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Proof step 55 from proof min-com (previous step)

∀ x1, x2. #(x1 + x2) < #((a + x3) + (a0 + x)) → true → x1 + x2 = [] ∨ min(x1 + x2) = min(x2 + x1) ⊦ min(x3 + a0 ' + x) = min((a0 ' + x) + a ' + x3), a < min(x3 + a0 ' + x)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 56:
⊦ x ≠ [] → min(a ' + x) = (a < min(x) ⊃ a;min(x))
Proof step 57:
∀ x1, x2. #(x1 + x2) < #((a + x3) + (a0 + x)) → true → x1 + x2 = [] ∨ min(x1 + x2) = min(x2 + x1) ⊦ x + a ' + x3 = [], min(x3 + a0 ' + x) = (a0 < min(x + a ' + x3) ⊃ a0;min(x + a ' + x3)), a < min(x3 + a0 ' + x)

Rule argument

spec: <toplevel> name: min-rec seq: ⊦ x ≠ [] → min(a ' + x) = (a < min(x) ⊃ a;min(x)) subst: [x, a] → [x + a ' + x3, a0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ x + y = [] ↔ x = [] ∧ y = []; (as)% ⊦ a ' + x ≠ []; (s)%a: ⊦ (x + y) + z = x + y + z; from specification list Used terms: x + a ' + x3, a0