1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
|
Require Import Coq.Program.Program Coq.Program.Equality.
Goal forall (H: forall n m : nat, n = m -> n = 0) x, x = tt.
intros.
dependent destruction x.
reflexivity.
Qed.
Parameter A : Set.
Inductive vector : nat -> Type := vnil : vector 0 | vcons : A -> forall {n}, vector n -> vector (S n).
Goal forall n, forall v : vector (S n), vector n.
Proof.
intros n H.
dependent destruction H.
assumption.
Qed.
Require Import ProofIrrelevance.
Goal forall n, forall v : vector (S n), exists v' : vector n, exists a : A, v = vcons a v'.
Proof.
intros n v.
dependent destruction v.
exists v ; exists a.
reflexivity.
Qed.
(* Extraction Unnamed_thm. *)
Inductive type : Type :=
| base : type
| arrow : type -> type -> type.
Notation " t --> t' " := (arrow t t') (at level 20, t' at next level).
Inductive ctx : Type :=
| empty : ctx
| snoc : ctx -> type -> ctx.
Bind Scope context_scope with ctx.
Delimit Scope context_scope with ctx.
Arguments snoc _%_context_scope.
Notation " Γ , τ " := (snoc Γ τ) (at level 25, τ at next level, left associativity) : context_scope.
Fixpoint conc (Δ Γ : ctx) : ctx :=
match Δ with
| empty => Γ
| snoc Δ' x => snoc (conc Δ' Γ) x
end.
Notation " Γ ; Δ " := (conc Δ Γ) (at level 25, left associativity) : context_scope.
Reserved Notation " Γ ⊢ τ " (at level 30, no associativity).
Generalizable All Variables.
Inductive term : ctx -> type -> Type :=
| ax : `(Γ, τ ⊢ τ)
| weak : `{Γ ⊢ τ -> Γ, τ' ⊢ τ}
| abs : `{Γ, τ ⊢ τ' -> Γ ⊢ τ --> τ'}
| app : `{Γ ⊢ τ --> τ' -> Γ ⊢ τ -> Γ ⊢ τ'}
where " Γ ⊢ τ " := (term Γ τ) : type_scope.
#[export] Hint Constructors term : lambda.
Local Open Scope context_scope.
Ltac eqns := subst ; reverse ; simplify_dep_elim ; simplify_IH_hyps.
Lemma weakening : forall Γ Δ τ, Γ ; Δ ⊢ τ ->
forall τ', Γ , τ' ; Δ ⊢ τ.
Proof with simpl in * ; eqns ; eauto with lambda.
intros Γ Δ τ H.
dependent induction H.
destruct Δ as [|Δ τ'']...
destruct Δ as [|Δ τ'']...
destruct Δ as [|Δ τ'']...
apply abs.
specialize (IHterm Γ (Δ, τ'', τ))...
intro. eapply app...
Defined.
Lemma weakening_ctx : forall Γ Δ τ, Γ ; Δ ⊢ τ ->
forall Δ', Γ ; Δ' ; Δ ⊢ τ.
Proof with simpl in * ; eqns ; eauto with lambda.
intros Γ Δ τ H.
dependent induction H.
destruct Δ as [|Δ τ'']...
induction Δ'...
destruct Δ as [|Δ τ'']...
induction Δ'...
destruct Δ as [|Δ τ'']...
apply abs.
specialize (IHterm Γ (empty, τ))...
apply abs.
specialize (IHterm Γ (Δ, τ'', τ))...
intro. eapply app...
Defined.
Lemma exchange : forall Γ Δ α β τ, term (Γ, α, β ; Δ) τ -> term (Γ, β, α ; Δ) τ.
Proof with simpl in * ; eqns ; eauto.
intros until 1.
dependent induction H.
destruct Δ ; eqns.
apply weak ; apply ax.
apply ax.
destruct Δ...
pose (weakening Γ (empty, α))...
apply weak...
apply abs...
specialize (IHterm Γ (Δ, τ))...
eapply app...
Defined.
(** Example by Andrew Kenedy, uses simplification of the first component of dependent pairs. *)
Set Implicit Arguments.
Inductive Ty :=
| Nat : Ty
| Prod : Ty -> Ty -> Ty.
Inductive Exp : Ty -> Type :=
| Const : nat -> Exp Nat
| Pair : forall t1 t2, Exp t1 -> Exp t2 -> Exp (Prod t1 t2)
| Fst : forall t1 t2, Exp (Prod t1 t2) -> Exp t1.
Inductive Ev : forall t, Exp t -> Exp t -> Prop :=
| EvConst : forall n, Ev (Const n) (Const n)
| EvPair : forall t1 t2 (e1:Exp t1) (e2:Exp t2) e1' e2',
Ev e1 e1' -> Ev e2 e2' -> Ev (Pair e1 e2) (Pair e1' e2')
| EvFst : forall t1 t2 (e:Exp (Prod t1 t2)) e1 e2,
Ev e (Pair e1 e2) ->
Ev (Fst e) e1.
Lemma EvFst_inversion : forall t1 t2 (e:Exp (Prod t1 t2)) e1, Ev (Fst e) e1 -> exists e2, Ev e (Pair e1 e2).
intros t1 t2 e e1 ev. dependent destruction ev. exists e2 ; assumption.
Qed.
|
