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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(cINV, 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(cINV, cstep, cext), aioaut), ¬ C_≤*T_A(IOAut(cINV, 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(aINV, astep, aext), ≤wB(IOAut(cINV, cstep, cext), aioaut), ¬ C_≤*T_A(IOAut(cINV, 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(cINV, cstep, cext), IOAut(aINV, astep, aext)), ¬ C_≤*T_A(IOAut(cINV, cstep, cext), IOAut(aINV, 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(cINV, cstep, cext), IOAut(aINV, astep, aext)]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: IOAut(cINV, cstep, cext), IOAut(aINV, 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(cINV, cstep, cext), IOAut(aINV, astep, aext)), ∃ el. ¬(trace*(IOAut(cINV, cstep, cext))(el) → trace*(IOAut(aINV, 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(cINV, cstep, cext))(el), ≤wB(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)), ¬ trace*(IOAut(aINV, 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] → [aINV, astep, aext, el]; ['e, 's, 'a] → ['e, 'as, 'aa], <all paths>

Additional information

Used terms: aINV, 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(cINV, cstep, cext))(el), ≤wB(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)), ∀ sl, al. ¬(el = aext !(al) ∧ exec*(IOAut(aINV, 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(cINV, cstep, cext))(el), ≤wB(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, name: weakbackwardsim-def, seq: ⊦ ≤wB(cioaut, aioaut) ↔ (∃ t, aINV. weakbwsim(cioaut, aioaut, t, aINV)), subst:[cioaut, aioaut] → [IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: IOAut(cINV, cstep, cext), IOAut(aINV, 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(cINV, cstep, cext))(el), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∃ t, aINV0. weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0) ⊦

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(cINV, cstep, cext))(el), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(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] → [cINV, cstep, cext, el]; ['e, 's, 'a] → ['e, 'cs, 'ca], <all paths>

Additional information

Used terms: cINV, 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(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. el ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∃ sl, al. el = cext !(al) ∧ exec*(IOAut(cINV, 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)

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∃ as. t(csl.last, as) ∧ aINV0(as)

Additional information

Used formulas: ∃ as. t(csl.last, as) ∧ aINV0(as)

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

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∃ as. t(csl.last, as) ∧ aINV0(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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, 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, [], aINV, cINV, t, csl[0], cext, csl.last, cext !(cal), aINV0]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e]

Additional information

Used terms: astep, aINV, t, cext !(cal), aext, cINV, cstep, [], as, csl.last, aINV0, csl[0] ', cext, csl[0]

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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]) ⊦ exec*(IOAut(cINV, cstep, cext))(csl[0] ', []) ∧ csl[0] '.last = csl[0] ∧ move(IOAut(cINV, 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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ¬ move(IOAut(cINV, cstep, cext))(csl[0], cext !(cal), csl.last), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(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(cINV, cstep, cext), csl[0], cext !(cal), csl.last]; ['s, 'e, 'a] → ['cs, 'e, 'ca], <all paths>

Additional information

Used terms: IOAut(cINV, 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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∀ al, sl. ¬(frag*(IOAut(cINV, cstep, cext))(sl, al) ∧ csl[0] = sl[0] ∧ csl.last = sl.last ∧ IOAut(cINV, 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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∀ al, sl. ¬ frag*(IOAut(cINV, 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)

aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∃ as1. move(IOAut(aINV, astep, aext))(as1, cext !(cal), as) ∧ t(csl[0], as1) ∧ aINV0(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)

aINV0(as1), aINV0(as), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), move(IOAut(aINV, astep, aext))(as1, cext !(cal), as), t(csl[0], as1), t(csl.last, as), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(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(aINV, astep, aext), as1, cext !(cal), as]; ['s, 'e, 'a] → ['as, 'e, 'aa], <all paths>

Additional information

Used terms: IOAut(aINV, 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)

aINV0(as), aINV0(as1), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), t(csl[0], as1), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∃ al, sl. frag*(IOAut(aINV, astep, aext))(sl, al) ∧ as1 = sl[0] ∧ as = sl.last ∧ IOAut(aINV, 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), aINV0(asl[0]), aINV0(asl.last), cINV(csl[0]), frag*(IOAut(aINV, astep, aext))(asl, aal), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(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), aINV0(asl.last), aINV0(asl[0]), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), frag*(IOAut(aINV, astep, aext))(asl, aal), t(csl.last, asl.last), t(csl[0], asl[0]), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), cext !(cal) ≠ aext !(aal) ∨ ¬ frag*(IOAut(aINV, astep, aext))(asl, aal) ∨ ¬ aINV(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), aINV0(asl[0]), aINV0(asl.last), cINV(csl[0]), frag*(IOAut(aINV, astep, aext))(asl, aal), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ¬ aINV(asl[0]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, 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, aINV0, aINV, cINV]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: astep, aINV, t, aext, cINV, cstep, aINV0, cext

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), aINV0(asl.last), aINV0(asl[0]), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), frag*(IOAut(aINV, astep, aext))(asl, aal), t(csl.last, asl.last), t(csl[0], asl[0]), ¬ aINV(asl[0]), ∀ csl, cal. exec*(IOAut(cINV, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ asl, aal. exec*(IOAut(aINV, astep, aext))(asl, aal) → aINV0(asl.last), ∀ csl, cal, ca, cs0, as0. exec*(IOAut(cINV, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, 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)

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

aext !(aal) = cext !(cal), aINV0(asl[0]), aINV0(asl.last), cINV(csl[0]), frag*(IOAut(aINV, astep, aext))(asl, aal), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), ¬ aINV(asl[0]), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ csl0, cal0, ca, cs0, as0. frag*(IOAut(cINV, cstep, cext))(csl0, cal0) ∧ cINV(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(csl0.last, as)), ∀ asl0, aal0. frag*(IOAut(aINV, astep, aext))(asl0, aal0) ∧ aINV(asl0[0]) → aINV0(asl0.last), ∀ csl0, cal0. frag*(IOAut(cINV, cstep, cext))(csl0, cal0) ∧ cINV(csl0[0]) → (∃ as. t(csl0.last, as) ∧ aINV0(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)

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]) ⊦ ∃ as. t(csl.last, as) ∧ aINV0(as)

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, 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, aINV0, aINV, cINV]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: astep, aINV, t, aext, cINV, cstep, aINV0, cext

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)

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∀ csl, cal. exec*(IOAut(cINV, cstep, cext))(csl, cal) → (∃ as. t(csl.last, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ asl, aal. exec*(IOAut(aINV, astep, aext))(asl, aal) → aINV0(asl.last), ∀ csl, cal, ca, cs0, as0. exec*(IOAut(cINV, cstep, cext))(csl, cal) ∧ cstep(csl.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(csl.last, as)) ⊦ ∃ as. t(csl.last, as) ∧ aINV0(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)

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

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ sl, al. cext !(cal) ≠ aext !(al) ∨ ¬ frag*(IOAut(aINV, astep, aext))(sl, al) ∨ ¬ aINV(sl[0]), ∀ csl0, cal0, ca, cs0, as0. frag*(IOAut(cINV, cstep, cext))(csl0, cal0) ∧ cINV(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(csl0.last, as)), ∀ asl, aal. frag*(IOAut(aINV, astep, aext))(asl, aal) ∧ aINV(asl[0]) → aINV0(asl.last), ∀ csl0, cal0. frag*(IOAut(cINV, cstep, cext))(csl0, cal0) ∧ cINV(csl0[0]) → (∃ as. t(csl0.last, as) ∧ aINV0(as)) ⊦ ∃ as. t(csl.last, as) ∧ aINV0(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)

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the proof tree