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

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

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

≤BI(cioaut, aioaut), ¬ ≤wB(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 bwsiminv-implies-weakbwsim (previous step) (back to tree)

cioaut = IOAut(cINV, cstep, cext), ≤BI(cioaut, aioaut), ¬ ≤wB(cioaut, aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

≤BI(IOAut(cINV, cstep, cext), aioaut), ¬ ≤wB(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 bwsiminv-implies-weakbwsim (previous step) (back to tree)

aioaut = IOAut(aINV, astep, aext), ≤BI(IOAut(cINV, cstep, cext), aioaut), ¬ ≤wB(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 bwsiminv-implies-weakbwsim (previous step) (back to tree)

≤BI(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)), ¬ ≤wB(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, 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 bwsiminv-implies-weakbwsim (previous step) (back to tree)

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

Lemma backwardsiminv-def

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

¬ ≤wB(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 bwsiminv-implies-weakbwsim (previous step) (back to tree)

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), ¬ ≤wB(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name: weakbackwardsim-def, seq: ⊦ ≤wB(cioaut, aioaut) ↔ (∃ t, aINV. weakbwsim(cioaut, aioaut, t, 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 10 from proof bwsiminv-implies-weakbwsim (previous step) (back to tree)

⊦ ≤wB(cioaut, aioaut) ↔ (∃ t, aINV. weakbwsim(cioaut, aioaut, t, aINV))

Lemma weakbackwardsim-def

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), ∀ t, aINV0. ¬ weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0) ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: <toplevel>, name:bwsiminv-implies-weakbwsim-aux, seq:bwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, cINV, aINV) ⊦ weakbwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), t, aINV), 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]

Additional information

Used terms: astep, aINV, t, aext, cINV, cstep, aINV0, cext, cINV0

Proof step 12 from proof bwsiminv-implies-weakbwsim (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)

Lemma bwsiminv-implies-weakbwsim-aux

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

bwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, cINV0, aINV0), weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0), ∀ t, aINV0. ¬ weakbwsim(IOAut(cINV, cstep, cext), IOAut(aINV, astep, aext), t, aINV0) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: t, aINV0

Additional information

Used terms: t, aINV0

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

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the proof tree