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

≤BI(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 bwinv-leqstar (previous step) (back to tree)

≤BI(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 bwinv-leqstar (previous step) (back to tree)

cioaut = IOAut(cINV, cstep, cext), ≤BI(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 bwinv-leqstar (previous step) (back to tree)

≤BI(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 bwinv-leqstar (previous step) (back to tree)

aioaut = IOAut(aINV, astep, aext), ≤BI(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 bwinv-leqstar (previous step) (back to tree)

≤BI(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: IObackward-INV, name: backwardsiminv-def, seq: ⊦ ≤BI(cioaut, aioaut) ↔ (∃ t, aINV, cINV. bwsim(cioaut, aioaut, t, cINV, 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 7 from proof bwinv-leqstar (previous step) (back to tree)

⊦ ≤BI(cioaut, aioaut) ↔ (∃ t, aINV, cINV. bwsim(cioaut, aioaut, t, cINV, aINV))

A spec-lemma

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

¬ C_≤*T_A(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)), ∃ t, aINV0, cINV0. bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), ¬ 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 10 from proof bwinv-leqstar (previous step) (back to tree)

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

A spec-lemma

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), ∃ 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 12 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), trace*(IOAut(cINV, cstep, cext))(el), ¬ 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] → [cINV, cstep, cext, el]; ['e, 's, 'a] → ['e, 'cs, 'ca], <all paths>

Additional information

Used terms: cINV, cstep, cext, el

Proof step 13 from proof bwinv-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 14 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), ¬ trace*(IOAut(aINV, astep, aext))(el), ∃ 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 15 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:preserve-INV-fin, seq:frag*(IOAut(start, step, ext))(sl, al), iNV(sl[0]), ∀ s, a, s0. iNV(s) ∧ step(s, a, s0) → iNV(s0) ⊦ iNV(sl.last), subst:[al, step, iNV, start, ext, sl] → [cal, cstep, cINV0, cINV, cext, csl]; ['s, 'a, 'e] → ['cs, 'ca, 'e]

Additional information

Used terms: cext, cal, cstep, cINV, cINV0, csl

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

frag*(IOAut(start, step, ext))(sl, al), iNV(sl[0]), ∀ s, a, s0. iNV(s) ∧ step(s, a, s0) → iNV(s0) ⊦ iNV(sl.last)

A spec-lemma

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦ cINV0(csl[0]) ∧ (∀ s, a, s0. cINV0(s) ∧ cstep(s, a, s0) → cINV0(s0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦ cINV0(csl[0]) ∧ (cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, name:proofobls-backwardsiminv, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, aINV, cINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV0, cINV0, 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, cINV0

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

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)))

A spec-lemma

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

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ cs. cINV(cs) → cINV0(cs), ∀ as. aINV(as) → aINV0(as), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)) ⊦ cINV0(csl[0]) ∧ (cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0))

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

cINV(csl[0]), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ cINV0(csl[0]), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as. aINV(as) → aINV0(as), ∀ cs. cINV(cs) → cINV0(cs) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: csl[0]

Additional information

Used terms: csl[0]

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

cINV(csl[0]), cINV0(cs), cstep(cs, ca, cs0), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ cINV0(cs0), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as. aINV(as) → aINV0(as), ∀ cs. cINV(cs) → cINV0(cs) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: cs, ca, cs0

Additional information

Used terms: cs, ca, cs0

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

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 24 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∃ as. t(csl.last, as) ∧ aINV0(as) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, name:lemma_3-15, seq:bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as), subst:[cstep, astep, aext, as0, cINV, astart, cstart, t, cs, cext, cs0, el, aINV] → [cstep, astep, aext, as, cINV0, 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, cext, csl[0], cINV0

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

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(cs), move(IOAut(cstart, cstep, cext))(cs, el, cs0), t(cs0, as0), aINV(as0) ⊦ ∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, el, as0) ∧ t(cs, as)

A spec-lemma

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

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦ cINV0(csl[0]) ∧ move(IOAut(cINV, cstep, cext))(csl[0], cext !(cal), csl.last)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

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

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ cINV0(csl[0]), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, name:proofobls-backwardsiminv, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, aINV, cINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV0, cINV0, 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, cINV0

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

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)))

A spec-lemma

Proof step 30 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ cINV0(csl[0]), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ cs. cINV(cs) → cINV0(cs), ∀ as. aINV(as) → aINV0(as), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: csl[0]

Additional information

Used terms: csl[0]

Proof step 31 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ move(IOAut(cINV, cstep, cext))(csl[0], cext !(cal), csl.last), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

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 32 from proof bwinv-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 33 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ 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 34 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ 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 8 terms: cal, csl

Additional information

Used terms: cal, csl

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

aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∃ as1. aINV0(as1) ∧ move(IOAut(aINV, astep, aext))(as1, cext !(cal), as) ∧ t(csl[0], as1) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

aINV0(as1), aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), move(IOAut(aINV, astep, aext))(as1, cext !(cal), as), t(csl[0], as1), t(csl.last, as), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦

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

Additional information

Used terms: aINV, astep, aext, cext !(cal)

Proof step 37 from proof bwinv-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 38 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), aINV0(as1), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), move(IOAut(aINV, astep, aext))(as1, cext !(cal), as), t(csl.last, as), t(csl[0], as1), ∀ sl, al. ¬(cext !(cal) = 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 39 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as1), aINV0(as), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), move(IOAut(aINV, astep, aext))(as1, cext !(cal), as), t(csl[0], as1), t(csl.last, as), ∀ 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 40 from proof bwinv-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 41 from proof bwinv-leqstar (previous step) (back to tree)

aINV0(as), aINV0(as1), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl.last, as), t(csl[0], as1), ∀ 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 42 from proof bwinv-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV0(asl[0]), aINV0(asl.last), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(aINV, astep, aext))(asl, aal), frag*(IOAut(cINV, cstep, cext))(csl, cal), t(csl[0], asl[0]), t(csl.last, asl.last), ∀ 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 11 terms: asl, aal abort: F discard: T computed: T

Additional information

Used terms: asl, aal

Proof step 43 from proof bwinv-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV0(asl.last), aINV0(asl[0]), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), frag*(IOAut(aINV, astep, aext))(asl, aal), t(csl.last, asl.last), t(csl[0], asl[0]), 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 44 from proof bwinv-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV0(asl[0]), aINV0(asl.last), bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), 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]) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IObackward-INV, name:proofobls-backwardsiminv, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, aINV, cINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV0, cINV0, 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, cINV0

Proof step 45 from proof bwinv-leqstar (previous step) (back to tree)

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)))

A spec-lemma

Proof step 46 from proof bwinv-leqstar (previous step) (back to tree)

aext !(aal) = cext !(cal), aINV0(asl.last), aINV0(asl[0]), cINV(csl[0]), cINV0(csl.last), 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]), ∀ cs. cINV(cs) → cINV0(cs), ∀ as. aINV(as) → aINV0(as), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 16 terms: csl[0], asl[0]

Additional information

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

Proof step 47 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)) ⊦ ∃ 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-backwardsiminv, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, aINV, cINV, astart, cstart] → [cstep, astep, aext, t, cext, aINV0, cINV0, 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, cINV0

Proof step 48 from proof bwinv-leqstar (previous step) (back to tree)

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ↔ (∀ cs. cstart(cs) → cINV(cs)) ∧ (∀ as. astart(as) → aINV(as)) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ∧ (∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as))) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cINV(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as. aINV(as) ∧ move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)))

A spec-lemma

Proof step 49 from proof bwinv-leqstar (previous step) (back to tree)

cINV(csl[0]), cINV0(csl.last), frag*(IOAut(cINV, cstep, cext))(csl, cal), ¬ trace*(IOAut(aINV, astep, aext))(cext !(cal)), ∀ cs. cINV(cs) → cINV0(cs), ∀ as. aINV(as) → aINV0(as), ∀ cs, ca, cs0. cINV0(cs) ∧ cstep(cs, ca, cs0) → cINV0(cs0), ∀ as, aa, as0. aINV0(as) ∧ astep(as, aa, as0) → aINV0(as0), ∀ cs. cINV0(cs) → (∃ as. t(cs, as) ∧ aINV0(as)), ∀ cs, as. cINV(cs) ∧ t(cs, as) ∧ aINV0(as) → aINV(as), ∀ cs, ca, cs0, as0. cINV0(cs) ∧ cstep(cs, ca, cs0) ∧ t(cs0, as0) ∧ aINV0(as0) → (∃ as. aINV0(as) ∧ move(IOAut(aINV, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as)) ⊦ ∃ as. t(csl.last, as) ∧ aINV0(as)

User applied rule all left

And closed the goal!

Rule argument

pos: left 9 terms: csl.last abort: F discard: T computed: T

Additional information

Used terms: csl.last

Proof Tree for bwinv-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