[Imprint][Data Privacy Statement][E-Mail]



View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 1 from proof fw-trans-aux (conclusion) (back to tree)

⊦ fwsim(cioaut, bioaut, rbc) ∧ fwsim(bioaut, aioaut, rab) → fwsim(cioaut, aioaut, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof fw-trans-aux (previous step) (back to tree)

fwsim(cioaut, bioaut, rbc), fwsim(bioaut, aioaut, rab) ⊦ fwsim(cioaut, aioaut, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cioaut

Additional information

Used simplifier rules: IOAutomaton('s, 'a, 'e) freely generated by IOAut;;

Proof step 3 from proof fw-trans-aux (previous step) (back to tree)

cioaut = IOAut(cstart, cstep, cext), fwsim(bioaut, aioaut, rab), fwsim(cioaut, bioaut, rbc) ⊦ fwsim(cioaut, aioaut, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 4 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(cstart, cstep, cext), bioaut, rbc), fwsim(bioaut, aioaut, rab) ⊦ fwsim(IOAut(cstart, cstep, cext), aioaut, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: aioaut

Additional information

Used simplifier rules: IOAutomaton('s, 'a, 'e) freely generated by IOAut;;

Proof step 5 from proof fw-trans-aux (previous step) (back to tree)

aioaut = IOAut(astart, astep, aext), fwsim(bioaut, aioaut, rab), fwsim(IOAut(cstart, cstep, cext), bioaut, rbc) ⊦ fwsim(IOAut(cstart, cstep, cext), aioaut, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(cstart, cstep, cext), bioaut, rbc), fwsim(bioaut, IOAut(astart, astep, aext), rab) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: bioaut

Additional information

Used simplifier rules: IOAutomaton('s, 'a, 'e) freely generated by IOAut;;

Proof step 7 from proof fw-trans-aux (previous step) (back to tree)

bioaut = IOAut(bstart, bstep, bext), fwsim(bioaut, IOAut(astart, astep, aext), rab), fwsim(IOAut(cstart, cstep, cext), bioaut, rbc) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 8 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(cstart, cstep, cext), IOAut(bstart, bstep, bext), rbc), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, astep, r, aext, cext, astart, cstart] → [cstep, bstep, rbc, bext, cext, bstart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'ba, 'bs, 'e], <all paths>

Additional information

Used terms: cstep, cext, bext, rbc, bstart, cstart, bstep

Proof step 9 from proof fw-trans-aux (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

Lemma proofobls-forwardsim

Proof step 10 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, cext !(ca '), bs0) ∧ rbc(bs0, cs0)) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 11 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, astep, r, aext, cext, astart, cstart] → [cstep, astep, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs), aext, cext, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cstep, cext, aext, λ as, cs. ∃ bs. rab(as, bs) ∧ rbc(bs, cs), astart, cstart, astep

Proof step 12 from proof fw-trans-aux (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

Lemma proofobls-forwardsim

Proof step 13 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)) ⊦ (∀ cs. cstart(cs) → (∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs0))))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 14 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ (cstart(cs) → (∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as))) ∧ ( rab(as, bs) ∧ rbc(bs, cs0) ∧ cstep(cs0, ca, cs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 15 from proof fw-trans-aux (previous step) (back to tree)

cstart(cs), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: cs abort: F discard: T computed: T

Additional information

Used terms: cs

Proof step 16 from proof fw-trans-aux (previous step) (back to tree)

cstart(cs), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 17 from proof fw-trans-aux (previous step) (back to tree)

bstart(bs), cstart(cs), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rbc(bs, cs), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:proofobls-forwardsim, seq:⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0))), subst:[cstep, astep, r, aext, cext, astart, cstart] → [bstep, astep, rab, aext, bext, astart, bstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ba, 'bs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: bstep, bext, aext, rab, astart, bstart, astep

Proof step 18 from proof fw-trans-aux (previous step) (back to tree)

⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ↔ (∀ cs. cstart(cs) → (∃ as. r(as, cs) ∧ astart(as))) ∧ (∀ cs, ca, cs0, as. r(as, cs) ∧ cstep(cs, ca, cs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ r(as0, cs0)))

Lemma proofobls-forwardsim

Proof step 19 from proof fw-trans-aux (previous step) (back to tree)

bstart(bs), cstart(cs), rbc(bs, cs), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ bs. bstart(bs) → (∃ as. rab(as, bs) ∧ astart(as)), ∀ bs, ba, bs0, as. rab(as, bs) ∧ bstep(bs, ba, bs0) → (∃ as0. move(IOAut(astart, astep, aext))(as, bext !(ba '), as0) ∧ rab(as0, bs0)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 20 from proof fw-trans-aux (previous step) (back to tree)

bstart(bs), cstart(cs), rbc(bs, cs), ∀ bs. bstart(bs) → (∃ as. rab(as, bs) ∧ astart(as)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ bs0, ba, bs1, as. rab(as, bs0) ∧ bstep(bs0, ba, bs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (bext(ba) = None ⊃ [];bext(ba).get '), as0) ∧ rab(as0, bs1)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: bs abort: F discard: T computed: T

Additional information

Used terms: bs

Proof step 21 from proof fw-trans-aux (previous step) (back to tree)

bstart(bs), cstart(cs), rbc(bs, cs), bstart(bs) → (∃ as. rab(as, bs) ∧ astart(as)), ∀ bs0, ba, bs1, as. rab(as, bs0) ∧ bstep(bs0, ba, bs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (bext(ba) = None ⊃ [];bext(ba).get '), as0) ∧ rab(as0, bs1)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 22 from proof fw-trans-aux (previous step) (back to tree)

astart(as), bstart(bs), cstart(cs), rab(as, bs), rbc(bs, cs), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ bs0, ba, bs1, as. rab(as, bs0) ∧ bstep(bs0, ba, bs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (bext(ba) = None ⊃ [];bext(ba).get '), as0) ∧ rab(as0, bs1)) ⊦ ∃ as. (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: as abort: F discard: T computed: T

Additional information

Used terms: as

Proof step 23 from proof fw-trans-aux (previous step) (back to tree)

astart(as), bstart(bs), cstart(cs), rab(as, bs), rbc(bs, cs), ∀ bs0, ba, bs1, as. rab(as, bs0) ∧ bstep(bs0, ba, bs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (bext(ba) = None ⊃ [];bext(ba).get '), as0) ∧ rab(as0, bs1)), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ (∃ bs. rab(as, bs) ∧ rbc(bs, cs)) ∧ astart(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 24 from proof fw-trans-aux (previous step) (back to tree)

astart(as), bstart(bs), cstart(cs), rab(as, bs), rbc(bs, cs), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ bs0, ba, bs1, as. rab(as, bs0) ∧ bstep(bs0, ba, bs1) → (∃ as0. move(IOAut(astart, astep, aext))(as, (bext(ba) = None ⊃ [];bext(ba).get '), as0) ∧ rab(as0, bs1)) ⊦ ∃ bs. rab(as, bs) ∧ rbc(bs, cs)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: bs

Additional information

Used terms: bs

Proof step 25 from proof fw-trans-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bs), rbc(bs, cs0), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)), ∀ cs. cstart(cs) → (∃ bs. rbc(bs, cs) ∧ bstart(bs)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 6

Additional information

Proof step 26 from proof fw-trans-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bs), rbc(bs, cs0), ∀ cs, ca, cs0, bs. rbc(bs, cs) ∧ cstep(cs, ca, cs0) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs0)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: cs0, ca, cs1, bs abort: F discard: T computed: T

Additional information

Used terms: cs0, ca, cs1, bs

Proof step 27 from proof fw-trans-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bs), rbc(bs, cs0), rbc(bs, cs0) ∧ cstep(cs0, ca, cs1) → (∃ bs0. move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0) ∧ rbc(bs0, cs1)) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof fw-trans-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), move(IOAut(bstart, bstep, bext))(bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0), rab(as, bs), rbc(bs, cs0), rbc(bs0, cs1) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(bstart, bstep, bext), bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0]; ['s, 'e, 'a] → ['bs, 'e, 'ba], <all paths>

Additional information

Used terms: IOAut(bstart, bstep, bext), bs, (cext(ca) = None ⊃ [];cext(ca).get '), bs0

Proof step 29 from proof fw-trans-aux (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

A spec-lemma

Proof step 30 from proof fw-trans-aux (previous step) (back to tree)

cstep(cs0, ca, cs1), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bs), rbc(bs0, cs1), rbc(bs, cs0), ∃ al, sl. frag*(IOAut(bstart, bstep, bext))(sl, al) ∧ bs = sl[0] ∧ bs0 = sl.last ∧ IOAut(bstart, bstep, bext).ext !(al) = (cext(ca) = None ⊃ [];cext(ca).get ') ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 31 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bsl[0]), rbc(bsl[0], cs0), rbc(bsl.last, cs1) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, bext !(bal), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:fw-frag, seq:fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), frag*(IOAut(cstart, cstep, cext))(csl, cal), r(as, csl[0]) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl.last), subst:[cstep, astep, r, cal, aext, as, astart, cstart, csl, cext] → [bstep, astep, rab, bal, aext, as, astart, bstart, bsl, bext]; ['ca, 'cs, 'aa, 'as, 'e] → ['ba, 'bs, 'aa, 'as, 'e]

Additional information

Used terms: bstep, bext, as, aext, rab, astart, bstart, bal, astep, bsl

Proof step 32 from proof fw-trans-aux (previous step) (back to tree)

fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), frag*(IOAut(cstart, cstep, cext))(csl, cal), r(as, csl[0]) ⊦ ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ cext !(cal) = aext !(aal) ∧ r(asl.last, csl.last)

Lemma fw-frag

Proof step 33 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(as, bsl[0]), rbc(bsl.last, cs1), rbc(bsl[0], cs0), ∃ asl, aal. asl[0] = as ∧ frag*(IOAut(astart, astep, aext))(asl, aal) ∧ bext !(bal) = aext !(aal) ∧ rab(asl.last, bsl.last) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(as, bext !(bal), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl.last, bsl.last), rab(asl[0], bsl[0]), rbc(bsl[0], cs0), rbc(bsl.last, cs1) ⊦ ∃ as0. move(IOAut(astart, astep, aext))(asl[0], aext !(aal), as0) ∧ (∃ bs. rab(as0, bs) ∧ rbc(bs, cs1))

User applied rule exists right

And obtained the following new goals

Rule argument

pos: right 1 terms: asl.last abort: F discard: T computed: F

Additional information

Used terms: asl.last

Proof step 35 from proof fw-trans-aux (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), bext !(bal) = aext !(aal), cstep(cs0, ca, cs1), frag*(IOAut(bstart, bstep, bext))(bsl, bal), frag*(IOAut(astart, astep, aext))(asl, aal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl[0], bsl[0]), rab(asl.last, bsl.last), rbc(bsl.last, cs1), rbc(bsl[0], cs0) ⊦ move(IOAut(astart, astep, aext))(asl[0], aext !(aal), asl.last) ∧ (∃ bs. rab(asl.last, bs) ∧ rbc(bs, cs1))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 36 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl.last, bsl.last), rab(asl[0], bsl[0]), rbc(bsl[0], cs0), rbc(bsl.last, cs1), ¬ move(IOAut(astart, astep, aext))(asl[0], aext !(aal), asl.last) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:move, seq:⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el), subst:[ioa, s1, el, s2] → [IOAut(astart, astep, aext), asl[0], aext !(aal), asl.last]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(astart, astep, aext), asl[0], aext !(aal), asl.last

Proof step 37 from proof fw-trans-aux (previous step) (back to tree)

⊦ move(ioa)(s1, el, s2) ↔ (∃ al, sl. frag*(ioa)(sl, al) ∧ s1 = sl[0] ∧ s2 = sl.last ∧ ioa.ext !(al) = el)

A spec-lemma

Proof step 38 from proof fw-trans-aux (previous step) (back to tree)

aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), bext !(bal) = aext !(aal), cstep(cs0, ca, cs1), frag*(IOAut(bstart, bstep, bext))(bsl, bal), frag*(IOAut(astart, astep, aext))(asl, aal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl[0], bsl[0]), rab(asl.last, bsl.last), rbc(bsl.last, cs1), rbc(bsl[0], cs0), ∀ al, sl. ¬(frag*(IOAut(astart, astep, aext))(sl, al) ∧ asl[0] = sl[0] ∧ asl.last = sl.last ∧ IOAut(astart, astep, aext).ext !(al) = aext !(aal)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ IOAut(start, step, ext).ext = ext; (s)

Proof step 39 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl.last, bsl.last), rab(asl[0], bsl[0]), rbc(bsl[0], cs0), rbc(bsl.last, cs1), ∀ al, sl. ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ asl[0] ≠ sl[0] ∨ asl.last ≠ sl.last ∨ aext !(al) ≠ aext !(aal) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: aal, asl

Additional information

Used terms: aal, asl

Proof step 40 from proof fw-trans-aux (previous step) (back to tree)

bext !(bal) = aext !(aal), aext !(aal) = (cext(ca) = None ⊃ [];cext(ca).get '), cstep(cs0, ca, cs1), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(bstart, bstep, bext))(bsl, bal), fwsim(IOAut(bstart, bstep, bext), IOAut(astart, astep, aext), rab), rab(asl.last, bsl.last), rab(asl[0], bsl[0]), rbc(bsl[0], cs0), rbc(bsl.last, cs1) ⊦ ∃ bs. rab(asl.last, bs) ∧ rbc(bs, cs1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: bsl.last

Additional information

Used terms: bsl.last

Proof Tree for fw-trans-aux (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