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Proof step 59 from proof validrefs-put-array-badindex (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 33:
⊦ 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 58:
is_array(str, r0, st[r -- i ', val]) ∧ (st[r -- i ', val][r0 -- _type]) .type ≤ ty, is_array(str, r0, st), i < 0, st[r0 -- _type] .type ≤ ty, is_valid_ref(str, r0, st[r -- i ', val]), ∀ i0, str0, ty0, st0, r1, val0. ¬ 0 ≤ i0 → (validrefs(str0, refs0, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, refs0, ty0, st0)) ⊦ refs0 = [] ∨ (∃ i0. 0 ≤ i0 ∧ i0 < (st[r -- i ', val][r0 -- _length]) .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ refs0 .first = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, refs0, (st[r -- i ', val][r0 -- _type]) .type .type, st[r -- i ', val])) ↔ validrefs(str, r0 ' + refs0, ty, st), r0 = jvmref

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[r -- i ', val], ty, refs0, r0, str] add preconditions?: F is rewrite lemma?: T current spec?: T rotate lemma?: F replace all?: T paths: ((1, 0, 0, 1, 0)) dummy: () ]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : javastore, , type-at, ⊦ (st[r -- i ', val] [r0 -- _type ] ) = st[r0 -- _type]], [CSIMPNAMEDSEQ : addfuns, , is_valid_ref-01, ⊦ ¬ is_array(str, r, st) → ( is_valid_ref(str, r, st) ↔ is_obj(str, r, st))], [CSIMPNAMEDSEQ : addfuns, , is_obj-put-index, ⊦ is_obj (str, r, st[r0 -- i ', val]) ↔ is_obj(str, r, st)], [CSIMPNAMEDSEQ : addfuns, , isarray-isobj-01, ⊦ is_array(str, r, st) → ¬ is_obj(str, r, st)])], [EXTRATERMS : (st[r -- i ', val], ty, refs0, r0, str)]]