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annotate src/coq/Semantics.v @ 1903:de3afd3ecf07

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author Adam Chlipala <adam@chlipala.net>
date Thu, 21 Nov 2013 16:36:22 -0500
parents 705cb41ac7d0
children
rev   line source
adam@1618 1 (* Copyright (c) 2009, 2011, Adam Chlipala
adamc@615 2 * All rights reserved.
adamc@615 3 *
adamc@615 4 * Redistribution and use in source and binary forms, with or without
adamc@615 5 * modification, are permitted provided that the following conditions are met:
adamc@615 6 *
adamc@615 7 * - Redistributions of source code must retain the above copyright notice,
adamc@615 8 * this list of conditions and the following disclaimer.
adamc@615 9 * - Redistributions in binary form must reproduce the above copyright notice,
adamc@615 10 * this list of conditions and the following disclaimer in the documentation
adamc@615 11 * and/or other materials provided with the distribution.
adamc@615 12 * - The names of contributors may not be used to endorse or promote products
adamc@615 13 * derived from this software without specific prior written permission.
adamc@615 14 *
adamc@615 15 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
adamc@615 16 * AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
adamc@615 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
adamc@615 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
adamc@615 19 * LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
adamc@615 20 * CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
adamc@615 21 * SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
adamc@615 22 * INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
adamc@615 23 * CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
adamc@615 24 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
adamc@615 25 * POSSIBILITY OF SUCH DAMAGE.
adamc@615 26 *)
adamc@615 27
adam@1618 28 Require Import Eqdep_dec.
adamc@618 29
adamc@616 30 Require Import Axioms.
adamc@615 31 Require Import Syntax.
adamc@615 32
adamc@615 33 Set Implicit Arguments.
adamc@615 34
adamc@615 35
adamc@617 36 Definition row (A : Type) : Type := name -> option A.
adamc@615 37
adamc@617 38 Definition record (r : row Set) := forall n, match r n with
adamc@617 39 | None => unit
adamc@617 40 | Some T => T
adamc@617 41 end.
adamc@615 42
adamc@615 43 Fixpoint kDen (k : kind) : Type :=
adamc@615 44 match k with
adamc@615 45 | KType => Set
adamc@615 46 | KName => name
adamc@615 47 | KArrow k1 k2 => kDen k1 -> kDen k2
adamc@615 48 | KRecord k1 => row (kDen k1)
adamc@615 49 end.
adamc@615 50
adamc@635 51 Definition disjoint T (r1 r2 : row T) :=
adamc@635 52 forall n, match r1 n, r2 n with
adamc@635 53 | Some _, Some _ => False
adamc@635 54 | _, _ => True
adamc@635 55 end.
adamc@620 56
adam@1618 57 Fixpoint cDen k (c : con kDen k) : kDen k :=
adam@1618 58 match c with
adamc@615 59 | CVar _ x => x
adamc@615 60 | Arrow c1 c2 => cDen c1 -> cDen c2
adamc@615 61 | Poly _ c1 => forall x, cDen (c1 x)
adamc@615 62 | CAbs _ _ c1 => fun x => cDen (c1 x)
adamc@615 63 | CApp _ _ c1 c2 => (cDen c1) (cDen c2)
adamc@615 64 | Name n => n
adamc@615 65 | TRecord c1 => record (cDen c1)
adamc@617 66 | CEmpty _ => fun _ => None
adamc@617 67 | CSingle _ c1 c2 => fun n => if name_eq_dec n (cDen c1) then Some (cDen c2) else None
adamc@617 68 | CConcat _ c1 c2 => fun n => match (cDen c1) n with
adamc@617 69 | None => (cDen c2) n
adamc@617 70 | v => v
adamc@617 71 end
adamc@617 72 | CMap k1 k2 => fun f r n => match r n with
adamc@617 73 | None => None
adamc@617 74 | Some T => Some (f T)
adamc@617 75 end
adamc@635 76 | TGuarded _ c1 c2 t => disjoint (cDen c1) (cDen c2) -> cDen t
adamc@615 77 end.
adamc@616 78
adamc@616 79 Theorem subs_correct : forall k1 (c1 : con kDen k1) k2 (c2 : _ -> con kDen k2) c2',
adamc@616 80 subs c1 c2 c2'
adamc@616 81 -> cDen (c2 (cDen c1)) = cDen c2'.
adamc@616 82 induction 1; simpl; intuition; try (apply ext_eq_forallS || apply ext_eq);
adamc@616 83 repeat match goal with
adamc@616 84 | [ H : _ |- _ ] => rewrite H
adamc@616 85 end; intuition.
adamc@616 86 Qed.
adamc@616 87
adamc@616 88 Definition dvar k (c1 c2 : con kDen (KRecord k)) :=
adamc@616 89 disjoint (cDen c1) (cDen c2).
adamc@616 90
adamc@616 91 Scheme deq_mut := Minimality for deq Sort Prop
adamc@616 92 with disj_mut := Minimality for disj Sort Prop.
adamc@616 93
adamc@618 94 Ltac deq_disj_correct scm :=
adamc@635 95 let t := repeat progress (simpl; intuition; subst) in
adamc@617 96
adamc@618 97 let rec use_disjoint' notDone E :=
adamc@617 98 match goal with
adamc@617 99 | [ H : disjoint _ _ |- _ ] =>
adamc@617 100 notDone H; generalize (H E); use_disjoint'
adamc@617 101 ltac:(fun H' =>
adamc@617 102 match H' with
adamc@617 103 | H => fail 1
adamc@617 104 | _ => notDone H'
adamc@617 105 end) E
adamc@617 106 | _ => idtac
adamc@618 107 end in
adamc@618 108 let use_disjoint := use_disjoint' ltac:(fun _ => idtac) in
adamc@616 109
adamc@618 110 apply (scm _ dvar
adamc@616 111 (fun k (c1 c2 : con kDen k) =>
adamc@616 112 cDen c1 = cDen c2)
adamc@616 113 (fun k (c1 c2 : con kDen (KRecord k)) =>
adamc@617 114 disjoint (cDen c1) (cDen c2))); t;
adamc@617 115 repeat ((unfold row; apply ext_eq)
adamc@617 116 || (match goal with
adamc@620 117 | [ H : _ |- _ ] => rewrite H; []
adamc@617 118 | [ H : subs _ _ _ |- _ ] => rewrite <- (subs_correct H)
adamc@617 119 end); t);
adamc@617 120 unfold disjoint; t;
adamc@616 121 repeat (match goal with
adamc@617 122 | [ |- context[match cDen ?C ?E with Some _ => _ | None => _ end] ] =>
adamc@617 123 use_disjoint E; destruct (cDen C E)
adamc@617 124 | [ |- context[if name_eq_dec ?N1 ?N2 then _ else _] ] =>
adamc@617 125 use_disjoint N1; use_disjoint N2; destruct (name_eq_dec N1 N2)
adamc@617 126 | [ _ : context[match cDen ?C ?E with Some _ => _ | None => _ end] |- _ ] =>
adamc@617 127 use_disjoint E; destruct (cDen C E)
adamc@620 128 | [ |- context[if ?E then _ else _] ] => destruct E
adamc@617 129 end; t).
adamc@618 130
adamc@620 131 Hint Unfold dvar.
adamc@620 132
adamc@618 133 Theorem deq_correct : forall k (c1 c2 : con kDen k),
adamc@618 134 deq dvar c1 c2
adamc@618 135 -> cDen c1 = cDen c2.
adamc@618 136 deq_disj_correct deq_mut.
adamc@616 137 Qed.
adamc@618 138
adamc@618 139 Theorem disj_correct : forall k (c1 c2 : con kDen (KRecord k)),
adamc@618 140 disj dvar c1 c2
adamc@618 141 -> disjoint (cDen c1) (cDen c2).
adamc@618 142 deq_disj_correct disj_mut.
adamc@618 143 Qed.
adamc@618 144
adamc@618 145 Definition tDen (t : con kDen KType) : Set := cDen t.
adamc@618 146
adamc@618 147 Theorem name_eq_dec_refl : forall n, name_eq_dec n n = left _ (refl_equal n).
adamc@618 148 intros; destruct (name_eq_dec n n); intuition; [
adamc@618 149 match goal with
adam@1618 150 | [ e : _ = _ |- _ ] => rewrite (UIP_dec name_eq_dec e (refl_equal _)); reflexivity
adamc@618 151 end
adamc@618 152 | elimtype False; tauto
adamc@618 153 ].
adamc@618 154 Qed.
adamc@618 155
adamc@618 156 Theorem cut_disjoint : forall n1 v r,
adamc@618 157 disjoint (fun n => if name_eq_dec n n1 then Some v else None) r
adamc@618 158 -> unit = match r n1 with
adamc@618 159 | Some T => T
adamc@618 160 | None => unit
adamc@618 161 end.
adamc@618 162 intros;
adamc@618 163 match goal with
adamc@618 164 | [ H : disjoint _ _ |- _ ] => generalize (H n1)
adamc@618 165 end; rewrite name_eq_dec_refl;
adamc@618 166 destruct (r n1); intuition.
adamc@618 167 Qed.
adamc@618 168
adamc@618 169 Implicit Arguments cut_disjoint [v r].
adamc@618 170
adam@1618 171 Fixpoint eDen t (e : exp dvar tDen t) : tDen t :=
adamc@618 172 match e in exp _ _ t return tDen t with
adamc@618 173 | Var _ x => x
adamc@618 174 | App _ _ e1 e2 => (eDen e1) (eDen e2)
adamc@618 175 | Abs _ _ e1 => fun x => eDen (e1 x)
adamc@618 176 | ECApp _ c _ _ e1 Hsub => match subs_correct Hsub in _ = T return T with
adamc@618 177 | refl_equal => (eDen e1) (cDen c)
adamc@618 178 end
adamc@618 179 | ECAbs _ _ e1 => fun X => eDen (e1 X)
adamc@618 180 | Cast _ _ Heq e1 => match deq_correct Heq in _ = T return T with
adamc@618 181 | refl_equal => eDen e1
adamc@618 182 end
adamc@618 183 | Empty => fun _ => tt
adam@1618 184 | Single c c' e1 => fun n => if name_eq_dec n (cDen c) as B
adam@1618 185 return (match (match (if B then _ else _) with Some _ => _ | None => _ end)
adamc@618 186 with Some _ => _ | None => unit end)
adamc@618 187 then eDen e1 else tt
adamc@618 188 | Proj c _ _ e1 =>
adamc@618 189 match name_eq_dec_refl (cDen c) in _ = B
adamc@618 190 return (match (match (if B then _ else _) with
adam@1618 191 | Some _ => _
adamc@618 192 | None => _ end)
adamc@618 193 return Set
adamc@618 194 with Some _ => _ | None => _ end) with
adamc@618 195 | refl_equal => (eDen e1) (cDen c)
adamc@618 196 end
adamc@618 197 | Cut c _ c' Hdisj e1 => fun n =>
adam@1618 198 match name_eq_dec n (cDen c) as B return (match (match (if B then Some _ else None) with Some _ => _ | None => (cDen c') n end)
adamc@618 199 with Some T => T | None => unit end
adamc@618 200 -> match (cDen c') n with
adamc@618 201 | None => unit
adamc@618 202 | Some T => T
adamc@618 203 end) with
adamc@618 204 | left Heq => fun _ =>
adamc@618 205 match sym_eq Heq in _ = n' return match cDen c' n' return Set with Some _ => _ | None => _ end with
adamc@618 206 | refl_equal =>
adamc@618 207 match cut_disjoint _ (disj_correct Hdisj) in _ = T return T with
adamc@618 208 | refl_equal => tt
adamc@618 209 end
adamc@618 210 end
adamc@618 211 | right _ => fun x => x
adamc@618 212 end ((eDen e1) n)
adamc@618 213
adamc@619 214 | Concat c1 c2 e1 e2 => fun n =>
adamc@619 215 match (cDen c1) n as D return match D with
adamc@619 216 | None => unit
adamc@619 217 | Some T => T
adamc@619 218 end
adamc@619 219 -> match (match D with
adamc@619 220 | None => (cDen c2) n
adam@1618 221 | Some v => Some v
adamc@619 222 end) with
adamc@619 223 | None => unit
adamc@619 224 | Some T => T
adamc@619 225 end with
adamc@619 226 | None => fun _ => (eDen e2) n
adamc@619 227 | _ => fun x => x
adamc@619 228 end ((eDen e1) n)
adamc@619 229
adamc@635 230 | Guarded _ _ _ _ e1 => fun pf => eDen (e1 pf)
adamc@635 231 | GuardedApp _ _ _ _ e1 Hdisj => (eDen e1) (disj_correct Hdisj)
adamc@618 232 end.