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Proof step 20 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))) * 10 ^ min(dec_j.scale, dec_i.scale)) ≤ dec_i.scale + dec_j.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))) * 10 ^ min(dec_j.scale, dec_i.scale)), dec_i.scale = dec_j.scale

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ 0 ≤ j ∧ i ≠ 0 → maxScale(i * 10 ^ j) = maxScale(i) + j
Proof step 22:
¬ ( 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), (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))) * 10 ^ min(dec_j.scale, dec_i.scale)) ≤ dec_i.scale + dec_j.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))) * 10 ^ min(dec_j.scale, dec_i.scale)), dec_i.scale = dec_j.scale
Proof step 46:
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, (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 ⊦ 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

Rule argument

spec: decimal inst: name: maxScale-pot10-02 seq: ⊦ 0 ≤ j ∧ i ≠ 0 → maxScale(i * 10 ^ j) = maxScale(i) + j subst: [j, i] → [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))] flags: rewrite

Additional information

Used terms: 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))