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Proof step 41 from proof add-scale-neq (previous step)

(dec_i.int - dec_i.int * 10 ^ (dec_j.scale - dec_i.scale)) % 10 = 0, 0 < dec_i.scale, 1 < dec_j.scale, 0 ≤ dec_j.scale, dec_i.scale < dec_j.scale, 0 ≤ dec_i.scale + dec_j.scale, 0 < dec_j.scale ⊦ dec_i.scale = 0, (dec_i.int * 10 ^ (dec_j.scale - dec_i.scale)) % 10 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: int-pot inst: name: Pot-pos seq: ⊦ 0 < i → j ^ i = j ^ i -1 * j subst: [i, j] → [dec_j.scale - dec_i.scale, 10] flags: rewrite

Additional information

Used simplifier rules: sub-pred: ⊦ i -1 = i - 1; from specification int-abs%sub-sub: ⊦ i - j - i0 = i - j + i0; from specification int-abs%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%mod-zero-num: i0 % j = 0, j ≠ 0 ⊦ (i * (* i0)) % (* j) = 0; from specification int-div%commutativity: ⊦ i * j = j * i; from specification int-mult%sub-ls-01: ⊦ i < j - i0 ↔ i + i0 < j; from specification int-abs%add-zero: ⊦ i + 0 = i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-c: ⊦ i + j = j + i; from specification int-basic2%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%lels-03: ⊦ i < j → (¬ ¬ j ≤ i ↔ false); from specification int-abs Used terms: dec_j.scale - dec_i.scale, 10