Fixed notion of values, and cached reduction
* Made the primitive form of elim (elim t_0 t_1), allowing this form to be considered a value when t_0 and t_1 are values. * Moved definition of values to reduction language, and fixed it. This fixed issues with unique decomposition, and thus fixed reduction of eliminators. * Enabled caching in apply-reduction-relation* to speed up results of recursive calls.
This commit is contained in:
William J. Bowman
parent
0807128f9e
commit
b95de692b9
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@ -35,14 +35,11 @@
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(i ::= natural)
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(U ::= (Unv i))
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(x ::= variable-not-otherwise-mentioned)
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(v ::= (Π (x : t) t) (λ (x : t) t) elim x U capp)
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(capp ::= (x v) (capp v))
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(t e ::= v (t t)))
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(t e ::= (Π (x : t) t) (λ (x : t) t) (elim t t) x U (t t)))
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(define x? (redex-match? cicL x))
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(define t? (redex-match? cicL t))
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(define e? (redex-match? cicL e))
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(define v? (redex-match? cicL v))
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(define U? (redex-match? cicL U))
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(module+ test
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@ -61,14 +58,8 @@
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(check-true (U? (term (Unv 0))))
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(check-true (U? (term Type)))
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(check-true (e? (term (λ (x_0 : (Unv 0)) x_0))))
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(check-true (v? (term (λ (x_0 : (Unv 0)) x_0))))
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(check-true (t? (term (λ (x_0 : (Unv 0)) x_0))))
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(check-true (t? (term (λ (x_0 : (Unv 0)) x_0))))
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(check-true
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(v? (term (refl Nat))))
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(check-true
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(v? (term ((refl Nat) z))))
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)
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(check-true (t? (term (λ (x_0 : (Unv 0)) x_0)))))
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;; 'A'
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;; Types of Universes
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@ -104,10 +95,10 @@
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#:contract (α-equivalent t t)
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[----------------- "α-x"
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(α-equivalent x x)]
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(α-equivalent x x)]
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[----------------- "α-U"
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(α-equivalent U U)]
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(α-equivalent U U)]
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[(α-equivalent t_1 (subst t_3 x_1 x_0))
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(α-equivalent t_0 t_2)
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@ -125,8 +116,8 @@
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(check-holds (α-equivalent (λ (x : A) x) (λ (y : A) y))))
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(define-metafunction cicL
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fresh-x : any ... -> x
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[(fresh-x any ...) ,(variable-not-in (term (any ...)) (term x))])
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fresh-x : any ... -> x
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[(fresh-x any ...) ,(variable-not-in (term (any ...)) (term x))])
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;; NB: Substitution is hard
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;; NB: Copy and pasted from Redex examples
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@ -156,7 +147,7 @@
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(term (x_0 t_0 x t t_1))
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(term x_0)))]
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[(subst (e_0 e_1) x t) ((subst e_0 x t) (subst e_1 x t))]
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[(subst elim x t) elim])
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[(subst (elim e_0 e_1) x t) (elim (subst e_0 x t) (subst e_1 x t))])
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(module+ test
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(check-true (t? (term (Π (a : A) (Π (b : B) ((and A) B))))))
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(check-holds
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@ -175,23 +166,32 @@
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;; TODO: I think a lot of things can be simplified if I rethink how
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;; TODO: model contexts, telescopes, and such.
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(define-extended-language cic-redL cicL
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;; NB: (in-hole Θv (elim x U)) is only a value when it's a partially applied elim.
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;; TODO: Perhaps (elim x U) should step to an eta-expanded version of elim
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(v ::= x U (Π (x : t) t) (λ (x : t) t) (elim x U) (in-hole Θv x) (in-hole Θv (elim x U)))
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;; call-by-value, plus reduce under Π (helps with typing checking)
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(E ::= hole (v E) (E e)
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(Π (x : (in-hole Θ x)) E)
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(E ::= hole (E e) (v E)
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(Π (x : v) E)
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(Π (x : E) e))
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;; TODO: Σ should probably be moved to cicL, since elim is there.
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;; Σ (signature). (inductive-name : type ((constructor : type) ...))
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(Σ ::= ∅ (Σ (x : t ((x : t) ...))))
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(Ξ Φ ::= hole (Π (x : t) Ξ)) ;;(Telescope)
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;; NB: Does an apply context correspond to a substitution (γ)?
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(Θ ::= hole (Θ e))) ;;(Apply context)
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(Θ ::= hole (Θ e)) ;;(Apply context)
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(Θv ::= hole (Θv v)))
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(define Σ? (redex-match? cic-redL Σ))
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(define Ξ? (redex-match? cic-redL Ξ))
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(define Φ? (redex-match? cic-redL Φ))
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(define Θ? (redex-match? cic-redL Θ))
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(define Θv? (redex-match? cic-redL Θv))
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(define E? (redex-match? cic-redL E))
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(define v? (redex-match? cic-redL v))
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(module+ test
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(check-true (v? (term (λ (x_0 : (Unv 0)) x_0))))
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(check-true (v? (term (refl Nat))))
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(check-true (v? (term ((refl Nat) z))))
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;; TODO: Rename these signatures, and use them in all future tests.
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(define Σ (term ((∅ (nat : (Unv 0) ((zero : nat) (s : (Π (x : nat) nat)))))
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(bool : (Unv 0) ((true : bool) (false : bool))))))
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@ -231,7 +231,6 @@
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((append-Σ Σ_2 Σ_1) (x : t ((x_c : t_c) ...)))])
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;; TODO: Test
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;; TODO: Isn't this just plug?
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;; TODO: Maybe this should be called "apply-to-telescope"
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(define-metafunction cic-redL
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apply-telescope : t Ξ -> t
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@ -274,7 +273,7 @@
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(in-hole Θ_i (hole (in-hole Θ_r x_ci)))
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(x_c0 ... x_ci x_c1 ...))
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((elim-recur x_D U e_P Θ Θ_m Θ_i (x_c0 ... x_ci x_c1 ...))
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(in-hole (Θ (in-hole Θ_r x_ci)) (((elim x_D) U) e_P)))]
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(in-hole (Θ (in-hole Θ_r x_ci)) ((elim x_D U) e_P)))]
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[(elim-recur x_D U e_P Θ Θ_i Θ_nr (x ...)) hole])
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(module+ test
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(check-true
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@ -285,7 +284,7 @@
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((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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(hole zero)
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(zero s)))
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(term (hole ((((((elim nat) Type) (λ (x : nat) nat))
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(term (hole (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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zero))))
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@ -295,7 +294,7 @@
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((hole (s zero)) (λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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(hole (s zero))
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(zero s)))
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(term (hole ((((((elim nat) Type) (λ (x : nat) nat))
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(term (hole (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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(s zero))))))
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@ -327,16 +326,16 @@
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(--> (Σ (in-hole E ((any (x : t_0) t_1) t_2)))
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(Σ (in-hole E (subst t_1 x t_2)))
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-->β)
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(--> (Σ (in-hole E (in-hole Θ (((elim x_D) U) e_P))))
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(Σ (in-hole E (in-hole Θ_r (in-hole Θ_i e_mi))))
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(--> (Σ (in-hole E (in-hole Θv ((elim x_D U) v_P))))
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(Σ (in-hole E (in-hole Θ_r (in-hole Θv_i v_mi))))
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#|
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| The elim form must appear applied like so:
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| (elim x_D U e_P m_0 ... m_i m_j ... m_n p ... (c_i a ...))
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| (elim x_D U v_P m_0 ... m_i m_j ... m_n p ... (c_i a ...))
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| Where:
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| x_D is the inductive being eliminated
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| U is the universe of the result of the motive
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| e_P is the motive
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| v_P is the motive
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| m_{0..n} are the methods
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| p ... are the parameters of x_D
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| c_i is a constructor of x_d
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@ -344,11 +343,11 @@
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| Unfortunately, Θ contexts turn all this inside out:
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| TODO: Write better abstractions for this notation
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|#
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(where (in-hole (Θ_p (in-hole Θ_i x_ci)) Θ_m)
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Θ)
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(where (in-hole (Θv_p (in-hole Θv_i x_ci)) Θv_m)
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Θv)
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;; Check that Θ_p actually matches the parameters of x_D, to ensure it doesn't capture other
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;; arguments.
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(judgment-holds (telescope-match Θ_p (parameters-of Σ x_D)))
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(judgment-holds (telescope-match Θv_p (parameters-of Σ x_D)))
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;; Ensure x_ci is actually a constructor for x_D
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(where ((x_c* : t_c*) ...)
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(constructors-for Σ x_D))
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@ -356,11 +355,11 @@
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(x_c* ...))
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;; There should be a number of methods equal to the number of constructors; to ensure E
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;; doesn't capture methods and Θ_m doesn't capture other arguments
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(judgment-holds (length-match Θ_m (x_c* ...)))
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(judgment-holds (length-match Θv_m (x_c* ...)))
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;; Find the method for constructor x_ci, relying on the order of the arguments.
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(where e_mi (method-lookup Σ x_D x_ci Θ_m))
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(where v_mi (method-lookup Σ x_D x_ci Θv_m))
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;; Generate the inductive recursion
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(where Θ_r (elim-recur x_D U e_P (in-hole Θ_p Θ_m) Θ_m Θ_i (x_c* ...)))
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(where Θ_r (elim-recur x_D U v_P (in-hole Θv_p Θv_m) Θv_m Θv_i (x_c* ...)))
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-->elim)))
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(define-metafunction cic-redL
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@ -373,7 +372,8 @@
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reduce : Σ e -> e
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[(reduce Σ e)
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e_r
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(where (_ e_r) ,(let ([r (apply-reduction-relation* cic--> (term (Σ e)))])
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(where (_ e_r) ,(let ([r (apply-reduction-relation* cic--> (term (Σ e))
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#:cache-all? #t)])
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(unless (null? (cdr r))
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(error "Church-rosser broken" r))
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(car r)))])
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@ -386,19 +386,19 @@
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(term (Π (x : t) (Unv 0))))
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(check-equal? (term (reduce ∅ (Π (x : t) ((Π (x_0 : t) (x_0 x)) x))))
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(term (Π (x : t) (x x))))
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(check-equal? (term (reduce ,Σ ((((((elim nat) Type) (λ (x : nat) nat))
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(check-equal? (term (reduce ,Σ (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat)
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(s (s x)))))
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zero)))
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(term (s zero)))
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(check-equal? (term (reduce ,Σ ((((((elim nat) Type) (λ (x : nat) nat))
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(check-equal? (term (reduce ,Σ (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat)
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(s (s x)))))
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(s zero))))
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(term (s (s zero))))
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(check-equal? (term (reduce ,Σ ((((((elim nat) Type) (λ (x : nat) nat))
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(check-equal? (term (reduce ,Σ (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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(s (s (s zero))))))
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@ -406,21 +406,21 @@
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(check-equal?
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(term (reduce ,Σ
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((((((elim nat) Type) (λ (x : nat) nat))
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(((((elim nat Type) (λ (x : nat) nat))
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(s (s zero)))
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(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
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(s (s zero)))))
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(term (s (s (s (s zero))))))
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(check-equal?
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(term (step ,Σ
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((((((elim nat) Type) (λ (x : nat) nat))
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(((((elim nat Type) (λ (x : nat) nat))
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(s (s zero)))
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(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
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(s (s zero)))))
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(term
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(((λ (x : nat) (λ (ih-x : nat) (s ih-x)))
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(s zero))
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((((((elim nat) Type) (λ (x : nat) nat))
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(((((elim nat Type) (λ (x : nat) nat))
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(s (s zero)))
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(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
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(s zero)))))
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@ -428,7 +428,7 @@
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(term (step ,Σ (step ,Σ
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(((λ (x : nat) (λ (ih-x : nat) (s ih-x)))
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(s zero))
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((((((elim nat) Type) (λ (x : nat) nat))
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(((((elim nat Type) (λ (x : nat) nat))
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(s (s zero)))
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(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
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(s zero))))))
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@ -437,7 +437,7 @@
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((λ (ih-x1 : nat) (s ih-x1))
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(((λ (x : nat) (λ (ih-x : nat) (s ih-x)))
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zero)
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((((((elim nat) Type) (λ (x : nat) nat))
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(((((elim nat Type) (λ (x : nat) nat))
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(s (s zero)))
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(λ (x : nat) (λ (ih-x : nat) (s ih-x))))
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zero))))))
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@ -804,8 +804,8 @@
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;; The types of the methods for this inductive.
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(where Ξ_m (methods-for x_D x_P (constructors-for Σ x_D)))
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----------------- "DTR-Elim_D"
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(type-infer Σ Γ ((elim x_D) U)
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;; The type of ((elim x_D) U) is something like:
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(type-infer Σ Γ (elim x_D U)
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;; The type of (elim x_D U) is something like:
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;; (∀ (P : (∀ a -> ... -> (D a ...) -> U))
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;; (method_ci ...) -> ... ->
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;; (a -> ... -> (D a ...) ->
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@ -858,8 +858,8 @@
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;; TODO: Clean up/Reorganize these tests
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(check-true
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(redex-match? cic-typingL
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(in-hole Θ_m ((((elim x_D) U) e_D) e_P))
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(term (((((elim truth) Type) T) (Π (x : truth) (Unv 1))) (Unv 0)))))
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(in-hole Θ_m (((elim x_D U) e_D) e_P))
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(term ((((elim truth Type) T) (Π (x : truth) (Unv 1))) (Unv 0)))))
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(define Σtruth (term (∅ (truth : (Unv 0) ((T : truth))))))
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(check-holds (type-infer ,Σtruth ∅ truth (in-hole Ξ U)))
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(check-holds (type-infer ,Σtruth ∅ T (in-hole Θ_ai truth)))
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@ -872,15 +872,15 @@
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(methods-for truth
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(λ (x : truth) (Unv 1))
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((T : truth)))))
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(check-holds (type-infer ,Σtruth ∅ ((elim truth) Type) t))
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(check-holds (type-infer ,Σtruth ∅ (elim truth Type) t))
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(check-holds (type-check (∅ (truth : (Unv 0) ((T : truth))))
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∅
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(((((elim truth) (Unv 2)) (λ (x : truth) (Unv 1))) (Unv 0))
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((((elim truth (Unv 2)) (λ (x : truth) (Unv 1))) (Unv 0))
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T)
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(Unv 1)))
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(check-not-holds (type-check (∅ (truth : (Unv 0) ((T : truth))))
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∅
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(((((elim truth) (Unv 1)) Type) Type) T)
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((((elim truth (Unv 1)) Type) Type) T)
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(Unv 1)))
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(check-holds
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(type-infer ∅ ∅ (Π (x2 : (Unv 0)) (Unv 0)) U))
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@ -898,7 +898,7 @@
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(check-holds (type-check ,Σ syn ...)))
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(nat-test ∅ (Π (x : nat) nat) (Unv 0))
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(nat-test ∅ (λ (x : nat) x) (Π (x : nat) nat))
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(nat-test ∅ ((((((elim nat) Type) (λ (x : nat) nat)) zero)
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(nat-test ∅ (((((elim nat Type) (λ (x : nat) nat)) zero)
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(λ (x : nat) (λ (ih-x : nat) x))) zero)
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nat)
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(nat-test ∅ nat (Unv 0))
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@ -907,38 +907,38 @@
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(nat-test ∅ (s zero) nat)
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;; TODO: Meta-function auto-currying and such
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(check-holds
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(type-infer ,Σ ∅ (((((elim nat) (Unv 0)) (λ (x : nat) nat))
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(type-infer ,Σ ∅ ((((elim nat (Unv 0)) (λ (x : nat) nat))
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zero)
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(λ (x : nat) (λ (ih-x : nat) x)))
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t))
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(nat-test ∅ ((((((elim nat) (Unv 0)) (λ (x : nat) nat))
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(nat-test ∅ (((((elim nat (Unv 0)) (λ (x : nat) nat))
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zero)
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(λ (x : nat) (λ (ih-x : nat) x)))
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zero)
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nat)
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(nat-test ∅ ((((((elim nat) (Unv 0)) (λ (x : nat) nat))
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(nat-test ∅ (((((elim nat (Unv 0)) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat) (s (s x)))))
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zero)
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nat)
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(nat-test ∅ ((((((elim nat) Type) (λ (x : nat) nat))
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(nat-test ∅ (((((elim nat Type) (λ (x : nat) nat))
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(s zero))
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(λ (x : nat) (λ (ih-x : nat) (s (s x))))) zero)
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nat)
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(nat-test (∅ n : nat)
|
||||
((((((elim nat) (Unv 0)) (λ (x : nat) nat)) zero) (λ (x : nat) (λ (ih-x : nat) x))) n)
|
||||
(((((elim nat (Unv 0)) (λ (x : nat) nat)) zero) (λ (x : nat) (λ (ih-x : nat) x))) n)
|
||||
nat)
|
||||
(check-holds
|
||||
(type-check (,Σ (bool : (Unv 0) ((btrue : bool) (bfalse : bool))))
|
||||
(∅ n2 : nat)
|
||||
((((((elim nat) (Unv 0)) (λ (x : nat) bool))
|
||||
(((((elim nat (Unv 0)) (λ (x : nat) bool))
|
||||
btrue)
|
||||
(λ (x : nat) (λ (ih-x : bool) bfalse)))
|
||||
n2)
|
||||
bool))
|
||||
(check-not-holds
|
||||
(type-check ,Σ ∅
|
||||
(((((elim nat) (Unv 0)) nat) (s zero)) zero)
|
||||
((((elim nat (Unv 0)) nat) (s zero)) zero)
|
||||
nat))
|
||||
(define lam (term (λ (nat : (Unv 0)) nat)))
|
||||
(check-equal?
|
||||
|
|
@ -997,7 +997,7 @@
|
|||
(in-hole Ξ (Π (x : (in-hole Θ_Ξ and)) U_P))))
|
||||
(check-holds
|
||||
(type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
|
||||
(((((((elim and) (Unv 0))
|
||||
((((((elim and (Unv 0))
|
||||
(λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
|
||||
true))))
|
||||
(λ (A : (Unv 0))
|
||||
|
|
@ -1034,7 +1034,7 @@
|
|||
(check-holds
|
||||
(type-check ,Σ4
|
||||
(((∅ P : (Unv 0)) Q : (Unv 0)) ab : ((and P) Q))
|
||||
(((((((elim and) (Unv 0))
|
||||
((((((elim and (Unv 0))
|
||||
(λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
|
||||
((and B) A)))))
|
||||
(λ (A : (Unv 0))
|
||||
|
|
@ -1054,7 +1054,7 @@
|
|||
t))
|
||||
(check-holds
|
||||
(type-check (,Σ4 (true : (Unv 0) ((tt : true)))) ∅
|
||||
(((((((elim and) (Unv 0))
|
||||
((((((elim and (Unv 0))
|
||||
(λ (A : Type) (λ (B : Type) (λ (x : ((and A) B))
|
||||
((and B) A)))))
|
||||
(λ (A : (Unv 0))
|
||||
|
|
@ -1086,17 +1086,17 @@
|
|||
(in-hole Ξ (Π (x : (in-hole Θ false)) U))))
|
||||
(check-true
|
||||
(redex-match? cic-typingL
|
||||
((in-hole Θ_m (((elim x_D) U) e_P)) e_D)
|
||||
(term ((((elim false) (Unv 1)) (λ (y : false) (Π (x : Type) x)))
|
||||
((in-hole Θ_m ((elim x_D U) e_P)) e_D)
|
||||
(term (((elim false (Unv 1)) (λ (y : false) (Π (x : Type) x)))
|
||||
x))))
|
||||
(check-holds
|
||||
(type-check ,sigma (,gamma x : false)
|
||||
((((elim false) (Unv 0)) (λ (y : false) (Π (x : Type) x))) x)
|
||||
(((elim false (Unv 0)) (λ (y : false) (Π (x : Type) x))) x)
|
||||
(Π (x : (Unv 0)) x)))
|
||||
|
||||
;; nat-equal? tests
|
||||
(define zero?
|
||||
(term (((((elim nat) Type) (λ (x : nat) bool))
|
||||
(term ((((elim nat Type) (λ (x : nat) bool))
|
||||
true)
|
||||
(λ (x : nat) (λ (x_ih : bool) false)))))
|
||||
(check-holds
|
||||
|
|
@ -1108,7 +1108,7 @@
|
|||
(term (reduce ,Σ (,zero? (s zero))))
|
||||
(term false))
|
||||
(define ih-equal?
|
||||
(term (((((elim nat) Type) (λ (x : nat) bool))
|
||||
(term ((((elim nat Type) (λ (x : nat) bool))
|
||||
false)
|
||||
(λ (x : nat) (λ (y : bool) (x_ih x))))))
|
||||
(check-holds
|
||||
|
|
@ -1122,7 +1122,7 @@
|
|||
(check-holds
|
||||
(type-infer ,Σ ∅ (λ (x : nat) (Π (x : nat) bool)) (Π (x : nat) (Unv 0))))
|
||||
(define nat-equal?
|
||||
(term (((((elim nat) Type) (λ (x : nat) (Π (x : nat) bool)))
|
||||
(term ((((elim nat Type) (λ (x : nat) (Π (x : nat) bool)))
|
||||
,zero?)
|
||||
(λ (x : nat) (λ (x_ih : (Π (x : nat) bool))
|
||||
,ih-equal?)))))
|
||||
|
|
@ -1143,7 +1143,7 @@
|
|||
(check-true (Σ? Σ=))
|
||||
|
||||
(define refl-elim
|
||||
(term ((((((((elim ==) (Unv 0)) (λ (A1 : (Unv 0)) (λ (x1 : A1) (λ (y1 : A1) (λ (p2 : (((==
|
||||
(term (((((((elim == (Unv 0)) (λ (A1 : (Unv 0)) (λ (x1 : A1) (λ (y1 : A1) (λ (p2 : (((==
|
||||
A1)
|
||||
x1)
|
||||
y1))
|
||||
|
|
@ -1154,7 +1154,7 @@
|
|||
(check-true
|
||||
(redex-match?
|
||||
cic-redL
|
||||
(Σ (in-hole E (in-hole Θ (((elim x_D) U) e_P))))
|
||||
(Σ (in-hole E (in-hole Θ ((elim x_D U) e_P))))
|
||||
(term (,Σ= ,refl-elim))))
|
||||
(check-true
|
||||
(redex-match?
|
||||
|
|
|
|||
|
|
@ -173,7 +173,7 @@
|
|||
[(elim t1 t2)
|
||||
(let* ([t1 (cur->datum #'t1)]
|
||||
[t2 (cur->datum #'t2)])
|
||||
(term ((elim ,t1) ,t2)))]
|
||||
(term (elim ,t1 ,t2)))]
|
||||
[(#%app e1 e2)
|
||||
(term (,(cur->datum #'e1) ,(cur->datum #'e2)))]))))
|
||||
(unless (and inner-expand? (type-infer/term reified-term))
|
||||
|
|
|
|||
|
|
@ -47,22 +47,20 @@
|
|||
(check-equal? (nat-equal? z (s z)) false)
|
||||
(check-equal? (nat-equal? (s z) (s z)) true))
|
||||
|
||||
#| TODO: Disabled until #20 fixed
|
||||
(define (even? (n : Nat))
|
||||
(elim Nat Type (lambda (x : Nat) Bool)
|
||||
false
|
||||
true
|
||||
(lambda* (n : Nat) (odd? : Bool)
|
||||
(not odd?))
|
||||
n))
|
||||
|
||||
(define (odd? (n : Nat))
|
||||
(and (not (even? n))
|
||||
(not (nat-equal? n z))))
|
||||
(not (even? n)))
|
||||
|
||||
(module+ test
|
||||
(check-equal?
|
||||
(even? z)
|
||||
false)
|
||||
true)
|
||||
(check-equal?
|
||||
(even? (s z))
|
||||
false)
|
||||
|
|
@ -81,4 +79,4 @@
|
|||
(check-equal?
|
||||
(odd? (s (s (s z))))
|
||||
true))
|
||||
|#
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue
Block a user