-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathfoformulas_vec.v
221 lines (179 loc) · 7.22 KB
/
foformulas_vec.v
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
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
(* First-Order Formulas *)
(* arity check based on vectors *)
#[local] Set Warnings "-stdlib-vector".
From Coq Require Vector.
From Quantifiers Require Export foterms_vec.
Set Implicit Arguments.
Notation vec := Vector.t.
(** * First-Order Formulas *)
Section Formulas.
Context {vatom : DecType} {tatom : Type} {tarity : tatom -> nat}.
Notation term := (@term vatom tatom tarity).
Arguments evar _ _ _ {T}.
Notation evar := (evar vatom tatom tarity).
Notation "r ;; s" := (fecomp r s) (at level 20, format "r ;; s").
Notation closed t := (tvars t = nil).
Notation "⇑" := fup.
Notation "⇑[ u ] r" := (felift u r) (at level 25, format "⇑[ u ] r").
Hint Rewrite (@tesubs_evar vatom tatom) : term_db.
Hint Rewrite (@tesubs_comp vatom tatom) : term_db.
Hint Rewrite (@tsubs_tsubs_eq vatom tatom) : term_db.
Hint Rewrite (@tsubs_tsubs vatom tatom)
using try (intuition; fail);
(try apply closed_notvars); intuition; fail : term_db.
Hint Rewrite (@tvars_tesubs_closed vatom tatom) using intuition; fail : term_db.
Hint Rewrite (@tvars_tsubs_closed vatom tatom) using intuition; fail : term_db.
Hint Rewrite (@notin_tsubs vatom tatom)
using try easy;
(try (intuition; fail));
(try apply closed_notvars); intuition; fail : term_db.
Hint Rewrite (@notin_tsubs_bivar vatom tatom)
using try easy;
(try (intuition; fail));
(try apply closed_notvars); intuition; fail : term_db.
Hint Rewrite (@tsubs_tesubs vatom tatom) using try (intuition; fail) : term_db.
Hint Resolve tesubs_ext : term_db.
Hint Resolve closed_notvars : term_db.
Context {fatom : Type}. (* relation symbols for [formula] *)
Context {farity : fatom -> nat}. (* arity of relation symbols *)
(* Generic sets of connectives *)
Context {NCon : Type}. (* nullary connectives *)
Context {UCon : Type}. (* unary connectives *)
Context {BCon : Type}. (* binary connectives *)
Context {QCon : Type}. (* quantifiers *)
(** formulas *)
(** first-order formulas *)
Inductive formula T :=
| fvar R : vec (term T) (farity R) -> formula T
| fnul : NCon -> formula T
| funa : UCon -> formula T -> formula T
| fbin : BCon -> formula T -> formula T -> formula T
| fqtf : QCon -> vatom -> formula T -> formula T.
Arguments fvar {T} R _.
Arguments fnul {T} _.
(* Nullary connectives in [NCon] and [fnul] are mostly redundant with [fvar f nil] *)
Ltac formula_induction A :=
(try intros until A) ;
let XX := fresh "X" in
let xx := fresh "x" in
let ncon := fresh "ncon" in
let ucon := fresh "ucon" in
let bcon := fresh "bcon" in
let qcon := fresh "qcon" in
let A1 := fresh A in
let A2 := fresh A in
let ll := fresh "l" in
let lll := fresh "l" in
let tt := fresh "t" in
let kk := fresh "k" in
let IHll := fresh "IHl" in
induction A as [ XX ll | ncon | ucon A ? | bcon A1 ? A2 ? | qcon xx A ]; simpl; intros;
[ rewrite ? flat_map_concat_map;
try (apply (f_equal (fvar _)));
try (induction ll as [ | tt kk lll IHll ]; simpl; intuition;
rewrite IHll; f_equal; intuition)
| try ((try f_equal); intuition; fail)
| try (apply (f_equal (funa _))); intuition
| try (apply (f_equal2 (fbin _))); intuition
| (try apply (f_equal (fqtf _ _))); repeat case_analysis; try (intuition; fail);
try (now rcauto) ];
try (now rcauto).
(** * Size of [formula] *)
Fixpoint fsize T (A : formula T) :=
match A with
| fvar _ _ => 1
| fnul _ => 1
| funa _ B => S (fsize B)
| fbin _ B C => S (fsize B + fsize C)
| fqtf _ _ B => S (fsize B)
end.
(** * Substitution of eigen variables in [formula] *)
(** substitutes the [term] [r n] for index [n] in [formula] [A] *)
(* capture is not avoided *)
Fixpoint esubs T1 T2 (r : T1 -> term T2) (A : formula T1) :=
match A with
| fvar X l => fvar X (Vector.map (tesubs r) l)
| fnul ncon => fnul ncon
| funa ucon B => funa ucon (esubs r B)
| fbin bcon B C => fbin bcon (esubs r B) (esubs r C)
| fqtf qcon x B => fqtf qcon x (esubs r B)
end.
Notation "A ⟦ r ⟧" := (esubs r A) (at level 8, left associativity, format "A ⟦ r ⟧").
Lemma fsize_esubs T1 T2 : forall (r : T1 -> term T2) A, fsize A⟦r⟧ = fsize A.
Proof. formula_induction A. Qed.
Hint Rewrite fsize_esubs : term_db.
Lemma esubs_evar T : forall (A : formula T),
A⟦evar⟧ = A.
Proof. formula_induction A. Qed.
Hint Rewrite esubs_evar : term_db.
Lemma esubs_comp T1 T2 T3 (r : T1 -> term T2) (s : T2 -> term T3) :
forall A, A⟦r⟧⟦s⟧ = A⟦r ;; s⟧.
Proof. formula_induction A. Qed.
Hint Rewrite esubs_comp : term_db.
(* the result of substitution depends extensionnaly on the substituting function *)
Lemma esubs_ext T1 T2 (r1 r2 : T1 -> term T2) : r1 ~ r2 -> forall A, A⟦r1⟧ = A⟦r2⟧.
Proof. formula_induction A. Qed.
(** * Formula substitution *)
(** substitutes [term] [u] for variable [x] in [formula] [A] *)
(* capture is not avoided *)
Fixpoint subs T x u (A : formula T) :=
match A with
| fvar X l => fvar X (Vector.map (tsubs x u) l)
| fnul ncon => fnul ncon
| funa ucon B => funa ucon (subs x u B)
| fbin bcon B C => fbin bcon (subs x u B) (subs x u C)
| fqtf qcon y B => fqtf qcon y (if eq_dt_dec y x then B else subs x u B)
end.
Notation "A [ u // x ]" := (subs x u A) (at level 8, format "A [ u // x ]").
Lemma fsize_subs T : forall u x (A : formula T), fsize A[u//x] = fsize A.
Proof. formula_induction A. Qed.
Hint Rewrite fsize_subs : term_db.
(** * Additional results with variable eigen type *)
Lemma subs_esubs T1 T2 (r : T1 -> term T2): forall x u A, (forall n, closed (r n)) ->
A[u//x]⟦r⟧ = A⟦r⟧[tesubs r u//x].
Proof. formula_induction A; simpl in H; rcauto. Qed.
Hint Rewrite subs_esubs using intuition; fail : term_db.
(* We restrict to [formula nat] *)
(** * Eigen variables *)
Notation "A ↑" := (A⟦⇑⟧) (at level 8, format "A ↑").
Notation "v ⇓" := (fesubs v) (at level 18, format "v ⇓").
Lemma esubs_fup v A : A↑⟦v⇓⟧ = A.
Proof. rcauto. Qed.
Hint Rewrite esubs_fup : term_db.
Lemma felift_esubs u r A : A↑⟦⇑[u]r⟧ = A⟦r⟧↑.
Proof. rcauto. Qed.
Hint Rewrite felift_esubs : term_db.
End Formulas.
(* Some sets of connectives *)
Inductive Nocon := .
Inductive Icon := imp_con.
Inductive Qcon := frl_con | exs_con.
Notation imp := (fbin imp_con).
Notation frl := (fqtf frl_con).
Notation exs := (fqtf exs_con).
Ltac formula_induction A :=
(try intros until A) ;
let XX := fresh "X" in
let xx := fresh "x" in
let ncon := fresh "ncon" in
let ucon := fresh "ucon" in
let bcon := fresh "bcon" in
let qcon := fresh "qcon" in
let A1 := fresh A in
let A2 := fresh A in
let ll := fresh "l" in
let lll := fresh "l" in
let tt := fresh "t" in
let kk := fresh "k" in
let IHll := fresh "IHl" in
induction A as [ XX ll | ncon | ucon A ? | bcon A1 ? A2 ? | qcon xx A ]; simpl; intros;
[ rewrite ? flat_map_concat_map;
try (apply (f_equal (fvar _)));
try (induction ll as [ | tt kk lll IHll ]; simpl; intuition;
rewrite IHll; f_equal; intuition)
| try ((try f_equal); intuition; fail)
| try (apply (f_equal (funa _))); intuition
| try (apply (f_equal2 (fbin _))); intuition
| (try apply (f_equal (fqtf _ _))); repeat case_analysis; try (intuition; fail);
try (now rcauto) ];
try (now rcauto).