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Proof step 40 from proof ex-modeq-half (previous step)

i0 % j = j0 * j + j, j0 < j, j0 < 0, odd(j), 0 < j, j / 2 < i0 % j, 0 ≤ i0 % j ⊦ j0 * j + j < 0, j = 0, ∃ i. ≡mod(i, i0, j) ∧ abs(i) ≤ j / 2, (i0 % j - j) |%| j = i0 % j ∧ j ≤ i0 % j + j / 2

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-mult inst: name: numsucc seq: ⊦ (* i0) * i + i = (i0 + 1) * i subst: [i, i0] → [j, j0] flags: rewrite

Additional information

Used simplifier rules: zero-ls: ⊦ 0 < i → (i * j < 0 ↔ j < 0); from specification int-mult%numls-03: ⊦ i + (* j) < (* j0) ↔ i < j0 - j; from specification int-abs%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%mone-08: ⊦ i < 0 → (~1 ≤ i ↔ i = ~1); 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%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%lel: ⊦ i < i0 ∧ i = i1 → i1 < i0; (forward) from specification oelem[intlista]%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%mone-06: ⊦ ~1 < i ↔ 0 ≤ i; from specification int-abs%lels-02: ⊦ i ≠ j → (i ≤ j ↔ i < j); from specification int-abs%div2-neg: ⊦ i / 2 < 0 → i < 0; (forward) from specification int-div%transitivity: ⊦ i < i0 ∧ i0 < i1 → i < i1; (forward) from specification oelem[intlista]%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%mod-less-01: ⊦ 0 < j → i % j < j; from specification int-div%ls-01: ⊦ i < j → ¬ j < i; from specification int-basic1 Used terms: j, j0