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Proof step 83 from proof add-scale-neq (previous step)

0 ≤ dec_i.scale, 0 ≤ dec_j.scale, dec_i.scale + dec_j.scale < maxScale ( (dec_i.int * 10 ^ (dec_j.scale - min(dec_j.scale, dec_i.scale)) + dec_j.int * 10 ^ (dec_i.scale - min(dec_j.scale, dec_i.scale))) * 10 ^ min(dec_j.scale, dec_i.scale)) ⊦ 0 = max(dec_i.scale, dec_j.scale), dec_i.scale = dec_j.scale, 0 ≤ min(dec_j.scale, dec_i.scale) ∧ dec_i.int * 10 ^ (dec_j.scale - min(dec_j.scale, dec_i.scale)) + dec_j.int * 10 ^ (dec_i.scale - min(dec_j.scale, dec_i.scale)) ≠ 0

User applied rule cut formula

And obtained the following new goals

Proof step 84:
0 ≤ dec_i.scale, 0 ≤ dec_j.scale, dec_i.scale + dec_j.scale < maxScale ( (dec_i.int * 10 ^ (dec_j.scale - min(dec_j.scale, dec_i.scale)) + dec_j.int * 10 ^ (dec_i.scale - min(dec_j.scale, dec_i.scale))) * 10 ^ min(dec_j.scale, dec_i.scale)) ⊦ 0 ≤ min(dec_j.scale, dec_i.scale), 0 = max(dec_i.scale, dec_j.scale), dec_i.scale = dec_j.scale, 0 ≤ min(dec_j.scale, dec_i.scale) ∧ dec_i.int * 10 ^ (dec_j.scale - min(dec_j.scale, dec_i.scale)) + dec_j.int * 10 ^ (dec_i.scale - min(dec_j.scale, dec_i.scale)) ≠ 0

Rule argument

fma: 0 ≤ min(dec_j.scale, dec_i.scale)

Additional information

Used simplifier rules: zero-gels-04: ⊦ i ≤ 0 ∧ 0 ≤ j → i ≤ j; from specification int-abs%forward-le-01: ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j; (forward) from specification int-abs%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%commutativity: ⊦ i * j = j * i; from specification int-mult%maxScale-zero: ⊦ maxScale(0) = 0; from specification decimal%zero-gs: ⊦ 0 ≤ i ∧ 0 ≤ j → ¬ i + j < 0; 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 Used formulas: 0 ≤ min(dec_j.scale, dec_i.scale)