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Proof step 1 from proof refines-product (conclusion) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), C_≤T_A(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦ C_≤T_A(Π(IOAut(start, step, ext), IOAut(cstart, cstep, cext), ren, cren), Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof refines-product (previous step) (back to tree)

C_≤T_A(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ C_≤T_A(Π(IOAut(start, step, ext), IOAut(cstart, cstep, cext), ren, cren), Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s)

Proof step 3 from proof refines-product (previous step) (back to tree)

C_≤T_A(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ C_≤T_A (IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)), IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-def, name: leq-def, seq: ⊦ C_≤T_A(cioaut, aioaut) ↔ (∀ β. trace(cioaut)(β) → trace(aioaut)(β)), subst:[cioaut, aioaut] → [IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)

Proof step 4 from proof refines-product (previous step) (back to tree)

⊦ C_≤T_A(cioaut, aioaut) ↔ (∀ β. trace(cioaut)(β) → trace(aioaut)(β))

A spec-lemma

Proof step 5 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ C_≤T_A (IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)), IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-def, name: leq-def, seq: ⊦ C_≤T_A(cioaut, aioaut) ↔ (∀ β. trace(cioaut)(β) → trace(aioaut)(β)), subst:[cioaut, aioaut] → [IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)), IOAut (Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))]; ['ca, 'cs, 'aa, 'as, 'e] → ['pa, ('s, 'cs), 'pa, ('s, 'as), 'e], <all paths>

Additional information

Used terms: IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)), IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))

Proof step 6 from proof refines-product (previous step) (back to tree)

⊦ C_≤T_A(cioaut, aioaut) ↔ (∀ β. trace(cioaut)(β) → trace(aioaut)(β))

A spec-lemma

Proof step 7 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∃ β. ¬ ( trace(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(β) → trace(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(β)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), trace(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(β), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ trace(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(β), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren), β]; ['e, 's, 'a] → ['e, ('s, 'as), 'pa], <all paths>

Additional information

Used terms: Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren), β

Proof step 9 from proof refines-product (previous step) (back to tree)

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

A spec-lemma

Proof step 10 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), trace(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(β), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∀ σ, α. ¬(β = Π(ext, aext, ren, aren) !(α) ∧ exec(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 11 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), trace(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(β), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∀ σ, α. β ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren), β]; ['e, 's, 'a] → ['e, ('s, 'cs), 'pa], <all paths>

Additional information

Used terms: Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren), β

Proof step 12 from proof refines-product (previous step) (back to tree)

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

A spec-lemma

Proof step 13 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ, α. β ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∃ σ, α. β = Π(ext, cext, ren, cren) !(α) ∧ exec(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(σ, α) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 14 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: project-right, name:proj-exec, seq:IOAut?(IOAut(lstart, lstep, lext)), IOAut?(IOAut(rstart, rstep, rext)), Πok(lext, rext, lren, rren), exec(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(pσ, pα) ⊦ exec(IOAut(rstart, rstep, rext)) (πrs(lstep, rstep, lren, rren, pσ, pα), πra(lstep, rstep, lren, rren, pσ, pα)), subst:[lstart, rstart, rext, lstep, pσ, rren, lren, lext, rstep, pα] → [start, cstart, cext, step, pcσ, cren, ren, ext, cstep, pα]; ['ls, 'ra, 'la, 'pa, 'rs, 'e] → ['s, 'ca, 'a, 'pa, 'cs, 'e]

Additional information

Used simplifier rules: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s) ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); (s) ⊦ IOAut(start, step, ext).start = start; (s) Used terms: pα, cext, cstep, cstart, ren, cren, start, step, ext, pcσ

Proof step 15 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(lstart, lstep, lext)), IOAut?(IOAut(rstart, rstep, rext)), Πok(lext, rext, lren, rren), exec(Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren))(pσ, pα) ⊦ exec(IOAut(rstart, rstep, rext))(πrs(lstep, rstep, lren, rren, pσ, pα), πra(lstep, rstep, lren, rren, pσ, pα))

A spec-lemma

Proof step 16 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), exec(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 17 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ β. trace(IOAut(cstart, cstep, cext))(β) → trace(IOAut(astart, astep, aext))(β), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 13 terms: cext !(πra(step, cstep, ren, cren, pcσ, pα)) abort: F discard: T computed: F

Additional information

Used terms: cext !(πra(step, cstep, ren, cren, pcσ, pα))

Proof step 18 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), trace(IOAut(cstart, cstep, cext))(cext !(πra(step, cstep, ren, cren, pcσ, pα))) → trace(IOAut(astart, astep, aext))(cext !(πra(step, cstep, ren, cren, pcσ, pα))), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 13

Additional information

Proof step 19 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ¬ trace(IOAut(cstart, cstep, cext))(cext !(πra(step, cstep, ren, cren, pcσ, pα))), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [cstart, cstep, cext, cext !(πra(step, cstep, ren, cren, pcσ, pα))]; ['e, 's, 'a] → ['e, 'cs, 'ca], <all paths>

Additional information

Used terms: cstart, cstep, cext, cext !(πra(step, cstep, ren, cren, pcσ, pα))

Proof step 20 from proof refines-product (previous step) (back to tree)

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

A spec-lemma

Proof step 21 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]), ∀ σ, α. ¬(cext !(πra(step, cstep, ren, cren, pcσ, pα)) = cext !(α) ∧ exec(IOAut(cstart, cstep, cext))(σ, α)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 22 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]), ∀ σ, α. cext !(πra(step, cstep, ren, cren, pcσ, pα)) ≠ cext !(α) ∨ ¬ frag(IOAut(cstart, cstep, cext))(σ, α) ∨ ¬ cstart(σ[0]) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 14 terms: πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)

Additional information

Used terms: πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)

Proof step 23 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), trace(IOAut(astart, astep, aext))(cext !(πra(step, cstep, ren, cren, pcσ, pα))), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [astart, astep, aext, cext !(πra(step, cstep, ren, cren, pcσ, pα))]; ['e, 's, 'a] → ['e, 'as, 'aa], <all paths>

Additional information

Used terms: astart, astep, aext, cext !(πra(step, cstep, ren, cren, pcσ, pα))

Proof step 24 from proof refines-product (previous step) (back to tree)

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

A spec-lemma

Proof step 25 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]), ∃ σ, α. cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(α) ∧ exec(IOAut(astart, astep, aext))(σ, α) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 26 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:repl-exec, seq:IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), exec(Π(IOAut(start, step, ext), IOAut(cstart, cstep, cext), ren, cren))(pcσ, pα), exec(IOAut(astart, astep, aext))(aσ, aα), cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) ⊦ exec(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), subst:[astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα] → [astep, cext, aα, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used simplifier rules: ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s) ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); (s) ⊦ IOAut(start, step, ext).start = start; (s) Used terms: aren, pα, ren, ext, astep, pcσ, astart, step, aext, cstart, cren, cstep, aα, start, aσ, cext

Proof step 27 from proof refines-product (previous step) (back to tree)

IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), exec(Π(IOAut(start, step, ext), IOAut(cstart, cstep, cext), ren, cren))(pcσ, pα), exec(IOAut(astart, astep, aext))(aσ, aα), cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) ⊦ exec(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα))

Lemma repl-exec

Proof step 28 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), exec(Π(IOAut(start, step, ext), IOAut(astart, astep, aext), ren, aren)) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec(ioa)(σ, α) ↔ frag(ioa)(σ, α) ∧ ioa.start(σ[0]); (s) ⊦ Π(IOAut(lstart, lstep, lext), IOAut(rstart, rstep, rext), lren, rren) = IOAut (Π(lstart, rstart), Π(lstep, rstep, lren, rren, lext, rext), Π(lext, rext, lren, rren)); (s)

Proof step 29 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∀ σ, α. Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(α) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren)))(σ, α) ∨ ¬ Π(start, astart)(σ[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 18 terms: repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα) abort: F discard: T computed: T

Additional information

Used terms: repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)

Proof step 30 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∨ ¬ frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)) ∨ ¬ Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 31 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:frag-repla-sametrace, seq:⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ sameshared(cren, aren, ren) ∧ frag (IOAut (Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren))) (pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) → Π(ext, aext, ren, aren) ! (repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pα), subst:[astep, cext, aα, as, start, pcσ, aσ, cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα] → [astep, cext, aα, hd(aσ), start, pcσ, tl(aσ), cren, ren, astart, cstart, cstep, aren, aext, step, ext, pα]; ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as] → ['ca, 'cs, 'a, 'aa, 'pa, 'e, 's, 'as]

Additional information

Used terms: aren, pα, ren, ext, hd(aσ), astep, pcσ, astart, step, aext, cstart, cren, cstep, aα, start, tl(aσ), cext

Proof step 32 from proof refines-product (previous step) (back to tree)

⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ sameshared(cren, aren, ren) ∧ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ, aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ, aα)) = Π(ext, cext, ren, cren) !(pα)

Lemma frag-repla-sametrace

Proof step 33 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ sameshared(cren, aren, ren) ∧ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα) → Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(pα) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 19

Additional information

Proof step 34 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ nounsharedextR(cren, cext, ren) ∧ nounsharedextR(aren, aext, ren) ∧ Πok(ext, cext, ren, cren) ∧ Πok(ext, aext, ren, aren) ∧ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ sameshared(cren, aren, ren) ∧ frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα) ∧ cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 35 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag(IOAut(astart, astep, aext))(⌜hd(aσ) '⌝ + tl(aσ), aα)

User applied rule apply elim lemma

And obtained the following new goals

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [aσ, as, aσ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, aσ0

Proof step 36 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), aσ = as + aσ0, Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), aσ ≠ [], IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦ oinj(cext) ∧ oinj(aext) ∧ oinj(ext) ∧ frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 37 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), IOAut?(IOAut(start, step, ext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(as), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), sameshared(cren, aren, ren), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦ oinj(cext) ∧ oinj(aext) ∧ oinj(ext)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [start, step, ext]; ['s, 'a, 'e] → ['s, 'a, 'e], <all paths>

Additional information

Used terms: start, step, ext

Proof step 38 from proof refines-product (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 39 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), astart(as), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∃ s. start(s) ∧ oinj(ext) ⊦ oinj(cext) ∧ oinj(aext) ∧ oinj(ext)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 40 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(astart, astep, aext)), astart(as), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(ext), sameshared(cren, aren, ren), start(s), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren) ⊦ oinj(cext) ∧ oinj(aext)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [cstart, cstep, cext]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used terms: cstart, cstep, cext

Proof step 41 from proof refines-product (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 42 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), IOAut?(IOAut(astart, astep, aext)), astart(as), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), oinj(ext), sameshared(cren, aren, ren), start(s), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren), ∃ cs. cstart(cs) ∧ oinj(cext) ⊦ oinj(cext) ∧ oinj(aext)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 43 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), IOAut?(IOAut(astart, astep, aext)), astart(as), cstart(cs), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)), frag(IOAut(astart, astep, aext))(⌜as '⌝ + aσ0, aα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), nounsharedextR(cren, cext, ren), nounsharedextR(aren, aext, ren), oinj(cext), oinj(ext), sameshared(cren, aren, ren), start(s), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜as '⌝ + aσ0, aα)[0]), Π(start, cstart)(pcσ[0]), Πok(ext, cext, ren, cren), Πok(ext, aext, ren, aren), ¬ oinj(aext) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IO-Automaton, name: isIOAut-def, seq: ⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext)), subst:[start, step, ext] → [astart, astep, aext]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used terms: astart, astep, aext

Proof step 44 from proof refines-product (previous step) (back to tree)

⊦ IOAut?(IOAut(start, step, ext)) ↔ (∃ s. start(s) ∧ oinj(ext))

A spec-lemma

Proof step 45 from proof refines-product (previous step) (back to tree)

cext !(πra(step, cstep, ren, cren, pcσ, pα)) = aext !(aα), Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, ⌜hd(aσ) '⌝ + tl(aσ), aα)) = Π(ext, cext, ren, cren) !(pα), Π(ext, cext, ren, cren) !(pα) ≠ Π(ext, aext, ren, aren) !(repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), IOAut?(IOAut(astart, astep, aext)), IOAut?(IOAut(cstart, cstep, cext)), IOAut?(IOAut(start, step, ext)), astart(aσ[0]), cstart(πrs(step, cstep, ren, cren, pcσ, pα)[0]), frag(IOAut(Π(start, cstart), Π(step, cstep, ren, cren, ext, cext), Π(ext, cext, ren, cren)))(pcσ, pα), frag(IOAut(cstart, cstep, cext))(πrs(step, cstep, ren, cren, pcσ, pα), πra(step, cstep, ren, cren, pcσ, pα)), frag(IOAut(astart, astep, aext))(aσ, aα), frag(IOAut(Π(start, astart), Π(step, astep, ren, aren, ext, aext), Π(ext, aext, ren, aren))) (repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα), repla(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)), nounsharedextR(aren, aext, ren), nounsharedextR(cren, cext, ren), sameshared(cren, aren, ren), Π(start, cstart)(pcσ[0]), Π(start, astart)(repls(step, cstep, cren, ren, aren, cext, aext, pcσ, pα, aσ, aα)[0]), Πok(ext, aext, ren, aren), Πok(ext, cext, ren, cren) ⊦

User applied rule apply elim lemma

And closed the goal!

Rule argument

spec: stream, name:elim-hd-tl, seq: ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0), subst:[σ, a, σ0] → [aσ, as, aσ0]; ['a] → ['as]

Additional information

Used simplifier rules: ⊦ a + σ = ⌜a '⌝ + σ; (s) ⊦ ¬ frag(IOAut(start, step, ext))([], α); (s) ⊦ σ ≠ [] → (a = hd(σ) ∧ σ0 = tl(σ) ↔ σ = a + σ0); (elim) Used terms: aσ, as, aσ0

Proof Tree for refines-product (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