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Proof step 1 from proof minmax-03 (conclusion) (back to tree)

⊦ 1 < # s → s.min < s.max

System applied rule normalize

And obtained the following new goals

Rule argument

Additional information

Proof step 2 from proof minmax-03 (previous step) (back to tree)

1 < # s, ¬ s.min < s.max ⊦

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: Max-in: ⊦ s ≠ ∅ → s.max ∈ s; Min-in: ⊦ s ≠ ∅ → s.min ∈ s; Min-min: ⊦ a ∈ s → ¬ a < s.min; Max-max: ⊦ a ∈ s → ¬ s.max < a; ⊦ ¬ a < a; (s) ⊦ ¬ b < a → (a < b ↔ a ≠ b); (ss) size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)

Proof step 3 from proof minmax-03 (previous step) (back to tree)

s.max = s.min, # s ≠ 1, # s ≠ 0, s ≠ ∅, s.min ∈ s ⊦

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: set-basic, name:size-two-03, seq:1 < # s ⊦ ∃ a, b. a ≠ b ∧ a ∈ s ∧ b ∈ s, subst:[s] → [s]

Additional information

Used simplifier rules: Max-max: ⊦ a ∈ s → ¬ s.max < a; Min-min: ⊦ a ∈ s → ¬ a < s.min; nnn: ⊦ ¬ c < b ∧ ¬ a < c → ¬ a < b; (forward) from specification oelem ⊦ ¬ a < a; (s) ⊦ ¬ b < a → (a < b ↔ a ≠ b); (ss) ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s) Used terms: s

Proof step 4 from proof minmax-03 (previous step) (back to tree)

1 < # s ⊦ ∃ a, b. a ≠ b ∧ a ∈ s ∧ b ∈ s

A spec-lemma

Proof Tree for minmax-03 (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