From Undecidability.L Require Import L Tactics.Computable Tactics.ComputableTactics Tactics.Extract.
From MetaCoq Require Import Template.All TemplateMonad.Core Template.Ast.
Require Import String List.
Export String.StringSyntax.
Import MonadNotation.
Local Open Scope string_scope.
Fixpoint mkLApp (s : term) (L : list term) :=
match L with
| [] => s
| t :: L => mkLApp (tApp (tConstruct (mkInd term_kn 0) 1 []) [s; t]) L
end.
Definition encode_arguments (B : term) (a i : nat) A_j :=
A <- tmUnquoteTyped Type A_j ;;
name <- (tmEval cbv (name_of A_j ++ "_term") >>= Core.tmFreshName) ;;
E <- tmTryInfer name None (registered A);;
t <- ret (@enc A E);;
l <- tmQuote t;;
ret (tApp l [tRel (a - i - 1) ]).
Definition mkMatch (t1 t2 d : Ast.term) (cases : nat -> list term -> Core.TemplateMonad term) :=
hs_num <- tmGetOption (split_head_symbol t1) "no head symbol found";;
let '(ind, Params) := hs_num in
let params := List.length Params in
L <- list_constructors ind >>= tmEval hnf ;;
body <- monad_map_i (fun i '(n, s, args) =>
l <- tmArgsOfConstructor ind i ;;
l' <- monad_map_i (insert_params FUEL Params) (skipn params l) ;;
t <- cases i l' ;; ret (args, t)) L ;;
ret (tCase (ind, params) (tLambda nAnon t1 t2) d
body).
Definition L_facts_mp := MPfile ["L_facts"; "Util"; "L"; "Undecidability"].
Definition tmMatchCorrect (A : Type) : Core.TemplateMonad Prop :=
t <- (tmEval hnf A >>= tmQuote) ;;
hs_num <- tmGetOption (split_head_symbol t) "no inductive";;
let '(ind, Params) := hs_num in
num <- tmNumConstructors (inductive_mind ind) ;;
x <- Core.tmFreshName "x" ;;
mtch <- mkMatch t tTerm (tRel (2 * num))
(fun i ctr_types =>
args <- tmEval cbv (|ctr_types|);;
C <- monad_map_i (encode_arguments t args) ctr_types ;;
ret (stack (map (tLambda (nAnon)) ctr_types)
(((fun s => mkAppList s C) (tRel (args + 2 * (num - i) - 1)))))
) ;;
E' <- Core.tmInferInstance None (registered A);;
E <- tmGetMyOption E' "failed" ;;
t' <- ret (@enc A E);;
l <- tmQuote t';;
encn <- ret (tApp l [tRel (2*num) ]) ;;
lhs <- ret (mkLApp encn ((fix f n := match n with 0 => [] | S n => tRel (2 * n + 1) :: f n end ) num)) ;;
ter <- ret (tProd nAnon t (it (fun s : term => tProd nAnon tTerm (tProd nAnon (tApp (tConst (L_facts_mp, "proc") []) [tRel 0]) s)) num ((tApp (tConst (L_facts_mp, "redLe") []) [mkNat num; lhs; mtch]))));;
ter <- tmEval cbv ter ;;
tmUnquoteTyped Prop ter.
Definition matchlem n A := (Core.tmBind (tmMatchCorrect A) (fun m => tmLemma n m ;; ret tt)).
Definition tmGenEncode (n : ident) (A : Type) : TemplateMonad unit :=
e <- tmEncode n A;;
modpath <- tmCurrentModPath tt ;;
e <- tmUnquoteTyped (encodable A) (tConst (modpath, n) []);;
p <- Core.tmLemma (n ++ "_proc") (forall x : A, proc (@enc_f A e x)) ;;
n2 <- tmEval cbv ((n ++ "_inj"));;
i <- Core.tmLemma n2 (injective (@enc_f _ e)) ;;
n3 <- tmEval cbv ("registered_" ++ n) ;;
d <- tmInstanceRed n3 None (@mk_registered A e p i);;
m <- tmMatchCorrect A;;
n4 <- tmEval cbv (n ++ "_correct") ;;
(Core.tmBind (tmMatchCorrect A) (fun m => tmLemma n4 m ;; ret tt)).
Arguments tmGenEncode _%string _%type.
Global Obligation Tactic := try fold (injective (enc_f)); match goal with
| [ |- forall x : ?X, proc ?f ] => try register_proc
| [ |- injective ?f ] => register_inj
| [ |- context [_ >(<= _) _] ] => extract match
end || Tactics.program_simpl.
From MetaCoq Require Import Template.All TemplateMonad.Core Template.Ast.
Require Import String List.
Export String.StringSyntax.
Import MonadNotation.
Local Open Scope string_scope.
Fixpoint mkLApp (s : term) (L : list term) :=
match L with
| [] => s
| t :: L => mkLApp (tApp (tConstruct (mkInd term_kn 0) 1 []) [s; t]) L
end.
Definition encode_arguments (B : term) (a i : nat) A_j :=
A <- tmUnquoteTyped Type A_j ;;
name <- (tmEval cbv (name_of A_j ++ "_term") >>= Core.tmFreshName) ;;
E <- tmTryInfer name None (registered A);;
t <- ret (@enc A E);;
l <- tmQuote t;;
ret (tApp l [tRel (a - i - 1) ]).
Definition mkMatch (t1 t2 d : Ast.term) (cases : nat -> list term -> Core.TemplateMonad term) :=
hs_num <- tmGetOption (split_head_symbol t1) "no head symbol found";;
let '(ind, Params) := hs_num in
let params := List.length Params in
L <- list_constructors ind >>= tmEval hnf ;;
body <- monad_map_i (fun i '(n, s, args) =>
l <- tmArgsOfConstructor ind i ;;
l' <- monad_map_i (insert_params FUEL Params) (skipn params l) ;;
t <- cases i l' ;; ret (args, t)) L ;;
ret (tCase (ind, params) (tLambda nAnon t1 t2) d
body).
Definition L_facts_mp := MPfile ["L_facts"; "Util"; "L"; "Undecidability"].
Definition tmMatchCorrect (A : Type) : Core.TemplateMonad Prop :=
t <- (tmEval hnf A >>= tmQuote) ;;
hs_num <- tmGetOption (split_head_symbol t) "no inductive";;
let '(ind, Params) := hs_num in
num <- tmNumConstructors (inductive_mind ind) ;;
x <- Core.tmFreshName "x" ;;
mtch <- mkMatch t tTerm (tRel (2 * num))
(fun i ctr_types =>
args <- tmEval cbv (|ctr_types|);;
C <- monad_map_i (encode_arguments t args) ctr_types ;;
ret (stack (map (tLambda (nAnon)) ctr_types)
(((fun s => mkAppList s C) (tRel (args + 2 * (num - i) - 1)))))
) ;;
E' <- Core.tmInferInstance None (registered A);;
E <- tmGetMyOption E' "failed" ;;
t' <- ret (@enc A E);;
l <- tmQuote t';;
encn <- ret (tApp l [tRel (2*num) ]) ;;
lhs <- ret (mkLApp encn ((fix f n := match n with 0 => [] | S n => tRel (2 * n + 1) :: f n end ) num)) ;;
ter <- ret (tProd nAnon t (it (fun s : term => tProd nAnon tTerm (tProd nAnon (tApp (tConst (L_facts_mp, "proc") []) [tRel 0]) s)) num ((tApp (tConst (L_facts_mp, "redLe") []) [mkNat num; lhs; mtch]))));;
ter <- tmEval cbv ter ;;
tmUnquoteTyped Prop ter.
Definition matchlem n A := (Core.tmBind (tmMatchCorrect A) (fun m => tmLemma n m ;; ret tt)).
Definition tmGenEncode (n : ident) (A : Type) : TemplateMonad unit :=
e <- tmEncode n A;;
modpath <- tmCurrentModPath tt ;;
e <- tmUnquoteTyped (encodable A) (tConst (modpath, n) []);;
p <- Core.tmLemma (n ++ "_proc") (forall x : A, proc (@enc_f A e x)) ;;
n2 <- tmEval cbv ((n ++ "_inj"));;
i <- Core.tmLemma n2 (injective (@enc_f _ e)) ;;
n3 <- tmEval cbv ("registered_" ++ n) ;;
d <- tmInstanceRed n3 None (@mk_registered A e p i);;
m <- tmMatchCorrect A;;
n4 <- tmEval cbv (n ++ "_correct") ;;
(Core.tmBind (tmMatchCorrect A) (fun m => tmLemma n4 m ;; ret tt)).
Arguments tmGenEncode _%string _%type.
Global Obligation Tactic := try fold (injective (enc_f)); match goal with
| [ |- forall x : ?X, proc ?f ] => try register_proc
| [ |- injective ?f ] => register_inj
| [ |- context [_ >(<= _) _] ] => extract match
end || Tactics.program_simpl.