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Proof step 1 from proof wbw-leqstar (conclusion) (back to tree)

≤wB(cioaut, aioaut) ⊦ C_≤*T_A(cioaut, aioaut)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof wbw-leqstar (previous step) (back to tree)

≤wB(cioaut, aioaut), ¬ C_≤*T_A(cioaut, aioaut) ⊦

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 wbw-leqstar (previous step) (back to tree)

cioaut = IOAut(cstart, cstep, cext), ≤wB(cioaut, aioaut), ¬ C_≤*T_A(cioaut, aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 4 from proof wbw-leqstar (previous step) (back to tree)

≤wB(IOAut(cstart, cstep, cext), aioaut), ¬ C_≤*T_A(IOAut(cstart, cstep, cext), aioaut) ⊦

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 wbw-leqstar (previous step) (back to tree)

aioaut = IOAut(astart, astep, aext), ≤wB(IOAut(cstart, cstep, cext), aioaut), ¬ C_≤*T_A(IOAut(cstart, cstep, cext), aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 6 from proof wbw-leqstar (previous step) (back to tree)

≤wB(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ¬ C_≤*T_A(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-def, name: leqstar-def, seq: ⊦ C_≤*T_A(cioaut, aioaut) ↔ (∀ el. trace*(cioaut)(el) → trace*(aioaut)(el)), 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 7 from proof wbw-leqstar (previous step) (back to tree)

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

A spec-lemma

Proof step 8 from proof wbw-leqstar (previous step) (back to tree)

≤wB(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∃ el. ¬(trace*(IOAut(cstart, cstep, cext))(el) → trace*(IOAut(astart, astep, aext))(el)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof wbw-leqstar (previous step) (back to tree)

trace*(IOAut(cstart, cstep, cext))(el), ≤wB(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ¬ trace*(IOAut(astart, astep, aext))(el) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [astart, astep, aext, el]; ['e, 's, 'a] → ['e, 'as, 'aa], <all paths>

Additional information

Used terms: astart, astep, aext, el

Proof step 10 from proof wbw-leqstar (previous step) (back to tree)

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

A spec-lemma

Proof step 11 from proof wbw-leqstar (previous step) (back to tree)

trace*(IOAut(cstart, cstep, cext))(el), ≤wB(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∀ sl, al. ¬(el = aext !(al) ∧ exec*(IOAut(astart, astep, aext))(sl, al)) ⊦

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)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

Proof step 12 from proof wbw-leqstar (previous step) (back to tree)

trace*(IOAut(cstart, cstep, cext))(el), ≤wB(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name: weakbackwardsim-def, seq: ⊦ ≤wB(cioaut, aioaut) ↔ (∃ t, aINV. weakbwsim(cioaut, aioaut, t, aINV)), 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 13 from proof wbw-leqstar (previous step) (back to tree)

⊦ ≤wB(cioaut, aioaut) ↔ (∃ t, aINV. weakbwsim(cioaut, aioaut, t, aINV))

A spec-lemma

Proof step 14 from proof wbw-leqstar (previous step) (back to tree)

trace*(IOAut(cstart, cstep, cext))(el), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∃ t, aINV. weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 15 from proof wbw-leqstar (previous step) (back to tree)

trace*(IOAut(cstart, cstep, cext))(el), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, 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] → [cstart, cstep, cext, el]; ['e, 's, 'a] → ['e, 'cs, 'ca], <all paths>

Additional information

Used terms: cstart, cstep, cext, el

Proof step 16 from proof wbw-leqstar (previous step) (back to tree)

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

A spec-lemma

Proof step 17 from proof wbw-leqstar (previous step) (back to tree)

weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∃ sl, al. el = cext !(al) ∧ exec*(IOAut(cstart, cstep, cext))(sl, al) ⊦

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)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

Proof step 18 from proof wbw-leqstar (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ as. t(csl.last, as) ∧ aINV(as)

Additional information

Used formulas: ∃ as. t(csl.last, as) ∧ aINV(as)

Proof step 19 from proof wbw-leqstar (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∃ as. t(csl.last, as) ∧ aINV(as) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 20 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name:weak-lemma_3-15, seq:weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), exec*(IOAut(cstart, cstep, cext))(csl0, cal0), csl0.last = cs, move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as) ∧ aINV(as), subst:[cstep, astep, aext, csl0, as0, cal0, astart, cstart, t, cs, cext, cs0, el, aINV] → [cstep, astep, aext, csl[0] ', as, [], astart, cstart, t, csl[0], cext, csl.last, cext !(cal), aINV]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e]

Additional information

Used terms: [], cstep, cext, aext, cext !(cal), t, astart, cstart, csl[0] ', as, astep, csl.last, csl[0], aINV

Proof step 21 from proof wbw-leqstar (previous step) (back to tree)

weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), exec*(IOAut(cstart, cstep, cext))(csl0, cal0), csl0.last = cs, move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as) ∧ aINV(as)

A spec-lemma

Proof step 22 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦ exec*(IOAut(cstart, cstep, cext))(csl[0] ', []) ∧ csl[0] '.last = csl[0] ∧ move(IOAut(cstart, cstep, cext))(csl[0], cext !(cal), csl.last)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

Proof step 23 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ¬ move(IOAut(cstart, cstep, cext))(csl[0], cext !(cal), csl.last), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

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(cstart, cstep, cext), csl[0], cext !(cal), csl.last]; ['s, 'e, 'a] → ['cs, 'e, 'ca], <all paths>

Additional information

Used terms: IOAut(cstart, cstep, cext), csl[0], cext !(cal), csl.last

Proof step 24 from proof wbw-leqstar (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 25 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∀ al, sl. ¬(frag*(IOAut(cstart, cstep, cext))(sl, al) ∧ csl[0] = sl[0] ∧ csl.last = sl.last ∧ IOAut(cstart, cstep, cext).ext !(al) = cext !(cal)) ⊦

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 26 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∀ al, sl. ¬ frag*(IOAut(cstart, cstep, cext))(sl, al) ∨ csl[0] ≠ sl[0] ∨ csl.last ≠ sl.last ∨ cext !(al) ≠ cext !(cal) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: cal, csl

Additional information

Used terms: cal, csl

Proof step 27 from proof wbw-leqstar (previous step) (back to tree)

aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∃ as1. move(IOAut(astart, astep, aext))(as1, cext !(cal), as) ∧ t(csl[0], as1) ∧ aINV(as1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 28 from proof wbw-leqstar (previous step) (back to tree)

aINV(as1), aINV(as), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), move(IOAut(astart, astep, aext))(as1, cext !(cal), as), t(csl[0], as1), t(csl.last, as), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

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), as1, cext !(cal), as]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(astart, astep, aext), as1, cext !(cal), as

Proof step 29 from proof wbw-leqstar (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 wbw-leqstar (previous step) (back to tree)

aINV(as), aINV(as1), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl.last, as), t(csl[0], as1), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∃ al, sl. frag*(IOAut(astart, astep, aext))(sl, al) ∧ as1 = sl[0] ∧ as = sl.last ∧ IOAut(astart, astep, aext).ext !(al) = cext !(cal) ⊦

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 wbw-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV(asl[0]), aINV(asl.last), cstart(csl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: asl, aal abort: F discard: T computed: T

Additional information

Used terms: asl, aal

Proof step 32 from proof wbw-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV(asl.last), aINV(asl[0]), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), frag*(IOAut(astart, astep, aext))(asl, aal), t(csl.last, asl.last), t(csl[0], asl[0]), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), cext !(cal) ≠ aext !(aal) ∨ ¬ frag*(IOAut(astart, astep, aext))(asl, aal) ∨ ¬ astart(asl[0]) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 33 from proof wbw-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV(asl[0]), aINV(asl.last), cstart(csl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ¬ astart(asl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name:proofobls-weakbackwardsim, seq:⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as))), subst:[cstep, astep, aext, t, cext, aINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cstep, cext, aext, t, astart, cstart, astep, aINV

Proof step 34 from proof wbw-leqstar (previous step) (back to tree)

⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)))

A spec-lemma

Proof step 35 from proof wbw-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV(asl.last), aINV(asl[0]), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), frag*(IOAut(astart, astep, aext))(asl, aal), t(csl.last, asl.last), t(csl[0], asl[0]), ¬ astart(asl[0]), ∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last), ∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)) ⊦

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)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s) ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 36 from proof wbw-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV(asl[0]), aINV(asl.last), cstart(csl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), frag*(IOAut(cstart, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), ¬ astart(asl[0]), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ csl0, cal0, ca, cs0, as0. frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as)), ∀ asl0, aal0. frag*(IOAut(astart, astep, aext))(asl0, aal0) ∧ astart(asl0[0]) → aINV(asl0.last), ∀ csl0, cal0. frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) → (∃ as. t(csl0.last, as) ∧ aINV(as)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 10 terms: csl[0], asl[0]

Additional information

Used terms: csl[0], asl[0]

Proof step 37 from proof wbw-leqstar (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, name:proofobls-weakbackwardsim, seq:⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as))), subst:[cstep, astep, aext, t, cext, aINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cstep, cext, aext, t, astart, cstart, astep, aINV

Proof step 38 from proof wbw-leqstar (previous step) (back to tree)

⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ↔ (∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last)) ∧ (∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)))

A spec-lemma

Proof step 39 from proof wbw-leqstar (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∀ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV(as)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ asl, aal. exec*(IOAut(astart, astep, aext))(asl, aal) → aINV(asl.last), ∀ csl, cal, ca, cs0, as0. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

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)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s) ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s)

Proof step 40 from proof wbw-leqstar (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(astart, astep, aext))(sl, al) ∨ ¬ astart(sl[0]), ∀ csl0, cal0, ca, cs0, as0. frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as)), ∀ asl, aal. frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last), ∀ csl0, cal0. frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) → (∃ as. t(csl0.last, as) ∧ aINV(as)) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 7 terms: csl, cal

Additional information

Used terms: csl, cal

Proof Tree for wbw-leqstar (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