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Proof step 1 from proof append-to-addsize-01 (conclusion)

⊦ ⌜x⌝ + σ to m + # x = ⌜x⌝ + (σ to m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 2:
⊦ # x < m → ⌜x⌝ + σ to m = ⌜x⌝ + (σ to (m - # x))

Rule argument

spec: stream inst: name: append-to-gr-01 seq: ⊦ # x < m → ⌜x⌝ + σ to m = ⌜x⌝ + (σ to (m - # x)) subst: [m, σ, x] → [m + # x, σ, x] flags: rewrite, current

Additional information

Used simplifier rules: append-to-size-01: ⊦ ⌜x⌝ + σ to # x = ⌜x⌝ ;%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2;%to-zero: ⊦ σ to 0 = [];%append-empty: ⊦ σ + [] = σ;%append-cancel-l: ⊦ ⌜x⌝ + σ0 = ⌜x⌝ + σ1 ↔ σ0 = σ1;%append-assoc: ⊦ (σ + σ1) + σ2 = σ + σ1 + σ2;%eq-01: ⊦ m + n - m = n; from specification nat%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic2%c: ⊦ m + n = n + m; from specification nat-basic2%lels: ⊦ ¬ m < n ↔ n ≤ m; from specification nat%le-add: ⊦ m + m0 ≤ m ↔ m0 = 0; from specification nat%add-base: ⊦ n + 0 = n; from specification nat-basic2%inj-02: ⊦ ⌜x⌝ = ⌜x0⌝ ↔ x = x0; from specification stream-basic Used terms: m + # x, σ, x