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Proof step 7 from proof noeth-modlep (previous step)

j0 < j2, 0 < j2, j < 0, i0 < i, j0 + j < 0, 0 ≤ j + j2, 0 ≤ j0, j * i0 ≤ j * i + j0, j0 < abs(j + 1) ∧ abs(j + 1) < j2 ∧ abs(j + 1) * i ≤ abs(j + 1) * i0 + j0 ∧ 0 ≤ j0 → i ≤ i0, ∀ j0, j, i, i0. j0 < abs(j) ∧ abs(j) < j2 ∧ abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 → i ≤ i0 ⊦ j = 0

User applied rule simplifier

And obtained the following new goals

Proof step 8:
j0 + j = ~1, i0 < i, j < 0, 0 < j2, j0 < j2, j * i0 + i0 ≤ i + j * i + j0, j * i0 ≤ j * i + j0, 0 ≤ j0, 0 ≤ j + j2, ∀ j0, j, i, i0. j0 < abs(j) ∧ abs(j) < j2 ∧ abs(j) * i ≤ abs(j) * i0 + j0 ∧ 0 ≤ j0 → i ≤ i0 ⊦ j = 0

Rule argument

Additional information

Used simplifier rules: minus-left: ⊦ ~ i * j = ~(i * j);%dist-lsplit: ⊦ (i + i0) * j = i * j + i0 * j;%one: ⊦ 1 * i = i;%associativity: ⊦ i * j * j0 = (i * j) * j0;%commutativity: ⊦ i * j = j * i;%forward-le-02: ⊦ j < i ∧ i ≤ 0 → j < 0; (forward) from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs% ⊦ i < 0 → (~1 ≤ i ↔ i = ~1); (s)% ⊦ i < j ↔ ¬ j ≤ i; (ss)% ⊦ i ≤ i1 ∧ i2 < i3 → (i1 + i3 < i + i2 ↔ false); (s)% ⊦ i ≤ j ↔ ¬ j < i; (ss)% ⊦ i < j → (j ≤ i ↔ false); (as)% ⊦ i + 0 = i; (s)% ⊦ i ≤ j - i0 ↔ i + i0 ≤ j; (s)% ⊦ i + (j - i0) = i + j - i0; (s)% ⊦ ~ i ≤ j ↔ 0 ≤ i + j; (s)% ⊦ i + ~ j = i - j; (s)% ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; (s)% ⊦ ~1 < i ↔ 0 ≤ i; (s)% ⊦ (* j) < i0 + (* j0) ↔ j - j0 < i0; (s)% ⊦ ~ i < j ↔ 0 < j + i; (s)% ⊦ i + (* j) < (* j0) ↔ i < j0 - j; (s)% ⊦ i < ~ j ↔ i + j < 0; (s)%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2% ⊦ ~ i - j = ~(i + j); (s)% ⊦ i < 0 → abs(i) = ~ i; (s)% ⊦ i < 0 → abs(i + 1) = abs(i) - 1; (s)