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Proof step 78 from proof btree-clean-fwd-strong (previous step)

# zbrar0 = BRANCH_SIZE, # asar = # asar0, # asar2 = # asar0, # asar1 = # asar0, # zbrar = BRANCH_SIZE, # asar0 = (n0 + 1) + n, asar0[n] = asar2[n], asar1[n] = asar2[n], asar[n] = asar2[n], zbrar0[n] = mkzbranch(key0, adr1, r4), zbrar[n] = mkzbranch(key0, adr1, r3), key-range(zbrar0, n, # asar0, ri0) = key-range(zbrar, n, # asar0, ri1), union(asar2, 0, # asar0) = alladrs1 -- adr, union(asar1, 0, # asar0) = alladrs1 -- adr, union(asar0, 0, # asar0) = alladrs0 -- adr, union(asar, 0, # asar0) = alladrs0 -- adr, ((btree(r2, r4, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast0, asar2[n], asar2[n], ok) * P2))(zh), ((btree(r, r3, adr1, height0, false, key-range(zbrar, n, # asar0, ri1), iast1, asar2[n], asar2[n], ok) * P))(zh3), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, # asar0)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, # asar0)))(zh5), disj(asar2, 0, # asar0), disj(asar1, 0, # asar0), disj(asar0, 0, # asar0), disj(asar, 0, # asar0), nsize(# asar0, nstat), sorted(zbrar0, 0, n0 + n), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar0), iast0), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar0), iast1), adr ∈ alladrs0, adr ∈ iast0, adr ∈ alladrs1, adr ∈ iast1, asar2[n] ⊆ dom iast0, asar2[n] ⊆ dom iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, zh4 ⊥ zh3, zh5 ⊥ zh, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0) ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar, n, # asar0, ri1) = key-range(zbrar0, n, # asar0, ri1)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: zbranch-array-ops name: key-range-replace-branches seq: ⊦ n < m ∧ m ≤ # zbrar0 ∧ # zbrar0 = # zbrar1 ∧ (∀ n0. n0 ≤ n ∧ n0 ≤ m → zbrar0[n0].key = zbrar1[n0].key) → key-range(zbrar0, n, m, ri) = key-range(zbrar1, n, m, ri) subst: [zbrar1, ri, m, n, zbrar0] → [zbrar0, ri1, # asar0, n, zbrar] flags: <all paths>

Additional information

Used simplifier rules: nat-le-eq: ⊦ n = n0 + n1 → (n ≤ n0 ↔ n1 = 0 ∧ n = n0);% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ nsize(m, nstat) → ¬ BRANCH_SIZE < m; (s)%n < nused, nused ≤ BRANCH_SIZE, # zbrar0 = # zbrar1, # zbrar0 = BRANCH_SIZE ⊦ iast0 ⊆ iast1 ∧ unmodified(adr, mkznode(rp0, zbrar0, bv, false, nused), iast0) ∧ unmodified(adr, mkznode(rp1, zbrar1, bv, false, nused), iast1) → (zbrar1[n].key = zbrar0[n].key ↔ true); (ss)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≠ 0; (s) Used terms: zbrar0, ri1, # asar0, n, zbrar