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Proof step 1 from proof bwsiminv-implies-weakbwsim-aux (conclusion) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), ¬ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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 3 from proof bwsiminv-implies-weakbwsim-aux (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)))

Lemma proofobls-weakbackwardsim

Proof step 4 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, 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)))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

Proof step 5 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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 6 from proof bwsiminv-implies-weakbwsim-aux (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)))

Lemma proofobls-backwardsiminv

Proof step 7 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

∀ 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)), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

User applied rule cut formula

And obtained the following new goals

Rule argument

fma: ∀ cs. (∃ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ csl.last = cs) → cINV(cs)

Additional information

Used formulas: ∀ cs. (∃ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ csl.last = cs) → cINV(cs)

Proof step 8 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

∀ 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), ∀ cs. (∃ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ csl.last = cs) → cINV(cs), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) ⊦ IOAut(start, step, ext).start = start; (s) ⊦ exec*(ioa)(sl, al) ↔ frag*(ioa)(sl, al) ∧ ioa.start(sl[0]); (s)

Proof step 9 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

∀ 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), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 9

Additional information

Proof step 10 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: csl.last abort: F discard: T computed: F

Additional information

Used terms: csl.last

Proof step 11 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), cINV(csl.last) → (∃ as. t(csl.last, as) ∧ aINV(as)), ∀ 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, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 12 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ cINV(csl.last), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(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) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: csl.last abort: F discard: T computed: T

Additional information

Used terms: csl.last

Proof step 13 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ cINV(csl.last), (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl.last) → cINV(csl.last), ∀ 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, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 14 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl[0]), frag*(IOAut(cstart, cstep, cext))(csl, cal), ¬ cINV(csl.last), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs, as. cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(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) ⊦ ∃ as. t(csl.last, as) ∧ aINV(as), ∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl.last

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: csl, cal

Additional information

Used terms: csl, cal

Proof step 15 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), cstart(cs), t(cs, as), ¬ astart(as), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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 7 terms: cs, as

Additional information

Used terms: cs, as

Proof step 16 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

astart(asl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), ¬ aINV(asl.last), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 10 terms: asl[0] abort: F discard: T computed: T

Additional information

Used terms: asl[0]

Proof step 17 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

astart(asl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), ¬ aINV(asl.last), astart(asl[0]) → aINV(asl[0]), ∀ cs. cstart(cs) → cINV(cs), ∀ 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), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 18 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(asl[0]), astart(asl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), ¬ aINV(asl.last), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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), ∀ cs. cstart(cs) → cINV(cs) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 2, left 5, left 6, left 7, left 8, left 10, left 11

Additional information

Proof step 19 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(asl[0]), frag*(IOAut(astart, astep, aext))(asl, aal), ¬ aINV(asl.last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: asl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 20 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV((as + asl0)[0]), frag*(IOAut(astart, astep, aext))(as + asl0, aal), ¬ aINV((as + asl0).last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ aINV, astart, astep, aext, aal. ¬(aINV(asl0[0]) ∧ frag*(IOAut(astart, astep, aext))(asl0, aal) ∧ ¬ aINV(asl0.last) ∧ (∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s)

Proof step 21 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), frag*(IOAut(astart, astep, aext))(as ' + asl0, aal), ¬ aINV((as ' + asl0).last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ aINV0, astart0, astep0, aext0, aal0. ¬ aINV0(asl0[0]) ∨ ¬ frag*(IOAut(astart0, astep0, aext0))(asl0, aal0) ∨ aINV0(asl0.last) ∨ ¬(∀ as, aa, as0. aINV0(as) ∧ astep0(as, aa, as0) → aINV0(as0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: aal

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s) ⊦ (a ' + x).head = a; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 22 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aal = aa0 + aal1, aINV(as), frag*(IOAut(astart, astep, aext))(as ' + asl0, aal), ¬ aINV((as ' + asl0).last), ∀ aINV0, astart0, astep0, aext0, aal0. ¬ aINV0(asl0[0]) ∨ ¬ frag*(IOAut(astart0, astep0, aext0))(asl0, aal0) ∨ aINV0(asl0.last) ∨ ¬(∀ as, aa, as0. aINV0(as) ∧ astep0(as, aa, as0) → aINV0(as0)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 23 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), frag*(IOAut(astart, astep, aext))(as ' + asl0, aa0 ' + aal1), ¬ aINV((as ' + asl0).last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ∀ aINV0, astart0, astep0, aext0, aal0. ¬ aINV0(asl0[0]) ∨ ¬ frag*(IOAut(astart0, astep0, aext0))(asl0, aal0) ∨ aINV0(asl0.last) ∨ ¬(∀ as, aa, as0. aINV0(as) ∧ astep0(as, aa, as0) → aINV0(as0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [aa0, aal1, astep, as, astart, aext, asl0]; ['s, 'a, 'e] → ['as, 'aa, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: aal1, as, aext, asl0, aa0, astart, astep

Proof step 24 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 25 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), astep(as, aa0, asl0.head), frag*(IOAut(astart, astep, aext))(asl0, aal1), ¬ aINV((as ' + asl0).last), ∀ aINV0, astart0, astep0, aext0, aal0. ¬ aINV0(asl0[0]) ∨ ¬ frag*(IOAut(astart0, astep0, aext0))(asl0, aal0) ∨ aINV0(asl0.last) ∨ ¬(∀ as, aa, as0. aINV0(as) ∧ astep0(as, aa, as0) → aINV0(as0)), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: aINV, astart, astep, aext, aal1 abort: F discard: T computed: T

Additional information

Used terms: aINV, astart, astep, aext, aal1

Proof step 26 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), astep(as, aa0, asl0.head), frag*(IOAut(astart, astep, aext))(asl0, aal1), ¬ aINV((as ' + asl0).last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ¬ aINV(asl0[0]) ∨ ¬ frag*(IOAut(astart, astep, aext))(asl0, aal1) ∨ aINV(asl0.last) ∨ ¬(∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 27 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), astep(as, aa0, asl0.head), frag*(IOAut(astart, astep, aext))(asl0, aal1), ¬ aINV((as ' + asl0).last), ∀ as, aa, as0. aINV(as) ∧ astep(as, aa, as0) → aINV(as0), ¬ aINV(asl0[0]) ∨ aINV(asl0.last) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: as, aa0, hd(⌜asl0⌝ ) abort: F discard: T computed: T

Additional information

Used terms: as, aa0, hd(⌜asl0⌝ )

Proof step 28 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), astep(as, aa0, asl0.head), frag*(IOAut(astart, astep, aext))(asl0, aal1), ¬ aINV((as ' + asl0).last), aINV(as) ∧ astep(as, aa0, hd(⌜asl0⌝ )) → aINV(hd(⌜asl0⌝ )), ¬ aINV(asl0[0]) ∨ aINV(asl0.last) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 29 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as), astep(as, aa0, asl0.head), frag*(IOAut(astart, astep, aext))(asl0, aal1), ¬ aINV((as ' + asl0).last), astep(as, aa0, hd(⌜asl0⌝ )) → aINV(hd(⌜asl0⌝ )), ¬ aINV(asl0[0]) ∨ aINV(asl0.last) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: asl0

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ (a ' + x).head = a; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ a + x = a ' + x; (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ hd(⌜a '⌝ + ⌜x⌝ ) = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 30 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 6 terms: csl0.last, ca, cs0, as0 abort: F discard: T computed: T

Additional information

Used terms: csl0.last, ca, cs0, as0

Proof step 31 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), cINV(csl0.last) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as2)), ∀ 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), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 32 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), cINV(csl0.last) → aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as2), ∀ cs1. (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = cs1) → cINV(cs1), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 7 terms: csl0.last abort: F discard: T computed: T

Additional information

Used terms: csl0.last

Proof step 33 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), cINV(csl0.last) → aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as2), (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl0.last) → cINV(csl0.last), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 34 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl0.last) → cINV(csl0.last), cINV(csl0.last) → aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as2), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 7

Additional information

Proof step 35 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), ¬ cINV(csl0.last), (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl0.last) → cINV(csl0.last), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 36 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), t(cs0, as0), ¬ cINV(csl0.last), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1), ∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl0.last

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: csl0, cal0

Additional information

Used terms: csl0, cal0

Proof step 37 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

aINV(as0), aINV(as2), cstart(csl0[0]), cstep(csl0.last, ca, cs0), frag*(IOAut(cstart, cstep, cext))(csl0, cal0), move(IOAut(astart, astep, aext))(as2, (cext(ca) = None ⊃ [];cext(ca).get '), as0), t(cs0, as0), t(csl0.last, as2), (∃ csl1, cal1. frag*(IOAut(cstart, cstep, cext))(csl1, cal1) ∧ cstart(csl1[0]) ∧ csl1.last = csl0.last) → cINV(csl0.last), ∀ 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) ⊦ ∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1)

Heuristic quantifier closing applied rule exists right

And closed the goal!

Rule argument

pos: right 1 terms: as2

Additional information

Used terms: as2

Proof step 38 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

∀ 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), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦ ∀ cs. (∃ csl, cal. exec*(IOAut(cstart, cstep, cext))(csl, cal) ∧ csl.last = cs) → cINV(cs)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) ⊦ 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 bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl1[0]), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), ¬ cINV(csl1.last), ∀ 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), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 4 terms: csl1[0] abort: F discard: T computed: T

Additional information

Used terms: csl1[0]

Proof step 40 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cstart(csl1[0]), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), ¬ cINV(csl1.last), cstart(csl1[0]) → cINV(csl1[0]), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ∀ 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), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 41 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(csl1[0]), cstart(csl1[0]), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), ¬ cINV(csl1.last), ∀ 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), ∀ cs1, ca0, cs2, as1. cINV(cs1) ∧ cstep(cs1, ca0, cs2) ∧ t(cs2, as1) ∧ aINV(as1) → (∃ as2. aINV(as2) ∧ move(IOAut(astart, astep, aext))(as2, (cext(ca0) = None ⊃ [];cext(ca0).get '), as1) ∧ t(cs1, as2)), ¬(frag*(IOAut(cstart, cstep, cext))(csl, cal) ∧ cstart(csl[0]) → (∃ as. t(csl.last, as) ∧ aINV(as))) ∨ ¬(cstart(cs) ∧ t(cs, as) ∧ aINV(as) → astart(as)) ∨ ¬(frag*(IOAut(astart, astep, aext))(asl, aal) ∧ astart(asl[0]) → aINV(asl.last)) ∨ ¬ ( frag*(IOAut(cstart, cstep, cext))(csl0, cal0) ∧ cstart(csl0[0]) ∧ cstep(csl0.last, ca, cs0) ∧ t(cs0, as0) ∧ aINV(as0) → (∃ as1. aINV(as1) ∧ move(IOAut(astart, astep, aext))(as1, (cext(ca) = None ⊃ [];cext(ca).get '), as0) ∧ t(csl0.last, as1))) ⊦

User applied rule weakening

And obtained the following new goals

Rule argument

poslist: left 2, left 5, left 7, left 8, left 9, left 10, left 11

Additional information

Proof step 42 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(csl1[0]), frag*(IOAut(cstart, cstep, cext))(csl1, cal1), ¬ cINV(csl1.last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0) ⊦

User applied rule structural induction

And obtained the following new goals

Rule argument

term: csl1

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 43 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV((cs + csl)[0]), frag*(IOAut(cstart, cstep, cext))(cs + csl, cal1), ¬ cINV((cs + csl).last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ cINV, cstart, cstep, cext, cal1. ¬(cINV(csl[0]) ∧ frag*(IOAut(cstart, cstep, cext))(csl, cal1) ∧ ¬ cINV(csl.last) ∧ (∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0))) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ (a ' + x)[0] = a; (s)

Proof step 44 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), ¬ cINV((cs ' + csl).last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ cINV0, cstart0, cstep0, cext0, cal. ¬ cINV0(csl[0]) ∨ ¬ frag*(IOAut(cstart0, cstep0, cext0))(csl, cal) ∨ cINV0(csl.last) ∨ ¬(∀ cs, ca, cs0. cINV0(cs) ∧ cstep0(cs, ca, cs0) → cINV0(cs0)) ⊦

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: cal1

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ frag*(IOAut(start, step, ext))(sl, []) ↔ sl = sl.head '; (s) ⊦ (a ' + x).head = a; (s) ⊦ a ' = a0 ' + x ↔ a = a0 ∧ x = []; (ws) ⊦ x + [] = x; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ a '.last = a; (s) ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof step 45 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cal1 = ca0 + cal0, cINV(cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, cal1), ¬ cINV((cs ' + csl).last), ∀ cINV0, cstart0, cstep0, cext0, cal. ¬ cINV0(csl[0]) ∨ ¬ frag*(IOAut(cstart0, cstep0, cext0))(csl, cal) ∨ cINV0(csl.last) ∨ ¬(∀ cs, ca, cs0. cINV0(cs) ∧ cstep0(cs, ca, cs0) → cINV0(cs0)), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ a + x = a ' + x; (s)

Proof step 46 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), frag*(IOAut(cstart, cstep, cext))(cs ' + csl, ca0 ' + cal0), ¬ cINV((cs ' + csl).last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ∀ cINV0, cstart0, cstep0, cext0, cal. ¬ cINV0(csl[0]) ∨ ¬ frag*(IOAut(cstart0, cstep0, cext0))(csl, cal) ∨ cINV0(csl.last) ∨ ¬(∀ cs, ca, cs0. cINV0(cs) ∧ cstep0(cs, ca, cs0) → cINV0(cs0)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: IOtrace, name:finfrag-rec, seq:⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al), subst:[a, al, step, s, start, ext, sl] → [ca0, cal0, cstep, cs, cstart, cext, csl]; ['s, 'a, 'e] → ['cs, 'ca, 'e], <all paths>

Additional information

Used simplifier rules: a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: cal0, cs, cext, csl, ca0, cstart, cstep

Proof step 47 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

⊦ frag*(IOAut(start, step, ext))(s ' + sl, a ' + al) ↔ step(s, a, sl.head) ∧ frag*(IOAut(start, step, ext))(sl, al)

A spec-lemma

Proof step 48 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), cstep(cs, ca0, csl.head), frag*(IOAut(cstart, cstep, cext))(csl, cal0), ¬ cINV((cs ' + csl).last), ∀ cINV0, cstart0, cstep0, cext0, cal. ¬ cINV0(csl[0]) ∨ ¬ frag*(IOAut(cstart0, cstep0, cext0))(csl, cal) ∨ cINV0(csl.last) ∨ ¬(∀ cs, ca, cs0. cINV0(cs) ∧ cstep0(cs, ca, cs0) → cINV0(cs0)), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: cINV, cstart, cstep, cext, cal0 abort: F discard: T computed: T

Additional information

Used terms: cINV, cstart, cstep, cext, cal0

Proof step 49 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), cstep(cs, ca0, csl.head), frag*(IOAut(cstart, cstep, cext))(csl, cal0), ¬ cINV((cs ' + csl).last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ¬ cINV(csl[0]) ∨ ¬ frag*(IOAut(cstart, cstep, cext))(csl, cal0) ∨ cINV(csl.last) ∨ ¬(∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 50 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), cstep(cs, ca0, csl.head), frag*(IOAut(cstart, cstep, cext))(csl, cal0), ¬ cINV((cs ' + csl).last), ∀ cs, ca, cs0. cINV(cs) ∧ cstep(cs, ca, cs0) → cINV(cs0), ¬ cINV(csl[0]) ∨ cINV(csl.last) ⊦

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 5 terms: cs, ca0, hd(⌜csl⌝ ) abort: F discard: T computed: T

Additional information

Used terms: cs, ca0, hd(⌜csl⌝ )

Proof step 51 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), cstep(cs, ca0, csl.head), frag*(IOAut(cstart, cstep, cext))(csl, cal0), ¬ cINV((cs ' + csl).last), cINV(cs) ∧ cstep(cs, ca0, hd(⌜csl⌝ )) → cINV(hd(⌜csl⌝ )), ¬ cINV(csl[0]) ∨ cINV(csl.last) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 52 from proof bwsiminv-implies-weakbwsim-aux (previous step) (back to tree)

cINV(cs), cstep(cs, ca0, csl.head), frag*(IOAut(cstart, cstep, cext))(csl, cal0), ¬ cINV((cs ' + csl).last), cstep(cs, ca0, hd(⌜csl⌝ )) → cINV(hd(⌜csl⌝ )), ¬ cINV(csl[0]) ∨ cINV(csl.last) ⊦

User applied rule constructor cut

And closed the goal!

Rule argument

var: csl

Additional information

Used simplifier rules: List('a) freely generated by [], +;; ⊦ (a ' + x).head = a; (s) ⊦ (b ' + a ' + x).last = (a ' + x).last; (s) ⊦ (a ' + x)[0] = a; (s) ⊦ a + x = a ' + x; (s) ⊦ ⌜x + y⌝ = ⌜x⌝ + ⌜y⌝ ; (s) a: ⊦ (x + y) + z = x + y + z; from specification list ⊦ hd(⌜a '⌝ + ⌜x⌝ ) = a; (s) append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2; from specification stream ⊦ ¬ frag*(IOAut(start, step, ext))([], al); (s)

Proof Tree for bwsiminv-implies-weakbwsim-aux (View conclusion)

The following image shows the proof tree of sequent calculus. The proved goal is at the bottom. You can click on the nodes to view additional rules. See here for details on colors in a proof tree.

the proof tree