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Proof step 2 from proof bmin-sub-05 (previous step)

s.max ∈ s, s.min ∈ s, s ⊆ s0, s.max ∈ s0, s.min ∈ s0, s0.min ∈ s0, s0.max ∈ s0 ⊦ s0 = ∅, # s = 0, s = ∅, # s0 = 0, s.min < s0.min, s0.max < s.min, s.max < s0.min, s0.max < s.max, s -- s.min ⊆ s0 -- s0.min

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-basic inst: name: Subset seq: ⊦ s1 ⊆ s2 ↔ (∀ a. a ∈ s1 → a ∈ s2) subst: [s2, s1] → [s0 -- s0.min, s -- s.min] flags: rewrite

Additional information

Used simplifier rules: Max-max: ⊦ a ∈ s → ¬ s.max < a;%Min-min: ⊦ a ∈ s → ¬ a < s.min;%min-less: ⊦ a ≠ s.min ∧ a ∈ s → s.min < a;%min-sub-01: ⊦ s ⊆ s0 ∧ s ≠ ∅ ∧ s0.min ∈ s → (s.min ≠ s0.min ↔ false);%Delete: ⊦ a ∈ s -- b ↔ a ≠ b ∧ a ∈ s; from specification set-union%del-rax: ⊦ a ∈ s -- b ↔ ¬ ¬(a ≠ b ∧ a ∈ s); from specification set-union%sub-in: ⊦ a ∈ s ∧ s ⊆ s0 → a ∈ s0; (forward) from specification set-basic%sls: ⊦ a ≠ b → (¬ a < b ↔ b < a); from specification oelem%nen: ⊦ ¬ b < a ∧ a = c → ¬ b < c; (forward) from specification oelem Used terms: s0 -- s0.min, s -- s.min