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Proof step 57 from proof pathex-put-pathex-array-refval (previous step)

¬ refs2 ≠ [], list2set(refs1) ∩ list2set(refs2) = ∅, refs1 + r0 ' = r1 ' + refs0, st[r0 -- _type] .type = mkarraytype(ty), validrefs(str, r3 ' + refs2, ty, st), validrefs(str, r ' + refs1 + r0 ' + r3 ' + refs2, st[r -- _type] .type, st[r0 -- i ', refval(r3)]), path∃(str, r, r0, st), validref(str, r3, ty, st), 0 < st[r0 -- _length] .intval, 0 ≤ i, validrefs(str, refs1 + r0 ' + r3 ' + refs2, st[r1 -- _type] .type, st[r0 -- i ', refval(r3)]), is_array(str, r0, st[r0 -- i ', refval(r3)]), i < st[r0 -- _length] .intval, r3 = jvmref → is_reftype(ty) ⊦ r = r3, r0 = r3, r = (r3 ' + refs2) .last, r0 = jvmref, r = r0, path∃(str, r, (r3 ' + refs2) .last, st), path∃(str, r3, (r3 ' + refs2) .last, st), r ∈ refs1, r ∈ refs2, r0 ∈ refs2, r0 ∈ refs1, r3 ∈ refs2, dups(refs2), dups(refs1), r3 ∈ refs1, (r3 ' + refs2) .last ∈ refs2, ¬ r1 ∈ refs1 ∧ r1 ≠ r0, r0 ≠ r1 ∧ ¬ r0 ∈ refs0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 53:
⊦ path∃(str, r, r, st) ↔ r = jvmref ∨ r ≠ jvmref ∧ is_valid_ref(str, r, st)
Proof step 54:
⊦ ¬ is_valid_ref(str, r1, st) ∧ r1 ≠ jvmref → ¬ path∃(str, r, r1, st)
Proof step 56:
refs1 + r0 ' = r1 ' + refs0, st[r0 -- _type] .type = mkarraytype(ty), is_array(str, r0, st[r0 -- i ', refval(r3)]), i < st[r0 -- _length] .intval, validrefs(str, r3 ', ty, st), validrefs(str, r ' + refs1 + r0 ' + r3 ', st[r -- _type] .type, st[r0 -- i ', refval(r3)]), path∃(str, r, r0, st), validref(str, r3, ty, st), 0 < st[r0 -- _length] .intval, 0 ≤ i, validrefs(str, refs1 + r0 ' + r3 ', st[r1 -- _type] .type, st[r0 -- i ', refval(r3)]) ⊦ r0 = jvmref, r = r0, r0 = r3, r = r3, r3 = jvmref, r ∈ refs1, r0 ∈ refs1, dups(refs1), r3 ∈ refs1, is_valid_ref(str, r3, st), ¬ r1 ∈ refs1 ∧ r1 ≠ r0, r0 ≠ r1 ∧ ¬ r0 ∈ refs0

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPNAMEDSEQ : list-dup, references, e-01, ⊦ ¬ dups([])], [CSIMPNAMEDSEQ : list, references, in-empty, ⊦ ¬ r ∈ []], [CSIMPNAMEDSEQ : list-last, references, l, ⊦ r ' .last = r], [CSIMPNAMEDSEQ : list, references, a, ⊦ (refs + refs0) + refs1 = refs + refs0 + refs1], [CSIMPNAMEDSEQ : list, references, app, ⊦ refs + [] = refs], [CSIMPNAMEDSEQ : set-union, references, int-c, ⊦ rset ∩ rset0 = rset0 ∩ rset], [CSIMPNAMEDSEQ : set-union, references, int-a, ⊦ (rset ∩ rset0) ∩ rset1 = rset ∩ rset0 ∩ rset1], [CSIMPNAMEDSEQ : set-union, references, int-e, ⊦ rset ∩ ∅ = ∅], [CSIMPNAMEDSEQ : setlist, references, list2set-empty, ⊦ list2set([]) = ∅])]]