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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(cstart, 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(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 bwinv-leqstar (previous step) (back to tree)

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

≤BI(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-conds, name: backwardsiminv-def, seq: ⊦ ≤BI(cioaut, aioaut) ↔ (∃ t, aINV, cINV. bwsim(cioaut, aioaut, t, cINV, 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 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(cstart, cstep, cext), IOAut(astart, astep, aext)), ∃ t, aINV, cINV. bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦

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(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), ¬ 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 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(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), ∃ 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 12 from proof bwinv-leqstar (previous step) (back to tree)

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

Additional information

Used terms: cstart, 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(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), ¬ trace*(IOAut(astart, astep, aext))(el), ∃ 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 15 from proof bwinv-leqstar (previous step) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, 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, cINV, cstart, cext, csl]; ['s, 'a, 'e] → ['cs, 'ca, 'e]

Additional information

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

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(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)) ⊦ cINV(csl[0]) ∧ (∀ s, a, s0. cINV(s) ∧ cstep(s, a, s0) → cINV(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(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)) ⊦ cINV(csl[0]) ∧ (cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0))

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, 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, aINV, cINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

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)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)), ∀ 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)) ⊦ cINV(csl[0]) ∧ (cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(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)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ cINV(csl[0]), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)), ∀ 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)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ as. astart(as) → aINV(as), ∀ cs. cstart(cs) → cINV(cs) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: csl[0]

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: csl[0]

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

cINV(cs), cstart(csl[0]), cstep(cs, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ cINV(cs0), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)), ∀ 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)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs. cINV(cs) → (∃ as. t(cs, as) ∧ aINV(as)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ as. astart(as) → aINV(as), ∀ cs. cstart(cs) → cINV(cs) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 11 terms: cs, ca, cs0

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: cs, ca, cs0

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

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

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

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

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

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, 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, cINV, 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, as, cINV, astep, csl.last, csl[0], aINV

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)

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, 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, aINV, cINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

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)

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

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 8 terms: csl[0]

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: csl[0]

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

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

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

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

Additional information

Used terms: cal, csl

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

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

aINV(as1), aINV(as), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl.last), 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), ¬ trace*(IOAut(astart, 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] → [astart, astep, aext, cext !(cal)]; ['e, 's, 'a] → ['e, 'as, 'aa], <all paths>

Additional information

Used terms: astart, 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)

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

aINV(as1), aINV(as), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl.last), 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), ∀ 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 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)

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

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

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

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOrefine-conds, 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, aINV, cINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

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), aINV(asl.last), aINV(asl[0]), cINV(csl.last), 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]), ∀ 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)) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

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

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: csl[0], asl[0]

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

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), cINV(csl.last), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)) ⊦ ∃ 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-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, aINV, cINV, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

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.last), cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ trace*(IOAut(astart, astep, aext))(cext !(cal)), ∀ 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)) ⊦ ∃ as. t(csl.last, as) ∧ aINV(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 simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: csl.last

Proof Tree for bwinv-leqstar (View conclusion)

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the proof tree