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Proof step 81 from proof btree-clean-fwd (previous step)

# asar2 = m + 1, # asar1 = m + 1, # zbrar = BRANCH_SIZE, # asar0 = m + 1, # asar = m + 1, # zbrar0 = BRANCH_SIZE, asar1[n] = asar2[n], union(asar1, 0, m + 1) = union(asar2, 0, m + 1), union(asar, 0, m + 1) = union(asar0, 0, m + 1), (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, m + 1, ri0), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P0) )(zh), (( ( ( btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar2[n], asar2[n], ok) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar1[n], asar2[n], ok) )⊗_ (0, n )) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar1[n], asar2[n], ok) )⊗_ (1 + n, m - n )) * P1) )(zh3), zbrar[n].branch?, m < BRANCH_SIZE, n < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, m + 1)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, m + 1)))(zh5), disj(asar2, 0, m + 1), disj(asar1, 0, m + 1), disj(asar0, 0, m + 1), disj(asar, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar, 0, m), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp0, zbrar0, false, false, m + 1), iast), unmodified(adr, mkznode(rp1, zbrar, false, false, m + 1), iast), adr ∈ iast, n ≤ m, asar2[n] ⊆ dom iast, zh4 ⊥ zh3, zh5 ⊥ zh, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1, ∀ n. ¬ m < n → asar2[n] ⊆ dom iast, ∀ n. ¬ m < n → asar0[n] ⊆ dom iast ⊦ asar0[n] = asar2[n], union(asar0, 0, m + 1) = union(asar2, 0, m + 1), adr ∈ union(asar0, 0, m + 1), adr ∈ union(asar2, 0, m + 1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 82:
# zbrar0 = BRANCH_SIZE, # asar = m + 1, # asar0 = m + 1, # zbrar = BRANCH_SIZE, # asar1 = m + 1, # asar2 = m + 1, asar1[n] = asar2[n], m = n0 + n, union(asar, 0, m + 1) = union(asar0, 0, m + 1), union(asar1, 0, m + 1) = union(asar2, 0, m + 1), (( ( ( btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar2[n], asar2[n], ok) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar1[n], asar2[n], ok) )⊗_ (0, n )) * (λ n. ⌜zbrar[n].branch?⌝ ∧ btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, m + 1, ri1), iast, asar1[n], asar2[n], ok) )⊗_ (1 + n, n0 )) * P1) )(zh3), (( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(r2, zbrar0[n].child, zbrar0[n].adr, height, false, key-range(zbrar0, n, m + 1, ri0), iast, asar[n], asar0[n], ok) )⊗_ (0, m + 1 ) * P0) )(zh), zbrar[n].branch?, n < BRANCH_SIZE, m < BRANCH_SIZE, ((r2 => mkznode(rp0, zbrar0, false, false, m + 1)))(zh5), ((r => mkznode(rp1, zbrar, false, false, m + 1)))(zh4), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), disj(asar1, 0, m + 1), disj(asar2, 0, m + 1), nsize(m + 1, nstat), sorted(zbrar0, 0, m), sorted(zbrar, 0, m), unmodified(adr, mkznode(rp1, zbrar, false, false, m + 1), iast), unmodified(adr, mkznode(rp0, zbrar0, false, false, m + 1), iast), adr ∈ iast, n ≤ m, asar2[n] ⊆ dom iast, zh5 ⊥ zh, zh4 ⊥ zh3, (r0 = null ⊃ P0(zh0);r2 = r0 ∧ zh5 ∪ zh = zh0), (r1 = null ⊃ P1(zh1);r = r1 ∧ zh4 ∪ zh3 = zh1), ∀ n. ¬ m < n → asar0[n] ⊆ dom iast, ∀ n. ¬ m < n → asar2[n] ⊆ dom iast, ∀ rp0, r0, adr, ri0, iast, as0, alladrs0, nstat, P0, zh0, rp1, r1, height1, ri1, as1, alladrs1, P1, zh1. ((btree(rp0, r0, adr, height, false, ri0, iast, as0, alladrs0, nstat) * P0))(zh0) ∧ ((btree(rp1, r1, adr, height1, false, ri1, iast, as1, alladrs1, nstat) * P1))(zh1) → height = height1 ∧ as0 = as1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ asar0[n] = asar2[n], union(asar0, 0, m + 1) = union(asar2, 0, m + 1), adr ∈ union(asar2, 0, m + 1), adr ∈ union(asar0, 0, m + 1)

Rule argument

spec: nat name: elim-sub seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [n, m, n0] → [n, m, n0]

Additional information

Used simplifier rules: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n, m, n0