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Proof step 90 from proof validrefs-subset-equiv (previous step)

is_array(str, r, st) ∧ st[r -- _type] .type ≤ ty, is_array(str, r, st0), st ⊆ st0, is_valid_ref(str, r, st), st0[r -- _type] .type ≤ ty, refs = [] ∨ 0 ≤ i ∧ i < st0[r -- _length] .intval ∧ is_referencevalue(st0[r -- i ']) ∧ refs .first = st0[r -- i '] .refval ∧ validrefs(str, refs, st0[r -- _type] .type .type, st0), ∀ st, st0, ty, r, str, refs0. # refs0 < # refs → (validrefs(str, r ' + refs0, ty, st0) ∧ st ⊆ st0 ∧ validref(str, r, st[r -- _type] .type, st) → validrefs(str, r ' + refs0, ty, st)) ⊦ r = jvmref, junkref(str, r, 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))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 89:
refs .first = st0[r -- i '] .refval, is_array(str, r, st), st[r -- _type] .type ≤ ty, st ⊆ st0, st0[r -- _type] .type ≤ ty, i < st0[r -- _length] .intval, is_referencevalue(st0[r -- i ']), validrefs(str, refs, st0[r -- _type] .type .type, st0), 0 ≤ i, 0 < st0[r -- _length] .intval, ∀ st, st0, ty, r, str, refs0. # refs0 < # refs → (validrefs(str, r ' + refs0, ty, st0) ∧ st ⊆ st0 ∧ validref(str, r, st[r -- _type] .type, st) → validrefs(str, r ' + refs0, ty, st)) ⊦ r = jvmref, refs = [], junkref(str, r, st), ∃ i0. ¬ i0 < 0 ∧ ¬ st[r -- _length] .intval ≤ i0 ∧ is_referencevalue(st[r -- i0 ']) ∧ refs .first = st[r -- i0 '] .refval ∧ validrefs(str, refs, st[r -- _type] .type .type, st)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPFORWARD : ⊦ i ≤ j ∧ 0 ≤ i → 0 ≤ j], [CSIMPFORWARD : ⊦ i < j ∧ 0 ≤ i → 0 < j], [CSIMPSEQ : ⊦ ¬ i < j ↔ j ≤ i], [CSIMPSEQ : ⊦ ¬ i ≤ j ↔ j < i], [CSIMPSEQ : ⊦ is_array(str, r, st) → is_valid_ref(str, r, st)], [CSIMPSEQ : ⊦ st ⊆ st1 ∧ is_array(str, r, st) → is_array(str, r, st1)])]]