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Proof step 10 from proof longin-mult-01 (previous step)

~2147483648 * j ≤ i * j, i * j < ~9223372036854775808, ~2147483648 ≤ i, ~2147483648 ≤ j, j ≤ 2147483647, i < 0, 0 < j ⊦

User applied rule cut formula

And obtained the following new goals

Proof step 11:
~2147483648 * j ≤ i * j, i * j < ~9223372036854775808, ~2147483648 ≤ i, ~2147483648 ≤ j, j ≤ 2147483647, i < 0, 0 < j ⊦ ~2147483648 * j < ~(2 ^ 31 * 2 ^ 32)

Rule argument

fma: ~2147483648 * j < ~(2 ^ 31 * 2 ^ 32)

Additional information

Used simplifier rules: numlsneg-02: i < 0 ⊦ (* i) * i0 < j ↔ 0 < ~ i * i0 + j; from specification int-mult%minus-add: i < 0 ⊦ (* i) + j = j - ~ i; from specification int-abs%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%nummult-ls-02: ⊦ 0 < i ∧ 0 ≤ j → ((* j) < (* i) * i0 ↔ j / i < i0); from specification int-div%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%le-pos: ⊦ 0 < i → (i * j ≤ i * i0 ↔ j ≤ i0); from specification int-mult%numleledel-01: i0 ≠ 0, i0 ≤ i ⊦ (* i) ≤ j → (* i0) ≤ j; from specification int-abs%forward-lsle: ⊦ i < j ∧ j ≤ j0 → i < j0; (forward) from specification int-abs%forward-pos: ⊦ i < j ∧ 0 < i → 0 < j ∧ 1 < j; (forward) from specification int-basic1 Used formulas: ~2147483648 * j < ~(2 ^ 31 * 2 ^ 32)