From 73e1afb4a3a11433e464d2671f4da5536228fff3 Mon Sep 17 00:00:00 2001 From: "github-actions[bot]" <41898282+github-actions[bot]@users.noreply.github.com> Date: Fri, 15 Mar 2024 09:44:39 +0000 Subject: [PATCH] Deploy to GitHub pages --- .nojekyll | 0 docs/coqdoc/ABS.abs_defs.html | 267 ++++ docs/coqdoc/ABS.abs_examples.html | 89 ++ .../coqdoc/ABS.abs_functional_metatheory.html | 825 ++++++++++++ docs/coqdoc/ABS.utils.html | 775 ++++++++++++ docs/coqdoc/config.js | 79 ++ docs/coqdoc/coqdoc.css | 197 +++ docs/coqdoc/coqdocjs.css | 239 ++++ docs/coqdoc/coqdocjs.js | 197 +++ docs/coqdoc/indexpage.html | 1101 +++++++++++++++++ docs/coqdoc/toc.html | 48 + docs/report/main.pdf | Bin 0 -> 167718 bytes index.html | 42 + 13 files changed, 3859 insertions(+) create mode 100644 .nojekyll create mode 100644 docs/coqdoc/ABS.abs_defs.html create mode 100644 docs/coqdoc/ABS.abs_examples.html create mode 100644 docs/coqdoc/ABS.abs_functional_metatheory.html create mode 100644 docs/coqdoc/ABS.utils.html create mode 100644 docs/coqdoc/config.js create mode 100644 docs/coqdoc/coqdoc.css create mode 100644 docs/coqdoc/coqdocjs.css create mode 100644 docs/coqdoc/coqdocjs.js create mode 100644 docs/coqdoc/indexpage.html create mode 100644 docs/coqdoc/toc.html create mode 100644 docs/report/main.pdf create mode 100644 index.html diff --git a/.nojekyll b/.nojekyll new file mode 100644 index 0000000..e69de29 diff --git a/docs/coqdoc/ABS.abs_defs.html b/docs/coqdoc/ABS.abs_defs.html new file mode 100644 index 0000000..621715f --- /dev/null +++ b/docs/coqdoc/ABS.abs_defs.html @@ -0,0 +1,267 @@ + + + + + + + + + + + + + +
+
+
+(* generated by Ott 0.33 from: src/abs.ott *)
+ +
+Require Import Arith.
+Require Import Bool.
+Require Import List.
+Require Import Ott.ott_list_core.
+ +
+Require Export Ascii.
+Require Export String.
+Require Export ZArith.
+Require Import Lia.
+From Equations Require Import Equations.
+ +
+Require Import FMapList OrderedTypeEx.
+Module Map <: FMapInterface.S := FMapList.Make String_as_OT.
+ +
+#[export] Hint Resolve bool_dec : ott_coq_equality.
+#[export] Hint Resolve ascii_dec : ott_coq_equality.
+#[export] Hint Resolve Pos.eq_dec : ott_coq_equality.
+ +
+
+ +
+

ABS Definitions

+ +
+
+ +
+Definition i : Set := nat. (*r index variables (subscripts) *)
+Lemma eq_i: forall (x y : i), {x = y} + {x <> y}.
+Proof.
+  decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_i : ott_coq_equality.
+Definition fn : Set := string. (*r function name *)
+Lemma eq_fn: forall (x y : fn), {x = y} + {x <> y}.
+Proof.
+  decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_fn : ott_coq_equality.
+Definition x : Set := string. (*r variable *)
+Lemma eq_x: forall (x' y : x), {x' = y} + {x' <> y}.
+Proof.
+  decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_x : ott_coq_equality.
+Definition b : Set := bool. (*r boolean *)
+Lemma eq_b: forall (x y : b), {x = y} + {x <> y}.
+Proof.
+  decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_b : ott_coq_equality.
+Definition z : Set := Z. (*r integer *)
+Lemma eq_z: forall (x y : z), {x = y} + {x <> y}.
+Proof.
+  decide equality; auto with ott_coq_equality arith.
+Defined.
+Hint Resolve eq_z : ott_coq_equality.
+ +
+Inductive T : Set := (*r ground type *)
+ | T_bool : T
+ | T_int : T.
+ +
+Inductive t : Set := (*r ground term *)
+ | t_b (b5:b) (*r boolean *)
+ | t_int (z5:z) (*r integer *).
+ +
+Inductive sig : Set :=
+ | sig_sig (_:list T) (T_5:T).
+ +
+Inductive e : Set := (*r expression *)
+ | e_t (t5:t) (*r term *)
+ | e_var (x5:x) (*r variable *)
+ | e_fn_call (fn5:fn) (_:list e) (*r function call *).
+ +
+Inductive ctxv : Set :=
+ | ctxv_T (T5:T)
+ | ctxv_sig (sig5:sig).
+ +
+Inductive F : Set := (*r function definition *)
+ | F_fn (T_5:T) (fn5:fn) (_:list (T*x)) (e5:e).
+ +
+Definition s : Type := Map.t t.
+ +
+Definition G : Type := Map.t ctxv.
+
+ +
+induction principles +
+
+Section e_rect.
+ +
+Variables
+  (P_e : e -> Prop)
+  (P_list_e : list e -> Prop).
+ +
+Hypothesis
+  (H_e_t : forall (t5:t), P_e (e_t t5))
+  (H_e_var : forall (x5:x), P_e (e_var x5))
+  (H_e_fn_call : forall (e_list:list e), P_list_e e_list -> forall (fn5:fn), P_e (e_fn_call fn5 e_list))
+  (H_list_e_nil : P_list_e nil)
+  (H_list_e_cons : forall (e0:e), P_e e0 -> forall (e_l:list e), P_list_e e_l -> P_list_e (cons e0 e_l)).
+ +
+Fixpoint e_ott_ind (n:e) : P_e n :=
+  match n as x return P_e x with
+  | (e_t t5) => H_e_t t5
+  | (e_var x5) => H_e_var x5
+  | (e_fn_call fn5 e_list) => H_e_fn_call e_list (((fix e_list_ott_ind (e_l:list e) : P_list_e e_l := match e_l as x return P_list_e x with nil => H_list_e_nil | cons e1 xl => H_list_e_cons e1(e_ott_ind e1)xl (e_list_ott_ind xl) end)) e_list) fn5
+end.
+ +
+End e_rect.
+ +
+Definition disjoint {A:Type} (l1 l2: list A): Prop :=
+ forall a, In a l1 -> ~ In a l2.
+ +
+Equations e_var_subst_one (e5:e) (x_ y_: x) : e := {
+ e_var_subst_one (e_t t) _ _ := e_t t;
+ e_var_subst_one (e_var x0) x_ y_ := if (eq_x x0 x_) then (e_var y_) else (e_var x0);
+ e_var_subst_one (e_fn_call fn0 arg_list) x_ y_ := e_fn_call fn0 (e_list_subst_one arg_list x_ y_) }
+where e_list_subst_one (es:list e) (x_ y_: x) : list e := {
+ e_list_subst_one nil _ _ := nil;
+ e_list_subst_one (e0::es) x_ y_ := e_var_subst_one e0 x_ y_ :: e_list_subst_one es x_ y_
+}.
+ +
+Definition e_var_subst (e5:e) (l:list (x*x)) : e := fold_right (fun '(x', y') e' => e_var_subst_one e' x' y') e5 l.
+ +
+Equations fresh_vars_e (l : list x) (e0 : e) : Prop := {
+ fresh_vars_e _ (e_t _) := True;
+ fresh_vars_e l (e_var x) := ~ In x l;
+ fresh_vars_e l (e_fn_call fn el) := fresh_vars_el l el }
+where fresh_vars_el (l : list x) (el0 : list e) : Prop := {
+ fresh_vars_el l nil := True;
+ fresh_vars_el l (e1::el0) := fresh_vars_e l e1 /\ fresh_vars_el l el0 }.
+ +
+Fixpoint fresh_vars_s (l : list x) (s0 : s): Prop :=
+  match l with
+  | nil => True
+  | (y::ys) => ~ Map.In y s0 /\ fresh_vars_s ys s0
+  end.
+ +
+Definition fresh_vars (l : list x) (e0: e) (s0: s) : Prop :=
+  fresh_vars_s l s0 /\ fresh_vars_e l e0.
+ +
+Definition well_formed (e0: e) (s0: s) (l:list x) : Prop := fresh_vars l e0 s0 /\ NoDup l.
+ +
+
+ +
+definitions +
+
+ +
+(* defns expression_well_typing *)
+Inductive typ_e : G -> e -> T -> Prop := (* defn e *)
+ | typ_bool : forall (G5:G) (b5:b),
+     typ_e G5 (e_t (t_b b5)) T_bool
+ | typ_int : forall (G5:G) (z5:z),
+     typ_e G5 (e_t (t_int z5)) T_int
+ | typ_var : forall (G5:G) (x5:x) (T5:T),
+      (Map.find x5 G5 = Some (ctxv_T T5 )) ->
+     typ_e G5 (e_var x5) T5
+ | typ_func_expr : forall (e_T_list:list (e*T)) (G5:G) (fn5:fn) (T_5:T),
+     (forall e_ T_, In (e_,T_) (map (fun (pat_: (e*T)) => match pat_ with (e_,T_) => (e_,T_) end) e_T_list) -> (typ_e G5 e_ T_)) ->
+      (Map.find fn5 G5 = Some (ctxv_sig (sig_sig (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => T_ end ) e_T_list) T_5) )) ->
+     typ_e G5 (e_fn_call fn5 (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => e_ end ) e_T_list)) T_5.
+
+ +
+definitions +
+
+ +
+(* defns function_well_typing *)
+Inductive typ_F : G -> F -> Prop := (* defn F *)
+ | typ_func_decl : forall (T_x_list:list (T*x)) (G5:G) (T_5:T) (fn5:fn) (e5:e),
+      (Map.find fn5 G5 = Some (ctxv_sig (sig_sig (map (fun (pat_:(T*x)) => match pat_ with (T_,x_) => T_ end ) T_x_list) T_5) )) ->
+     typ_e (fold_right (fun (ax : x * T) (G5 : G) => Map.add (fst ax) (ctxv_T (snd ax)) G5) G5 (map (fun (pat_:(T*x)) => match pat_ with (T_,x_) => (x_,T_) end ) T_x_list) ) e5 T_5 ->
+      (NoDup (map (fun (pat_:(T*x)) => match pat_ with (T_,x_) => x_ end ) T_x_list) ) ->
+     typ_F G5 (F_fn T_5 fn5 T_x_list e5).
+
+ +
+definitions +
+
+ +
+(* defns evaluation_reduction *)
+Inductive red_e : list F -> s -> e -> s -> e -> Prop := (* defn e *)
+ | red_var : forall (F_list:list F) (s5:s) (x5:x) (t5:t),
+      (Map.find x5 s5 = Some ( t5 )) ->
+     red_e F_list s5 (e_var x5) s5 (e_t t5)
+ | red_fun_exp : forall (e'_list e_list:list e) (F_list:list F) (s5:s) (fn5:fn) (e_5:e) (s':s) (e':e),
+     red_e F_list s5 e_5 s' e' ->
+     red_e F_list s5 (e_fn_call fn5 ((app e_list (app (cons e_5 nil) (app e'_list nil))))) s' (e_fn_call fn5 ((app e_list (app (cons e' nil) (app e'_list nil)))))
+ | red_fun_ground : forall (T_x_t_y_list:list (T*x*t*x)) (F'_list F_list:list F) (T_5:T) (fn5:fn) (e5:e) (s5:s),
+      (well_formed e5 s5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => y_ end ) T_x_t_y_list) ) ->
+      (disjoint (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => y_ end ) T_x_t_y_list) (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => x_ end ) T_x_t_y_list) ) ->
+     red_e ((app F_list (app (cons (F_fn T_5 fn5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (T_,x_) end ) T_x_t_y_list) e5) nil) (app F'_list nil)))) s5 (e_fn_call fn5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (e_t t_) end ) T_x_t_y_list)) (fold_right (fun (xt : x * t) (s5 : s) => Map.add (fst xt) (snd xt) s5) s5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (y_,t_) end ) T_x_t_y_list) ) (e_var_subst e5 (map (fun (pat_:(T*x*t*x)) => match pat_ with (T_,x_,t_,y_) => (x_,y_) end ) T_x_t_y_list) ) .
+ +
+
+
+ +
+ + + diff --git a/docs/coqdoc/ABS.abs_examples.html b/docs/coqdoc/ABS.abs_examples.html new file mode 100644 index 0000000..9a433f3 --- /dev/null +++ b/docs/coqdoc/ABS.abs_examples.html @@ -0,0 +1,89 @@ + + + + + + + + + + + + + +
+
+
+From ABS Require Import abs_defs.
+From stdpp Require Import prelude base.
+ +
+
+ +
+

ABS Examples

+ +
+
+ +
+Definition e_const_5 : e :=
+  e_t (t_int 5%Z).
+ +
+Definition func_const_5 : F :=
+  F_fn T_int "const_5"%string [] e_const_5.
+ +
+Definition e_call_const_5 : e :=
+  e_fn_call "const_5"%string [].
+ +
+Definition Ctx : G :=
+  Map.add ("const_5"%string) (ctxv_sig (sig_sig [] T_int)) (Map.empty ctxv).
+ +
+Lemma e_const_5_T_int :
+ typ_e Ctx e_const_5 T_int.
+Proof. by apply typ_int. Qed.
+ +
+Lemma e_call_const_typ_F :
+ typ_F Ctx func_const_5.
+Proof.
+  apply typ_func_decl.
+  - easy.
+  - apply e_const_5_T_int.
+  - simpl; constructor.
+Qed.
+ +
+Lemma e_call_const_5_T_int :
+ typ_e Ctx e_call_const_5 T_int.
+Proof.
+  unfold e_call_const_5.
+  replace [] with (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => e_ end ) [])
+    by auto.
+  by apply typ_func_expr.
+Qed.
+
+
+ +
+ + + diff --git a/docs/coqdoc/ABS.abs_functional_metatheory.html b/docs/coqdoc/ABS.abs_functional_metatheory.html new file mode 100644 index 0000000..5d94fff --- /dev/null +++ b/docs/coqdoc/ABS.abs_functional_metatheory.html @@ -0,0 +1,825 @@ + + + + + + + + + + + + + +
+
+
+From ABS Require Import abs_defs
+  utils.
+ +
+From Coq Require Import List
+  FSets.FMapFacts
+  Lia.
+ +
+From Equations Require Import Equations.
+ +
+Require Import FMapList.
+Module MapFacts := FSets.FMapFacts.Facts Map.
+Import ListNotations.
+ +
+
+ +
+

ABS Functional Metatheory

+ +
+
+ +
+Section FunctionalMetatheory.
+ +
+Hypothesis (vars_fs_distinct: forall (x_:x) (fn_:fn), x_ <> fn_).
+Hypothesis (vars_well_typed: forall (x_:x) (G0: G) T0,
+             Map.find x_ G0 = Some T0 ->
+             exists T_, T0 = ctxv_T T_).
+ +
+Definition subG (G1 G2 : G) : Prop :=
+  forall (key : string) (elt : ctxv),
+    Map.find key G1 = Some elt ->
+    Map.find key G2 = Some elt.
+ +
+Notation "G ⊢ e : T" := (typ_e G e T) (at level 5).
+Notation "G F⊢ F" := (typ_F G F) (at level 5).
+Notation "G1 ⊆ G2" := (subG G1 G2) (at level 5).
+ +
+Lemma subG_refl: forall G0,
+  subG G0 G0.
+Proof. easy. Qed.
+ +
+Lemma subG_add: forall G0 G1 y T_,
+  subG G0 G1 ->
+  ~ Map.In y G0 ->
+  subG G0 (Map.add y T_ G1).
+Proof.
+  intros.
+  intros ?x_ ?T_ ?.
+ +
+  destruct (eq_x y x_); subst.
+  - exfalso.
+    apply H0.
+    apply MapFacts.in_find_iff.
+    intro.
+    rewrite H1 in H2.
+    discriminate.
+  - apply Map.find_1.
+    apply MapFacts.add_neq_mapsto_iff; auto.
+    pose proof H x_ T_0 H1.
+    apply Map.find_2.
+    assumption.
+Qed.
+ +
+Lemma subG_add_2: forall G0 G1 y T_,
+  subG G0 G1 ->
+  Map.find y G0 = Some T_ ->
+  subG G0 (Map.add y T_ G1).
+Proof.
+  intros.
+  intros ?x_ ?T_ ?.
+ +
+  destruct (eq_x y x_); subst.
+  - rewrite H0 in H1.
+    inv H1.
+    apply Map.find_1.
+    now apply Map.add_1.
+  - apply Map.find_1.
+    apply MapFacts.add_neq_mapsto_iff; auto.
+    pose proof H x_ T_0 H1.
+    apply Map.find_2.
+    assumption.
+Qed.
+ +
+(* this is stricter than the ABS paper *)
+(* we require that any variables in the typing context also exist in the state *)
+Definition G_vdash_s (G5 : G) (s5 : s) :=
+ forall (x5 : x) (T5 : T),
+  Map.find x5 G5 = Some (ctxv_T T5) ->
+  exists t5,
+  Map.find x5 s5 = Some t5 /\ typ_e G5 (e_t t5) T5.
+ +
+Notation "G1 G⊢ s1" := (G_vdash_s G1 s1) (at level 5).
+ +
+Lemma fresh_subG: forall G0 s0 (sub_list: list (T*x*t*x)),
+  G_vdash_s G0 s0 ->
+  fresh_vars_s (map (fun '(_,_,_,y)=>y) sub_list) s0 ->
+  NoDup (map (fun '(_,_,_,y)=>y) sub_list) ->
+  subG G0 (fold_right (fun '(T_,_,_,y_) G0 => Map.add y_ (ctxv_T T_) G0) G0 sub_list).
+Proof.
+  intros.
+  induction sub_list.
+  - easy.
+  - destruct a as (((?T_, ?x_), ?t_), y).
+    inv H1.
+    inv H0.
+    pose proof IHsub_list H2 H5.
+    cbn.
+    apply subG_add; auto.
+    intro.
+ +
+    apply MapFacts.in_find_iff in H3.
+    apply MapFacts.not_find_in_iff in H1.
+    pose proof map_neq_none_is_some _ _ H3 as (?T_, ?).
+    pose proof vars_well_typed _ _ _ H6 as (?T_, ->).
+    pose proof H _ _ H6 as (?t_, (?, _)).
+    rewrite H1 in H7.
+    discriminate.
+Qed.
+ +
+Lemma fresh_consistent: forall G0 s0 (sub_list: list (T*x*t*x)),
+  G_vdash_s G0 s0 ->
+  (forall T_ t_, In (T_, t_) (map (fun '(T_, _, t_, _) => (T_, t_)) sub_list) ->
+            typ_e G0 (e_t t_) T_) ->
+  fresh_vars_s (map (fun '(_,_,_,y)=>y) sub_list) s0 ->
+  NoDup (map (fun '(_,_,_,y)=>y) sub_list) ->
+  G_vdash_s (fold_right (fun '(T_,_,_,y_) G0 => Map.add y_ (ctxv_T T_) G0) G0 sub_list)
+    (fold_right (fun '(_,_,t_,y_) s0 => Map.add y_ t_ s0) s0 sub_list).
+Proof.
+  intros.
+  induction sub_list.
+  * easy.
+  * destruct a as (((?T_, ?x_), ?t_), y).
+    inv H1.
+    inv H2.
+ +
+    cbn.
+    intros ?y ?T_ ?.
+    destruct (eq_x y y0); subst.
+  - exists t_.
+    split...
+    + apply Map.find_1.
+      now apply Map.add_1.
+    + assert (T_ = T_0).
+      {
+        apply Map.find_2 in H1.
+        apply MapFacts.add_mapsto_iff in H1.
+        destruct H1 as [(_, ?) | (?, ?)].
+        + now inv H1.
+        + exfalso.
+          now apply H1.
+      }
+      assert (In (T_0, t_) (map (fun '(T_, _, t_, _) => (T_, t_)) ((T_, x_, t_, y0) :: sub_list)))
+        by (left; now rewrite H2).
+      specialize (H0 _ _ H5).
+      inv H0; constructor.
+  - apply Map.find_2 in H1.
+    apply Map.add_3 in H1; auto.
+    apply Map.find_1 in H1.
+    assert (forall (T_ : T) (t_ : t),
+             In (T_, t_) (map (fun '(T_0, _, t_0, _) => (T_0, t_0)) sub_list) ->
+             typ_e G0 (e_t t_) T_).
+    {
+      intros.
+      apply H0.
+      right.
+      assumption.
+    }
+    pose proof IHsub_list H2 H4 H7 _ _ H1 as (?t_, (?, ?)).
+    exists t_0.
+    split...
+    + apply Map.find_1.
+      apply Map.add_2; auto.
+      apply Map.find_2; assumption.
+    + inv H8; constructor.
+Qed.
+ +
+Lemma subG_consistent: forall G1 G2 s0,
+  subG G1 G2 -> G_vdash_s G2 s0 -> G_vdash_s G1 s0.
+Proof.
+  intros.
+  intros x_ t_ ?.
+ +
+  pose proof H _ _ H1.
+  pose proof H0 _ _ H2 as (?t, (?, ?)).
+  exists t.
+  split; auto.
+  inv H4;constructor.
+Qed.
+ +
+Lemma subG_type: forall G1 G2 e0 T0,
+  subG G1 G2 -> typ_e G1 e0 T0 -> typ_e G2 e0 T0.
+Proof.
+  intros.
+  induction H0.
+  - constructor.
+  - constructor.
+  - apply H in H0.
+    constructor.
+    apply H0.
+  - constructor.
+    + intros.
+      apply H1; auto.
+    + apply H in H2.
+      apply H2.
+Qed.
+ +
+(* this is 9.3.8 from Types and Programming Languages *)
+(* it has not yet proven useful *)
+Lemma subst_lemma_one_alt: forall (x0 x1:x) G0 e0 T0 T1,
+  typ_e (Map.add x0 (ctxv_T T1) G0) e0 T0 ->
+  typ_e G0 (e_var x1) T1 ->
+  typ_e G0 (e_var_subst_one e0 x0 x1) T0.
+Proof.
+  intros.
+  generalize dependent T0.
+  generalize dependent T1.
+  generalize dependent G0.
+  eapply e_ott_ind with
+      (n:=e0)
+      (P_list_e:= fun e_list => forall e0,
+                    In e0 e_list ->
+                    forall G0 T0 T1,
+                    typ_e (Map.add x0 (ctxv_T T1) G0) e0 T0 ->
+                    typ_e G0 (e_var x1) T1 ->
+                    typ_e G0 (e_var_subst e0 [(x0, x1)]) T0)
+  ;cbn; intros.
+  - inv H; simp e_var_subst_one; constructor.
+  - inv H.
+    simp e_var_subst_one.
+    destruct (eq_x x5 x0); subst.
+    + apply Map.find_2 in H3.
+      apply MapFacts.add_mapsto_iff in H3.
+      destruct H3 as [(?&?) | (?&?)]; subst.
+      * inv H1.
+        assumption.
+      * contradiction.
+    + constructor.
+      apply Map.find_1.
+      eapply Map.add_3.
+      * apply not_eq_sym.
+        apply n.
+      * apply Map.find_2 in H3.
+        apply H3.
+  - inv H1.
+    simp e_var_subst_one.
+    rewrite <- subst_fst_commute.
+    replace (map (fun x : e * T => let (e_, _) := let '(e_, T_) := x in (e_var_subst_one e_ x0 x1, T_) in e_) e_T_list)
+      with
+        (map (fun pat_ : e * T => let (e_, _) := pat_ in e_)
+           (map (fun '(e_, T_) => (e_var_subst_one e_ x0 x1, T_)) e_T_list))
+    by apply map_map.
+    constructor.
+    + rewrite map_map.
+      intros.
+      apply in_map_iff in H1.
+      destruct H1 as ((?e_&?T_) & ? & ?).
+      inv H1.
+      eapply H.
+      * apply in_map_iff.
+        exists (e_0, T_).
+        split; eauto.
+      * apply H5.
+        apply in_map_iff.
+        exists (e_0, T_).
+        split; auto.
+      * apply H0.
+    + apply Map.find_1.
+      apply Map.find_2 in H7.
+      rewrite subst_snd_commutes.
+      eapply Map.add_3; eauto.
+  - inv H.
+  - inv H1.
+    + eapply H; eauto.
+    + eapply H0; eauto.
+Qed.
+ +
+Lemma same_type_sub: forall (sub_list: list (x*x)) G0 e0 T0,
+    typ_e G0 e0 T0 ->
+    (forall x0 y0 T0,
+      In (x0, y0) sub_list ->
+      Map.find x0 G0 = Some T0 ->
+      Map.find y0 G0 = Some T0) ->
+    typ_e G0 (e_var_subst e0 sub_list) T0.
+Proof.
+  {
+  intros.
+  generalize dependent T0.
+  eapply e_ott_ind with
+      (n:= e0)
+      (P_list_e:= fun e_list => forall e0 T0,
+                    In e0 e_list ->
+                    typ_e G0 e0 T0 ->
+                    typ_e G0 (e_var_subst e0 sub_list) T0)
+  ; intros.
+  - rewrite subst_term.
+    simp e_var_subst_one.
+  - rewrite subst_var.
+    constructor.
+    induction sub_list.
+    + inv H.
+      now simpl.
+    + destruct a.
+      simpl.
+      destruct (eq_x (replace_var x5 sub_list) x); subst.
+      * eapply H0; eauto.
+        -- now left.
+        -- apply IHsub_list; eauto.
+           intros.
+           eapply H0; eauto.
+           now right.
+      * eapply IHsub_list; eauto.
+        intros.
+        eapply H0; eauto.
+        now right.
+  - inv H1.
+    rewrite subst_fn.
+    rewrite map_map.
+    replace (map (fun x : e * T => e_var_subst (let (e_, _) := x in e_) sub_list) e_T_list)
+      with (map (fun pat_:e*T => let (e, _) := pat_ in e) (map (fun '(e, T) => (e_var_subst e sub_list, T)) e_T_list))
+      by (rewrite map_map; apply map_ext; now intros (?, ?)).
+    constructor.
+    + intros.
+      apply in_map_iff in H1.
+      destruct H1 as ((?e, ?T), (?, ?)).
+      inv H1.
+      apply in_map_iff in H2.
+      destruct H2 as ((?e_, ?T_), (?, ?)).
+      inv H1.
+      apply H.
+      * apply in_map_iff.
+        exists (e_0, T_).
+        split; eauto.
+      * apply H5.
+        apply in_map_iff.
+        exists (e_0, T_).
+        split; eauto.
+    + now replace (map (fun pat_ : e * T => let (_, T_) := pat_ in T_) (map (fun '(e, T) => (e_var_subst e sub_list, T)) e_T_list))
+        with(map (fun pat_ : e * T => let (_, T_) := pat_ in T_) e_T_list)
+      by (rewrite map_map; apply map_ext; now intros (?, ?)).
+  - inv H.
+  - inv H2.
+    + now apply H.
+    + now apply H1.
+  }
+
+Qed.
+ +
+Lemma subst_lemma_one: forall G0 x0 y0 e0 Ai A,
+  typ_e (Map.add x0 Ai G0) e0 A ->
+  fresh_vars_e [y0] e0 ->
+  typ_e (Map.add y0 Ai G0) (e_var_subst_one e0 x0 y0) A.
+Proof.
+  {
+    intros.
+    generalize dependent A.
+    revert H0.
+    apply e_ott_ind with
+        (n:=e0)
+        (P_list_e:= fun e_list =>
+                      forall e0 A,
+                      In e0 e_list ->
+                      fresh_vars_e [y0] e0 ->
+                      typ_e (Map.add x0 Ai G0) e0 A ->
+                      typ_e (Map.add y0 Ai G0) (e_var_subst_one e0 x0 y0) A);
+      intros.
+    - simp e_var_subst_one.
+      inv H; constructor.
+    - simp e_var_subst_one.
+      inv H.
+      apply Map.find_2 in H3.
+      apply MapFacts.add_mapsto_iff in H3.
+      destruct H3 as [(?,?)|(?,?)]; subst.
+      + destruct (eq_x x5 x5); subst.
+        * constructor.
+          apply Map.find_1.
+          now apply Map.add_1.
+        * contradiction.
+ +
+      + destruct (eq_x x5 x0); subst.
+        * contradiction.
+        * destruct (eq_x x5 y0); subst.
+          -- simp fresh_vars_e in H0.
+             exfalso.
+             apply H0.
+             now left.
+          -- constructor.
+             apply Map.find_1.
+             apply Map.add_2; eauto.
+    - simp e_var_subst_one.
+      inv H1.
+      replace (e_list_subst_one (map (fun pat_ : e * T => let (e_, _) := pat_ in e_) e_T_list) x0 y0)
+        with (map (fun pat_:e*T => let (e, _) := pat_ in e) (map (fun '(e, T) => (e_var_subst_one e x0 y0, T)) e_T_list))
+        by (rewrite e_list_subst_map;
+            rewrite 2 map_map;
+            apply map_ext;
+            now intros (?,?)).
+      constructor.
+      + intros.
+        apply in_map_iff in H1.
+        destruct H1 as ((?e, ?T), (?, ?)).
+        inv H1.
+        apply in_map_iff in H2.
+        destruct H2 as ((?e_, ?T_), (?, ?)).
+        inv H1.
+        apply H.
+        * apply in_map_iff.
+          exists (e_0, T_).
+          split; eauto.
+        * simp fresh_vars_e in H0.
+          eapply e_list_fresh.
+          apply H0.
+          apply in_map_iff.
+          eexists (e_0, T_).
+          split; auto.
+        * apply H5.
+          apply in_map_iff.
+          exists (e_0, T_).
+          split; eauto.
+      + replace (map (fun pat_ : e * T => let (_, T_) := pat_ in T_) (map (fun '(e, T) => (e_var_subst_one e x0 y0, T)) e_T_list))
+          with(map (fun pat_ : e * T => let (_, T_) := pat_ in T_) e_T_list)
+          by (rewrite map_map; apply map_ext; now intros (?, ?)).
+ +
+        apply Map.find_2 in H7.
+        apply MapFacts.add_mapsto_iff in H7.
+        destruct H7 as [(?,?)|(?,?)].
+        * exfalso.
+          apply (vars_fs_distinct _ _ H1).
+        * apply Map.find_1.
+          apply Map.add_2; eauto.
+    - inv H.
+    - destruct H1; subst.
+      + apply H; eauto.
+      + apply H0; eauto.
+  }
+Qed.
+ +
+Lemma typ_e_G_Equal_equiv_imp :
+ forall G1 G2 e T, Map.Equal G1 G2 -> typ_e G1 e T -> typ_e G2 e T.
+Proof.
+intros G1 G2 e0 T0 Heq.
+intros Ht.
+induction Ht.
+- apply typ_bool.
+- apply typ_int.
+- apply typ_var.
+  symmetry in Heq.
+  rewrite <- H.
+  apply Heq.
+- apply typ_func_expr.
+  * intros.
+    apply H0; auto.
+  * rewrite <- H1.
+    symmetry in Heq.
+    apply Heq.
+Qed.
+ +
+Lemma typ_e_G_Equal_equiv :
+ forall G1 G2 e T, Map.Equal G1 G2 -> (typ_e G1 e T <-> typ_e G2 e T).
+Proof.
+intros G1 G2 e0 T0 Heq; split; apply typ_e_G_Equal_equiv_imp; auto.
+symmetry; assumption.
+Qed.
+ +
+Lemma fold_map4 {A X Y Z W: Type}: forall (f : _ -> _ -> A -> A) (l: list (X*Y*Z*W)) a0,
+    fold_right (fun '(z, w, _, _) a => f z w a) a0 l =
+      fold_right (fun '(z, w) a => f z w a) a0
+        (map (fun '(z, w, _, _) => (z, w)) l).
+Proof.
+  induction l; intros; eauto.
+  destruct a as [[[? ?] ?] ?].
+  simpl.
+  now rewrite IHl.
+Qed.
+ +
+Lemma fold_map5 {A X Y Z W: Type}: forall (f : _ -> _ -> A -> A) (l: list (X*Y*Z*W)) a0,
+    fold_right (fun '(z, _, _, w) a => f z w a) a0 l =
+      fold_right (fun '(z, w) a => f z w a) a0
+        (map (fun '(z, _, _, w) => (z, w)) l).
+Proof.
+  induction l; intros; eauto.
+  destruct a as [[[? ?] ?] ?].
+  simpl.
+  now rewrite IHl.
+Qed.
+ +
+Lemma subst_lemma: forall (T_x_t_y_list:list (T*x*t*x)) G0 e0 A,
+  typ_e (fold_right (fun '(Ai, x_, _, _) (G0 : G) => Map.add x_ (ctxv_T Ai) G0) G0 T_x_t_y_list) e0 A ->
+  NoDup (map (fun '(_, _, _, y) => y) T_x_t_y_list) ->
+  NoDup (map (fun '(_, x, _, _) => x) T_x_t_y_list) ->
+  fresh_vars_e (map (fun '(_, _, _, y) => y) T_x_t_y_list) e0 ->
+  disjoint (map (fun '(_, _, _, y) => y) T_x_t_y_list) (map (fun '(_, x, _, _) => x) T_x_t_y_list) ->
+  typ_e (fold_right (fun '(Ai, _, _, y) (G0 : G) => Map.add y (ctxv_T Ai) G0) G0 T_x_t_y_list)
+    (e_var_subst e0 (map (fun '(_, x_, _, y_) => (x_, y_)) T_x_t_y_list)) A.
+Proof.
+  intros.
+  generalize dependent G0.
+  generalize dependent e0.
+  induction T_x_t_y_list; intros.
+  - easy.
+  - destruct a as (((?T_,?x_),?t_),?y).
+    unfold e_var_subst.
+    simpl in *.
+    inv H0.
+    inv H1.
+    pose proof fresh_first_e _ _ _ H2.
+    pose proof fresh_monotone_e _ _ _ H2.
+    epose proof disjoint_monotone _ _ _ _ H3.
+    assert (~ In x_ (map (fun '(_, y) => y) (map (fun '(Ai, _, _, y1) => (Ai, y1)) T_x_t_y_list))).
+    { intro.
+      apply (H3 x_).
+      - right.
+        rewrite map_map in H9.
+        replace (map (fun '(_, _, _, y1) => y1) T_x_t_y_list)
+          with
+            (map (fun x0 : T * x * t * x => let '(_, y) := let '(Ai, _, _, y1) := x0 in (Ai, y1) in y) T_x_t_y_list).
+        apply H9.
+        apply map_eq.
+        now intros (((?, ?), ?), ?).
+      - now left.
+    }
+    assert (~ In x_ (map (fun '(_, y) => y) (map (fun '(Ai, x1, _, _) => (Ai, x1)) T_x_t_y_list))) as Hinx.
+    {
+      intro.
+      apply H5.
+      rewrite map_map in H10.
+      revert H10.
+      clear.
+      induction T_x_t_y_list; auto.
+      simpl.
+      intros Hx.
+      destruct Hx.
+      - destruct a, p, p.
+        subst.
+        left; reflexivity.
+      - right.
+        apply IHT_x_t_y_list.
+        assumption.
+    }
+    assert (Map.Equal
+     (Map.add x_ (ctxv_T T_) (fold_right (fun '(Ai, x_0, _, _) (G0 : G) => Map.add x_0 (ctxv_T Ai) G0) G0 T_x_t_y_list))
+     (fold_right (fun '(Ai, x_, _, _) (G0 : G) => Map.add x_ (ctxv_T Ai) G0) (Map.add x_ (ctxv_T T_) G0) T_x_t_y_list))
+      as HMEqualx.
+    {
+      epose proof fold_add_comm G0 x_ T_ (map (fun '(Ai, x1, _, _) => (Ai, x1)) T_x_t_y_list) Hinx.
+      rewrite 2 fold_map4.
+      assumption.
+    }
+    assert (Map.Equal
+     (fold_right (fun '(Ai, _, _, y) (G0 : G) => Map.add y (ctxv_T Ai) G0) (Map.add x_ (ctxv_T T_) G0) T_x_t_y_list)
+     (Map.add x_ (ctxv_T T_) (fold_right (fun '(Ai, _, _, y) (G0 : G) => Map.add y (ctxv_T Ai) G0) G0 T_x_t_y_list)))
+      as HMEqualy.
+    {
+      epose proof fold_add_comm G0 x_ T_ (map (fun '(Ai, _, _, y1) => (Ai, y1)) T_x_t_y_list) H9.
+      rewrite fold_map5.
+      set (fd := fold_right _ _ _).
+      rewrite fold_map5.
+      unfold fd; clear fd.
+      symmetry.
+      assumption.
+    }
+    apply (typ_e_G_Equal_equiv _ _ _ _ HMEqualx) in H.
+    pose proof IHT_x_t_y_list H7 H8 H4 _ H1 _ H as IH.
+    apply (typ_e_G_Equal_equiv _ _ _ _ HMEqualy) in IH.
+    assert (fresh_vars_e [y] (e_var_subst e0 (map (fun '(_, x_, _, y_) => (x_, y_)) T_x_t_y_list))).
+    {
+      assert (map snd (map (fun '(_, x_0, _, y_) => (x_0, y_)) T_x_t_y_list) = map (fun '(_, _, _, y) => y) T_x_t_y_list).
+      {
+        rewrite map_map.
+        apply map_eq.
+        now intros (((?, ?), ?), ?).
+      }
+      apply fresh_var_subst; try rewrite H10; auto.
+    }
+    now apply (subst_lemma_one _ _ _ _ _ _ IH H10).
+Qed.
+ +
+Lemma type_preservation : forall (Flist : list F) (G5 : G) (s5 : s),
+  G_vdash_s G5 s5 ->
+  Forall (typ_F G5) Flist ->
+  forall (e5 : e) (T5 : T) (s' : s) (e' : e),
+    typ_e G5 e5 T5 ->
+    red_e Flist s5 e5 s' e' ->
+    exists G', subG G5 G' /\ G_vdash_s G' s' /\ typ_e G' e' T5.
+Proof.
+  intros Fs G5 s0 s_well_typed F_well_typed.
+  intros e0 T0 s' e' e0_type reduction.
+  generalize dependent T0.
+  generalize dependent G5.
+  induction reduction.
+  - intros.
+    exists G5; splits.
+    1,2: easy.
+    inv e0_type.
+    destruct (s_well_typed x5 T0 H2) as (?, (?, ?)).
+    rewrite H in H0.
+    inv H0.
+    assumption.
+ +
+  - (* RED_FUN_EXPR *)
+    intros.
+    inv e0_type.
+    rewrite app_nil_r in H0.
+    pose proof split_corresponding_list e_T_list e_list e_5 e'_list H0
+      as (T_list & T_5 & T'_list & ? & ? & ?).
+    assert (In (e_5, T_5) (map (fun pat_ : e * T => let (e_0, T_0) := pat_ in (e_0, T_0)) e_T_list)). {
+      rewrite H3.
+      apply in_map_iff.
+      exists (e_5, T_5).
+      split; auto.
+      apply in_combine_split with (2:=eq_refl); auto.
+    }
+    pose proof H2 e_5 T_5 H5.
+    destruct (IHreduction G5 s_well_typed F_well_typed _ H6) as (G' & G'extends & G'consistent & ?).
+    exists G'.
+    splits.
+    1,2: easy.
+    replace (e_list ++ [e'] ++ e'_list ++ []) with
+        (map (fun (pat_:(e*T)) => match pat_ with (e_,T_) => e_ end )
+           (combine (e_list ++ [e'] ++ e'_list ++[]) (map (fun pat_ : e * T => let (_, T_) := pat_ in T_) e_T_list))).
+    apply typ_func_expr.
+      * intros.
+        rewrite pat2_id in H8.
+        rewrite H3 in H8.
+        rewrite combine_snd in H8.
+        rewrite combine_app in H8...
+        apply in_app_iff in H8.
+        destruct H8.
+        -- apply subG_type with (G1:=G5); auto.
+           apply H2.
+           apply in_map_iff.
+           exists (e_, T_).
+           split; auto.
+           rewrite H3.
+           rewrite combine_app; auto.
+           {
+             eapply in_app_iff.
+             now left.
+           }
+           rewrite 2 app_length.
+           simpl.
+           rewrite H1.
+           reflexivity.
+        -- rewrite combine_app in H8; auto.
+           apply in_app_iff in H8.
+           destruct H8.
+           ++ inv H8;[|contradiction].
+              inv H9.
+              apply H7.
+           ++ apply subG_type with (G1:=G5); auto.
+              apply H2.
+              apply in_map_iff.
+              exists (e_, T_).
+              split; auto.
+              rewrite H3.
+              rewrite combine_app; auto.
+              {
+                eapply in_app_iff.
+                right.
+                right.
+                rewrite app_nil_r in H8.
+                apply H8.
+              }
+           rewrite 2 app_length.
+           simpl.
+           rewrite H1.
+           reflexivity.
+           ++ rewrite app_nil_r.
+              apply H1.
+        -- auto.
+        -- rewrite app_nil_r.
+           simpl.
+           rewrite H1.
+           reflexivity.
+        -- rewrite 4 app_length.
+           simpl.
+           lia.
+      * rewrite combine_snd.
+        ++ apply G'extends.
+           apply H4.
+        ++ rewrite map_length.
+           rewrite H3.
+           rewrite 2 combine_app; auto.
+           ** repeat (rewrite app_length).
+              rewrite 3 combine_length.
+              simpl.
+              rewrite H, H1.
+              lia.
+           ** rewrite 2 app_length.
+              simpl.
+              rewrite H1.
+              reflexivity.
+      * rewrite app_nil_r.
+        rewrite combine_fst; auto.
+        rewrite map_length.
+        rewrite H3.
+        rewrite combine_length.
+        repeat (rewrite app_length).
+        rewrite H, H1.
+        simpl.
+        lia.
+ +
+  - (* RED_FUN_GROUND *)
+    intros.
+    set (fn_def:=(F_fn T_5 fn5 (map (fun '(T_, x_, _, _) => (T_, x_)) T_x_t_y_list) e5)).
+    pose proof utils.in_split F_list F'_list fn_def .
+    rewrite app_nil_r in *.
+    pose proof Forall_forall (typ_F G5) (F_list ++ [fn_def] ++ F'_list) as (? & _).
+    pose proof H2 F_well_typed fn_def H1 as fn_typed.
+    inv fn_typed.
+    inv e0_type.
+    destruct H as ((?, ?), ?).
+    assert (e_T_list = map (fun '(T_, _, t_, _) => (e_t t_, T_)) T_x_t_y_list).
+    {
+      (* inv H11. *)
+      eapply map_split'.
+      - replace (map fst e_T_list) with (map (fun pat_ : e * T => let (e_, _) := pat_ in e_) e_T_list) by easy.
+        rewrite H4.
+        apply map_ext.
+        easy.
+      - replace (map snd e_T_list) with (map (fun pat_ : e * T => let (_, T_) := pat_ in T_) e_T_list) by easy.
+        rewrite H7 in H11.
+        inv H11.
+        rewrite map_map.
+        apply map_ext.
+        intros.
+        now destruct a as (((?, ?), ?), ?).
+    }
+    exists (fold_right (fun '(T_, _, _, y) G0 => Map.add y (ctxv_T T_) G0) G5 T_x_t_y_list).
+    splits.
+    + eapply fresh_subG; eauto.
+    + rewrite <- fold_map.
+      apply fresh_consistent; eauto.
+      intros.
+      apply in_map_iff in H12.
+      destruct H12 as (Txty, (?, ?)).
+      apply H6.
+      inv H12.
+      rewrite map_map.
+      apply in_map_iff.
+      exists Txty.
+      split; auto.
+      destruct Txty as (((?, ?), ?), ?).
+      now inv H15.
+ +
+    + rewrite H7 in H11.
+      inv H11.
+      apply subst_lemma; auto.
+      * rewrite <- fold_map_reshuffle; eauto.
+      * rewrite map_xs in H10.
+        apply H10.
+Qed.
+End FunctionalMetatheory.
+
+
+ +
+ + + diff --git a/docs/coqdoc/ABS.utils.html b/docs/coqdoc/ABS.utils.html new file mode 100644 index 0000000..39bad59 --- /dev/null +++ b/docs/coqdoc/ABS.utils.html @@ -0,0 +1,775 @@ + + + + + + + + + + + + + +
+
+
+From Coq Require Import List
+  FSets.FMapFacts
+  Lia.
+ +
+Import ListNotations.
+ +
+Ltac splits := repeat split.
+Ltac inv H :=inversion H ; subst; clear H.
+Tactic Notation "intros*" := repeat intro.
+ +
+Ltac forward H :=
+  match type of H with
+  | ?x -> _ =>
+      let name := fresh in assert x as name; [| specialize (H name); clear name]
+  end.
+Tactic Notation "forward" ident(H) :=
+  forward H.
+ +
+Section ListLemmas.
+  Context {X Y: Type}.
+ +
+  Lemma map_eq: forall (l: list X) (f g: X -> Y),
+      (forall x, f x = g x) ->
+      map f l = map g l.
+  Proof.
+    induction l; intros; auto.
+    simpl.
+    rewrite (H a).
+    erewrite (IHl); eauto.
+  Qed.
+ +
+  Lemma map_split {Z W: Type}: forall (l: list (Z*X)) (l2: list (X*Y*Z*W)),
+      map fst l = map (fun '(_, _, z, _) => z) l2 ->
+      map snd l = map (fun '(x, _, _, _) => x) l2 ->
+      l = map (fun '(x, _, z,_) => (z, x)) l2.
+  Proof.
+    induction l; intros.
+    - inv H.
+      symmetry in H2.
+      apply map_eq_nil in H2.
+      now subst.
+    - destruct a.
+      destruct l2.
+      + inv H.
+      + destruct p as (((?, ?), ?), ?).
+        inv H.
+        inv H0.
+        rewrite (IHl l2); auto.
+  Qed.
+ +
+  Lemma map_xs {Z W: Type}: forall (T_x_t_y_list: list (X*Y*Z*W)),
+      map (fun pat_ : X * Y => let (_, x_) := pat_ in x_) (map (fun '(T_, x_, _, _) => (T_, x_)) T_x_t_y_list) =
+        map (fun '(_, x, _, _) => x) T_x_t_y_list.
+  Proof.
+    induction T_x_t_y_list.
+    - easy.
+    - destruct a as (((?, ?), ?), ?).
+      simpl.
+      now rewrite IHT_x_t_y_list.
+  Qed.
+ +
+  Lemma map_split' {Z Z' W: Type}: forall (l: list (Z'*X)) (l2: list (X*Y*Z*W)) (f: Z -> Z'),
+      map fst l = map (fun '(_, _, z, _) => f z) l2 ->
+      map snd l = map (fun '(x, _, _, _) => x) l2 ->
+      l = map (fun '(x, _, z,_) => (f z, x)) l2.
+  Proof.
+    induction l; intros.
+    - inv H.
+      symmetry in H2.
+      apply map_eq_nil in H2.
+      now subst.
+    - destruct a.
+      destruct l2.
+      + inv H.
+      + destruct p as (((?, ?), ?), ?).
+        inv H.
+        inv H0.
+        rewrite (IHl l2 f); auto.
+  Qed.
+ +
+  Lemma combine_fst: forall (l1: list X) (l2: list Y),
+      length l1 = length l2 ->
+      map (fun pat_: (X*Y) => let (e_, _) := pat_ in e_) (combine l1 l2) = l1.
+  Proof.
+    induction l1;intros.
+    - easy.
+    - destruct l2.
+      + inv H.
+      + inv H.
+        simpl.
+        rewrite (IHl1 l2 H1).
+        reflexivity.
+  Qed.
+ +
+  Lemma combine_nil: forall (l1: list X),
+      @combine X Y l1 [] = [].
+  Proof. destruct l1; easy. Qed.
+ +
+  Lemma combine_snd: forall (l1: list X) (l2: list Y),
+      length l1 = length l2 ->
+      map (fun pat_: (X*Y) => let (_, e_) := pat_ in e_) (combine l1 l2) = l2.
+  Proof.
+    intros.
+    generalize dependent l1.
+    induction l2; intros.
+    - rewrite combine_nil.
+      auto.
+    - destruct l1.
+      + inv H.
+      + inv H.
+        simpl.
+        rewrite (IHl2 l1 H1).
+        reflexivity.
+  Qed.
+ +
+  Lemma in_combine: forall (l1: list X) (l2: list Y) x y,
+      In (x,y) (combine l1 l2) -> In x l1 /\ In y l2.
+  Proof.
+    induction l1;intros.
+    - inv H.
+    - destruct l2.
+      + inv H.
+      + destruct H.
+        * inv H.
+          split; left; auto.
+        * destruct (IHl1 l2 x y H).
+          split; right; auto.
+  Qed.
+ +
+  Lemma combine_app: forall (l1 l2: list X) (l1' l2': list Y),
+      length l1 = length l1' ->
+      length l2 = length l2' ->
+      combine (l1 ++ l2) (l1' ++ l2') = combine l1 l1' ++ combine l2 l2'.
+  Proof.
+    induction l1; intros.
+    - inv H.
+      symmetry in H2.
+      apply length_zero_iff_nil in H2.
+      rewrite H2.
+      easy.
+    - destruct l1'.
+      + inv H.
+      + inv H.
+        simpl.
+        rewrite IHl1; easy.
+  Qed.
+ +
+  Lemma in_split: forall (l1 l2: list X) x,
+      In x (l1 ++ [x] ++ l2).
+  Proof.
+    induction l1; intros.
+    - left; auto.
+    - right.
+      apply IHl1.
+  Qed.
+ +
+  Lemma in_combine_split: forall (l: list (X*Y)) l1 l1' l2 l2' x y,
+      length l1 = length l2 ->
+      l = combine (l1 ++ [x] ++ l1') (l2 ++ [y] ++ l2') ->
+      In (x, y) l.
+  Proof.
+    induction l; intros; subst.
+    - destruct l1, l2; easy.
+    - destruct a.
+      destruct l1,l2.
+      + inv H0.
+        left;auto.
+      + inv H.
+      + inv H.
+      + inv H.
+        right.
+        inv H0.
+        eapply IHl.
+        apply H2.
+        reflexivity.
+  Qed.
+ +
+  Lemma in_fst_iff: forall (l : list (X*Y)) x,
+      (exists y, In (x, y) l) <-> In x (map (fun pat_ : X * Y => let (e_, _) := pat_ in e_) l).
+  Proof.
+    split; intros.
+    - induction l.
+      + destruct H.
+        inv H.
+      + destruct H as (?y & H).
+        inv H.
+        * left; auto.
+        * right.
+          apply IHl.
+          exists y; auto.
+    - induction l.
+      + inv H.
+      + destruct a as (?x & ?y).
+        inv H.
+        * exists y.
+          left;auto.
+        * destruct (IHl H0) as (?y & ?).
+          exists y0.
+          right.
+          apply H.
+  Qed.
+ +
+  Lemma pat2_id: forall l: list (X*Y),
+      map (fun pat_ : X*Y => let (x_, y_) := pat_ in (x_,y_)) l = l.
+  Proof.
+    induction l;auto.
+    destruct a; cbn.
+    rewrite IHl.
+    reflexivity.
+  Qed.
+ +
+  Lemma fold_map {A Z W: Type}: forall (f: W -> Z -> A -> A) (l: list (X*Y*Z*W)) a0,
+      fold_right (fun '(_, _, z, w) a => f w z a) a0 l =
+        fold_right (fun (xt : W * Z) a => f (fst xt) (snd xt) a) a0
+          (map (fun pat_ : X*Y*Z*W => let (p, y_) := pat_ in let (p0, t_) := p in let (_, _) := p0 in (y_, t_)) l).
+  Proof.
+    induction l; intros; eauto.
+    destruct a as [[[? ?] ?] ?].
+    simpl.
+    now rewrite IHl.
+  Qed.
+ +
+  Lemma fold_map1 {A Z W: Type}: forall (f: W -> X -> A -> A) (l: list (X*Y*Z*W)) a0,
+      fold_right (fun '(z, _, _, w) a => f w z a) a0 l =
+        fold_right (fun '(z, w) a => f z w a) a0
+          (map (fun '(z, _, _, w) => (w, z)) l).
+  Proof.
+    induction l; intros; eauto.
+    destruct a as [[[? ?] ?] ?].
+    simpl.
+    now rewrite IHl.
+  Qed.
+ +
+  Lemma fold_map2 {A Z W: Type}: forall (f: Y -> X -> A -> A) (l: list (X*Y*Z*W)) a0,
+      fold_right (fun '(z, w, _, _) a => f w z a) a0 l =
+        fold_right (fun '(z, w) a => f z w a) a0
+          (map (fun '(z, w, _, _) => (w, z)) l).
+  Proof.
+    induction l; intros; eauto.
+    destruct a as [[[? ?] ?] ?].
+    simpl.
+    now rewrite IHl.
+  Qed.
+ +
+  Lemma fold_map3 {A Z W: Type}: forall (f: W -> Y -> A -> A) (l: list (X*Y*Z*W)) a0,
+      fold_right (fun '(_, z, _, w) a => f w z a) a0 l =
+        fold_right (fun '(z, w) a => f z w a) a0
+          (map (fun '(_, z, _, w) => (w, z)) l).
+  Proof.
+    induction l; intros; eauto.
+    destruct a as [[[? ?] ?] ?].
+    simpl.
+    now rewrite IHl.
+  Qed.
+ +
+  Lemma combine_map: forall (l: list (X*Y)),
+      combine (map fst l) (map snd l) = l.
+  Proof.
+    induction l; auto.
+    destruct a; simpl.
+    rewrite IHl.
+    reflexivity.
+  Qed.
+ +
+  Lemma split_corresponding_list: forall (l: list (X*Y)) left_list mid right_list,
+      map (fun '(x, _) => x) l = left_list ++ [mid] ++ right_list ->
+      exists left_list' mid' right_list',
+        length left_list = length left_list'
+        /\ length right_list = length right_list'
+        /\ l = combine (left_list ++ [mid] ++ right_list) (left_list' ++ [mid'] ++ right_list').
+  Proof with auto.
+    induction l;intros.
+    - apply app_cons_not_nil in H.
+      contradiction.
+    - destruct a.
+      destruct left_list, right_list;
+        inv H.
+      + exists [], y, [].
+        apply map_eq_nil in H2.
+        rewrite H2.
+        splits...
+      + destruct (IHl [] x0 right_list H2)
+          as (left_list' & mid' & right_list' & ? & ? & ?).
+        exists [], y, (left_list' ++ [mid'] ++ right_list').
+        splits...
+        * inv H.
+          symmetry in H4.
+          apply length_zero_iff_nil in H4.
+          rewrite H4.
+          rewrite 2 app_nil_l.
+          rewrite map_length.
+          rewrite combine_length.
+          rewrite 2 app_length.
+          simpl.
+          rewrite H0.
+          lia.
+        * simpl.
+          rewrite H1.
+          rewrite combine_fst.
+          { easy. }
+          rewrite app_nil_l.
+          rewrite 3 app_length.
+          rewrite <- H.
+          rewrite H0.
+          easy.
+      + destruct (IHl left_list mid [] H2)
+          as (left_list' & mid' & right_list' & ? & ? & ?).
+        exists (y::left_list'), mid', right_list'.
+        splits...
+        * inv H0.
+          simpl.
+          rewrite H.
+          easy.
+        * rewrite H1.
+          easy.
+      + destruct (IHl left_list mid (x1::right_list) H2)
+          as (left_list' & mid' & right_list' & ? & ? & ?).
+        exists (y::left_list'), mid', (right_list').
+        splits...
+        * simpl.
+          rewrite H.
+          easy.
+        * rewrite H1.
+          easy.
+  Qed.
+ +
+  (* I thought this would be useful, but turns out we usually need the length *)
+  Corollary split_corresponding_list_no_length: forall (l: list (X*Y)) left_list mid right_list,
+      map (fun '(x, _) => x) l = left_list ++ [mid] ++ right_list ->
+      exists left_list' mid' right_list', l = combine (left_list ++ [mid] ++ right_list) (left_list' ++ [mid'] ++ right_list').
+  Proof.
+    intros.
+    destruct (split_corresponding_list _ _ _ _ H)
+      as (left_list' & mid' & right_list' & _ & _ & EQ).
+    exists left_list', mid', right_list'.
+    apply EQ.
+  Qed.
+ +
+End ListLemmas.
+ +
+Require Import FMapList OrderedTypeEx.
+From ABS Require Import abs_defs.
+From Equations Require Import Equations.
+ +
+Module MapFacts := FSets.FMapFacts.Facts Map.
+ +
+(* some slightly ad-hoc equalities *)
+Lemma subst_fst_commute: forall x0 x1 e_T_list,
+  map (fun x : e * T => let (e_, _) := let '(e_, T_) := x in (e_var_subst_one e_ x0 x1, T_) in e_) e_T_list =
+  (fix e_list_subst_one (es : list e) (x_ y_ : x) {struct es} : list e :=
+     match es with
+     | [] => fun _ _ : x => []
+     | e :: l => fun x_0 y_0 : x => e_var_subst_one e x_0 y_0 :: e_list_subst_one l x_0 y_0
+     end x_ y_) (map (fun pat_ : e * T => let (e_, _) := pat_ in e_) e_T_list) x0 x1.
+Proof.
+  induction e_T_list; auto.
+  destruct a.
+  simpl.
+  rewrite <- IHe_T_list.
+  reflexivity.
+Qed.
+ +
+Lemma subst_snd_commutes: forall x0 s e_T_list,
+  map (fun pat_ : e * T => let (_, T_) := pat_ in T_)
+    (map (fun pat_ : e * T => let (e_, T_) := pat_ in (e_var_subst_one e_ x0 s, T_)) e_T_list) =
+    map (fun pat_ : e * T => let (_, T_) := pat_ in T_) e_T_list.
+Proof.
+  intros.
+  induction e_T_list; auto.
+  destruct a;cbn in *.
+  now rewrite IHe_T_list.
+Qed.
+ +
+(* Characterizing substitution *)
+ +
+Lemma subst_term: forall t sub,
+  e_var_subst (e_t t) sub = (e_t t).
+Proof.
+  induction sub.
+  - trivial.
+  - destruct a.
+    simpl.
+    rewrite IHsub.
+    now simp e_var_subst_one.
+Qed.
+ +
+Definition replace_var (x0:x) (sub:list(x*x)) :=
+ fold_right (fun '(x_, y_) x0 => if (eq_x x0 x_) then y_ else x0) x0 sub.
+ +
+Lemma subst_var: forall x0 sub,
+  e_var_subst (e_var x0) sub = e_var (replace_var x0 sub).
+Proof.
+  induction sub.
+  - trivial.
+  - destruct a; simpl.
+    rewrite IHsub.
+    simp e_var_subst_one.
+    destruct (eq_x (replace_var x0 sub)); subst; eauto.
+Qed.
+ +
+Lemma e_list_subst_map: forall x0 y0 e_list,
+  e_list_subst_one e_list x0 y0 = map (fun e_ => e_var_subst_one e_ x0 y0) e_list.
+Proof.
+  induction e_list; [easy|];
+  destruct a;
+    simpl;
+    now rewrite IHe_list.
+Qed.
+ +
+Lemma subst_fn: forall fn0 sub e_list,
+  e_var_subst (e_fn_call fn0 e_list) sub = (e_fn_call fn0 (map (fun e' => e_var_subst e' sub) e_list)).
+Proof.
+  induction sub; intros.
+  - simpl.
+    now rewrite map_id.
+  - destruct a.
+    simpl.
+    rewrite IHsub.
+    simp e_var_subst_one.
+    now rewrite e_list_subst_map, map_map.
+Qed.
+ +
+Lemma e_list_subst: forall el x0 y0,
+  e_list_subst_one el x0 y0 = map (fun e => e_var_subst_one e x0 y0) el.
+Proof.
+  induction el; intros.
+  - trivial.
+  - simpl.
+    now rewrite IHel.
+Qed.
+ +
+Lemma e_list_fresh: forall e0 ys el,
+  fresh_vars_el ys el ->
+  In e0 el ->
+  fresh_vars_e ys e0.
+Proof.
+  induction el.
+  - easy.
+  - simpl; intros.
+    destruct H0; subst.
+    + now destruct H.
+    + destruct H.
+      apply IHel; eauto.
+Qed.
+ +
+(* Properties of fresh_e*)
+Lemma fresh_first_e: forall e0 y ys,
+  fresh_vars_e (y::ys) e0 -> fresh_vars_e [y] e0.
+Proof.
+  induction e0 using e_ott_ind
+    with (P_list_e := fun e_list =>
+                        forall e0 y ys,
+                        In e0 e_list ->
+                        fresh_vars_e (y::ys) e0 -> fresh_vars_e [y] e0);
+     intros; try easy.
+  - simp fresh_vars_e in *.
+    intro.
+    inv H0.
+    + apply H.
+    now left.
+    + inv H1.
+  - simp fresh_vars_e in *.
+    induction e_list.
+    + easy.
+    + inv H.
+      split.
+      * eapply IHe0; eauto.
+        now left.
+      * eapply IHe_list; eauto.
+        intros; eapply IHe0; eauto.
+        now right.
+  - inv H.
+    + eapply IHe0; eauto.
+    + eapply IHe1; eauto.
+Qed.
+ +
+Lemma fresh_monotone_e: forall e0 y ys,
+  fresh_vars_e (y::ys) e0 -> fresh_vars_e ys e0.
+Proof.
+  induction e0 using e_ott_ind
+    with (P_list_e := fun e_list =>
+                        forall e0 y ys,
+                        In e0 e_list ->
+                        fresh_vars_e (y::ys) e0 -> fresh_vars_e ys e0);
+     intros; try easy.
+  - simp fresh_vars_e in *.
+    intro.
+    apply H.
+    right.
+    assumption.
+  - simp fresh_vars_e in *.
+    induction e_list.
+    + easy.
+    + inv H.
+      split.
+      * eapply IHe0; eauto.
+        now left.
+      * apply IHe_list; eauto.
+        intros; eapply IHe0; eauto.
+        now right.
+  - inv H.
+    + eapply IHe0; eauto.
+    + eapply IHe1; eauto.
+Qed.
+ +
+Lemma fresh_first_el: forall el y ys,
+    fresh_vars_el (y::ys) el -> fresh_vars_el [y] el.
+Proof.
+  induction el; intros.
+  - now inv H.
+  - inv H.
+    split.
+    + now apply fresh_first_e in H0.
+    + eapply IHel; eauto.
+Qed.
+ +
+Lemma fresh_monotone_el: forall el y ys,
+  fresh_vars_el (y::ys) el -> fresh_vars_el ys el.
+Proof.
+  induction el; intros.
+  - now inv H.
+  - inv H.
+    split.
+    + now apply fresh_monotone_e in H0.
+    + eapply IHel; eauto.
+Qed.
+ +
+Lemma fresh_var_subst: forall e0 y sub,
+  fresh_vars_e (map snd sub) e0 ->
+  fresh_vars_e [y] e0 ->
+  ~ In y (map snd sub) ->
+  fresh_vars_e [y] (e_var_subst e0 sub).
+Proof.
+  intros.
+  generalize dependent y.
+  generalize dependent e0.
+  induction e0 using e_ott_ind
+    with
+    (P_list_e := fun e_list => forall y,
+                     fresh_vars_el (map snd sub) e_list ->
+                     fresh_vars_el [y] e_list ->
+                     ~ In y (map snd sub) ->
+                     fresh_vars_el [y] (map (fun e' => e_var_subst e' sub) e_list))
+  ; intros; auto.
+  - rewrite subst_term.
+    now simp e_var_subst_one.
+  - induction sub; auto.
+    destruct a as (?x_, ?y); simpl.
+    rewrite subst_var.
+    simp e_var_subst_one.
+    destruct (eq_x (replace_var x5 sub) x_).
+    + simp fresh_vars_e.
+      intro.
+      apply H1.
+      left.
+      now inv H2.
+    + rewrite <- subst_var.
+      apply Decidable.not_or in H1.
+      destruct H1.
+      apply fresh_monotone_e in H.
+      eapply IHsub; auto.
+  - rewrite subst_fn.
+    simp fresh_vars_e in *.
+  - destruct H,H0.
+    split.
+    + apply IHe0; auto.
+    + apply IHe1; eauto.
+Qed.
+ +
+Fact disjoint_empty{A:Type}: @disjoint A [] [].
+Proof. easy. Qed.
+ +
+Lemma disjoint_monotone {A:Type}: forall (l1 l2: list A) a1 a2,
+  disjoint (a1::l1) (a2::l2) -> disjoint l1 l2.
+Proof.
+  intros.
+  intros ? ? ?.
+    specialize (H a).
+    apply H; now right.
+Qed.
+ +
+Section MapLemmas.
+ +
+  Lemma map_neq_none_is_some{elt:Type}: forall m x,
+      Map.find x m <> None ->
+      exists y, Map.find (elt:=elt) x m = Some y.
+  Proof.
+    intros.
+    destruct (MapFacts.In_dec m x).
+    - inv i.
+      exists x0.
+      apply Map.find_1.
+      apply H0.
+    - apply MapFacts.not_find_in_iff in n.
+      rewrite n in H.
+      contradiction.
+  Qed.
+ +
+  Lemma fold_map_reshuffle: forall (l: list (T*x*t*x)) G0,
+      (fold_right (fun (ax : x * T) (G' : G) => Map.add (fst ax) (ctxv_T (snd ax)) G') G0
+         (map (fun pat_ : T * x => let (T_, x_) := pat_ in (x_, T_)) (map (fun '(T_0, x_, _, _) => (T_0, x_)) l)))
+      = (fold_right (fun '(T1, x_, _, _) (G' : G) => Map.add x_ (ctxv_T T1) G') G0 l).
+  Proof.
+    induction l;intros;auto.
+    destruct a as (((?T_ & ?x_) & ?t_) & ?y).
+    simpl.
+    rewrite IHl.
+    reflexivity.
+  Qed.
+ +
+  Lemma map_add_comm{E:Type}: forall m key key' elt elt',
+      key <> key' -> Map.Equal (Map.add (elt:=E) key elt (Map.add key' elt' m)) (Map.add key' elt' (Map.add key elt m)).
+  Proof.
+    intros.
+    intro.
+    destruct (Map.find y m) eqn:?.
+    - destruct (eq_x y key), (eq_x y key'); subst;
+        erewrite 2 Map.find_1; eauto;
+        (* this should just be "eauto with map", but I can't seem to import the hint database *)
+        try (apply Map.add_2; eauto; now apply Map.add_1);
+        try (now apply Map.add_1; eauto);
+        try (repeat (apply Map.add_2; eauto); now apply Map.find_2; eauto).
+    - destruct (eq_x y key), (eq_x y key'); subst.
+      + contradiction.
+      + erewrite 2 Map.find_1; eauto.
+        * apply Map.add_2; eauto.
+          now apply Map.add_1.
+        * now apply Map.add_1.
+      + erewrite 2 Map.find_1; eauto.
+        * now apply Map.add_1.
+        * apply Map.add_2; eauto.
+          now apply Map.add_1.
+ +
+      (* there has to be a cleaner way to do this case?? *)
+      + replace (Map.find (elt:=E) y (Map.add key elt (Map.add key' elt' m))) with (@None E).
+        replace (Map.find (elt:=E) y (Map.add key' elt' (Map.add key elt m))) with (@None E).
+        reflexivity.
+        * symmetry.
+          apply MapFacts.not_find_in_iff in Heqo.
+          apply MapFacts.not_find_in_iff.
+          intro.
+          apply Heqo.
+          inv H0.
+          exists x.
+          apply Map.add_3 with (x:= key) (e':=elt); auto.
+          apply Map.add_3 with (x:=key') (e':=elt'); auto.
+        * symmetry.
+          apply MapFacts.not_find_in_iff in Heqo.
+          apply MapFacts.not_find_in_iff.
+          intro.
+          inv H0.
+          apply Heqo.
+          exists x.
+          apply Map.add_3 with (x:=key') (e':=elt'); auto.
+          apply Map.add_3 with (x:= key) (e':=elt); auto.
+  Qed.
+ +
+  Lemma fold_not_in: forall G0 y (T_x_t_y_list: list (T*x*t*x)),
+      ~ Map.In y G0 ->
+      ~ In y (map (fun '(_, _, _, y') => y') T_x_t_y_list) ->
+      ~ Map.In y (fold_right (fun '(T_, _, _, y) (G0 : G) => Map.add y (ctxv_T T_) G0) G0 T_x_t_y_list).
+  Proof.
+    induction T_x_t_y_list as [ | (((?T_, ?x_), ?t_), ?y) T_x_t_y_list IH ]; [easy|].
+    simpl; intros ? ? ?.
+    destruct (eq_x y y0); subst.
+    - apply H0.
+      now left.
+    - apply MapFacts.add_in_iff in H1.
+      inv H1.
+      + apply H0.
+        now left.
+      + apply IH; eauto.
+  Qed.
+ +
+  Lemma fold_add_comm: forall G0 y T_ (upd: list (T*x)),
+      ~ In y (map (fun '(_, y) => y) upd) ->
+      Map.Equal (Map.add y (ctxv_T T_) (fold_right (fun '(T_, y) G0 => Map.add y (ctxv_T T_) G0) G0 upd))
+        (fold_right (fun '(T_, y) G0 => Map.add y (ctxv_T T_) G0) (Map.add y (ctxv_T T_) G0) upd).
+  Proof.
+    induction upd; intros.
+    - easy.
+    - destruct a; simpl in *.
+      apply Decidable.not_or in H.
+      destruct H.
+      now rewrite map_add_comm, IHupd; auto.
+  Qed.
+End MapLemmas.
+
+
+ +
+ + + diff --git a/docs/coqdoc/config.js b/docs/coqdoc/config.js new file mode 100644 index 0000000..1902b36 --- /dev/null +++ b/docs/coqdoc/config.js @@ -0,0 +1,79 @@ +var coqdocjs = coqdocjs || {}; + +coqdocjs.repl = { + "forall": "∀", + "exists": "∃", + "~": "¬", + "/\\": "∧", + "\\/": "∨", + "->": "→", + "<-": "←", + "<->": "↔", + "=>": "⇒", + "<>": "≠", + "<=": "≤", + ">=": "≥", + "el": "∈", + "nel": "∉", + "<<=": "⊆", + "|-": "⊢", + ">>": "»", + "<<": "⊆", + "++": "⧺", + "===": "≡", + "=/=": "≢", + "=~=": "≅", + "==>": "⟹", + "<==": "⟸", + "False": "⊥", + "True": "⊤", + ":=": "≔", + "-|": "⊣", + "*": "×", + "::": "∷", + "lhd": "⊲", + "rhd": "⊳", + "nat": "ℕ", + "alpha": "α", + "beta": "β", + "gamma": "γ", + "delta": "δ", + "epsilon": "ε", + "eta": "η", + "iota": "ι", + "kappa": "κ", + "lambda": "λ", + "mu": "μ", + "nu": "ν", + "omega": "ω", + "phi": "ϕ", + "pi": "π", + "psi": "ψ", + "rho": "ρ", + "sigma": "σ", + "tau": "τ", + "theta": "θ", + "xi": "ξ", + "zeta": "ζ", + "Delta": "Δ", + "Gamma": "Γ", + "Pi": "Π", + "Sigma": "Σ", + "Omega": "Ω", + "Xi": "Ξ" +}; + +coqdocjs.subscr = { + "0" : "₀", + "1" : "₁", + "2" : "₂", + "3" : "₃", + "4" : "₄", + "5" : "₅", + "6" : "₆", + "7" : "₇", + "8" : "₈", + "9" : "₉", +}; + +coqdocjs.replInText = ["==>","<=>", "=>", "->", "<-", ":="]; diff --git a/docs/coqdoc/coqdoc.css b/docs/coqdoc/coqdoc.css new file mode 100644 index 0000000..18dad89 --- /dev/null +++ b/docs/coqdoc/coqdoc.css @@ -0,0 +1,197 @@ +@import url(https://fonts.googleapis.com/css?family=Open+Sans:400,700); + +body{ + font-family: 'Open Sans', sans-serif; + font-size: 14px; + color: #2D2D2D +} + +a { + text-decoration: none; + border-radius: 3px; + padding-left: 3px; + padding-right: 3px; + margin-left: -3px; + margin-right: -3px; + color: inherit; + font-weight: bold; +} + +#main .code a, #main .inlinecode a, #toc a { + font-weight: inherit; +} + +a[href]:hover, [clickable]:hover{ + background-color: rgba(0,0,0,0.1); + cursor: pointer; +} + +h, h1, h2, h3, h4, h5 { + line-height: 1; + color: black; + text-rendering: optimizeLegibility; + font-weight: normal; + letter-spacing: 0.1em; + text-align: left; +} + +div + br { + display: none; +} + +div:empty{ display: none;} + +#main h1 { + font-size: 2em; +} + +#main h2 { + font-size: 1.667rem; +} + +#main h3 { + font-size: 1.333em; +} + +#main h4, #main h5, #main h6 { + font-size: 1em; +} + +#toc h2 { + padding-bottom: 0; +} + +#main .doc { + margin: 0; + text-align: justify; +} + +.inlinecode, .code, #main pre { + font-family: monospace; +} + +.code > br:first-child { + display: none; +} + +.doc + .code{ + margin-top:0.5em; +} + +.block{ + display: block; + margin-top: 5px; + margin-bottom: 5px; + padding: 10px; + text-align: center; +} + +.block img{ + margin: 15px; +} + +table.infrule { + border: 0px; + margin-left: 50px; + margin-top: 10px; + margin-bottom: 10px; +} + +td.infrule { + font-family: "Droid Sans Mono", "DejaVu Sans Mono", monospace; + text-align: center; + padding: 0; + line-height: 1; +} + +tr.infrulemiddle hr { + margin: 1px 0 1px 0; +} + +.infrulenamecol { + color: rgb(60%,60%,60%); + padding-left: 1em; + padding-bottom: 0.1em +} + +.id[type="constructor"], .id[type="projection"], .id[type="method"], +.id[title="constructor"], .id[title="projection"], .id[title="method"] { + color: #A30E16; +} + +.id[type="var"], .id[type="variable"], +.id[title="var"], .id[title="variable"] { + color: inherit; +} + +.id[type="definition"], .id[type="record"], .id[type="class"], .id[type="instance"], .id[type="inductive"], .id[type="library"], +.id[title="definition"], .id[title="record"], .id[title="class"], .id[title="instance"], .id[title="inductive"], .id[title="library"] { + color: #A6650F; +} + +.id[type="lemma"], +.id[title="lemma"]{ + color: #188B0C; +} + +.id[type="keyword"], .id[type="notation"], .id[type="abbreviation"], +.id[title="keyword"], .id[title="notation"], .id[title="abbreviation"]{ + color : #2874AE; +} + +.comment { + color: #808080; +} + +/* TOC */ + +#toc h2{ + letter-spacing: 0; + font-size: 1.333em; +} + +/* Index */ + +#index { + margin: 0; + padding: 0; + width: 100%; +} + +#index #frontispiece { + margin: 1em auto; + padding: 1em; + width: 60%; +} + +.booktitle { font-size : 140% } +.authors { font-size : 90%; + line-height: 115%; } +.moreauthors { font-size : 60% } + +#index #entrance { + text-align: center; +} + +#index #entrance .spacer { + margin: 0 30px 0 30px; +} + +ul.doclist { + margin-top: 0em; + margin-bottom: 0em; +} + +#toc > * { + clear: both; +} + +#toc > a { + display: block; + float: left; + margin-top: 1em; +} + +#toc a h2{ + display: inline; +} diff --git a/docs/coqdoc/coqdocjs.css b/docs/coqdoc/coqdocjs.css new file mode 100644 index 0000000..959b42e --- /dev/null +++ b/docs/coqdoc/coqdocjs.css @@ -0,0 +1,239 @@ +/* replace unicode */ + +.id[repl] .hidden { + font-size: 0; +} + +.id[repl]:before{ + content: attr(repl); +} + +/* folding proofs */ + +@keyframes show-proof { + 0% { + max-height: 1.2em; + opacity: 1; + } + 99% { + max-height: 1000em; + } + 100%{ + } +} + +@keyframes hide-proof { + from { + visibility: visible; + max-height: 10em; + opacity: 1; + } + to { + max-height: 1.2em; + } +} + +.proof { + cursor: pointer; +} +.proof * { + cursor: pointer; +} + +.proof { + overflow: hidden; + position: relative; + transition: opacity 1s; + display: inline-block; +} + +.proof[show="false"] { + max-height: 1.2em; + visibility: visible; + opacity: 0.3; +} + +.proof[show="false"][animate] { + animation-name: hide-proof; + animation-duration: 0.25s; +} + +.proof[show="true"] { + animation-name: show-proof; + animation-duration: 10s; +} + +.proof[show="true"]:before { + content: "\25BC"; /* arrow down */ +} +.proof[show="false"]:before { + content: "\25B6"; /* arrow right */ +} + +.proof[show="false"]:hover { + visibility: visible; + opacity: 0.5; +} + +#toggle-proofs[proof-status="no-proofs"] { + display: none; +} + +#toggle-proofs[proof-status="some-hidden"]:before { + content: "Show Proofs"; +} + +#toggle-proofs[proof-status="all-shown"]:before { + content: "Hide Proofs"; +} + + +/* page layout */ + +html, body { + height: 100%; + margin:0; + padding:0; +} + +@media only screen { /* no div with internal scrolling to allow printing of whole content */ + body { + display: flex; + flex-direction: column + } + + #content { + flex: 1; + overflow: auto; + display: flex; + flex-direction: column; + } +} + +#content:focus { + outline: none; /* prevent glow in OS X */ +} + +#main { + display: block; + padding: 16px; + padding-top: 1em; + padding-bottom: 2em; + margin-left: auto; + margin-right: auto; + max-width: 60em; + flex: 1 0 auto; +} + +.libtitle { + display: none; +} + +/* header */ +#header { + width:100%; + padding: 0; + margin: 0; + display: flex; + align-items: center; + background-color: rgb(21,57,105); + color: white; + font-weight: bold; + overflow: hidden; +} + + +.button { + cursor: pointer; +} + +#header * { + text-decoration: none; + vertical-align: middle; + margin-left: 15px; + margin-right: 15px; +} + +#header > .right, #header > .left { + display: flex; + flex: 1; + align-items: center; +} +#header > .left { + text-align: left; +} +#header > .right { + flex-direction: row-reverse; +} + +#header a, #header .button { + color: white; + box-sizing: border-box; +} + +#header a { + border-radius: 0; + padding: 0.2em; +} + +#header .button { + background-color: rgb(63, 103, 156); + border-radius: 1em; + padding-left: 0.5em; + padding-right: 0.5em; + margin: 0.2em; +} + +#header a:hover, #header .button:hover { + background-color: rgb(181, 213, 255); + color: black; +} + +#header h1 { padding: 0; + margin: 0;} + +/* footer */ +#footer { + text-align: center; + opacity: 0.5; + font-size: 75%; +} + +/* hyperlinks */ + +@keyframes highlight { + 50%{ + background-color: black; + } +} + +:target * { + animation-name: highlight; + animation-duration: 1s; +} + +a[name]:empty { + float: right; +} + +/* Proviola */ + +div.code { + width: auto; + float: none; +} + +div.goal { + position: fixed; + left: 75%; + width: 25%; + top: 3em; +} + +div.doc { + clear: both; +} + +span.command:hover { + background-color: inherit; +} diff --git a/docs/coqdoc/coqdocjs.js b/docs/coqdoc/coqdocjs.js new file mode 100644 index 0000000..727da8c --- /dev/null +++ b/docs/coqdoc/coqdocjs.js @@ -0,0 +1,197 @@ +var coqdocjs = coqdocjs || {}; +(function(){ + +function replace(s){ + var m; + if (m = s.match(/^(.+)'/)) { + return replace(m[1])+"'"; + } else if (m = s.match(/^([A-Za-z]+)_?(\d+)$/)) { + return replace(m[1])+m[2].replace(/\d/g, function(d){ + if (coqdocjs.subscr.hasOwnProperty(d)) { + return coqdocjs.subscr[d]; + } else { + return d; + } + }); + } else if (coqdocjs.repl.hasOwnProperty(s)){ + return coqdocjs.repl[s] + } else { + return s; + } +} + +function toArray(nl){ + return Array.prototype.slice.call(nl); +} + +function replInTextNodes() { + // Get all the nodes up front. + var nodes = Array.from(document.querySelectorAll(".code, .inlinecode")) + .flatMap(elem => Array.from(elem.childNodes) + .filter(e => e.nodeType == Node.TEXT_NODE) + ); + + // Create a replacement template node to clone from. + var replacementTemplate = document.createElement("span"); + replacementTemplate.setAttribute("class", "id"); + replacementTemplate.setAttribute("type", "keyword"); + + // Do the replacements. + coqdocjs.replInText.forEach(function(toReplace){ + var replacement = replacementTemplate.cloneNode(true); + replacement.appendChild(document.createTextNode(toReplace)); + + nodes.forEach(node => { + var fragments = node.textContent.split(toReplace); + node.textContent = fragments[fragments.length-1]; + for (var k = 0; k < fragments.length - 1; ++k) { + fragments[k] && node.parentNode.insertBefore(document.createTextNode(fragments[k]),node); + node.parentNode.insertBefore(replacement.cloneNode(true), node); + } + }); + }); +} + +function replNodes() { + toArray(document.getElementsByClassName("id")).forEach(function(node){ + if (["var", "variable", "keyword", "notation", "definition", "inductive"].indexOf(node.getAttribute("type"))>=0){ + var text = node.textContent; + var replText = replace(text); + if(text != replText) { + node.setAttribute("repl", replText); + node.setAttribute("title", text); + var hidden = document.createElement("span"); + hidden.setAttribute("class", "hidden"); + while (node.firstChild) { + hidden.appendChild(node.firstChild); + } + node.appendChild(hidden); + } + } + }); +} + +function isVernacStart(l, t){ + t = t.trim(); + for(var s of l){ + if (t == s || t.startsWith(s+" ") || t.startsWith(s+".")){ + return true; + } + } + return false; +} + +function isProofStart(n){ + return isVernacStart(["Proof"], n.textContent) || + (isVernacStart(["Next"], n.textContent) && isVernacStart(["Obligation"], n.nextSibling.nextSibling.textContent)); +} + +function isProofEnd(s){ + return isVernacStart(["Qed", "Admitted", "Defined", "Abort"], s); +} + +function proofStatus(){ + var proofs = toArray(document.getElementsByClassName("proof")); + if(proofs.length) { + for(var proof of proofs) { + if (proof.getAttribute("show") === "false") { + return "some-hidden"; + } + } + return "all-shown"; + } + else { + return "no-proofs"; + } +} + +function updateView(){ + document.getElementById("toggle-proofs").setAttribute("proof-status", proofStatus()); +} + +function foldProofs() { + var hasCommands = true; + var nodes = document.getElementsByClassName("command"); + if(nodes.length == 0) { + hasCommands = false; + console.log("no command tags found") + nodes = document.getElementsByClassName("id"); + } + toArray(nodes).forEach(function(node){ + if(isProofStart(node)) { + var proof = document.createElement("span"); + proof.setAttribute("class", "proof"); + + node.parentNode.insertBefore(proof, node); + if(proof.previousSibling.nodeType === Node.TEXT_NODE) + proof.appendChild(proof.previousSibling); + while(node && !isProofEnd(node.textContent)) { + proof.appendChild(node); + node = proof.nextSibling; + } + if (proof.nextSibling) proof.appendChild(proof.nextSibling); // the Qed + if (!hasCommands && proof.nextSibling) proof.appendChild(proof.nextSibling); // the dot after the Qed + + proof.addEventListener("click", function(proof){return function(e){ + if (e.target.parentNode.tagName.toLowerCase() === "a") + return; + proof.setAttribute("show", proof.getAttribute("show") === "true" ? "false" : "true"); + proof.setAttribute("animate", ""); + updateView(); + };}(proof)); + proof.setAttribute("show", "false"); + } + }); +} + +function toggleProofs(){ + var someProofsHidden = proofStatus() === "some-hidden"; + toArray(document.getElementsByClassName("proof")).forEach(function(proof){ + proof.setAttribute("show", someProofsHidden); + proof.setAttribute("animate", ""); + }); + updateView(); +} + +function repairDom(){ + // pull whitespace out of command + toArray(document.getElementsByClassName("command")).forEach(function(node){ + while(node.firstChild && node.firstChild.textContent.trim() == ""){ + console.log("try move"); + node.parentNode.insertBefore(node.firstChild, node); + } + }); + toArray(document.getElementsByClassName("id")).forEach(function(node){ + node.setAttribute("type", node.getAttribute("title")); + }); + toArray(document.getElementsByClassName("idref")).forEach(function(ref){ + toArray(ref.childNodes).forEach(function(child){ + if (["var", "variable"].indexOf(child.getAttribute("type")) > -1) + ref.removeAttribute("href"); + }); + }); + +} + +function fixTitle(){ + var url = "/" + window.location.pathname; + var basename = url.substring(url.lastIndexOf('/')+1, url.lastIndexOf('.')); + if (basename === "toc") {document.title = "Table of Contents";} + else if (basename === "indexpage") {document.title = "Index";} + else {document.title = basename;} +} + +function postprocess(){ + repairDom(); + replInTextNodes() + replNodes(); + foldProofs(); + document.getElementById("toggle-proofs").addEventListener("click", toggleProofs); + updateView(); +} + +fixTitle(); +document.addEventListener('DOMContentLoaded', postprocess); + +coqdocjs.toggleProofs = toggleProofs; +})(); diff --git a/docs/coqdoc/indexpage.html b/docs/coqdoc/indexpage.html new file mode 100644 index 0000000..6a6182f --- /dev/null +++ b/docs/coqdoc/indexpage.html @@ -0,0 +1,1101 @@ + + + + + + + + + + + + + +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(170 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(64 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(54 entries)
+
+

Global Index

+

A

+abs_functional_metatheory [library]
+abs_defs [library]
+abs_examples [library]
+

B

+b [definition, in ABS.abs_defs]
+

C

+combine_map [lemma, in ABS.utils]
+combine_app [lemma, in ABS.utils]
+combine_snd [lemma, in ABS.utils]
+combine_nil [lemma, in ABS.utils]
+combine_fst [lemma, in ABS.utils]
+Ctx [definition, in ABS.abs_examples]
+ctxv [inductive, in ABS.abs_defs]
+ctxv_sind [definition, in ABS.abs_defs]
+ctxv_rec [definition, in ABS.abs_defs]
+ctxv_ind [definition, in ABS.abs_defs]
+ctxv_rect [definition, in ABS.abs_defs]
+ctxv_sig [constructor, in ABS.abs_defs]
+ctxv_T [constructor, in ABS.abs_defs]
+

D

+disjoint [definition, in ABS.abs_defs]
+disjoint_monotone [lemma, in ABS.utils]
+disjoint_empty [lemma, in ABS.utils]
+

E

+e [inductive, in ABS.abs_defs]
+eq_z [lemma, in ABS.abs_defs]
+eq_b [lemma, in ABS.abs_defs]
+eq_x [lemma, in ABS.abs_defs]
+eq_fn [lemma, in ABS.abs_defs]
+eq_i [lemma, in ABS.abs_defs]
+e_list_fresh [lemma, in ABS.utils]
+e_list_subst [lemma, in ABS.utils]
+e_list_subst_map [lemma, in ABS.utils]
+e_call_const_5_T_int [lemma, in ABS.abs_examples]
+e_call_const_typ_F [lemma, in ABS.abs_examples]
+e_const_5_T_int [lemma, in ABS.abs_examples]
+e_call_const_5 [definition, in ABS.abs_examples]
+e_const_5 [definition, in ABS.abs_examples]
+e_var_subst [definition, in ABS.abs_defs]
+e_list_subst_one [definition, in ABS.abs_defs]
+e_var_subst_one [definition, in ABS.abs_defs]
+e_ott_ind [definition, in ABS.abs_defs]
+e_rect.H_list_e_cons [variable, in ABS.abs_defs]
+e_rect.H_list_e_nil [variable, in ABS.abs_defs]
+e_rect.H_e_fn_call [variable, in ABS.abs_defs]
+e_rect.H_e_var [variable, in ABS.abs_defs]
+e_rect.H_e_t [variable, in ABS.abs_defs]
+e_rect.P_list_e [variable, in ABS.abs_defs]
+e_rect.P_e [variable, in ABS.abs_defs]
+e_rect [section, in ABS.abs_defs]
+e_sind [definition, in ABS.abs_defs]
+e_rec [definition, in ABS.abs_defs]
+e_ind [definition, in ABS.abs_defs]
+e_rect [definition, in ABS.abs_defs]
+e_fn_call [constructor, in ABS.abs_defs]
+e_var [constructor, in ABS.abs_defs]
+e_t [constructor, in ABS.abs_defs]
+

F

+F [inductive, in ABS.abs_defs]
+fn [definition, in ABS.abs_defs]
+fold_add_comm [lemma, in ABS.utils]
+fold_not_in [lemma, in ABS.utils]
+fold_map_reshuffle [lemma, in ABS.utils]
+fold_map3 [lemma, in ABS.utils]
+fold_map2 [lemma, in ABS.utils]
+fold_map1 [lemma, in ABS.utils]
+fold_map [lemma, in ABS.utils]
+fold_map5 [lemma, in ABS.abs_functional_metatheory]
+fold_map4 [lemma, in ABS.abs_functional_metatheory]
+fresh_var_subst [lemma, in ABS.utils]
+fresh_monotone_el [lemma, in ABS.utils]
+fresh_first_el [lemma, in ABS.utils]
+fresh_monotone_e [lemma, in ABS.utils]
+fresh_first_e [lemma, in ABS.utils]
+fresh_consistent [lemma, in ABS.abs_functional_metatheory]
+fresh_subG [lemma, in ABS.abs_functional_metatheory]
+fresh_vars [definition, in ABS.abs_defs]
+fresh_vars_s [definition, in ABS.abs_defs]
+fresh_vars_el [definition, in ABS.abs_defs]
+fresh_vars_e [definition, in ABS.abs_defs]
+FunctionalMetatheory [section, in ABS.abs_functional_metatheory]
+FunctionalMetatheory.vars_well_typed [variable, in ABS.abs_functional_metatheory]
+FunctionalMetatheory.vars_fs_distinct [variable, in ABS.abs_functional_metatheory]
+_ G⊢ _ [notation, in ABS.abs_functional_metatheory]
+_ ⊆ _ [notation, in ABS.abs_functional_metatheory]
+_ F⊢ _ [notation, in ABS.abs_functional_metatheory]
+_ ⊢ _ : _ [notation, in ABS.abs_functional_metatheory]
+func_const_5 [definition, in ABS.abs_examples]
+F_sind [definition, in ABS.abs_defs]
+F_rec [definition, in ABS.abs_defs]
+F_ind [definition, in ABS.abs_defs]
+F_rect [definition, in ABS.abs_defs]
+F_fn [constructor, in ABS.abs_defs]
+

G

+G [definition, in ABS.abs_defs]
+G_vdash_s [definition, in ABS.abs_functional_metatheory]
+

I

+i [definition, in ABS.abs_defs]
+in_fst_iff [lemma, in ABS.utils]
+in_combine_split [lemma, in ABS.utils]
+in_split [lemma, in ABS.utils]
+in_combine [lemma, in ABS.utils]
+

L

+ListLemmas [section, in ABS.utils]
+

M

+Map [module, in ABS.abs_defs]
+MapFacts [module, in ABS.utils]
+MapFacts [module, in ABS.abs_functional_metatheory]
+MapLemmas [section, in ABS.utils]
+map_add_comm [lemma, in ABS.utils]
+map_neq_none_is_some [lemma, in ABS.utils]
+map_split' [lemma, in ABS.utils]
+map_xs [lemma, in ABS.utils]
+map_split [lemma, in ABS.utils]
+map_eq [lemma, in ABS.utils]
+

P

+pat2_id [lemma, in ABS.utils]
+

R

+red_e_sind [definition, in ABS.abs_defs]
+red_e_ind [definition, in ABS.abs_defs]
+red_fun_ground [constructor, in ABS.abs_defs]
+red_fun_exp [constructor, in ABS.abs_defs]
+red_var [constructor, in ABS.abs_defs]
+red_e [inductive, in ABS.abs_defs]
+replace_var [definition, in ABS.utils]
+

S

+s [definition, in ABS.abs_defs]
+same_type_sub [lemma, in ABS.abs_functional_metatheory]
+sig [inductive, in ABS.abs_defs]
+sig_sind [definition, in ABS.abs_defs]
+sig_rec [definition, in ABS.abs_defs]
+sig_ind [definition, in ABS.abs_defs]
+sig_rect [definition, in ABS.abs_defs]
+sig_sig [constructor, in ABS.abs_defs]
+split_corresponding_list_no_length [lemma, in ABS.utils]
+split_corresponding_list [lemma, in ABS.utils]
+subG [definition, in ABS.abs_functional_metatheory]
+subG_type [lemma, in ABS.abs_functional_metatheory]
+subG_consistent [lemma, in ABS.abs_functional_metatheory]
+subG_add_2 [lemma, in ABS.abs_functional_metatheory]
+subG_add [lemma, in ABS.abs_functional_metatheory]
+subG_refl [lemma, in ABS.abs_functional_metatheory]
+subst_fn [lemma, in ABS.utils]
+subst_var [lemma, in ABS.utils]
+subst_term [lemma, in ABS.utils]
+subst_snd_commutes [lemma, in ABS.utils]
+subst_fst_commute [lemma, in ABS.utils]
+subst_lemma [lemma, in ABS.abs_functional_metatheory]
+subst_lemma_one [lemma, in ABS.abs_functional_metatheory]
+subst_lemma_one_alt [lemma, in ABS.abs_functional_metatheory]
+

T

+t [inductive, in ABS.abs_defs]
+T [inductive, in ABS.abs_defs]
+type_preservation [lemma, in ABS.abs_functional_metatheory]
+typ_e_G_Equal_equiv [lemma, in ABS.abs_functional_metatheory]
+typ_e_G_Equal_equiv_imp [lemma, in ABS.abs_functional_metatheory]
+typ_F_sind [definition, in ABS.abs_defs]
+typ_F_ind [definition, in ABS.abs_defs]
+typ_func_decl [constructor, in ABS.abs_defs]
+typ_F [inductive, in ABS.abs_defs]
+typ_e_sind [definition, in ABS.abs_defs]
+typ_e_ind [definition, in ABS.abs_defs]
+typ_func_expr [constructor, in ABS.abs_defs]
+typ_var [constructor, in ABS.abs_defs]
+typ_int [constructor, in ABS.abs_defs]
+typ_bool [constructor, in ABS.abs_defs]
+typ_e [inductive, in ABS.abs_defs]
+t_sind [definition, in ABS.abs_defs]
+t_rec [definition, in ABS.abs_defs]
+t_ind [definition, in ABS.abs_defs]
+t_rect [definition, in ABS.abs_defs]
+t_int [constructor, in ABS.abs_defs]
+t_b [constructor, in ABS.abs_defs]
+T_sind [definition, in ABS.abs_defs]
+T_rec [definition, in ABS.abs_defs]
+T_ind [definition, in ABS.abs_defs]
+T_rect [definition, in ABS.abs_defs]
+T_int [constructor, in ABS.abs_defs]
+T_bool [constructor, in ABS.abs_defs]
+

U

+utils [library]
+

W

+well_formed [definition, in ABS.abs_defs]
+

X

+x [definition, in ABS.abs_defs]
+

Z

+z [definition, in ABS.abs_defs]
+

+

Notation Index

+

F

+_ G⊢ _ [in ABS.abs_functional_metatheory]
+_ ⊆ _ [in ABS.abs_functional_metatheory]
+_ F⊢ _ [in ABS.abs_functional_metatheory]
+_ ⊢ _ : _ [in ABS.abs_functional_metatheory]
+

+

Module Index

+

M

+Map [in ABS.abs_defs]
+MapFacts [in ABS.utils]
+MapFacts [in ABS.abs_functional_metatheory]
+

+

Variable Index

+

E

+e_rect.H_list_e_cons [in ABS.abs_defs]
+e_rect.H_list_e_nil [in ABS.abs_defs]
+e_rect.H_e_fn_call [in ABS.abs_defs]
+e_rect.H_e_var [in ABS.abs_defs]
+e_rect.H_e_t [in ABS.abs_defs]
+e_rect.P_list_e [in ABS.abs_defs]
+e_rect.P_e [in ABS.abs_defs]
+

F

+FunctionalMetatheory.vars_well_typed [in ABS.abs_functional_metatheory]
+FunctionalMetatheory.vars_fs_distinct [in ABS.abs_functional_metatheory]
+

+

Library Index

+

A

+abs_functional_metatheory
+abs_defs
+abs_examples
+

U

+utils
+

+

Lemma Index

+

C

+combine_map [in ABS.utils]
+combine_app [in ABS.utils]
+combine_snd [in ABS.utils]
+combine_nil [in ABS.utils]
+combine_fst [in ABS.utils]
+

D

+disjoint_monotone [in ABS.utils]
+disjoint_empty [in ABS.utils]
+

E

+eq_z [in ABS.abs_defs]
+eq_b [in ABS.abs_defs]
+eq_x [in ABS.abs_defs]
+eq_fn [in ABS.abs_defs]
+eq_i [in ABS.abs_defs]
+e_list_fresh [in ABS.utils]
+e_list_subst [in ABS.utils]
+e_list_subst_map [in ABS.utils]
+e_call_const_5_T_int [in ABS.abs_examples]
+e_call_const_typ_F [in ABS.abs_examples]
+e_const_5_T_int [in ABS.abs_examples]
+

F

+fold_add_comm [in ABS.utils]
+fold_not_in [in ABS.utils]
+fold_map_reshuffle [in ABS.utils]
+fold_map3 [in ABS.utils]
+fold_map2 [in ABS.utils]
+fold_map1 [in ABS.utils]
+fold_map [in ABS.utils]
+fold_map5 [in ABS.abs_functional_metatheory]
+fold_map4 [in ABS.abs_functional_metatheory]
+fresh_var_subst [in ABS.utils]
+fresh_monotone_el [in ABS.utils]
+fresh_first_el [in ABS.utils]
+fresh_monotone_e [in ABS.utils]
+fresh_first_e [in ABS.utils]
+fresh_consistent [in ABS.abs_functional_metatheory]
+fresh_subG [in ABS.abs_functional_metatheory]
+

I

+in_fst_iff [in ABS.utils]
+in_combine_split [in ABS.utils]
+in_split [in ABS.utils]
+in_combine [in ABS.utils]
+

M

+map_add_comm [in ABS.utils]
+map_neq_none_is_some [in ABS.utils]
+map_split' [in ABS.utils]
+map_xs [in ABS.utils]
+map_split [in ABS.utils]
+map_eq [in ABS.utils]
+

P

+pat2_id [in ABS.utils]
+

S

+same_type_sub [in ABS.abs_functional_metatheory]
+split_corresponding_list_no_length [in ABS.utils]
+split_corresponding_list [in ABS.utils]
+subG_type [in ABS.abs_functional_metatheory]
+subG_consistent [in ABS.abs_functional_metatheory]
+subG_add_2 [in ABS.abs_functional_metatheory]
+subG_add [in ABS.abs_functional_metatheory]
+subG_refl [in ABS.abs_functional_metatheory]
+subst_fn [in ABS.utils]
+subst_var [in ABS.utils]
+subst_term [in ABS.utils]
+subst_snd_commutes [in ABS.utils]
+subst_fst_commute [in ABS.utils]
+subst_lemma [in ABS.abs_functional_metatheory]
+subst_lemma_one [in ABS.abs_functional_metatheory]
+subst_lemma_one_alt [in ABS.abs_functional_metatheory]
+

T

+type_preservation [in ABS.abs_functional_metatheory]
+typ_e_G_Equal_equiv [in ABS.abs_functional_metatheory]
+typ_e_G_Equal_equiv_imp [in ABS.abs_functional_metatheory]
+

+

Constructor Index

+

C

+ctxv_sig [in ABS.abs_defs]
+ctxv_T [in ABS.abs_defs]
+

E

+e_fn_call [in ABS.abs_defs]
+e_var [in ABS.abs_defs]
+e_t [in ABS.abs_defs]
+

F

+F_fn [in ABS.abs_defs]
+

R

+red_fun_ground [in ABS.abs_defs]
+red_fun_exp [in ABS.abs_defs]
+red_var [in ABS.abs_defs]
+

S

+sig_sig [in ABS.abs_defs]
+

T

+typ_func_decl [in ABS.abs_defs]
+typ_func_expr [in ABS.abs_defs]
+typ_var [in ABS.abs_defs]
+typ_int [in ABS.abs_defs]
+typ_bool [in ABS.abs_defs]
+t_int [in ABS.abs_defs]
+t_b [in ABS.abs_defs]
+T_int [in ABS.abs_defs]
+T_bool [in ABS.abs_defs]
+

+

Inductive Index

+

C

+ctxv [in ABS.abs_defs]
+

E

+e [in ABS.abs_defs]
+

F

+F [in ABS.abs_defs]
+

R

+red_e [in ABS.abs_defs]
+

S

+sig [in ABS.abs_defs]
+

T

+t [in ABS.abs_defs]
+T [in ABS.abs_defs]
+typ_F [in ABS.abs_defs]
+typ_e [in ABS.abs_defs]
+

+

Section Index

+

E

+e_rect [in ABS.abs_defs]
+

F

+FunctionalMetatheory [in ABS.abs_functional_metatheory]
+

L

+ListLemmas [in ABS.utils]
+

M

+MapLemmas [in ABS.utils]
+

+

Definition Index

+

B

+b [in ABS.abs_defs]
+

C

+Ctx [in ABS.abs_examples]
+ctxv_sind [in ABS.abs_defs]
+ctxv_rec [in ABS.abs_defs]
+ctxv_ind [in ABS.abs_defs]
+ctxv_rect [in ABS.abs_defs]
+

D

+disjoint [in ABS.abs_defs]
+

E

+e_call_const_5 [in ABS.abs_examples]
+e_const_5 [in ABS.abs_examples]
+e_var_subst [in ABS.abs_defs]
+e_list_subst_one [in ABS.abs_defs]
+e_var_subst_one [in ABS.abs_defs]
+e_ott_ind [in ABS.abs_defs]
+e_sind [in ABS.abs_defs]
+e_rec [in ABS.abs_defs]
+e_ind [in ABS.abs_defs]
+e_rect [in ABS.abs_defs]
+

F

+fn [in ABS.abs_defs]
+fresh_vars [in ABS.abs_defs]
+fresh_vars_s [in ABS.abs_defs]
+fresh_vars_el [in ABS.abs_defs]
+fresh_vars_e [in ABS.abs_defs]
+func_const_5 [in ABS.abs_examples]
+F_sind [in ABS.abs_defs]
+F_rec [in ABS.abs_defs]
+F_ind [in ABS.abs_defs]
+F_rect [in ABS.abs_defs]
+

G

+G [in ABS.abs_defs]
+G_vdash_s [in ABS.abs_functional_metatheory]
+

I

+i [in ABS.abs_defs]
+

R

+red_e_sind [in ABS.abs_defs]
+red_e_ind [in ABS.abs_defs]
+replace_var [in ABS.utils]
+

S

+s [in ABS.abs_defs]
+sig_sind [in ABS.abs_defs]
+sig_rec [in ABS.abs_defs]
+sig_ind [in ABS.abs_defs]
+sig_rect [in ABS.abs_defs]
+subG [in ABS.abs_functional_metatheory]
+

T

+typ_F_sind [in ABS.abs_defs]
+typ_F_ind [in ABS.abs_defs]
+typ_e_sind [in ABS.abs_defs]
+typ_e_ind [in ABS.abs_defs]
+t_sind [in ABS.abs_defs]
+t_rec [in ABS.abs_defs]
+t_ind [in ABS.abs_defs]
+t_rect [in ABS.abs_defs]
+T_sind [in ABS.abs_defs]
+T_rec [in ABS.abs_defs]
+T_ind [in ABS.abs_defs]
+T_rect [in ABS.abs_defs]
+

W

+well_formed [in ABS.abs_defs]
+

X

+x [in ABS.abs_defs]
+

Z

+z [in ABS.abs_defs]
+

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(170 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(64 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(4 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(54 entries)
+
+ +
+ + + diff --git a/docs/coqdoc/toc.html b/docs/coqdoc/toc.html new file mode 100644 index 0000000..3510c38 --- /dev/null +++ b/docs/coqdoc/toc.html @@ -0,0 +1,48 @@ + + + + + + + + + + + + + +
+
+ +
+ +
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ABS Metatheory

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About

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Welcome to the ABS Metatheory project website!

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Formal metatheory in Coq for the ABS language.

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Documentation

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Help and contact

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