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

≤B(cioaut, aioaut) ⊦ ≤BI(cioaut, aioaut)

System applied rule normalize

And obtained the following new goals

Proof step 2:
≤B(cioaut, aioaut), ¬ ≤BI(cioaut, aioaut) ⊦

Rule argument

Additional information

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

≤B(cioaut, aioaut), ¬ ≤BI(cioaut, aioaut) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ ≤B(cioaut, aioaut) ↔ (∃ t. bwsim(cioaut, aioaut, t))
Proof step 4:
¬ ≤BI(cioaut, aioaut), ∃ t. bwsim(cioaut, aioaut, t) ⊦

Rule argument

spec: <toplevel>, name: backwardsim-def, seq: ⊦ ≤B(cioaut, aioaut) ↔ (∃ t. bwsim(cioaut, aioaut, t)), subst:[cioaut, aioaut] → [cioaut, aioaut]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cioaut, aioaut

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

⊦ ≤B(cioaut, aioaut) ↔ (∃ t. bwsim(cioaut, aioaut, t))

Lemma backwardsim-def

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

¬ ≤BI(cioaut, aioaut), ∃ t. bwsim(cioaut, aioaut, t) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
bwsim(cioaut, aioaut, t), ¬ ≤BI(cioaut, aioaut) ⊦

Rule argument

Additional information

Proof step 5 from proof bwsim-bwsiminv (previous step) (back to tree)

bwsim(cioaut, aioaut, t), ¬ ≤BI(cioaut, aioaut) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ ≤BI(cioaut, aioaut) ↔ (∃ t, aINV, cINV. bwsim(cioaut, aioaut, t, cINV, aINV))
Proof step 7:
bwsim(cioaut, aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(cioaut, aioaut, t, cINV, aINV) ⊦

Rule argument

spec: <toplevel>, name: backwardsiminv-def, seq: ⊦ ≤BI(cioaut, aioaut) ↔ (∃ t, aINV, cINV. bwsim(cioaut, aioaut, t, cINV, aINV)), subst:[cioaut, aioaut] → [cioaut, aioaut]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e], <all paths>

Additional information

Used terms: cioaut, aioaut

Proof step 6 from proof bwsim-bwsiminv (previous step) (back to tree)

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

Lemma backwardsiminv-def

Proof step 7 from proof bwsim-bwsiminv (previous step) (back to tree)

bwsim(cioaut, aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(cioaut, aioaut, t, cINV, aINV) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 8:
cioaut = IOAut(cstart, cstep, cext), bwsim(cioaut, aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(cioaut, aioaut, t, cINV, aINV) ⊦

Rule argument

var: cioaut

Additional information

Used simplifier rules: IOAutomaton('s, 'a, 'e) freely generated by IOAut;;

Proof step 8 from proof bwsim-bwsiminv (previous step) (back to tree)

cioaut = IOAut(cstart, cstep, cext), bwsim(cioaut, aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(cioaut, aioaut, t, cINV, aINV) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
bwsim(IOAut(cstart, cstep, cext), aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), aioaut, t, cINV, aINV) ⊦

Rule argument

Additional information

Proof step 9 from proof bwsim-bwsiminv (previous step) (back to tree)

bwsim(IOAut(cstart, cstep, cext), aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), aioaut, t, cINV, aINV) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 10:
aioaut = IOAut(astart, astep, aext), bwsim(IOAut(cstart, cstep, cext), aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), aioaut, t, cINV, aINV) ⊦

Rule argument

var: aioaut

Additional information

Used simplifier rules: IOAutomaton('s, 'a, 'e) freely generated by IOAut;;

Proof step 10 from proof bwsim-bwsiminv (previous step) (back to tree)

aioaut = IOAut(astart, astep, aext), bwsim(IOAut(cstart, cstep, cext), aioaut, t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), aioaut, t, cINV, aINV) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦

Rule argument

Additional information

Proof step 11 from proof bwsim-bwsiminv (previous step) (back to tree)

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

User applied rule apply lemma

And obtained the following new goals

Proof step 12:
⊦ 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)
Proof step 13:
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), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦

Rule argument

spec: <toplevel>, name:bwsim-bwsiminv-aux, seq:⊦ 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), 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]

Additional information

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

Proof step 12 from proof bwsim-bwsiminv (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)

Lemma bwsim-bwsiminv-aux

Proof step 13 from proof bwsim-bwsiminv (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), bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t), ∀ 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

Proof step 14:
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), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦

Rule argument

Additional information

Proof step 14 from proof bwsim-bwsiminv (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), ∀ t, aINV, cINV. ¬ bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: t, λ as. true, λ cs. true

Additional information

Used terms: t, λ as. true, λ cs. true

Proof Tree for bwsim-bwsiminv (View conclusion)

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