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Proof step 17 from proof modmod-add-case (previous step)

i + i0 % j < 0, i < 0, 0 < j, 0 < i0, 0 ≤ i0, 0 ≤ i + i0, j ≤ i0 ⊦ (i + i0 % j) % j = (i + i0) % j + j, (i + i0 % j) % j = (i + i0) % j, (i + i0) % j = (i + i0 % j) % j + j, 0 = j, j = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: modmod-add2-negpos-pos-01 seq: ⊦ i + i0 % j < 0 ∧ 0 ≤ i + i0 ∧ i < 0 ∧ 0 ≤ i0 ∧ (i + i0 % j) % j ≠ 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j - j subst: [j, i0, i] → [j, i0, i] flags: <all paths>

Additional information

Used simplifier rules: modmod-add2-negpos-pos-02: ⊦ i + i0 % j < 0 ∧ 0 ≤ i + i0 ∧ i < 0 ∧ 0 ≤ i0 ∧ (i + i0 % j) % j = 0 ∧ 0 < j → (i + i0 % j) % j = (i + i0) % j;%modmod-add-zero: 0 < j ⊦ (i + i0 % j) % j = 0 → (i + i0) % j = 0;% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + i = 2 * i; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i = i + j ↔ j = 0; (s)% ⊦ 2 * i = 0 ↔ i = 0; (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ i + (j - i) = j; (s)% ⊦ i < 0 ∧ 0 ≤ j → i < j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≠ j → (i ≤ j ↔ i < j); (s)% ⊦ i < j → ¬ j < i; (s) Used terms: j, i0, i