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Proof step 3 from proof mod-neg (previous step)

⊦ i % ~ j = i % j, j = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
j ≠ 0 ⊦ i % j = i - i / j * j

Rule argument

spec: <toplevel> name: mod-rew seq: j ≠ 0 ⊦ i % j = i - i / j * j subst: [j, i] → [~ j, i] flags: <all paths>

Additional information

Used simplifier rules: Div-right: ⊦ j ≠ 0 → i / ~ j = ~(i / j);%divmod-rew: j ≠ 0 ⊦ i / j * j + i % j = i;% ⊦ ~ i = 0 ↔ i = 0; (s)% ⊦ ~ i * j = ~(i * j); (s)%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult% ⊦ ~ ~ i = i; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2 Used terms: ~ j, i