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Proof step 178 from proof onlymod-shift (previous step)

rk ∈ st ⊦ (∀ rk0. rk0 ∈ st ↔ rk0 ∈ st1) ∧ (∀ r0, sk. r0 ≠ r → ((r0 -- sk = rk ⊃ val; st[r0 -- sk])) = st1[r0 -- sk]) ∧ (r -- _type ∈ st → ((r -- _type = rk ⊃ val; st[r -- _type])) = st1[r -- _type]) ∧ (r -- _length ∈ st → ((r -- _length = rk ⊃ val; st[r -- _length])) = st1[r -- _length]) ↔ (∀ rk0. rk0 ∈ st ↔ rk0 ∈ st1[rk, st[rk]]) ∧ (∀ r0, sk. r0 ≠ r → st[r0 -- sk] = (st1[rk, st[rk]][r0 -- sk])) ∧ (r -- _type ∈ st → st[r -- _type] = (st1[rk, st[rk]][r -- _type])) ∧ (r -- _length ∈ st → st[r -- _length] = (st1[rk, st[rk]][r -- _length])) ∧ rk ∈ st1 ∧ (rk = rk .ref -- _length ∨ r ≠ rk .ref ∨ rk = rk .ref -- _type → st1[rk] = val)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 177:
rk ∈ st ⊦ (∀ rk0. rk0 ∈ st ↔ rk0 ∈ st1) ∧ (∀ r0, sk. r0 ≠ r → ((r0 -- sk = rk ⊃ val; st[r0 -- sk])) = st1[r0 -- sk]) ∧ (r -- _type ∈ st → ((r -- _type = rk ⊃ val; st[r -- _type])) = st1[r -- _type]) ∧ (r -- _length ∈ st → ((r -- _length = rk ⊃ val; st[r -- _length])) = st1[r -- _length]) ↔ (∀ rk0. rk0 ∈ st ↔ rk0 ∈ st1) ∧ (∀ r0, sk. r0 ≠ r → st[r0 -- sk] = ((r0 -- sk = rk ⊃ st[r0 -- sk]; st1[r0 -- sk]))) ∧ (r -- _type ∈ st → st[r -- _type] = ((r -- _type = rk ⊃ st[r -- _type]; st1[r -- _type]))) ∧ (r -- _length ∈ st → st[r -- _length] = ((r -- _length = rk ⊃ st[r -- _length]; st1[r -- _length]))) ∧ rk ∈ st1 ∧ (rk = rk .ref -- _length ∨ r ≠ rk .ref ∨ rk = rk .ref -- _type → st1[rk] = val)

Rule argument

[emptyarg]

Additional information

[[CONCRETESIMPRULES : ([CSIMPSEQ : ⊦ (st[rk0, val][rk]) = ((rk = rk0 ⊃ val; st[rk]))], [CSIMPSEQ : ⊦ rk ∈ st[rk0, val] ↔ rk = rk0 ∨ rk ∈ st])]]