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Proof step 1 from proof equivomega-both-leqomega (conclusion) (back to tree)

⊦ C_≡ωT_A(cioaut, aioaut) ↔ C_≤ωT_A(cioaut, aioaut) ∧ C_≤ωT_A(aioaut, cioaut)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof equivomega-both-leqomega (previous step) (back to tree)

¬(C_≡ωT_A(cioaut, aioaut) ↔ C_≤ωT_A(cioaut, aioaut) ∧ C_≤ωT_A(aioaut, cioaut)) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: equivomega-def: ⊦ C_≡ωT_A(cioaut, aioaut) ↔ (∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef)); leqomega-def: ⊦ C_≤ωT_A(cioaut, aioaut) ↔ (∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef));

Proof step 3 from proof equivomega-both-leqomega (previous step) (back to tree)

⊦ (∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef)) ↔ (∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef)) ∧ (∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef))

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 4 from proof equivomega-both-leqomega (previous step) (back to tree)

∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦ (∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef)) ∧ (∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 5 from proof equivomega-both-leqomega (previous step) (back to tree)

∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦ (traceω(cioaut)(ef) → traceω(aioaut)(ef)) ∧ (traceω(aioaut)(ef0) → traceω(cioaut)(ef0))

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Proof step 6 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(cioaut)(ef), ¬ traceω(aioaut)(ef), ∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 7 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(cioaut)(ef), ¬ traceω(aioaut)(ef), ∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: ef

Additional information

Used terms: ef

Proof step 8 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(aioaut)(ef0), ¬ traceω(cioaut)(ef0), ∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 9 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(aioaut)(ef0), ¬ traceω(cioaut)(ef0), ∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: ef0

Additional information

Used terms: ef0

Proof step 10 from proof equivomega-both-leqomega (previous step) (back to tree)

∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef), ∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef) ⊦ ∀ ef. traceω(cioaut)(ef) ↔ traceω(aioaut)(ef)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Proof step 11 from proof equivomega-both-leqomega (previous step) (back to tree)

¬(traceω(cioaut)(ef) ↔ traceω(aioaut)(ef)), ∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef), ∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef) ⊦

Heuristic pl case distinction applied rule case distinction

And obtained the following new goals

Rule argument

pos: left 1

Additional information

Proof step 12 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(cioaut)(ef), ¬ traceω(aioaut)(ef), ∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef), ∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 3 terms: ef

Additional information

Used terms: ef

Proof step 13 from proof equivomega-both-leqomega (previous step) (back to tree)

traceω(aioaut)(ef), ¬ traceω(cioaut)(ef), ∀ ef. traceω(cioaut)(ef) → traceω(aioaut)(ef), ∀ ef. traceω(aioaut)(ef) → traceω(cioaut)(ef) ⊦

Heuristic quantifier closing applied rule all left

And closed the goal!

Rule argument

pos: left 4 terms: ef

Additional information

Used terms: ef

Proof Tree for equivomega-both-leqomega (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