[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 14 from proof btree-clean-fwd-strong (previous step)

((btree(rp0, r0, adr, height + 1, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0), (( r => mkznode(rp1, zbrar, height1 = 0, false, nused) * bnode(r, zbrar, 0, nused, nused, height1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1) )(zh), nsize(nused, nstat), unmodified(adr, mkznode(rp1, zbrar, height1 = 0, false, nused), iast1), adr ∈ alladrs1, adr ∈ iast1, iast0 ⊆ iast1, alladrs0 ⊆ dom iast0, alladrs1 ⊆ dom iast1, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh = zh1), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1 ⊦ # alladrs1 = 0, alladrs1 = ∅, height + 1 = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ ((btree(rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P))(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode(r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh))
Proof step 16:
(( r => mkznode(rp1, zbrar, height1 = 0, false, nused) * bnode(r, zbrar, 0, nused, nused, height1, false, ri1, iast1, alladrs1 -- adr, alladrs1 -- adr) * P1) )(zh), nsize(nused, nstat), unmodified(adr, mkznode(rp1, zbrar, height1 = 0, false, nused), iast1), adr ∈ iast1, adr ∈ alladrs1, alladrs1 ⊆ dom iast1, alladrs0 ⊆ dom iast0, iast0 ⊆ iast1, (r1 = null ⊃ P1(zh1);r = r1 ∧ zh = zh1), ∀ rp1, r1, adr, height1, ri1, iast1, alladrs1, nstat, P1, zh1, rp0, r0, ri0, iast0, alladrs0, P0, zh0. ((btree(rp1, r1, adr, height1, false, ri1, iast1, alladrs1, alladrs1, nstat) * P1))(zh1) ∧ ((btree(rp0, r0, adr, height, false, ri0, iast0, alladrs0, alladrs0, nstat) * P0))(zh0) ∧ alladrs1 ⊆ dom iast1 ∧ alladrs0 ⊆ dom iast0 ∧ iast0 ⊆ iast1 → height = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1, ∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P0(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → false) ∧ (adr ∈ alladrs0 ↔ ¬ dirty) ∧ adr ∈ alladrs0 ∧ nsize(nused, nstat) ∧ unmodified(adr, mkznode(rp0, zbrar, height + 1 = 0, dirty, nused), iast0) ∧ (( r => mkznode(rp0, zbrar, height + 1 = 0, dirty, nused) * bnode(r, zbrar, 0, nused, nused, height + 1, dirty, ri0, iast0, alladrs0 -- adr, alladrs0 -- adr) * P0) )(zh) ⊦ # alladrs1 = 0, alladrs1 = ∅, height + 1 = height1 ∧ ri0 = ri1 ∧ alladrs0 = alladrs1

Rule argument

spec: <toplevel> name: btree seq: ⊦ (( btree (rp, r0, adr, height, dirty0, ri, iast, as, alladrs, nstat) * P) )(zh0) ↔ (∃ r, zh, zbrar, dirty, nused. (r0 = null ⊃ ¬ dirty ∧ P(zh0);r = r0 ∧ zh = zh0) ∧ (dirty → dirty0) ∧ (adr ∈ as ↔ ¬ dirty) ∧ adr ∈ alladrs ∧ nsize(nused, nstat) ∧ unmodified (adr, mkznode(rp, zbrar, height = 0, dirty, nused), iast) ∧ (( r => mkznode(rp, zbrar, height = 0, dirty, nused) * bnode (r, zbrar, 0, nused, nused, height, dirty, ri, iast, as -- adr, alladrs -- adr) * P) )(zh)) subst: [zh0, P, nstat, alladrs, as, iast, ri, dirty0, height, adr, r0, rp] → [zh0, P0, nstat, alladrs0, alladrs0, iast0, ri0, false, height + 1, adr, r0, rp0] flags: <all paths>

Additional information

Used terms: zh0, P0, nstat, alladrs0, alladrs0, iast0, ri0, false, height + 1, adr, r0, rp0