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Step * 2 2 of Lemma values-for-distinct-property


1. [A] : Type
2. [V] : Type
3. eq : EqDecider(A)
4. L : (A × V) List
5. ||values-for-distinct(eq;L)|| = ||remove-repeats(eq;map(λp.(fst(p));L))|| ∈ ℤ
6. i : ℕ||remove-repeats(eq;map(λp.(fst(p));L))||
7. (remove-repeats(eq;map(λp.(fst(p));L))[i] ∈ map(λp.(fst(p));L))
⊢ (<remove-repeats(eq;map(λp.(fst(p));L))[i], values-for-distinct(eq;L)[i]> ∈ L)
BY
{ ((InstLemma `isl-apply-alist` [⌜V⌝;⌜A⌝;⌜eq⌝;⌜remove-repeats(eq;map(λp.(fst(p));L))[i]⌝;⌜L⌝]⋅ THENA Auto)
   THEN ExRepD
   THEN RepeatFor 2 (ThinTrivial)) }

1
1. [A] : Type
2. [V] : Type
3. eq : EqDecider(A)
4. L : (A × V) List
5. ||values-for-distinct(eq;L)|| = ||remove-repeats(eq;map(λp.(fst(p));L))|| ∈ ℤ
6. i : ℕ||remove-repeats(eq;map(λp.(fst(p));L))||
7. (remove-repeats(eq;map(λp.(fst(p));L))[i] ∈ map(λp.(fst(p));L))
8. ↑isl(apply-alist(eq;L;remove-repeats(eq;map(λp.(fst(p));L))[i]))
9. (<remove-repeats(eq;map(λp.(fst(p));L))[i], outl(apply-alist(eq;L;remove-repeats(eq;map(λp.(fst(p));L))[i]))> ∈ L)
⊢ (<remove-repeats(eq;map(λp.(fst(p));L))[i], values-for-distinct(eq;L)[i]> ∈ L)

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Latex:

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Latex:

1.  [A]  :  Type
2.  [V]  :  Type
3.  eq  :  EqDecider(A)
4.  L  :  (A  \mtimes{}  V)  List
5.  ||values-for-distinct(eq;L)||  =  ||remove-repeats(eq;map(\mlambda{}p.(fst(p));L))||
6.  i  :  \mBbbN{}||remove-repeats(eq;map(\mlambda{}p.(fst(p));L))||
7.  (remove-repeats(eq;map(\mlambda{}p.(fst(p));L))[i]  \mmember{}  map(\mlambda{}p.(fst(p));L))
\mvdash{}  (<remove-repeats(eq;map(\mlambda{}p.(fst(p));L))[i],  values-for-distinct(eq;L)[i]>  \mmember{}  L)


By

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Latex:
((InstLemma  `isl-apply-alist`  [\mkleeneopen{}V\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}eq\mkleeneclose{};\mkleeneopen{}remove-repeats(eq;map(\mlambda{}p.(fst(p));L))[i]\mkleeneclose{};\mkleeneopen{}L\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  ExRepD
  THEN  RepeatFor  2  (ThinTrivial))

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