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Proof step 9 from proof mod-add-01 (previous step)

i * (i0 + 1) + j0 = 0, 0 < i, i0 * i + j0 < 0, i0 < i, j0 < i, i0 < i + j0, j0 < 0, 0 < i + j0, i0 ≤ 0 ⊦ 0 = j0

User applied rule insert equation

And closed the goal!

Rule argument

pos: left 1 flags: rotate, discard

Additional information

Used simplifier rules: ⊦ i = i + j ↔ j = 0; (s)% ⊦ i * j = 0 ↔ i = 0 ∨ j = 0; (ws)% ⊦ (* j) = i0 + (* j0) ↔ j - j0 = i0; (s)%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% ⊦ i * (j + 1) + j1 < i ↔ i * j + j1 < 0; (s)% ⊦ i + j < i + i0 ↔ j < i0; (s)% ⊦ i < i + j ↔ 0 < j; (s)%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1%forward-le-05: ⊦ i < j ∧ 0 ≤ i → 0 < j; (forward) from specification int-abs%ls-forward: ⊦ i * j < i * j0 → i ≠ 0; (forward) from specification int-mult% ⊦ 0 < i → (0 < i * j ↔ 0 < j); (s)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ 0 < i → (i * j < i ↔ j < 1); (s)% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s)% ⊦ i ≤ j → (j < i ↔ false); (as)