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Proof step 53 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, r2 ' + refs, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, r2 ' + refs, ty0, st0)) ⊦ r0 = jvmref, (∃ i0. 0 ≤ i0 ∧ i0 < st[r0 -- _length] .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ r2 = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, (st[r -- i ', val][r0 -- i0 ']) .refval ' + refs, st[r0 -- _type] .type .type, st[r -- i ', val])) ↔ (∃ i0. ¬ i0 < 0 ∧ ¬ st[r0 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r0 -- i0 ']) ∧ r2 = st[r0 -- i0 '] .refval ∧ validrefs(str, st[r0 -- i0 '] .refval ' + refs, st[r0 -- _type] .type .type, st))

User applied rule case distinction

And obtained the following new goals

Proof step 43:
∃ i0. 0 ≤ i0 ∧ i0 < st[r0 -- _length] .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ r2 = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, (st[r -- i ', val][r0 -- i0 ']) .refval ' + refs, st[r0 -- _type] .type .type, st[r -- i ', val]), 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, r2 ' + refs, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, r2 ' + refs, ty0, st0)) ⊦ ∃ i0. ¬ i0 < 0 ∧ ¬ st[r0 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r0 -- i0 ']) ∧ r2 = st[r0 -- i0 '] .refval ∧ validrefs(str, st[r0 -- i0 '] .refval ' + refs, st[r0 -- _type] .type .type, st), r0 = jvmref
Proof step 52:
∃ i0. ¬ i0 < 0 ∧ ¬ st[r0 -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r0 -- i0 ']) ∧ r2 = st[r0 -- i0 '] .refval ∧ validrefs(str, st[r0 -- i0 '] .refval ' + refs, st[r0 -- _type] .type .type, st), 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, r2 ' + refs, ty0, st0[r1 -- i0 ', val0]) ↔ validrefs(str0, r2 ' + refs, ty0, st0)) ⊦ ∃ i0. 0 ≤ i0 ∧ i0 < st[r0 -- _length] .intval ∧ is_referencevalue(st[r -- i ', val][r0 -- i0 ']) ∧ r2 = (st[r -- i ', val][r0 -- i0 ']) .refval ∧ validrefs(str, (st[r -- i ', val][r0 -- i0 ']) .refval ' + refs, st[r0 -- _type] .type .type, st[r -- i ', val]), r0 = jvmref

Rule argument

[CASEDARG : pos: right 2, ()]

Additional information

[[CONCRETESIMPRULES : ()]]