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Proof step 38 from proof swap-to-from-app (previous step)

#((σ to n0 + m) + σ0 from m) = ⌜n1⌝ + # σ0, #((σ to m + n0) from m) = ⌜n1⌝ , ⌜n1⌝ < ⌜n0⌝ + # σ0, ⌜n2⌝ < # σ0, ⌜n2⌝ + ⌜n1⌝ < ⌜n0⌝ + # σ0 ⊦ σ0 = [], ((σ to n0 + m) + σ0)[m + n2 + n1] = σ0[n2], # σ < ⌜n0⌝ + ⌜m⌝ , # σ < ⌜m⌝ + ⌜n0⌝

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 39:
⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n]
Proof step 40:
¬ ¬ ⌜m + n2 + n1⌝ < #(σ to n0 + m), #((σ to n0 + m) + σ0 from m) = ⌜n1⌝ + # σ0, #((σ to m + n0) from m) = ⌜n1⌝ , ⌜n1⌝ < ⌜n0⌝ + # σ0, ⌜n2⌝ < # σ0, ⌜n2⌝ + ⌜n1⌝ < ⌜n0⌝ + # σ0 ⊦ σ0 = [], ((σ to n0 + m) + σ0)[m + n2 + n1] = σ0[n2], # σ < ⌜n0⌝ + ⌜m⌝ , # σ < ⌜m⌝ + ⌜n0⌝
Proof step 42:
¬ ⌜m + n2 + n1⌝ < #(σ to n0 + m), #((σ to n0 + m) + σ0 from m) = ⌜n1⌝ + # σ0, #((σ to m + n0) from m) = ⌜n1⌝ , ⌜n1⌝ < ⌜n0⌝ + # σ0, ⌜n2⌝ < # σ0, ⌜n2⌝ + ⌜n1⌝ < ⌜n0⌝ + # σ0 ⊦ σ0 = [], σ0[m + n2 + n1 - (#(σ to n0 + m)).n] = σ0[n2], # σ < ⌜n0⌝ + ⌜m⌝ , # σ < ⌜m⌝ + ⌜n0⌝

Rule argument

spec: inst: name: get-append-notls seq: ⊦ ¬ ⌜m⌝ < # σ → (σ + σ1)[m] = σ1[m - (# σ).n] subst: [m, σ1, σ] → [m + n2 + n1, σ0, σ to n0 + m] flags: rewrite, current

Additional information

Used terms: m + n2 + n1, σ0, σ to n0 + m