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Proof step 12 from proof payload-copy-right-upper (previous step)

n0 + 1 < # zbrar, n0 + nused0 + 1 < # zbrar, ∀ n, zbrar. n + nused0 < # zbrar → payload(copy(zbrar, n, zbrar, n + 1, nused0), n + 1, nused0) = payload(zbrar, n, nused0) ⊦ (zbrar[n0].entry? ⊃ payload(copy(zbrar, n0 + 1, zbrar, 2 + n0, nused0)[n0 + 1, zbrar[n0]], 2 + n0, nused0)[zbrar[n0].key, zbrar[n0].adr] ;payload(copy(zbrar, n0 + 1, zbrar, 2 + n0, nused0)[n0 + 1, zbrar[n0]], 2 + n0, nused0) ) = payload(zbrar, n0, nused0 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ m < # zbrar → payload(zbrar, m, nused + 1) = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused)[zbrar[m].key, zbrar[m].adr];payload(zbrar, m + 1, nused))
Proof step 14:
n0 + nused0 + 1 < # zbrar, n0 + 1 < # zbrar, n0 < # zbrar, ∀ n, zbrar. n + nused0 < # zbrar → payload(copy(zbrar, n, zbrar, n + 1, nused0), n + 1, nused0) = payload(zbrar, n, nused0) ⊦ (zbrar[n0].entry? ⊃ payload(copy(zbrar, n0 + 1, zbrar, 2 + n0, nused0)[n0 + 1, zbrar[n0]], 2 + n0, nused0)[zbrar[n0].key, zbrar[n0].adr] ;payload(copy(zbrar, n0 + 1, zbrar, 2 + n0, nused0)[n0 + 1, zbrar[n0]], 2 + n0, nused0) ) = (zbrar[n0].entry? ⊃ payload(zbrar, n0 + 1, nused0)[zbrar[n0].key, zbrar[n0].adr];payload(zbrar, n0 + 1, nused0))

Rule argument

spec: <toplevel> name: payload-rec seq: ⊦ m < # zbrar → payload(zbrar, m, nused + 1) = (zbrar[m].entry? ⊃ payload(zbrar, m + 1, nused)[zbrar[m].key, zbrar[m].adr] ;payload(zbrar, m + 1, nused) ) subst: [nused, m, zbrar] → [nused0, n0, zbrar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ¬ m < n → ¬ m + m0 < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused0, n0, zbrar