[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 83 from proof btree-clean-fwd (previous step)

# zbrar0 = BRANCH_SIZE, # asar = (n0 + n) + 1, # asar0 = # asar, # zbrar = BRANCH_SIZE, # asar1 = # asar, # asar2 = # asar, asar1[n] = asar2[n], union(asar, 0, # asar) = union(asar0, 0, # asar), union(asar1, 0, # asar) = union(asar2, 0, # asar), (( ( ( btree(r, zbrar[n].child, zbrar[n].adr, height0, false, key-range(zbrar, n, # asar, ri1), iast, asar2[n], asar2[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri1), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri1), iast, asar1[n1], asar2[n1], ok) )⊗_ (1 + n, n0 )) * P1) )(zh3), (( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(r2, zbrar0[n1].child, zbrar0[n1].adr, height, false, key-range(zbrar0, n1, # asar, ri0), iast, asar[n1], asar0[n1], ok) )⊗_ (0, # asar ) * P0) )(zh), zbrar[n].branch?, n < BRANCH_SIZE, n0 + n < BRANCH_SIZE, ((r2 => mkznode(rp0, zbrar0, false, false, # asar)))(zh5), ((r => mkznode(rp1, zbrar, false, false, # asar)))(zh4), disj(asar, 0, # asar), disj(asar0, 0, # asar), disj(asar1, 0, # asar), disj(asar2, 0, # asar), nsize(# asar, nstat), sorted(zbrar0, 0, n0 + n), sorted(zbrar, 0, n0 + n), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar), iast), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar), iast), adr ∈ iast, 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), ∀ n1. ¬ n0 + n < n1 → asar0[n1] ⊆ dom iast, ∀ n1. ¬ n0 + n < n1 → asar2[n1] ⊆ 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, # asar) = union(asar2, 0, # asar), adr ∈ union(asar2, 0, # asar), adr ∈ union(asar0, 0, # asar)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 84:
# asar2 = # asar, # asar1 = # asar, # zbrar = BRANCH_SIZE, # asar0 = # asar, # asar = (n0 + n) + 1, # zbrar0 = BRANCH_SIZE, asar1[n] = asar2[n], zbrar[n] = mkzbranch(key, adr0, r3), union(asar1, 0, # asar) = union(asar2, 0, # asar), union(asar, 0, # asar) = union(asar0, 0, # asar), (( (λ n1. ⌜zbrar0[n1].branch?⌝ ∧ btree(r2, zbrar0[n1].child, zbrar0[n1].adr, height, false, key-range(zbrar0, n1, # asar, ri0), iast, asar[n1], asar0[n1], ok) )⊗_ (0, # asar ) * P0) )(zh), (( ( ( btree(r, r3, adr0, height0, false, key-range(zbrar, n, # asar, ri1), iast, asar2[n], asar2[n], ok) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri1), iast, asar1[n1], asar2[n1], ok) )⊗_ (0, n )) * (λ n1. ⌜zbrar[n1].branch?⌝ ∧ btree(r, zbrar[n1].child, zbrar[n1].adr, height0, false, key-range(zbrar, n1, # asar, ri1), iast, asar1[n1], asar2[n1], ok) )⊗_ (1 + n, n0 )) * P1) )(zh3), zbrar[n].branch?, n0 + n < BRANCH_SIZE, n < BRANCH_SIZE, ((r => mkznode(rp1, zbrar, false, false, # asar)))(zh4), ((r2 => mkznode(rp0, zbrar0, false, false, # asar)))(zh5), disj(asar2, 0, # asar), disj(asar1, 0, # asar), disj(asar0, 0, # asar), disj(asar, 0, # asar), nsize(# asar, nstat), sorted(zbrar, 0, n0 + n), sorted(zbrar0, 0, n0 + n), unmodified(adr, mkznode(rp0, zbrar0, false, false, # asar), iast), unmodified(adr, mkznode(rp1, zbrar, false, false, # asar), iast), adr ∈ iast, 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, ∀ n1. ¬ n0 + n < n1 → asar2[n1] ⊆ dom iast, ∀ n1. ¬ n0 + n < n1 → asar0[n1] ⊆ dom iast ⊦ asar0[n] = asar2[n], union(asar0, 0, # asar) = union(asar2, 0, # asar), adr ∈ union(asar0, 0, # asar), adr ∈ union(asar2, 0, # asar)

Rule argument

spec: zbranch name: elim-01 seq: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)) subst: [zbr, key, adr, r] → [zbrar[n], key, adr0, r3]

Additional information

Used simplifier rules: ⊦ zbr.branch? → ( (key = zbr.key ∧ adr = zbr.adr) ∧ r = zbr.child ↔ zbr = mkzbranch(key, adr, r)); (elim) Used terms: zbrar[n], key, adr0, r3