[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 41 from proof btree-ri-unique (previous step)

# asar = # asar2, # zbrar0 = BRANCH_SIZE, # asar0 = # asar2, # asar1 = # asar2, # asar2 = (n0 + n) + 1, adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, zbrar0[n] = mkzbranch(key, adr0, r), union(asar0, 0, # asar2) = alladrs1 -- adr, union(asar, 0, # asar2) = as1 -- adr, union(asar1, 0, # asar2) = as0 -- adr, union(asar2, 0, # asar2) = alladrs0 -- adr, (( ( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, # asar2 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P0) )(zh), (( ( ( ( btree(rp0, r, adr0, height0, dirty1, key-range(zbrar0, n, # asar2, ri1), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P1) )(zh), # asar2 < BRANCH_SIZE, n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, # asar2), disj(asar, 0, # asar2), disj(asar1, 0, # asar2), disj(asar2, 0, # asar2), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar2), iast), adr ∈ alladrs0, adr ∈ alladrs1, as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, asar[n] ⊆ dom iast, asar[n] ⊆ asar0[n], dirty1 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null ⊦ # alladrs1 = 0, # alladrs0 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar0, n, # asar2, ri0) = key-range(zbrar0, n, # asar2, ri1)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1)
Proof step 43:
# asar2 = (n0 + n) + 1, # asar1 = # asar2, # asar0 = # asar2, # zbrar0 = BRANCH_SIZE, # asar = # asar2, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, zbrar0[n] = mkzbranch(key, adr0, r), union(asar2, 0, # asar2) = alladrs0 -- adr, union(asar1, 0, # asar2) = as0 -- adr, union(asar, 0, # asar2) = as1 -- adr, union(asar0, 0, # asar2) = alladrs1 -- adr, (( ( ( ( btree(rp0, r, adr0, height0, dirty1, key-range(zbrar0, n, # asar2, ri1), iast, asar[n], asar0[n], ok) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri1), iast, asar[n1], asar0[n1], ok) )⊗_ (1 + n, n0 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P1) )(zh), (( ( ( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, n ) * ( ⌜zbrar0[0 + n].branch?⌝ ∧ btree (rp0, zbrar0[0 + n].child, zbrar0[0 + n].adr, height0, dirty1, key-range(zbrar0, 0 + n, # asar2, ri0), iast, asar1[0 + n], asar2[0 + n], ok)) * (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok) )⊗_ (0 + n + 1, # asar2 - n - 1 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar2)) * P0) )(zh), n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, # asar2 < BRANCH_SIZE, n < # asar2, disj(asar2, 0, # asar2), disj(asar1, 0, # asar2), disj(asar, 0, # asar2), disj(asar0, 0, # asar2), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar2), iast), adr ∈ alladrs1, adr ∈ alladrs0, asar[n] ⊆ asar0[n], asar[n] ⊆ dom iast, as1 ⊆ dom iast, as1 ⊆ alladrs1, as0 ⊆ dom iast, as0 ⊆ alladrs0, dirty1 → dirty, ∀ rp, rp0, adr, dirty, ri0, iast, as0, alladrs0, P0, zh, ri1, as1, alladrs1, P1. ((btree(rp, rp0, adr, height0, dirty, ri0, iast, as0, alladrs0, ok) * P0))(zh) ∧ ((btree(rp, rp0, adr, height0, dirty, ri1, iast, as1, alladrs1, ok) * P1))(zh) ∧ as0 ⊆ alladrs0 ∧ as0 ⊆ dom iast ∧ as1 ⊆ alladrs1 ∧ as1 ⊆ dom iast → ri0 = ri1 ∨ rp0 = null ⊦ # alladrs0 = 0, # alladrs1 = 0, alladrs0 = ∅, alladrs1 = ∅, key-range(zbrar0, n, # asar2, ri0) = key-range(zbrar0, n, # asar2, ri1)

Rule argument

spec: iter-n, inst: zheap name: Sep-unfold seq: ⊦ m0 < m → nP0 ⊗_(n, m) = nP0 ⊗_(n, m0) * nP0(n + m0) * nP0 ⊗_(n + m0 + 1, m - m0 - 1) subst: [m0, m, n, nP0] → [n, # asar2, 0, λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok)] 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)%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: n, # asar2, 0, λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree (rp0, zbrar0[n1].child, zbrar0[n1].adr, height0, dirty1, key-range(zbrar0, n1, # asar2, ri0), iast, asar1[n1], asar2[n1], ok)