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Proof step 125 from proof validrefs-put-notin-03 (previous step)

is_array(str, r1, st) ∧ st[r1 -- _type] .type ≤ ty, refs0 + r ' = r4 ' + refs2, is_valid_ref(str, r, st) ↔ is_valid_ref(str, r, st[r -- sk, val]), list2set(refs1) ∩ list2set(refs0) = ∅, st[r1 -- _type] .type ≤ ty, is_array(str, r1, st), is_valid_ref(r1, st), ∀ str0, r2, st0, sk0, val0, refs, r3, ty0. (is_valid_ref(str0, r2, st0) ↔ is_valid_ref(str0, r2, st0[r2 -- sk0, val0])) ∧ list2set(refs0) ∩ list2set(refs) = ∅ → val0 .refval = r3 ∨ r3 = st0[r2 -- sk0] .refval ∨ sk0 = _type ∨ sk0 = _length ∨ (validrefs(str0, refs0 + r2 ' + r3 ' + refs, ty0, st0[r2 -- sk0, val0]) ↔ validrefs(str0, refs0 + r2 ' + r3 ' + refs, ty0, st0)) ∨ r2 = r3 ∨ r3 ∈ refs0 ∨ r2 ∈ refs ∨ r2 ∈ refs0 ∨ dups(refs) ∨ r3 ∈ refs ⊦ (∃ i. 0 ≤ i ∧ i < st[r1 -- _length] .intval ∧ is_referencevalue(st[r1 -- i ']) ∧ r4 = st[r1 -- i '] .refval ∧ validrefs(str, refs0 + r ' + r0 ' + refs1, st[r1 -- _type] .type .type, st[r -- sk, val])) ↔ refs0 + r ' + r0 ' + refs1 = [] ∨ (∃ i. 0 ≤ i ∧ i < st[r1 -- _length] .intval ∧ is_referencevalue(st[r1 -- i ']) ∧ (refs0 + r ' + r0 ' + refs1) .first = st[r1 -- i '] .refval ∧ validrefs(str, refs0 + r ' + r0 ' + refs1, st[r1 -- _type] .type .type, st)), r0 = r1, r = r1, r1 = jvmref, r = r0, val .refval = r0, r0 = st[r -- sk] .refval, sk = _type, sk = _length, dups(refs1), r0 ∈ refs1, r1 ∈ refs1, dups(refs0), r1 ∈ refs0, r0 ∈ refs0, r ∈ refs1, r ∈ refs0, r ≠ r4 ∧ ¬ r ∈ refs2, ¬ r4 ∈ refs0 ∧ r4 ≠ r

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 124:
list2set(refs1) ∩ list2set(refs0) = ∅, is_valid_ref(str, r, st) ↔ is_valid_ref(str, r, st[r -- sk, val]), refs0 + r ' = r4 ' + refs2, is_valid_ref(r1, st), is_array(str, r1, st), st[r1 -- _type] .type ≤ ty, ∀ str0, r2, st0, sk0, val0, refs, r3, ty0. (is_valid_ref(str0, r2, st0) ↔ is_valid_ref(str0, r2, st0[r2 -- sk0, val0])) ∧ list2set(refs0) ∩ list2set(refs) = ∅ → val0 .refval = r3 ∨ r3 = st0[r2 -- sk0] .refval ∨ sk0 = _type ∨ sk0 = _length ∨ (validrefs(str0, refs0 + r2 ' + r3 ' + refs, ty0, st0[r2 -- sk0, val0]) ↔ validrefs(str0, refs0 + r2 ' + r3 ' + refs, ty0, st0)) ∨ r2 = r3 ∨ r3 ∈ refs0 ∨ r2 ∈ refs ∨ r2 ∈ refs0 ∨ dups(refs) ∨ r3 ∈ refs ⊦ sk = _length, sk = _type, r0 = st[r -- sk] .refval, val .refval = r0, r = r0, r1 = jvmref, r = r1, r0 = r1, (∃ i. 0 ≤ i ∧ i < st[r1 -- _length] .intval ∧ is_referencevalue(st[r1 -- i ']) ∧ r4 = st[r1 -- i '] .refval ∧ validrefs(str, refs0 + r ' + r0 ' + refs1, st[r1 -- _type] .type .type, st[r -- sk, val])) ↔ (∃ i. ¬ i < 0 ∧ ¬ st[r1 -- _length] .intval ≤ i ∧ is_referencevalue(st[r1 -- i ']) ∧ r4 = st[r1 -- i '] .refval ∧ validrefs(str, refs0 + r ' + r0 ' + refs1, st[r1 -- _type] .type .type, st)), r ∈ refs0, r ∈ refs1, r0 ∈ refs0, r1 ∈ refs0, dups(refs0), r1 ∈ refs1, r0 ∈ refs1, dups(refs1), ¬ r4 ∈ refs0 ∧ r4 ≠ r, r ≠ r4 ∧ ¬ r ∈ refs2

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : list, references, c-08, ⊦ (r ' + refs) .first = r], [CSIMPNAMEDSEQ : list, references, first-02, ⊦ (refs + r ' + refs0) .first = (refs + r ') .first], [CSIMPNAMEDSEQ : int-abs, , lels, ⊦ ¬ i < j ↔ j ≤ i], [CSIMPNAMEDSEQ : int-abs, , lels-01, ⊦ ¬ i ≤ j ↔ j < i], [CSIMPNAMEDSEQ : list, references, a, ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPNAMEDSEQ : list, references, e-03, ⊦ refs + r ' + refs0 ≠ []])]]