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Proof step 23 from proof exdivmod (previous step)

i = i0 * ~ j + j0, 0 < i, j < 0, j0 < ~ j, 0 ≤ i0, 0 ≤ j0, i ≤ 0 → i + j1 = j * i1 ∧ 0 ≤ i1 ∧ 0 ≤ j1 ∧ j1 + j < 0 ⊦ j = 0, ∃ i0, j0. i = i0 * j + j0 ∧ abs(j0) + j < 0

User applied rule exists right

And closed the goal!

Rule argument

pos: right 2 terms: ~ i0, j0 abort: F discard: F computed: F

Additional information

Used simplifier rules: minus-left: ⊦ ~ i * j = ~(i * j);%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;% ⊦ i + ~ j = i - j; (s)%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2% ⊦ i < j - i0 ↔ i + i0 < j; (s)% ⊦ i + 0 = i; (s)% ⊦ i < ~ j ↔ i + j < 0; (s)% ⊦ i - j = i0 ↔ i = i0 + j; (s)% ⊦ i + (j - i) = j; (s)% ⊦ 0 ≤ i → abs(i) = i; (s) Used terms: ~ i0, j0