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Proof step 47 from proof add-scale-neq (previous step)

(dec_i.int + dec_j.int) % 10 = 0, 0 ≤ dec_j.scale, 0 ≤ dec_i.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))) + min(dec_j.scale, dec_i.scale) ≤ dec_i.scale + dec_j.scale, 0 ≤ min(dec_j.scale, dec_i.scale) ⊦ dec_j.int % 10 = 0, dec_i.int % 10 = 0, dec_i.scale + dec_j.scale = max(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))) + min(dec_j.scale, dec_i.scale), dec_i.scale = dec_j.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 insert rewrite lemma

And obtained the following new goals

Proof step 48:
⊦ i % 10 ≠ 0 ∧ i ≠ 0 → maxScale(i) = 0
Proof step 49:
¬ ( (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 ≠ 0 ∧ 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), (dec_i.int + dec_j.int) % 10 = 0, 0 ≤ dec_j.scale, 0 ≤ dec_i.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))) + min(dec_j.scale, dec_i.scale) ≤ dec_i.scale + dec_j.scale, 0 ≤ min(dec_j.scale, dec_i.scale) ⊦ dec_j.int % 10 = 0, dec_i.int % 10 = 0, dec_i.scale + dec_j.scale = max(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))) + min(dec_j.scale, dec_i.scale), dec_i.scale = dec_j.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
Proof step 58:
(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 ≠ 0 ∧ 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, (dec_i.int + dec_j.int) % 10 = 0, 0 ≤ dec_j.scale, 0 ≤ dec_i.scale, 0 + min(dec_j.scale, dec_i.scale) ≤ dec_i.scale + dec_j.scale, 0 ≤ min(dec_j.scale, dec_i.scale) ⊦ dec_j.int % 10 = 0, dec_i.int % 10 = 0, dec_i.scale + dec_j.scale = max(dec_i.scale, dec_j.scale) + 0 + min(dec_j.scale, dec_i.scale), dec_i.scale = dec_j.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

spec: decimal inst: name: maxScale-base seq: ⊦ i % 10 ≠ 0 ∧ i ≠ 0 → maxScale(i) = 0 subst: [i] → [ 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))] flags: rewrite

Additional information

Used terms: 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))