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Step * 1 1 1 1 2 of Lemma simple-swap-test2_wf


1. n : ℕ
2. AType : array{i:l}(ℤ;n)
3. prog : A-map Unit
4. i : ℕn
5. j : ℕn
6. k : ℕn
7. A-fetch'(array-model(AType)) i ∈ A-map'(array-model(AType)) ℤ
8. A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) i) ∈ A-map ℤ
9. A-fetch'(array-model(AType)) j ∈ A-map'(array-model(AType)) ℤ
10. A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j) ∈ A-map ℤ
11. A-fetch'(array-model(AType)) k ∈ A-map'(array-model(AType)) ℤ
12. A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j) ∈ A-map ℤ
13. in@i : ℤ@i
14. λin@j.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k)) 
           (λin@k.(A-bind(array-model(AType)) prog 
                   (λx.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) i)) 
                        (λout@i.(A-bind(array-model(AType)) 
                                 (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j)) 
                                 (λout@j.(A-bind(array-model(AType)) 
                                          (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k)) 
                                          (λout@k.(A-return(array-model(AType)) 
                                                   (((out@i =z in@j) ∧b (out@j =z in@i))
                                                   ∧b ((¬b(k =z i)) ∧b (¬b(k =z j)) ⇒b (out@k =z in@k)))))))))))))) ∈ ℤ
    ⟶ (A-map 𝔹)
⊢ A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j)) 
  (λin@j.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k)) 
          (λin@k.(A-bind(array-model(AType)) prog 
                  (λx.(A-bind(array-model(AType)) (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) i)) 
                       (λout@i.(A-bind(array-model(AType)) 
                                (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) j)) 
                                (λout@j.(A-bind(array-model(AType)) 
                                         (A-coerce(array-model(AType)) (A-fetch'(array-model(AType)) k)) 
                                         (λout@k.(A-return(array-model(AType)) 
                                                  (((out@i =z in@j) ∧b (out@j =z in@i))
                                                  ∧b ((¬b(k =z i)) ∧b (¬b(k =z j)) ⇒b (out@k =z in@k)))))))))))))))
  ∈ A-map 𝔹
BY
{ (InstLemma `A-bind_wf` [⌜ℤ⌝;⌜n⌝;⌜AType⌝;⌜ℤ⌝;⌜𝔹⌝]⋅ THEN Auto)⋅ }

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Latex:

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Latex:

1.  n  :  \mBbbN{}
2.  AType  :  array\{i:l\}(\mBbbZ{};n)
3.  prog  :  A-map  Unit
4.  i  :  \mBbbN{}n
5.  j  :  \mBbbN{}n
6.  k  :  \mBbbN{}n
7.  A-fetch'(array-model(AType))  i  \mmember{}  A-map'(array-model(AType))  \mBbbZ{}
8.  A-coerce(array-model(AType))  (A-fetch'(array-model(AType))  i)  \mmember{}  A-map  \mBbbZ{}
9.  A-fetch'(array-model(AType))  j  \mmember{}  A-map'(array-model(AType))  \mBbbZ{}
10.  A-coerce(array-model(AType))  (A-fetch'(array-model(AType))  j)  \mmember{}  A-map  \mBbbZ{}
11.  A-fetch'(array-model(AType))  k  \mmember{}  A-map'(array-model(AType))  \mBbbZ{}
12.  A-coerce(array-model(AType))  (A-fetch'(array-model(AType))  j)  \mmember{}  A-map  \mBbbZ{}
13.  in@i  :  \mBbbZ{}@i
14.  \mlambda{}in@j.(A-bind(array-model(AType)) 
                      (A-coerce(array-model(AType))  (A-fetch'(array-model(AType))  k)) 
                      ...)  \mmember{}  \mBbbZ{}  {}\mrightarrow{}  (A-map  \mBbbB{})
\mvdash{}  A-bind(array-model(AType))  (A-coerce(array-model(AType))  (A-fetch'(array-model(AType))  j)) 
    (\mlambda{}in@j....)  \mmember{}  A-map  \mBbbB{}


By

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Latex:
(InstLemma  `A-bind\_wf`  [\mkleeneopen{}\mBbbZ{}\mkleeneclose{};\mkleeneopen{}n\mkleeneclose{};\mkleeneopen{}AType\mkleeneclose{};\mkleeneopen{}\mBbbZ{}\mkleeneclose{};\mkleeneopen{}\mBbbB{}\mkleeneclose{}]\mcdot{}  THEN  Auto)\mcdot{}

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