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Proof step 3 from proof orderedls-all (previous step)

ordered<(x0) ↔ (∀ m, n. m < n ∧ n < # x0 → x0[m] < x0[n]) ⊦ ordered<(a ' + x0) ↔ (∀ m, n. m < n ∧ ¬ # x0 < n → (a ' + x0)[m] < (a ' + x0)[n])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ x ≠ [] ∧ x.rest ≠ [] → (ordered<(x) ↔ x.first < x.rest.first ∧ ordered<(x.rest))
Proof step 5:
a ' + x0 ≠ [] ∧ (a ' + x0).rest ≠ [], ordered<(x0) ↔ (∀ m, n. m < n ∧ n < # x0 → x0[m] < x0[n]) ⊦ (a ' + x0).first < (a ' + x0).rest.first ∧ ordered<((a ' + x0).rest) ↔ (∀ m, n. m < n ∧ ¬ # x0 < n → (a ' + x0)[m] < (a ' + x0)[n])

Rule argument

spec: olist inst: name: ols-rec seq: ⊦ x ≠ [] ∧ x.rest ≠ [] → ( ordered<(x) ↔ x.first < x.rest.first ∧ ordered<(x.rest)) subst: [x] → [a ' + x0] flags: rewrite, current

Additional information

Used simplifier rules: ls-e: ⊦ ordered<([]);%ls-o: ⊦ ordered<(a ');%c-07: ⊦ (a ' + x).rest = x; from specification list%a: ⊦ (x + y) + z = x + y + z; from specification list%app-02-ant: ⊦ ¬ x + y ≠ [] ↔ x = [] ∧ y = []; from specification list%e: ⊦ a ' ≠ []; from specification list%size-01: ⊦ # [] = 0; from specification list-data%app: ⊦ x + [] = x; from specification list%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%lez-01: ⊦ m ≤ 0 ↔ m = 0; from specification nat%f: ⊦ m < n → n ≠ 0; from specification nat-basic1 Used terms: a ' + x0