Step
*
5
1
of Lemma
simple_more_general_fan_theorem
1. T : ℕ ⟶ Type
2. ∀i:ℕ. T[i]
3. ∀i:ℕ. ∀K:T[i] ⟶ ℕ. (∃B:ℕ [(∀t:T[i]. ((K t) ≤ B))])
4. X : n:ℕ ⟶ (i:ℕn ⟶ T[i]) ⟶ ℙ
5. ∀f:i:ℕ ⟶ T[i]. (↓∃n:ℕ. X[n;f])
6. ∀n:ℕ. ∀s:i:ℕn ⟶ T[i]. Dec(X[n;s])
7. ∀x:Top
∃k:ℕ [(∀f:ℕ ⟶ (i:ℕ × T[i])
∃m:ℕk. X[0 + m;project-seq(seq-append(0;m;x;f))] supposing ∀i:ℕ. ((fst((f i))) = (i + 0) ∈ ℤ))]
supposing ∀i:ℕ0. ((fst((x i))) = i ∈ ℤ)
8. k : ℕ
9. ∀f:ℕ ⟶ (i:ℕ × T[i]). ∃m:ℕk. X[0 + m;project-seq(seq-append(0;m;⊥;f))] supposing ∀i:ℕ. ((fst((f i))) = (i + 0) ∈ ℤ)
10. f : i:ℕ ⟶ T[i]
⊢ ∃n:ℕk. X[n;f]
BY
{ ((D -2 With ⌜λi.<i, f i>⌝ THENA Auto)
THEN Reduce -1
THEN (D -1 THENA Auto)
THEN ParallelLast
THEN (NthHypEq (-1) THEN EqCD)
THEN Auto) }
1
.....subterm..... T:t
2:n
1. T : ℕ ⟶ Type
2. ∀i:ℕ. T[i]
3. ∀i:ℕ. ∀K:T[i] ⟶ ℕ. (∃B:ℕ [(∀t:T[i]. ((K t) ≤ B))])
4. X : n:ℕ ⟶ (i:ℕn ⟶ T[i]) ⟶ ℙ
5. ∀f:i:ℕ ⟶ T[i]. (↓∃n:ℕ. X[n;f])
6. ∀n:ℕ. ∀s:i:ℕn ⟶ T[i]. Dec(X[n;s])
7. ∀x:Top
∃k:ℕ [(∀f:ℕ ⟶ (i:ℕ × T[i])
∃m:ℕk. X[0 + m;project-seq(seq-append(0;m;x;f))] supposing ∀i:ℕ. ((fst((f i))) = (i + 0) ∈ ℤ))]
supposing ∀i:ℕ0. ((fst((x i))) = i ∈ ℤ)
8. k : ℕ
9. f : i:ℕ ⟶ T[i]
10. m : ℕk
11. X[0 + m;project-seq(seq-append(0;m;⊥;λi.<i, f i>))]
⊢ f = project-seq(seq-append(0;m;⊥;λi.<i, f i>)) ∈ (i:ℕm ⟶ T[i])
Latex:
Latex:
1. T : \mBbbN{} {}\mrightarrow{} Type
2. \mforall{}i:\mBbbN{}. T[i]
3. \mforall{}i:\mBbbN{}. \mforall{}K:T[i] {}\mrightarrow{} \mBbbN{}. (\mexists{}B:\mBbbN{} [(\mforall{}t:T[i]. ((K t) \mleq{} B))])
4. X : n:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{}n {}\mrightarrow{} T[i]) {}\mrightarrow{} \mBbbP{}
5. \mforall{}f:i:\mBbbN{} {}\mrightarrow{} T[i]. (\mdownarrow{}\mexists{}n:\mBbbN{}. X[n;f])
6. \mforall{}n:\mBbbN{}. \mforall{}s:i:\mBbbN{}n {}\mrightarrow{} T[i]. Dec(X[n;s])
7. \mforall{}x:Top
\mexists{}k:\mBbbN{} [(\mforall{}f:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{} \mtimes{} T[i])
\mexists{}m:\mBbbN{}k. X[0 + m;project-seq(seq-append(0;m;x;f))]
supposing \mforall{}i:\mBbbN{}. ((fst((f i))) = (i + 0)))]
supposing \mforall{}i:\mBbbN{}0. ((fst((x i))) = i)
8. k : \mBbbN{}
9. \mforall{}f:\mBbbN{} {}\mrightarrow{} (i:\mBbbN{} \mtimes{} T[i])
\mexists{}m:\mBbbN{}k. X[0 + m;project-seq(seq-append(0;m;\mbot{};f))] supposing \mforall{}i:\mBbbN{}. ((fst((f i))) = (i + 0))
10. f : i:\mBbbN{} {}\mrightarrow{} T[i]
\mvdash{} \mexists{}n:\mBbbN{}k. X[n;f]
By
Latex:
((D -2 With \mkleeneopen{}\mlambda{}i.<i, f i>\mkleeneclose{} THENA Auto)
THEN Reduce -1
THEN (D -1 THENA Auto)
THEN ParallelLast
THEN (NthHypEq (-1) THEN EqCD)
THEN Auto)
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