∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0)))

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, ⌜ef⌝ ]; ['e, 's, 'a] → ['e, 's, 'a], <all paths>

Additional information

Used terms: start, step, ext, ⌜ef⌝

Proof step 5 from proof inftrace (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 6 from proof inftrace (previous step) (back to tree)

⊦ (∃ σ, α. ⌜ef⌝ = ext !(α) ∧ exec(IOAut(start, step, ext))(σ, α)) ↔ (∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0)))

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 7 from proof inftrace (previous step) (back to tree)

⊦ (∃ σ, α. ⌜ef⌝ = ext !(α) ∧ frag(IOAut(start, step, ext))(σ, α) ∧ start(σ[0])) ↔ (∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0)))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

∃ σ, α. ⌜ef⌝ = ext !(α) ∧ frag(IOAut(start, step, ext))(σ, α) ∧ start(σ[0]) ⊦ ∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

⌜ef⌝ = ext !(α), frag(IOAut(start, step, ext))(σ, α), start(σ[0]) ⊦ ∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: α

Additional information

Used simplifier rules: Stream('a) freely generated by ⌜ ⌝, ⌜ ⌝;; ⊦ ext !(⌜al⌝ ) = ⌜ext !(al)⌝ ; (s) ⊦ ⌜x⌝ ≠ ⌜nf⌝ ; (s)

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

α = ⌜af0⌝ , ⌜ef⌝ = ext !(α), frag(IOAut(start, step, ext))(σ, α), start(σ[0]) ⊦ ∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: frag-inf-al: ⊦ frag(ioa)(σ, ⌜af⌝ ) ↔ fragω(ioa)(σ.f, af) ∧ ¬ finp(σ); ⊦ ⌜nf⌝ [n] = nf(n); (s) ⊦ ⌜nf⌝ .f = nf; (s) ⊦ finp(σ) ↔ ¬(∃ nf. ⌜nf⌝ = σ); (ss)

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

⌜ef⌝ = ext !(⌜af0⌝ ), fragω(IOAut(start, step, ext))(sf, af0), start(sf(0)) ⊦ ∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0))

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: sf, af0

Additional information

Used terms: sf, af0

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

∃ sf, af. ⌜ef⌝ = ext !(⌜af⌝ ) ∧ fragω(IOAut(start, step, ext))(sf, af) ∧ start(sf(0)) ⊦ ∃ σ, α. ⌜ef⌝ = ext !(α) ∧ frag(IOAut(start, step, ext))(σ, α) ∧ start(σ[0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

⌜ef⌝ = ext !(⌜af⌝ ), fragω(IOAut(start, step, ext))(sf, af), start(sf(0)) ⊦ ∃ σ, α. ⌜ef⌝ = ext !(α) ∧ frag(IOAut(start, step, ext))(σ, α) ∧ start(σ[0])

User applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: ⌜sf⌝ , ⌜af⌝ abort: F discard: T computed: F

Additional information

Used simplifier rules: fragomega-def: ⊦ frag(ioa)(⌜sf⌝ , ⌜af⌝ ) ↔ fragω(ioa)(sf, af); ⊦ ⌜nf⌝ [n] = nf(n); (s) Used terms: ⌜sf⌝ , ⌜af⌝

Proof Tree for inftrace (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