Step
*
1
1
of Lemma
Escardo-Xu
1. M : F:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℕ
2. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀g:ℕ ⟶ ℕ. ((∀i:ℕM F. ((g i) = 0 ∈ ℕ))
⇒ ((F (λi.0)) = (F g) ∈ ℕ))
⊢ False
BY
{ ((GenConcl ⌜(M (λf.0)) = J ∈ ℕ⌝⋅ THENA Auto)
THEN (GenConcl ⌜(λg.(M (λf.(g (f J))))) = D ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)⌝⋅ THENA Auto)
THEN (GenConcl ⌜(M D) = K ∈ ℕ⌝⋅ THENA Auto)
THEN (GenConcl ⌜(λf.if f J <z K then 0 else 1 fi ) = G ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)⌝⋅ THENA Auto)
THEN (GenConcl ⌜(λf.(f J)) = H ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)⌝⋅ THENA Auto)
THEN (GenConcl ⌜(λt,n,i. if i <z t then 0 else n fi ) = z ∈ (ℕ ⟶ ℕ ⟶ ℕ ⟶ ℕ)⌝⋅ THENA Auto)) }
1
1. M : F:((ℕ ⟶ ℕ) ⟶ ℕ) ⟶ ℕ
2. ∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀g:ℕ ⟶ ℕ. ((∀i:ℕM F. ((g i) = 0 ∈ ℕ))
⇒ ((F (λi.0)) = (F g) ∈ ℕ))
3. J : ℕ
4. (M (λf.0)) = J ∈ ℕ
5. D : (ℕ ⟶ ℕ) ⟶ ℕ
6. (λg.(M (λf.(g (f J))))) = D ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)
7. K : ℕ
8. (M D) = K ∈ ℕ
9. G : (ℕ ⟶ ℕ) ⟶ ℕ
10. (λf.if f J <z K then 0 else 1 fi ) = G ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)
11. H : (ℕ ⟶ ℕ) ⟶ ℕ
12. (λf.(f J)) = H ∈ ((ℕ ⟶ ℕ) ⟶ ℕ)
13. z : ℕ ⟶ ℕ ⟶ ℕ ⟶ ℕ
14. (λt,n,i. if i <z t then 0 else n fi ) = z ∈ (ℕ ⟶ ℕ ⟶ ℕ ⟶ ℕ)
⊢ False
Latex:
Latex:
1. M : F:((\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
2. \mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((\mforall{}i:\mBbbN{}M F. ((g i) = 0)) {}\mRightarrow{} ((F (\mlambda{}i.0)) = (F g)))
\mvdash{} False
By
Latex:
((GenConcl \mkleeneopen{}(M (\mlambda{}f.0)) = J\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(\mlambda{}g.(M (\mlambda{}f.(g (f J))))) = D\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(M D) = K\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(\mlambda{}f.if f J <z K then 0 else 1 fi ) = G\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(\mlambda{}f.(f J)) = H\mkleeneclose{}\mcdot{} THENA Auto)
THEN (GenConcl \mkleeneopen{}(\mlambda{}t,n,i. if i <z t then 0 else n fi ) = z\mkleeneclose{}\mcdot{} THENA Auto))
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