On a Logical Foundation for Explicit Substitutions

ing Meta-Variables • Propositional reflection of meta-variables ∆, u:A[Ψ]; Γ ⊢ M : B ∆;Γ ⊢ λU.M : [Ψ]A → B →I ∆;Γ ⊢ M : [Ψ]A → B ∆;Ψ ⊢ N : A ∆;Γ ⊢ M (Ψ. N) : B →E • New reduction (λU.M) (Ψ. N) → [(Ψ. N)/U ]M RDP’07, Paris, June 07 – p.34 Incomplete Proofs, Rerevisited • Now open leaves have form ∆;Γ ⊢ ? : A • Need meta-variables U • New meta-hypothesis rule U : A[Σ;Ψ] ∈ ∆ ∆; ∆; Γ ⊢ (σ;σ) : (Σ;Ψ) ∆; ∆; Γ ⊢ U[σ;σ] : A mhyp • Not practical • Not expressively complete unless we close system under formation of meta-variables at any level RDP’07, Paris, June 07 – p.35 A Multi-Level System • Unify in a multi-level system • Models open derivations at any level • Variables x at level k ≥ 0 • Ordinary variables x for k = 0 • Meta-variables x for k = 1 (so far: U ) • Unified contexts ∆ ::= • | ∆, x : A[Ψ] • Ψ means n < k for all declarations x : A[Γ] in Ψ • For declarations x : A[Ψ], Ψ = (•) is forced! RDP’07, Paris, June 07 – p.36 Variables and Substitutions • Unified hypothesis rule x:A[Ψ] ∈ ∆ ∆ ⊢ σ : Ψ ∆ ⊢ x[σ] : A hyp • Substitution typing ∆ ⊢ (•) : (•) ∆ ⊢ σ : Ψ ∆|n,Γ n ⊢ M : A (n < k) ∆ ⊢ (σ, (Γ.M)) : (Ψ, x:A[Γ]) • ∆|n keeps only y for m ≥ n. • Enforces categorical restriction RDP’07, Paris, June 07 – p.37 Abstraction and Applicationion and Application • Typing rules ∆, x:A[Ψ] ⊢ M : B ∆ ⊢ λx.M : [Ψ]A → B →I ∆ ⊢ M : [Ψ]A → B ∆|k,Ψ k ⊢ N : A ∆ ⊢ M (Ψ. N) : B →E • [(•)]A → B as A ⊃ B • [(•)]A → B as A ⊃ B in IS4 • [(•)]A → B as A ⊃ B where A true if A true without using assumptions about truth or validity RDP’07, Paris, June 07 – p.38 Substitution Principle • Write σ if ∆ ⊢ σ : Ψ • M [σ] substitutes • for all variables in M of level n < k • for no variables in M of level n ≥ k • Typing guide ∆|k,Ψ k ⊢ M : A ∆ ⊢ σ : Ψ ∆ ⊢ M [σ] : A RDP’07, Paris, June 07 – p.39 Substitution Definition • Critical cases, extended compositionally (x[τ])[σ] = M [τ[σ]] for n < k, M/x ∈ σ = x[τ[σ]] for n ≥ k (λx.M)[σ] = λx.M [σ, x/x] for n < k = λx.M [σ] for n ≥ k (M (Γ. N))[σ] = (M [σ]) (Γ. N [σ|n, id n Γ ]) for n < k = (M [σ]) (Γ. N) for n ≥ k RDP’07, Paris, June 07 – p.40 Substitution Composition • Typing guide ∆|k,Ψ k ⊢ τ : Θ ∆ ⊢ σ : Ψ ∆ ⊢ τ [σ] : Θ • Definition (τ, (Γ.M)/x)[σ] = (τ [σ], (Γ.M [σ|n, id n Γ ])/x) for n < k = (τ [σ], (Γ.M)/x) for n ≥ k RDP’07, Paris, June 07 – p.41 Single Substitutions, Rerevisited • Typing guide ∆|k,Ψ k ⊢ N : B ∆, x:B[Ψ] ⊢ M : A ∆ ⊢ [(Ψ. N)/x]M : A • Compositional, similar to simultaneous substitution • Show only one case [(Ψ. N)/x](x[σ]) = N [σ 1 /Ψ] for σ 1 = [(Ψ. N)/x](σ) RDP’07, Paris, June 07 – p.42 Example, Modified and Revisited • Omit suspension [(•)] and closure (•). s : B[u:A ∧B], t : C[v:A,w:A ⊃ C] ⊢ λx. 〈s[x], λy. t[π1x , y]〉 : (A ∧B) ⊃ B ∧ ((A ⊃ C) ⊃ C) • Simultaneous substitution at level 2 σ = ((u. π2u )/s, (v, w. w v)/t) • Crucial part (t1[π1x , y0])[σ , x/x, y/y] = (w v0)[π1x /v, y/w] = y (π1x ) RDP’07, Paris, June 07 – p.43 Summary, Multi-Level System • Uniform system of meta-variables x • Contextual type x : A[Ψ] • Closed with respect to variables y for n < k • Suspensions x[σ] where σ : Ψ • Level 0: ordinary variables • Level 1: meta-variables • Variables at all levels can be abstracted and applied • Satisfies α-conversion, subject reduction RDP’07, Paris, June 07 – p.44 Ongoing Work, Theory • Identity principle, subject expansion • Extension to dependent types • In ∆, x:A[Ψ], A can depend on variables in ∆|k and Ψ • If ∆ ctx then ∆|k ctx • Conjecture substitution and identity properties • Checked for k = 2 (contextual modal type theory) • Polymorphism? Substitution variables? • Structural vs nominal contexts RDP’07, Paris, June 07 – p.45 Ongoing Work, Pragmatics • Integrating single-variable and simultaneous substitution • De Bruijn representation • Uniform numbering of all levels(?) • ∆|k marks variables x for n < k as invisible • Level annotations and reconstruction RDP’07, Paris, June 07 – p.46