Proof step 1 from proof branchinfw-fw (conclusion) (back to tree)

≤bF(cioaut, aioaut) ⊦ ≤F(cioaut, aioaut)

System applied rule normalize

And obtained the following new goals

Proof step 2:
≤bF(cioaut, aioaut), ¬ ≤F(cioaut, aioaut) ⊦

Rule argument

Additional information

Proof step 2 from proof branchinfw-fw (previous step) (back to tree)

≤bF(cioaut, aioaut), ¬ ≤F(cioaut, aioaut) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 3:
cioaut = IOAut(cstart, cstep, cext), ≤bF(cioaut, aioaut), ¬ ≤F(cioaut, aioaut) ⊦

Rule argument

var: cioaut

Additional information

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

Proof step 3 from proof branchinfw-fw (previous step) (back to tree)

cioaut = IOAut(cstart, cstep, cext), ≤bF(cioaut, aioaut), ¬ ≤F(cioaut, aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 4:
≤bF(IOAut(cstart, cstep, cext), aioaut), ¬ ≤F(IOAut(cstart, cstep, cext), aioaut) ⊦

Rule argument

Additional information

Proof step 4 from proof branchinfw-fw (previous step) (back to tree)

≤bF(IOAut(cstart, cstep, cext), aioaut), ¬ ≤F(IOAut(cstart, cstep, cext), aioaut) ⊦

User applied rule constructor cut

And obtained the following new goals

Proof step 5:
aioaut = IOAut(astart, astep, aext), ≤bF(IOAut(cstart, cstep, cext), aioaut), ¬ ≤F(IOAut(cstart, cstep, cext), aioaut) ⊦

Rule argument

var: aioaut

Additional information

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

Proof step 5 from proof branchinfw-fw (previous step) (back to tree)

aioaut = IOAut(astart, astep, aext), ≤bF(IOAut(cstart, cstep, cext), aioaut), ¬ ≤F(IOAut(cstart, cstep, cext), aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
≤bF(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

Rule argument

Additional information

Proof step 6 from proof branchinfw-fw (previous step) (back to tree)

≤bF(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ ≤bF(cioaut, aioaut) ↔ (∃ r. bfwsim(cioaut, aioaut, r))
Proof step 8:
¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∃ r. bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦

Rule argument

spec: IOrefine-conds, name: branchingforwardsim-def, seq: ⊦ ≤bF(cioaut, aioaut) ↔ (∃ r. bfwsim(cioaut, aioaut, r)), 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 branchinfw-fw (previous step) (back to tree)

⊦ ≤bF(cioaut, aioaut) ↔ (∃ r. bfwsim(cioaut, aioaut, r))

A spec-lemma

Proof step 8 from proof branchinfw-fw (previous step) (back to tree)

¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∃ r. bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

Rule argument

Additional information

Proof step 9 from proof branchinfw-fw (previous step) (back to tree)

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

User applied rule apply lemma

And obtained the following new goals

Proof step 10:
bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r)
Proof step 11:
bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

Rule argument

spec: <toplevel>, name:branchinfw-fw-aux, seq:bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), subst:[cstep, astep, r, aext, cext, astart, cstart] → [cstep, astep, r, aext, cext, astart, cstart]; ['ca, 'cs, 'aa, 'as, 'e] → ['ca, 'cs, 'aa, 'as, 'e]

Additional information

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

Proof step 10 from proof branchinfw-fw (previous step) (back to tree)

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r)

Lemma branchinfw-fw-aux

Proof step 11 from proof branchinfw-fw (previous step) (back to tree)

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ¬ ≤F(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

User applied rule apply rewrite lemma

And obtained the following new goals

Proof step 12:
⊦ ≤F(cioaut, aioaut) ↔ (∃ r. fwsim(cioaut, aioaut, r))
Proof step 13:
bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ∀ r. ¬ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦

Rule argument

spec: IOrefine-conds, name: forwardsim-def, seq: ⊦ ≤F(cioaut, aioaut) ↔ (∃ r. fwsim(cioaut, aioaut, r)), 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 12 from proof branchinfw-fw (previous step) (back to tree)

⊦ ≤F(cioaut, aioaut) ↔ (∃ r. fwsim(cioaut, aioaut, r))

A spec-lemma

Proof step 13 from proof branchinfw-fw (previous step) (back to tree)

bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), ∀ r. ¬ fwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: r

Additional information

Used terms: r

Proof Tree for branchinfw-fw (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.