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Proof step 2 from proof inv-io-save (previous step)

adr0 = adr1, istore1 = istore, ((btree(rp, r, adr0, height, dirty, ri, nodes(istore), as, alladrs, ok) * P))(zh), zh[r].dirty, pstate.index?, btree-sub-clean(r, zh[r].usedsize, zh), inv-index(istore, istore0), inv-journal(nx, gstore, gstore0, ginval), inv-state(pstate, nx, gstore, istore, syn), refsize-cons(istore, as0 ∪ as), r ∈ zh, alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore) ⊦ ⟪btree_io_save#(r, adr, istore0, ginval, gstore, gstore0, nx, maxino, ginval0; adr1, zh, istore, pstate, syn; err)⟫ ( refsize-cons(istore, as0 ∪ (err = ESUCCESS ⊃ as ++ adr1;as)) ∧ inv-index(istore, istore0) ∧ inv-state(pstate, nx, gstore, istore, syn) ∧ istore1 ⊑ istore ∧ r ∈ zh ∧ (err = ESUCCESS → ¬ adr1 ∈ as) ∧ (∃ as0, alladrs0. as0 = (err = ESUCCESS ⊃ as ++ adr1;as) ∧ alladrs0 = (err = ESUCCESS ⊃ (alladrs -- adr0) ++ adr1;alladrs) ∧ dom nodes(istore) = dom nodes(istore1) ∪ (err = ESUCCESS ⊃ {adr1} ;∅) ∧ (err = ESUCCESS → ¬ adr1 ∈ istore1) ∧ ((btree(rp, r, (err = ESUCCESS ⊃ adr1;adr0), height, err ≠ ESUCCESS, ri, nodes(istore), as0, alladrs0, ok) * P))(zh)) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err = ESUCCESS → pstate.index? ∧ ¬ zh[r].dirty))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
adr0 = adr1, istore1 = istore, remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, elems nx ∩ ginval = ∅, dom gstore0 ∪ elems nx = dom gstore ∪ ginval, pstate ≠ unbuffered, ((btree(rp, r, adr1, height, dirty, ri, nodes(istore), as, alladrs, ok) * P))(zh), zh[r].dirty, pstate.index?, btree-sub-clean(r, zh[r].usedsize, zh), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), refsize-cons(istore, as0 ∪ as), pstate.lnum ∈ istore, r ∈ zh, as ⊆ alladrs, as ⊆ dom nodes(istore), alladrs ⊆ dom nodes(istore), as0 ⊆ dom nodes(istore), istore0 ⊑ istore, ¬ dups(nx), pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore ⊦ ⟪btree_io_save#(r, adr, istore0, ginval, gstore, gstore0, nx, maxino, ginval0; adr1, zh, istore, pstate, syn; err)⟫ ( refsize-cons(istore, as0 ∪ (err = ESUCCESS ⊃ as ++ adr1;as)) ∧ inv(istore) ∧ istore0 ⊑ istore ∧ (pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn) ∧ istore1 ⊑ istore ∧ r ∈ zh ∧ (err = ESUCCESS → ¬ adr1 ∈ as) ∧ dom nodes(istore) = dom nodes(istore1) ∪ (err = ESUCCESS ⊃ ∅ ++ adr1;∅) ∧ (err = ESUCCESS → ¬ adr1 ∈ istore1) ∧ (( btree (rp, r, (err = ESUCCESS ⊃ adr1;adr0), height, err ≠ ESUCCESS, ri, nodes(istore), (err = ESUCCESS ⊃ as ++ adr1;as), (err = ESUCCESS ⊃ (alladrs -- adr0) ++ adr1;alladrs), ok) * P) )(zh) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ (err = ESUCCESS → pstate.index? ∧ ¬ zh[r].dirty))

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ unbuffered.index?; (s)%btree-adrs: ⊦ (( btree (rp, rp0, adr0, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ dom iast; (forward) from specification btree-inv%sep-com: ⊦ P * P2 = P2 * P; from specification heap-sep[zheap]%sep-ass: ⊦ (P * P3) * P4 = P * P3 * P4; from specification heap-sep[zheap]%btree-refed-subseteq-alladrs: ⊦ (( btree(rp, r, adr, height, dirty, fi, iast, as, alladrs, nstat) * P) )(zh) → as ⊆ alladrs; (forward) from specification btree-inv% ⊦ {adr} = ∅ ++ adr; (s)%⊦ inv-index(istore, istore0) ↔ inv(istore) ∧ inv(istore0) ∧ istore0 ⊑ istore ∧ remove-refsize0(istore0) = istore0; (s)%⊦ inv-journal(nx, gstore, gstore0, ginval) ↔ inv(gstore \ ginval) ∧ inv(gstore0) ∧ elems nx ∩ ginval = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ remove-refsize0(gstore0) = gstore0; (s)%⊦ inv-state(pstate, nx, gstore, istore, syn) ↔ ( pstate.journal? → nx ≠ [] ∧ pstate.lnum = nx.last ∧ pstate.lnum ∈ gstore) ∧ (pstate.index? → pstate.lnum ∈ istore) ∧ (pstate = unbuffered → syn); (s)