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Proof step 32 from proof noeth-modlep (previous step)

i1 * i ≤ i1 * i0 + i1, 0 < j2, 0 ≤ i1, i0 < i, i1 < j2, i1 * i + i ≤ i1 * i0 + i0 + i1 ⊦

User applied rule insert spec-lemma

And obtained the following new goals

Rule argument

spec: int-abs inst: name: ple-01 seq: i < j, i0 ≤ j0 ⊦ i + i0 < j + j0 subst: [i, j, i0, j0] → [i0, i, (i0 + 1) * i1, i * i1] flags: []

Additional information

Used simplifier rules: ls-forward: ⊦ i * j < i * j0 → i ≠ 0;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%ls-pos: ⊦ 0 < i → (i * j < i * j0 ↔ j < j0);%commutativity: ⊦ i * j = j * i;%ls-add-02: ⊦ i * (j + j1) + i0 < i * j + j0 ↔ i * j1 + i0 < j0;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%add-c: ⊦ i + j = j + i; from specification int-basic2%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%ls-01: ⊦ i + j < i + i0 ↔ j < i0; from specification int-basic2%irref: ⊦ ¬ i < i; from specification int-basic1%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%succ-02: ⊦ i < j + 1 ↔ i ≤ j; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used terms: i0, i, (i0 + 1) * i1, i * i1