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Proof step 5 from proof append-to-gr-01 (previous step)

1 + # x0 < m, ∀ m, σ. # x0 < m → ⌜x0⌝ + σ to m = ⌜x0⌝ + (σ to (m - # x0)) ⊦ (⌜a '⌝ + ⌜x0⌝ ) + σ to m = (⌜a '⌝ + ⌜x0⌝ ) + (σ to (m - 1 + # x0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ n ≠ 0 → ⌜a '⌝ + σ to (* n) = ⌜a '⌝ + (σ to (n - 1))
Proof step 7:
1 + # x0 < m, ∀ m, σ. # x0 < m → ⌜x0⌝ + σ to m = ⌜x0⌝ + (σ to (m - # x0)) ⊦ ⌜a '⌝ + (⌜x0⌝ + σ to (m - 1)) = (⌜a '⌝ + ⌜x0⌝ ) + (σ to (m - 1 + # x0)), m = 0

Rule argument

spec: <toplevel> name: cons-to-num seq: ⊦ n ≠ 0 → ⌜a '⌝ + σ to (* n) = ⌜a '⌝ + (σ to (n - 1)) subst: [n, σ, a] → [m, ⌜x0⌝ + σ, a] flags: <all paths>

Additional information

Used simplifier rules: append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2;%to-zero: ⊦ σ to 0 = [];%n0 < n ⊦ ¬ m + (* n) < (* n0); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: m, ⌜x0⌝ + σ, a