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Proof step 36 from proof btree-ri-unique (previous step)

# asar2 = m + 1, # asar1 = m + 1, # asar0 = m + 1, # zbrar0 = BRANCH_SIZE, # asar = m + 1, adr ∈ as0 ↔ adr ∈ as1, adr ∈ as1 ↔ ¬ dirty1, union(asar2, 0, m + 1) = alladrs0 -- adr, union(asar1, 0, m + 1) = as0 -- adr, union(asar, 0, m + 1) = as1 -- adr, union(asar0, 0, m + 1) = alladrs1 -- adr, (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri0), iast, asar1[n], asar2[n], ok) )⊗_ (0, m + 1 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P0) )(zh), (( ( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, n ) * ( ⌜zbrar0[0 + n].branch?⌝ ∧ btree (rp0, zbrar0[0 + n].child, zbrar0[0 + n].adr, height0, dirty1, key-range(zbrar0, 0 + n, m + 1, ri1), iast, asar[0 + n], asar0[0 + n], ok)) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0 + n + 1, m + 1 - n - 1 )) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P1) )(zh), n < BRANCH_SIZE, m < BRANCH_SIZE, m + 1 < BRANCH_SIZE, n < m + 1, disj(asar2, 0, m + 1), disj(asar1, 0, m + 1), disj(asar, 0, m + 1), disj(asar0, 0, m + 1), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, m + 1), iast), adr ∈ alladrs1, adr ∈ alladrs0, n ≤ m, 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, m + 1, ri0) = key-range(zbrar0, n, m + 1, ri1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 37:
# asar = m + 1, # zbrar0 = BRANCH_SIZE, # asar0 = m + 1, # asar1 = m + 1, # asar2 = m + 1, adr ∈ as1 ↔ ¬ dirty1, adr ∈ as0 ↔ adr ∈ as1, union(asar0, 0, m + 1) = alladrs1 -- adr, union(asar, 0, m + 1) = as1 -- adr, union(asar1, 0, m + 1) = as0 -- adr, union(asar2, 0, m + 1) = alladrs0 -- adr, (( ( (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri0), iast, asar1[n], asar2[n], ok) )⊗_ (0, m + 1 ) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P0) )(zh), (( ( ( ( btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (0, n )) * (λ n. ⌜zbrar0[n].branch?⌝ ∧ btree(rp0, zbrar0[n].child, zbrar0[n].adr, height0, dirty1, key-range(zbrar0, n, m + 1, ri1), iast, asar[n], asar0[n], ok) )⊗_ (1 + n, m - n )) * rp0 => mkznode(rp, zbrar0, false, dirty1, m + 1)) * P1) )(zh), zbrar0[n].branch?, m + 1 < BRANCH_SIZE, m < BRANCH_SIZE, n < BRANCH_SIZE, disj(asar0, 0, m + 1), disj(asar, 0, m + 1), disj(asar1, 0, m + 1), disj(asar2, 0, m + 1), sorted(zbrar0, 0, m), unmodified(adr, mkznode(rp, zbrar0, false, dirty1, m + 1), iast), adr ∈ alladrs0, adr ∈ alladrs1, n ≤ m, 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, m + 1, ri0) = key-range(zbrar0, n, m + 1, ri1)

Rule argument

Additional information

Used simplifier rules: btree-refed-subseteq-alladrs: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ alladrs;%btree-adrs: ⊦ (( btree (rp, rp0, adr0, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ dom iast;% ⊦ ⌜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]%and-ass: ⊦ ((P ∧ P3) ∧ P4) = (P ∧ P3 ∧ P4); from specification heap-sep[zheap]% ⊦ (⌜bv⌝ ∧ P) * P2 = (⌜bv⌝ ∧ P * P2); (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)