[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 57 from proof validrefs-put-array-badindex (previous step)

is_array(str, r0, st), i < 0, is_array(str, r0, st[r -- i ', val]), st[r0 -- _type] .type ≤ ty, ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ r0 = jvmref, refs0 = [] ∨ (∃ i0. 0 ≤ i0 ∧ i0 < st[r0 -- _length] .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ refs0 .first = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, refs0, st[r0 -- _type] .type .type, st[r -- i ', val])) ↔ validrefs(str, r0 ' + refs0, ty, st)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 34:
⊦ is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)))
Proof step 56:
is_array(str, r0, st) ∧ st[r0 -- _type] .type ≤ ty, is_array(str, r0, st), i < 0, is_array(str, r0, st[r -- i ', val]), st[r0 -- _type] .type ≤ ty, ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ r0 = jvmref, refs0 = [] ∨ (∃ i0. 0 ≤ i0 ∧ i0 < st[r0 -- _length] .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ refs0 .first = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, refs0, st[r0 -- _type] .type .type, st[r -- i ', val])) ↔ refs0 = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r0 -- _length] .intval ∧ is_referencevalue(st[r0 -- i ']) ∧ refs0 .first = st[r0 -- i '] .refval ∧ validrefs(str, refs0, st[r0 -- _type] .type .type, st))

Rule argument

[x0lemmaarg: spec-name: instance-name: lemma-name: validrefs-array antecedent: succedent: is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty → ( validrefs(str, r ' + refs, ty, st) ↔ refs = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r -- _length] .intval ∧ is_referencevalue(st[r -- i ']) ∧ refs .first = st[r -- i '] .refval ∧ validrefs (str, refs, st[r -- _type] .type .type, st))) substlist: [st, ty, refs, r, str] → [st, ty, refs0, r0, str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((1, 0, 1, 1, 1)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ()], [EXTRATERMS : (st, ty, refs0, r0, str)]]