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

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

User applied rule insert rewrite lemma

And obtained the following new goals

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

Rule argument

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

Additional information

Used simplifier rules: le-e: ⊦ ordered≤([]);%le-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%z: ⊦ 0 < n ↔ n ≠ 0; from specification nat-basic1%lez-01: ⊦ m ≤ 0 ↔ m = 0; from specification nat%one-03: ⊦ a '[0] = a; from specification list-del%irreflexivity: ⊦ ¬ a < a; from specification oelem Used terms: a ' + x0