Proof step 1 from proof normedfw-branchingfw (conclusion) (back to tree)

System applied rule normalize

And obtained the following new goals

• Proof step 2:

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

Rule argument

Additional information

Proof step 2 from proof normedfw-branchingfw (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 3:

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

Rule argument

Additional information

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

Proof step 3 from proof normedfw-branchingfw (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 4:

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

Rule argument

Additional information

Proof step 4 from proof normedfw-branchingfw (previous step) (back to tree)

User applied rule constructor cut

And obtained the following new goals

• Proof step 5:

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

Rule argument

Additional information

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

Proof step 5 from proof normedfw-branchingfw (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 6:

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

Rule argument

Additional information

Proof step 6 from proof normedfw-branchingfw (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 7:

  ⊦ ≤nF(cioaut, aioaut) ↔ (∃ r, size01f. nfwsim(cioaut, aioaut, r, size01f))

• Proof step 8:

  ¬ ≤bF(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), ∃ r, size01f. nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f) ⊦

Rule argument

Additional information

Used terms: IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)

Proof step 7 from proof normedfw-branchingfw (previous step) (back to tree)

A spec-lemma

Proof step 8 from proof normedfw-branchingfw (previous step) (back to tree)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

• Proof step 9:

  nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), ¬ ≤bF(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)) ⊦

Rule argument

Additional information

Proof step 9 from proof normedfw-branchingfw (previous step) (back to tree)

User applied rule apply lemma

And obtained the following new goals

• Proof step 10:

  nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f) ⊦ bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ as0. move(IOAut(astart, astep, aext))(as, [], as0) ∧ r(as0, cs))

• Proof step 11:

  nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), ¬ ≤bF(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)), bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ as0. move(IOAut(astart, astep, aext))(as, [], as0) ∧ r(as0, cs)) ⊦

Rule argument

Additional information

Used terms: cstep, cext, aext, r, astart, cstart, astep, size01f

Proof step 10 from proof normedfw-branchingfw (previous step) (back to tree)

Lemma normedfw-branchingfw-aux

Proof step 11 from proof normedfw-branchingfw (previous step) (back to tree)

User applied rule apply rewrite lemma

And obtained the following new goals

• Proof step 12:

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

• Proof step 13:

  nfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r, size01f), ∀ r. ¬ bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), r), bfwsim(IOAut(cstart, cstep, cext), IOAut(astart, astep, aext), λ as, cs. ∃ as0. move(IOAut(astart, astep, aext))(as, [], as0) ∧ r(as0, cs)) ⊦

Rule argument

Additional information

Used terms: IOAut(cstart, cstep, cext), IOAut(astart, astep, aext)

Proof step 12 from proof normedfw-branchingfw (previous step) (back to tree)

A spec-lemma

Proof step 13 from proof normedfw-branchingfw (previous step) (back to tree)

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

Additional information

Used terms: λ as, cs. ∃ as0. move(IOAut(astart, astep, aext))(as, [], as0) ∧ r(as0, cs)