Proof of Nuprl Lemma : simple-comb2-concat-classrel

Step * 2 of Lemma simple-comb2-concat-classrel

1. Info : Type
2. A : Type
3. B : Type
4. C : Type
5. f : A ─→ B ─→ bag(C)
6. X : EClass(A)
7. Y : EClass(B)
8. es : EO+(Info)
9. e : E
10. v : C
11. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].

      uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ ((λn.[A; B][n]) k)

                                                                            ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                            ∧ v ↓∈ (λw.(f (w 0) (w 1))) vs))

⊢ uiff(v ∈ λa,b.concat-lifting2(f;a;b)|X;Y|(e);↓∃a:A. ∃b:B. (a ∈ X(e) ∧ b ∈ Y(e) ∧ v ↓∈ f a b))

{ (RepUR ``simple-comb2`` 0 THEN Reduce (-1) THEN (InstHyp [⌈es⌉; ⌈e⌉; ⌈v⌉] (-1)⋅ THENA Auto) THEN D 0 THEN UnivCD) }

1
1. Info : Type
2. A : Type
3. B : Type
4. C : Type
5. f : A ─→ B ─→ bag(C)
6. X : EClass(A)
7. Y : EClass(B)
8. es : EO+(Info)
9. e : E
10. v : C
11. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].

      uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                            ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                            ∧ v ↓∈ f (vs 0) (vs 1)))

12. uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                          ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                          ∧ v ↓∈ f (vs 0) (vs 1)))

13. v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λz.[X; Y][z])(e)
⊢ ↓∃a:A. ∃b:B. (a ∈ X(e) ∧ b ∈ Y(e) ∧ v ↓∈ f a b)

2
.....wf.....
1. Info : Type
2. A : Type
3. B : Type
4. C : Type
5. f : A ─→ B ─→ bag(C)
6. X : EClass(A)
7. Y : EClass(B)
8. es : EO+(Info)
9. e : E
10. v : C
11. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].

      uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                            ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                            ∧ v ↓∈ f (vs 0) (vs 1)))

12. uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                          ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                          ∧ v ↓∈ f (vs 0) (vs 1)))

⊢ v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λz.[X; Y][z])(e) ∈ ℙ

3
1. Info : Type
2. A : Type
3. B : Type
4. C : Type
5. f : A ─→ B ─→ bag(C)
6. X : EClass(A)
7. Y : EClass(B)
8. es : EO+(Info)
9. e : E
10. v : C
11. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].

      uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                            ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                            ∧ v ↓∈ f (vs 0) (vs 1)))

12. uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                          ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                          ∧ v ↓∈ f (vs 0) (vs 1)))

13. ↓∃a:A. ∃b:B. (a ∈ X(e) ∧ b ∈ Y(e) ∧ v ↓∈ f a b)
⊢ v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λz.[X; Y][z])(e)

4
.....wf.....
1. Info : Type
2. A : Type
3. B : Type
4. C : Type
5. f : A ─→ B ─→ bag(C)
6. X : EClass(A)
7. Y : EClass(B)
8. es : EO+(Info)
9. e : E
10. v : C
11. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].

      uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                            ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                            ∧ v ↓∈ f (vs 0) (vs 1)))

12. uiff(v ∈ simple-comb(λw.concat-lifting2(f;w 0;w 1);λn.[X; Y][n])(e);↓∃vs:k:ℕ2 ─→ [A; B][k]

                                                                          ((∀k:ℕ2. vs[k] ∈ λn.[X; Y][n][k](e))

                                                                          ∧ v ↓∈ f (vs 0) (vs 1)))

⊢ ↓∃a:A. ∃b:B. (a ∈ X(e) ∧ b ∈ Y(e) ∧ v ↓∈ f a b) ∈ ℙ

Latex:

Latex:
1. Info : Type 2. A : Type 3. B : Type 4. C : Type 5. f : A {}\mrightarrow{} B {}\mrightarrow{} bag(C) 6. X : EClass(A) 7. Y : EClass(B) 8. es : EO+(Info) 9. e : E 10. v : C 11. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:C]. uiff(v \mmember{} simple-comb(\mlambda{}w.concat-lifting2(f;w 0;w 1);\mlambda{}n.[X; Y][n])( e);\mdownarrow{}\mexists{}vs:k:\mBbbN{}2 {}\mrightarrow{} ((\mlambda{}n.[A; B][n]) k) ((\mforall{}k:\mBbbN{}2. vs[k] \mmember{} \mlambda{}n.[X; Y][n][k](e)) \mwedge{} v \mdownarrow{}\mmember{} (\mlambda{}w.(f (w 0) (w 1))) vs)) \mvdash{} uiff(v \mmember{} \mlambda{}a,b.concat-lifting2(f;a;b)|X;Y|(e);\mdownarrow{}\mexists{}a:A. \mexists{}b:B. (a \mmember{} X(e) \mwedge{} b \mmember{} Y(e) \mwedge{} v \mdownarrow{}\mmember{} f a b)) By Latex: (RepUR ``simple-comb2`` 0 THEN Reduce (-1) THEN (InstHyp [\mkleeneopen{}es\mkleeneclose{}; \mkleeneopen{}e\mkleeneclose{}; \mkleeneopen{}v\mkleeneclose{}] (-1)\mcdot{} THENA Auto) THEN D 0 THEN UnivCD) Home Index