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

⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t) ↔ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, λ cs. true, λ as. true)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

¬ ( bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t) ↔ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, λ cs. true, λ as. true)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: proofobls-backwardsim, seq:⊦ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t) ↔ (∀ cs. ∃ as. t(cs, as)) ∧ (∀ cs, as. cstart(cs) ∧ t(cs, as) → astart(as)) ∧ (∀ cs, ca, cs0, as0. cstep(cs, ca, cs0) ∧ t(cs0, as0) → (∃ as. move(IOAut(astart, astep, aext))(as, cext !(ca '), as0) ∧ t(cs, as))), subst:[cstep, astep, aext, t, cext, astart, cstart] → [cstep, astep, aext, t, cext, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

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

Proof step 3 from proof bwsim-bwsiminv-aux (previous step) (back to tree)

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

Lemma proofobls-backwardsim

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

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

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, λ as. true, λ cs. true, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used simplifier rules: ⊦ ext !(a ') = (ext(a) = None ⊃ [];ext(a).get '); (s) Used terms: cstep, t, astart, cext, cstart, λ cs. true, astep, λ as. true, aext

Proof step 5 from proof bwsim-bwsiminv-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 Tree for bwsim-bwsiminv-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