+
+definition swap_l ≝
+ λalpha:FinSet.λfoo:alpha.
+ mk_TM alpha (swap_states alpha)
+ (λp.let 〈q,a〉 ≝ p in
+ let 〈q',b〉 ≝ q in
+ let q' ≝ pi1 nat (λi.i<4) q' in
+ match a with
+ [ None ⇒ 〈〈swap3,foo〉,None ?,N〉 (* if tape is empty then stop *)
+ | Some a' ⇒
+ match q' with
+ [ O ⇒ (* q0 *) 〈〈swap1,a'〉,Some ? a',L〉 (* save in register and move L *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q1 *) 〈〈swap2,a'〉,Some ? b,R〉 (* swap with register and move R *)
+ | S q' ⇒ match q' with
+ [ O ⇒ (* q2 *) 〈〈swap3,foo〉,Some ? b,N〉 (* copy from register and stay *)
+ | S q' ⇒ (* q3 *) 〈〈swap3,foo〉,None ?,N〉 (* final state *)
+ ]
+ ]
+ ]])
+ 〈swap0,foo〉
+ (λq.\fst q == swap3).
+
+definition Rswap_l ≝
+ λalpha,t1,t2.
+ (∀b,rs.
+ t1 = midtape alpha [ ] b rs →
+ t2 = leftof ? b rs) ∧
+ (∀a,b,ls,rs.
+ t1 = midtape alpha (a::ls) b rs →
+ t2 = midtape alpha (b::ls) a rs).
+
+lemma sem_swap_l : ∀alpha,foo.
+ swap_l alpha foo ⊨ Rswap_l alpha.
+#alpha #foo *
+ [@(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈swap3,foo〉 (niltape ?)))
+ % [% |% [#b #rs | #a #b #ls #rs] #H destruct]
+ |#l0 #lt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈swap3,foo〉 (leftof ? l0 lt0)))
+ % [% | % [#b #rs | #a #b #ls #rs] #H destruct]
+ |#r0 #rt0 @(ex_intro ?? 2) @(ex_intro … (mk_config ?? 〈swap3,foo〉 (rightof ? r0 rt0)))
+ % [% |% [#b #rs | #a #b #ls #rs] #H destruct]
+ | #lt #c #rt @(ex_intro ?? 4) cases lt
+ [@ex_intro [|% [ % | %
+ [#b #rs #H destruct normalize // |#a #b #ls #rs #H destruct]]]
+ |#r0 #rt0 @ex_intro [| % [ % | %
+ [#b #rs #H destruct | #a #b #ls #rs #H destruct normalize //
+ ]
+ ]
+qed.
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