Step
*
2
3
1
of Lemma
simple-comb1-concat-classrel
1. Info : Type
2. B : Type
3. C : Type
4. f : B ─→ bag(C)
5. X : EClass(B)
6. es : EO+(Info)
7. e : E
8. v : C
9. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].
uiff(v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e);↓∃vs:k:ℕ1 ─→ [B][k]
((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0)))
10. ↓∃vs:k:ℕ1 ─→ [B][k]. ((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0))
supposing v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e)
11. v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e)
supposing ↓∃vs:k:ℕ1 ─→ [B][k]. ((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0))
12. ↓∃b:B. (b ∈ X(e) ∧ v ↓∈ f b)
⊢ v ∈ simple-comb(λw.concat-lifting1(f;w 0);λz.[X][z])(e)
BY
{ (D (-2) THEN Try (Complete (Auto)) THEN RepeatFor 3 (D (-1)) THEN D 0 THEN InstConcl [⌈λz.[b][z]⌉]⋅) }
1
.....wf.....
1. Info : Type
2. B : Type
3. C : Type
4. f : B ─→ bag(C)
5. X : EClass(B)
6. es : EO+(Info)
7. e : E
8. v : C
9. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].
uiff(v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e);↓∃vs:k:ℕ1 ─→ [B][k]
((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0)))
10. ↓∃vs:k:ℕ1 ─→ [B][k]. ((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0))
supposing v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e)
11. b : B
12. b ∈ X(e)
13. v ↓∈ f b
⊢ λz.[b][z] ∈ k:ℕ1 ─→ [B][k]
2
1. Info : Type
2. B : Type
3. C : Type
4. f : B ─→ bag(C)
5. X : EClass(B)
6. es : EO+(Info)
7. e : E
8. v : C
9. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].
uiff(v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e);↓∃vs:k:ℕ1 ─→ [B][k]
((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0)))
10. ↓∃vs:k:ℕ1 ─→ [B][k]. ((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0))
supposing v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e)
11. b : B
12. b ∈ X(e)
13. v ↓∈ f b
⊢ (∀k:ℕ1. λz.[b][z][k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f ((λz.[b][z]) 0)
3
.....wf.....
1. Info : Type
2. B : Type
3. C : Type
4. f : B ─→ bag(C)
5. X : EClass(B)
6. es : EO+(Info)
7. e : E
8. v : C
9. ∀[es:EO+(Info)]. ∀[e:E]. ∀[v:C].
uiff(v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e);↓∃vs:k:ℕ1 ─→ [B][k]
((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0)))
10. ↓∃vs:k:ℕ1 ─→ [B][k]. ((∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0))
supposing v ∈ simple-comb(λw.concat-lifting1(f;w 0);λn.[X][n])(e)
11. b : B
12. b ∈ X(e)
13. v ↓∈ f b
14. vs : k:ℕ1 ─→ [B][k]
⊢ (∀k:ℕ1. vs[k] ∈ λn.[X][n][k](e)) ∧ v ↓∈ f (vs 0) ∈ ℙ
Latex:
Latex:
1. Info : Type
2. B : Type
3. C : Type
4. f : B {}\mrightarrow{} bag(C)
5. X : EClass(B)
6. es : EO+(Info)
7. e : E
8. v : C
9. \mforall{}[es:EO+(Info)]. \mforall{}[e:E]. \mforall{}[v:C].
uiff(v \mmember{} simple-comb(\mlambda{}w.concat-lifting1(f;w 0);\mlambda{}n.[X][n])(e);\mdownarrow{}\mexists{}vs:k:\mBbbN{}1 {}\mrightarrow{} [B][k]
((\mforall{}k:\mBbbN{}1
vs[k] \mmember{} \mlambda{}n.[X][n][k](e))
\mwedge{} v \mdownarrow{}\mmember{} f (vs 0)))
10. \mdownarrow{}\mexists{}vs:k:\mBbbN{}1 {}\mrightarrow{} [B][k]. ((\mforall{}k:\mBbbN{}1. vs[k] \mmember{} \mlambda{}n.[X][n][k](e)) \mwedge{} v \mdownarrow{}\mmember{} f (vs 0))
supposing v \mmember{} simple-comb(\mlambda{}w.concat-lifting1(f;w 0);\mlambda{}n.[X][n])(e)
11. v \mmember{} simple-comb(\mlambda{}w.concat-lifting1(f;w 0);\mlambda{}n.[X][n])(e)
supposing \mdownarrow{}\mexists{}vs:k:\mBbbN{}1 {}\mrightarrow{} [B][k]. ((\mforall{}k:\mBbbN{}1. vs[k] \mmember{} \mlambda{}n.[X][n][k](e)) \mwedge{} v \mdownarrow{}\mmember{} f (vs 0))
12. \mdownarrow{}\mexists{}b:B. (b \mmember{} X(e) \mwedge{} v \mdownarrow{}\mmember{} f b)
\mvdash{} v \mmember{} simple-comb(\mlambda{}w.concat-lifting1(f;w 0);\mlambda{}z.[X][z])(e)
By
Latex:
(D (-2) THEN Try (Complete (Auto)) THEN RepeatFor 3 (D (-1)) THEN D 0 THEN InstConcl [\mkleeneopen{}\mlambda{}z.[b][z]\mkleeneclose{}]\mcdot{})
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