Step
*
1
2
3
2
2
1
of Lemma
member-fpf-vals
1. [A] : Type
2. eq : EqDecider(A)@i
3. [B] : A ─→ Type
4. P : A ─→ 𝔹@i
5. d : A List@i
6. f1 : x:{x:A| (x ∈ d)} ─→ B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. ∀g:x:{x:A| (x ∈ v1)} ─→ B[x]
((<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])})@i
12. g : x:{x:A| (x ∈ [u / v1])} ─→ B[x]@i
13. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
14. (u ∈ [u / v1])
15. zip(filter(P;v1);map(g;filter(P;v1))) ∈ (x:A × B[x]) List
16. ↑(P u)
⊢ (<x, v> ∈ [<u, g u> / zip(filter(P;v1);map(g;filter(P;v1)))])
⇐⇒ {((↑((eq u x) ∨bx ∈b v1))) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
BY
{ (Unfold `and` 0 THEN Fold `cand` 0 THEN Auto) }
1
1. A : Type
2. eq : EqDecider(A)@i
3. B : A ─→ Type
4. P : A ─→ 𝔹@i
5. d : A List@i
6. f1 : x:{x:A| (x ∈ d)} ─→ B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. ∀g:x:{x:A| (x ∈ v1)} ─→ B[x]
((<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])})@i
12. g : x:{x:A| (x ∈ [u / v1])} ─→ B[x]@i
13. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
14. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
15. (u ∈ [u / v1])
16. zip(filter(P;v1);map(g;filter(P;v1))) ∈ (x:A × B[x]) List
17. ↑(P u)
18. (<x, v> ∈ [<u, g u> / zip(filter(P;v1);map(g;filter(P;v1)))])@i
⊢ {((↑((eq u x) ∨bx ∈b v1))) c∧ (↑(P x))) c∧ (v = (g x) ∈ B[x])}
2
1. A : Type
2. eq : EqDecider(A)@i
3. B : A ─→ Type
4. P : A ─→ 𝔹@i
5. d : A List@i
6. f1 : x:{x:A| (x ∈ d)} ─→ B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. ∀g:x:{x:A| (x ∈ v1)} ─→ B[x]
((<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])})@i
12. g : x:{x:A| (x ∈ [u / v1])} ─→ B[x]@i
13. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
14. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
15. (u ∈ [u / v1])
16. zip(filter(P;v1);map(g;filter(P;v1))) ∈ (x:A × B[x]) List
17. ↑(P u)
18. (<x, v> ∈ [<u, g u> / zip(filter(P;v1);map(g;filter(P;v1)))])@i
19. (↑((eq u x) ∨bx ∈b v1))) c∧ (↑(P x))
20. v = (g x) ∈ B[x]
⊢ (u = x ∈ A) ∨ (x ∈ v1)
3
1. [A] : Type
2. eq : EqDecider(A)@i
3. [B] : A ─→ Type
4. P : A ─→ 𝔹@i
5. d : A List@i
6. f1 : x:{x:A| (x ∈ d)} ─→ B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. ∀g:x:{x:A| (x ∈ v1)} ─→ B[x]
((<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])})@i
12. g : x:{x:A| (x ∈ [u / v1])} ─→ B[x]@i
13. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
14. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
15. (u ∈ [u / v1])
16. zip(filter(P;v1);map(g;filter(P;v1))) ∈ (x:A × B[x]) List
17. ↑(P u)
18. {((↑((eq u x) ∨bx ∈b v1))) c∧ (↑(P x))) c∧ (v = (g x) ∈ B[x])}@i
⊢ (<x, v> ∈ [<u, g u> / zip(filter(P;v1);map(g;filter(P;v1)))])
4
1. A : Type
2. eq : EqDecider(A)@i
3. B : A ─→ Type
4. P : A ─→ 𝔹@i
5. d : A List@i
6. f1 : x:{x:A| (x ∈ d)} ─→ B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. ∀g:x:{x:A| (x ∈ v1)} ─→ B[x]
((<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])})@i
12. g : x:{x:A| (x ∈ [u / v1])} ─→ B[x]@i
13. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇒ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
14. (<x, v> ∈ zip(filter(P;v1);map(g;filter(P;v1)))) ⇐ {((↑x ∈b v1)) ∧ (↑(P x))) ∧ (v = (g x) ∈ B[x])}
15. (u ∈ [u / v1])
16. zip(filter(P;v1);map(g;filter(P;v1))) ∈ (x:A × B[x]) List
17. ↑(P u)
18. (↑((eq u x) ∨bx ∈b v1))) c∧ (↑(P x))
⊢ x ∈ {x:A| (x ∈ [u / v1])}
Latex:
1. [A] : Type
2. eq : EqDecider(A)@i
3. [B] : A {}\mrightarrow{} Type
4. P : A {}\mrightarrow{} \mBbbB{}@i
5. d : A List@i
6. f1 : x:\{x:A| (x \mmember{} d)\} {}\mrightarrow{} B[x]@i
7. x : A@i
8. v : B[x]@i
9. u : A@i
10. v1 : A List@i
11. \mforall{}g:x:\{x:A| (x \mmember{} v1)\} {}\mrightarrow{} B[x]
((<x, v> \mmember{} zip(filter(P;v1);map(g;filter(P;v1))))
\mLeftarrow{}{}\mRightarrow{} \{((\muparrow{}x \mmember{}\msubb{} v1)) \mwedge{} (\muparrow{}(P x))) \mwedge{} (v = (g x))\})@i
12. g : x:\{x:A| (x \mmember{} [u / v1])\} {}\mrightarrow{} B[x]@i
13. (<x, v> \mmember{} zip(filter(P;v1);map(g;filter(P;v1)))) \mLeftarrow{}{}\mRightarrow{} \{((\muparrow{}x \mmember{}\msubb{} v1)) \mwedge{} (\muparrow{}(P x))) \mwedge{} (v = (g x))\}
14. (u \mmember{} [u / v1])
15. zip(filter(P;v1);map(g;filter(P;v1))) \mmember{} (x:A \mtimes{} B[x]) List
16. \muparrow{}(P u)
\mvdash{} (<x, v> \mmember{} [<u, g u> / zip(filter(P;v1);map(g;filter(P;v1)))])
\mLeftarrow{}{}\mRightarrow{} \{((\muparrow{}((eq u x) \mvee{}\msubb{}x \mmember{}\msubb{} v1))) \mwedge{} (\muparrow{}(P x))) \mwedge{} (v = (g x))\}
By
(Unfold `and` 0 THEN Fold `cand` 0 THEN Auto)
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