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

⊦ C_≤ωT_A(cioaut, bioaut) ∧ C_≤ωT_A(bioaut, aioaut) → C_≤ωT_A(cioaut, aioaut)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

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

C_≤ωT_A(cioaut, bioaut), C_≤ωT_A(bioaut, aioaut), ¬ C_≤ωT_A(cioaut, aioaut) ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

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

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

User applied rule all left

And obtained the following new goals

Rule argument

pos: left 3 terms: ef abort: F discard: T computed: T

Additional information

Used terms: ef

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

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

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

traceω(cioaut)(ef), ¬ traceω(aioaut)(ef), ¬ traceω(bioaut)(ef), ∀ ef. traceω(cioaut)(ef) → traceω(bioaut)(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 leqomega-trans (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