[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 9 from proof nonnullrefs-dups (previous step)

((sk1 × refval(r1)) ' + fis0)[sk0] = refval(r), fis0[sk] = refval(r), sk0 ∈ (sk1 × refval(r1)) ' + fis0, sk ∈ fis0, ∀ sk2, r0, sk3. fis0[sk2] = refval(r0) ∧ fis0[sk3] = refval(r0) ∧ sk2 ∈ fis0 ∧ sk3 ∈ fis0 → sk3 = sk2 ∨ r0 = jvmref ⊦ sk = sk0, r = jvmref, r1 = jvmref, sk = sk1, dups(fis0 .vals .nonnullrefs), r1 ∈ fis0 .vals .nonnullrefs

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ sk ≠ sk0 → ((sk0 × val) ' + fis)[sk] = fis[sk]
Proof step 6:
⊦ fis[sk] = refval(r) ∧ sk ∈ fis ∧ r ≠ jvmref → r ∈ fis .vals .nonnullrefs
Proof step 8:
sk0 ≠ sk1, fis0[sk0] = refval(r), fis0[sk] = refval(r), sk0 ∈ (sk1 × refval(r1)) ' + fis0, sk ∈ fis0, ∀ sk2, r0, sk3. fis0[sk2] = refval(r0) ∧ fis0[sk3] = refval(r0) ∧ sk2 ∈ fis0 ∧ sk3 ∈ fis0 → sk3 = sk2 ∨ r0 = jvmref ⊦ sk = sk0, r = jvmref, r1 = jvmref, sk = sk1, dups(fis0 .vals .nonnullrefs), r1 ∈ fis0 .vals .nonnullrefs

Rule argument

[x0lemmaarg: spec-name: assoclist instance-name: fieldinit lemma-name: assoc-no antecedent: succedent: sk ≠ sk0 → ((sk0 × val) ' + fis)[sk] = fis[sk] substlist: [sk, fis, val, sk0] → [sk0, fis0, refval(r1), sk1] add preconditions?: F is rewrite lemma?: T current spec?: F rotate lemma?: F replace all?: T paths: ((0, 0, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : assoclist, fieldinit, assoc-yes, ⊦ ((sk × val) ' + fis)[sk] = val], [CSIMPNAMEDSEQ : list, fieldinit-assoclista, a, ⊦ (fis + fis0) + fis1 = fis + fis0 + fis1], [CSIMPNAMEDSEQ : javavalue, , inj-03, ⊦ refval(r) = refval(r0) ↔ r = r0], [CSIMPNAMEDSEQ : assoclist, fieldinit, infirst, ⊦ sk ∈ (sk × val) ' + fis])], [EXTRATERMS : (sk0, fis0, refval(r1), sk1)]]