[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 25 from proof btree-ri-unique (previous step)

adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, (( rp0 => mkznode(rp, zbrar0, false, dirty1, nused0) * bnode(rp0, zbrar0, 0, nused0, nused0, height0 + 1, dirty1, ri0, iast, as0 -- adr, alladrs0 -- adr) * P0) )(zh), (( rp0 => mkznode(rp, zbrar0, false, dirty1, nused0) * ( ⌜0 + nused0 ≤ nused0 ∧ nused0 ≤ # zbrar0 ∧ # zbrar0 = BRANCH_SIZE⌝ ∧ ⌜nused0 ≠ 0 ∧ sorted(zbrar0, 0, nused0 - 1)⌝ ∧ _∃ (λ asar. _∃ (λ asar0. ⌜ # asar = nused0 ∧ disj(asar, 0, nused0) ∧ union(asar, 0, nused0) = as1 -- adr ∧ # asar0 = nused0 ∧ disj(asar0, 0, nused0) ∧ union(asar0, 0, nused0) = alladrs1 -- adr ⌝ ∧ (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, nused0, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, nused0 )))) * P1) )(zh), nused0 < BRANCH_SIZE, unmodified(adr, mkznode(rp, zbrar0, false, dirty1, nused0), iast), adr ∈ alladrs0, adr ∈ alladrs1, as0 ⊆ alladrs0, as0 ⊆ dom iast, as1 ⊆ alladrs1, as1 ⊆ dom iast, 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 = ∅, ri0 = ri1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 26:
# zbrar0 = BRANCH_SIZE, # asar = # asar0, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, union(asar, 0, # asar0) = as1 -- adr, union(asar0, 0, # asar0) = alladrs1 -- adr, (( rp0 => mkznode(rp, zbrar0, false, dirty1, # asar0) * bnode(rp0, zbrar0, 0, # asar0, # asar0, height0 + 1, dirty1, ri0, iast, as0 -- adr, alladrs0 -- adr) * P0) )(zh), (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, # asar0, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, # asar0 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, # asar0)) * P1) )(zh), # asar0 < BRANCH_SIZE, disj(asar, 0, # asar0), disj(asar0, 0, # asar0), sorted(zbrar0, 0, # asar0 - 1), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, # asar0), iast), adr ∈ alladrs1, adr ∈ alladrs0, 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, # asar0 = 0, alladrs0 = ∅, alladrs1 = ∅, ri0 = ri1

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%eq-objs: ⊦ ((r => znd * P))(zh) ∧ ((r => znd0 * P2))(zh) → znd = znd0; (forward) from specification maplet[zheap]% ⊦ ((_∃ Pa0 * P0))(zh) ↔ (∃ asar. ((Pa0(asar) * P0))(zh)); (s)% ⊦ ⌜bv⌝ (zh) ↔ bv; (s)% ⊦ ((P ∧ P2))(zh) ↔ P(zh) ∧ P2(zh); (s)%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%and-com: ⊦ (P ∧ P2) = (P2 ∧ P); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ m ≤ m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)