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Proof step 1 from proof two-ls (conclusion) (back to tree)

⊦ 0 ≤ i → (2 < 2 ^ i ↔ 1 < i)

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof two-ls (previous step) (back to tree)

¬(2 < 2 ^ i ↔ 1 < i), 0 ≤ i ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ ~1 < i ↔ 0 ≤ i; (s) numlef-01: ⊦ (* i) ≤ j → i - 1 < j; (forward) from specification int-abs

Proof step 3 from proof two-ls (previous step) (back to tree)

¬(2 < 2 ^ i ↔ 1 < i), 0 ≤ i ⊦

User applied rule case distinction

And closed the goal!

Rule argument

pos: left 1

Additional information

Used simplifier rules: pot2-pos-01: ⊦ 1 < 2 ^ i ↔ 0 < i; pot2-pos: ⊦ 0 < 2 ^ i ↔ 0 ≤ i; two-le-01: ⊦ 0 ≤ i → (2 ^ i ≤ 2 ↔ i ≤ 1); ⊦ i < j ↔ ¬ j ≤ i; (ss) ⊦ i < j → (j ≤ i ↔ false); (as) forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1 forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s) ⊦ 0 < i → (i ≤ 1 ↔ i = 1); (s) forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs

Proof Tree for two-ls (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