Lambda-Calculus and Combinators, an Introduction

11B The system TA → C
Example 11.6 The term SKK, which behaves like I (since SKKX ⊲
X), also has all the types that I has. That is, for all σ,
⊢TA → C
SKK : σ → σ.
Proof Let σ be any type. In axiom-scheme (→ S), take ρ, σ, τ to be σ,
σ →σ, σ respectively; this gives us an axiom
S :(σ → (σ →σ) → σ) → (σ → σ →σ) → σ → σ.
Also, from axiom-scheme (→K) we can get the following two axioms:
K : σ → (σ →σ) → σ,
K : σ → σ → σ.
Using these three axioms and rule (→e), we make the TA → C -proof below:
(→S)
S :(σ→(σ→σ) →σ) →(σ →σ →σ) →σ →σ
SK :(σ→σ→σ) →σ →σ
(→K)
K : σ→(σ →σ) →σ
SKK : σ →σ.
(→K)
K : σ→σ →σ
Example 11.7 Recall that B ≡ S(KS)K. Then, for all ρ, σ, τ:
⊢TA → C
B :(σ → τ) → (ρ → σ) → ρ → τ.
Proof The required proof is shown below. To make it fit into the width
of the page, it uses the following abbreviations:
θ ≡ (ρ → σ → τ) → (ρ → σ) → ρ → τ,
µ ≡ σ → τ,
ν ≡ ρ → σ → τ,
π ≡ (ρ → σ) → ρ → τ.
(We have ν ≡ ρ → µ, so the formula K : µ → ν is an axiom under
axiom-scheme (→K). Also ν →π ≡ θ, so µ→ν →π ≡ µ→θ.)
(→S)
S :(µ→ν →π)→(µ→ν)→µ→π
S(KS) :(µ→ν)→µ→π
(→K)
K : θ →µ→θ
KS : µ→θ
S(KS)K : µ→π.
(→S)
S : θ
(→K)
K : µ→ν
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