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Proof step 50 from proof add-scale-neq (previous step)

(dec_i.int + dec_j.int) % 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))) % 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 51:
⊦ ¬ j < i → min(j, i) = i
Proof step 52:
¬ ¬ dec_j.scale < dec_i.scale, (dec_i.int + dec_j.int) % 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))) % 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 55:
¬ dec_j.scale < dec_i.scale, (dec_i.int + dec_j.int) % 10 = 0, (dec_i.int * 10 ^ (dec_j.scale - dec_i.scale) + dec_j.int * 10 ^ (dec_i.scale - dec_i.scale)) % 10 = 0, 0 ≤ dec_j.scale, 0 ≤ dec_i.scale, maxScale(dec_i.int * 10 ^ (dec_j.scale - dec_i.scale) + dec_j.int * 10 ^ (dec_i.scale - dec_i.scale)) + dec_i.scale ≤ dec_i.scale + dec_j.scale, 0 ≤ 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 - dec_i.scale) + dec_j.int * 10 ^ (dec_i.scale - dec_i.scale)) + dec_i.scale, dec_i.scale = dec_j.scale, dec_i.int * 10 ^ (dec_j.scale - dec_i.scale) + dec_j.int * 10 ^ (dec_i.scale - dec_i.scale) = 0

Rule argument

spec: int-min inst: name: Min-right seq: ⊦ ¬ j < i → min(j, i) = i subst: [i, j] → [dec_i.scale, dec_j.scale] flags: rewrite

Additional information

Used terms: dec_i.scale, dec_j.scale