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Proof step 51 from proof add-def-any (previous step)

∀ i1, i2, j, j0. i→n(i1) + i→n(i2) < i→n(j1 * 10) + i→n(i0) → true → mkDecimal(i1, j) + mkDecimal(i2, j0) = mkDecimal(i1 * 10 ^ abs(j0) + i2 * 10 ^ abs(j), abs(j) + abs(j0)) ⊦ i→n(j1) + i→n(i0) < i→n(j1 * 10) + i→n(i0), abs(j) + abs(j0) = 0, j = 0, mkDecimal(j1, abs(j) - 1) + mkDecimal(i0, j0) = mkDecimal(j1 * 10 ^ abs(j0) + i0 * 10 ^ (abs(j) - 1), abs(j) + abs(j0) - 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: add-zero-01: ⊦ dec_i + mkDecimal(0, j) = dec_i;%add-c: ⊦ dec_i + dec_j = dec_j + dec_i;%add-c: ⊦ i + j = j + i; from specification int-basic2%add-a: ⊦ i + j + i0 = (i + j) + i0; from specification int-basic2%add-zero: ⊦ i + 0 = i; from specification int-basic2%Mult-zero: ⊦ i * 0 = 0; from specification int-mult%c: ⊦ m + n = n + m; from specification nat-basic2%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%le-same-04: ⊦ i * abs(j) ≤ abs(j) ↔ j = 0 ∨ i ≤ 1; from specification int-mult%lels: ⊦ ¬ i < j ↔ j ≤ i; from specification int-abs%commutativity: ⊦ i * j = j * i; from specification int-mult%associativity: ⊦ i * j * j0 = (i * j) * j0; from specification int-mult%abs-eqall: ⊦ abs(i * j) = abs(i) * abs(j); from specification int-mult%labs: ⊦ i→n(i) < i→n(j) ↔ abs(i) < abs(j); from specification intnat%l: ⊦ m + n < m + n0 ↔ n < n0; from specification nat-basic2