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Proof step 5 from proof payload-drop-upper (previous step)

m + nused0 < # zbrar, m + nused0 < n, n ≤ # zbrar, ∀ m, n, zbrar, zbr. m + nused0 ≤ n ∧ n ≤ # zbrar ∧ m + nused0 ≤ # zbrar → payload(zbrar[n, zbr], m, nused0) = payload(zbrar, m, nused0) ⊦ payload(zbrar[n, zbr], m, nused0 + 1) = payload(zbrar, m, nused0 + 1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ 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 7:
m + nused0 < n, m + nused0 < # zbrar, m < # zbrar[n, zbr], n ≤ # zbrar, ∀ m, n, zbrar, zbr. m + nused0 ≤ n ∧ n ≤ # zbrar ∧ m + nused0 ≤ # zbrar → payload(zbrar[n, zbr], m, nused0) = payload(zbrar, m, nused0) ⊦ (zbrar[n, zbr][m].entry? ⊃ payload(zbrar[n, zbr], m + 1, nused0)[zbrar[n, zbr][m].key, zbrar[n, zbr][m].adr] ;payload(zbrar[n, zbr], m + 1, nused0) ) = payload(zbrar, m, nused0 + 1)

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, m, zbrar[n, zbr]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # zbrar[n, zbr] = # zbrar; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ ¬ m + n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: nused0, m, zbrar[n, zbr]