[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 19 from proof fits-put-other (previous step)

st[r -- sk0] = refval(r1), ∀ sks3. sks3 + sk ' ⊑ sks → st[r1 -- sks3] .refval ≠ st[r1 -- sks] .refval, is_classtype(st[r -- _type] .type) → sk0 ∈ instfields(st[r -- _type] .type .class), r ≠ st[r1 -- sks] .refval ∨ sk0 ≠ sk ⊦ r = jvmref, sk = _length, sk = _type, (is_arraytype(st[r -- _type] .type) → is_indexkey(sk0) ∧ 0 ≤ sk0 .index ∧ sk0 .index < st[r -- _length] .intval) ∧ fits(r1, sks + sk ' + sks1, st[st[r1 -- sks] .refval -- sk, val]) ↔ (is_arraytype(st[r -- _type] .type) → ¬ is_fskey(sk0) ∧ ¬ sk0 .index < 0 ∧ ¬ st[r -- _length] .intval ≤ sk0 .index) ∧ fits(r1, sks + sk ' + sks1, st), fits(r1, sks + sk ' + sks1, st[st[r1 -- sks] .refval -- sk, val]) ↔ fits(r1, sks + sk ' + sks1, st), st[r -- (sk0 ' + sks)] .refval = st[r1 -- sks] .refval ∧ (sks0 + sk ' ⊑ sk0 ' + sks → st[r -- sks0] .refval ≠ st[r -- (sk0 ' + sks)] .refval)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks]
Proof step 18:
sks ≠ [], st[r -- sk0] = refval(r1), ∀ sks3. sks3 + sk ' ⊑ sks → st[r1 -- sks3] .refval ≠ st[r1 -- sks] .refval, is_classtype(st[r -- _type] .type) → sk0 ∈ instfields(st[r -- _type] .type .class), r ≠ st[r1 -- sks] .refval ∨ sk0 ≠ sk ⊦ r = jvmref, sk = _length, sk = _type, (is_arraytype(st[r -- _type] .type) → is_indexkey(sk0) ∧ 0 ≤ sk0 .index ∧ sk0 .index < st[r -- _length] .intval) ∧ fits(r1, sks + sk ' + sks1, st[st[r1 -- sks] .refval -- sk, val]) ↔ (is_arraytype(st[r -- _type] .type) → ¬ is_fskey(sk0) ∧ ¬ sk0 .index < 0 ∧ ¬ st[r -- _length] .intval ≤ sk0 .index) ∧ fits(r1, sks + sk ' + sks1, st), fits(r1, sks + sk ' + sks1, st[st[r1 -- sks] .refval -- sk, val]) ↔ fits(r1, sks + sk ' + sks1, st), st[st[r -- sk0] .refval -- sks] .refval = st[r1 -- sks] .refval ∧ (sks0 + sk ' ⊑ sk0 ' + sks → st[r -- sks0] .refval ≠ st[st[r -- sk0] .refval -- sks] .refval)

Rule argument

[x0lemmaarg: spec-name: storefuns instance-name: lemma-name: lookup-rec-01 antecedent: succedent: sks ≠ [] → st[r -- (sk ' + sks)] = st[st[r -- sk] .refval -- sks] substlist: [sks, sk, r, st] → [sks, sk0, r, st] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((1, 0, 5, 1, 0, 1, 0, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : list, storekeys, Nil, ⊦ [] + sks = sks], [CSIMPNAMEDSEQ : list, storekeys, a, ⊦ (sks + sks0) + sks1 = sks + sks0 + sks1], [CSIMPNAMEDSEQ : storefuns, , lookup-empty, ⊦ st[r -- []] = refval(r)], [CSIMPNAMEDSEQ : javavalue, , sel-09, ⊦ refval(r) .refval = r], [CSIMPNAMEDSEQ : list-last, storekeys, eq-01-01, ⊦ sks ⊑ [] ↔ sks = []], [CSIMPNAMEDSEQ : list, storekeys, e-02, ⊦ sks + sk ' ≠ []], [CSIMPNAMEDSEQ : list, storekeys, app, ⊦ sks + [] = sks], [CSIMPNAMEDSEQ : storefuns, , lookup-one, ⊦ st[r -- sk '] = st[r -- sk]], [CSIMPNAMEDSEQ : list-last, storekeys, ins-nil, ⊦ sks ⊑ sk ' ↔ ¬ ¬ (sks = [] ∨ sks = sk ')], [CSIMPNAMEDSEQ : list, storekeys, app-02-ant, ⊦ ¬ sks + sks0 ≠ [] ↔ sks = [] ∧ sks0 = []], [CSIMPNAMEDSEQ : list, storekeys, e, ⊦ sk ' ≠ []], [CSIMPNAMEDSEQ : list, storekeys, c-05, ⊦ sk ' = sks + sk0 ' ↔ sk = sk0 ∧ sks = []], [CSIMPNAMEDSEQ : storekey, , test-01, ⊦ ¬ is_fskey(sk) ↔ is_indexkey(sk)], [CSIMPNAMEDSEQ : int-abs, , lels, ⊦ ¬ i < j ↔ j ≤ i], [CSIMPNAMEDSEQ : int-abs, , lels-01, ⊦ ¬ i ≤ j ↔ j < i], [CSIMPNAMEDSEQ : storekey, , test, ⊦ ¬ is_indexkey(sk) ↔ is_fskey(sk)])], [EXTRATERMS : (sks, sk0, r, st)]]