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Proof step 19 from proof fac-le (previous step)

(i1 + 1) * fac(i1) ≤ (i2 + 1) * fac(i2), 0 ≤ i1, 0 ≤ i2, i2 < i1, fac(i2) < fac(i1), 0 < i1, ∀ i0, j. i→n(i0) < i→n(i1 + 1) → 0 < i0 ∧ 0 < j → (fac(i0) ≤ fac(j) ↔ i0 ≤ j) ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: pnle-pos seq: i ≤ i0, j < j0, i0 ≠ 0, 0 ≤ i, 0 ≤ j ⊦ ¬ i0 * j0 ≤ i * j subst: [j, i, j0, i0] → [fac(i2), i2 + 1, fac(i1), i1 + 1] flags: rewrite

Additional information

Used simplifier rules: fac-not-neg: ⊦ ¬ fac(i) < 0;%le-add: ⊦ i + j ≤ i0 + j ↔ i ≤ i0; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%zero-04: ⊦ 0 < i ∧ 0 < j → 0 ≠ i + j; from specification int-basic2%numle-02: ⊦ (* j) ≤ i0 + (* j0) ↔ j - j0 ≤ i0; from specification int-abs%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%lels-01: ⊦ ¬ i ≤ j ↔ j < i; from specification int-abs%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: fac(i2), i2 + 1, fac(i1), i1 + 1