1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 definition cast ≝ λA:CProp.λa:A.a.
17 notation < "[ a ] \sup H" with precedence 19 for @{ 'ass $a $H }.
18 interpretation "assumption" 'ass a H = (cast a H).
20 inductive Imply (A,B:CProp) : CProp ≝
21 Imply_intro: (A → B) → Imply A B.
23 notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
24 interpretation "Imply" 'Imply a b = (Imply a b).
26 notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i) " with precedence 19 for @{ 'Imply_intro $ab (λ${ident H}:$p.$b) }.
27 interpretation "Imply_intro" 'Imply_intro ab \eta.b = (cast ab (Imply_intro _ _ b)).
29 definition Imply_elim ≝ λA,B.λf:Imply A B.λa:A.match f with [ Imply_intro g ⇒ g a].
31 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19 for @{ 'Imply_elim $ab $a $b }.
32 interpretation "Imply_elim" 'Imply_elim ab a b = (cast b (Imply_elim _ _ ab a)).
34 inductive And (A,B:CProp) : CProp ≝
35 And_intro: A → B → And A B.
37 interpretation "constructive and" 'and x y = (And x y).
39 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)" with precedence 19 for @{ 'And_intro $a $b $ab }.
40 interpretation "And_intro" 'And_intro a b ab = (cast ab (And_intro _ _ a b)).
42 definition And_elim_l ≝
43 λA,B.λc:A∧B.match c with [ And_intro a b ⇒ a ].
45 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub\e\sup\l)" with precedence 19 for @{ 'And_elim_l $ab $a }.
46 interpretation "And_elim_l" 'And_elim_l ab a = (cast a (And_elim_l _ _ ab)).
48 definition And_elim_r ≝
49 λA,B.λc:A∧B.match c with [ And_intro a b ⇒ b ].
51 notation < "\infrule hbox(\emsp ab \emsp) b (∧\sub\e\sup\r)" with precedence 19 for @{ 'And_elim_r $ab $b }.
52 interpretation "And_elim_r" 'And_elim_r ab b = (cast b (And_elim_r _ _ ab)).
54 inductive Or (A,B:CProp) : CProp ≝
55 | Or_intro_l: A → Or A B
56 | Or_intro_r: B → Or A B.
58 interpretation "constructive or" 'or x y = (Or x y).
60 notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub\i\sup\l)" with precedence 19 for @{ 'Or_intro_l $a $ab }.
61 interpretation "Or_intro_l" 'Or_intro_l a ab = (cast ab (Or_intro_l _ _ a)).
63 notation < "\infrule hbox(\emsp b \emsp) ab (∨\sub\i\sup\l)" with precedence 19 for @{ 'Or_intro_r $b $ab }.
64 interpretation "Or_intro_l" 'Or_intro_r b ab = (cast ab (Or_intro_r _ _ b)).
67 λA,B,C:CProp.λc:A∨B.λfa: A → C.λfb: B → C.
68 match c with [ Or_intro_l a ⇒ fa a | Or_intro_r b ⇒ fb b].
70 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e)" with precedence 19 for @{ 'Or_elim $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
71 interpretation "Or_elim" 'Or_elim ab ac bc c = (cast c (Or_elim _ _ _ ab ac bc)).
73 inductive Exists (A:Type) (P:A→CProp) : CProp ≝
74 Exists_intro: ∀w:A. P w → Exists A P.
76 interpretation "constructive ex" 'exists \eta.x = (Exists _ x).
78 notation < "\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)" with precedence 19 for @{ 'Exists_intro $Pn $Px }.
79 interpretation "Exists_intro" 'Exists_intro Pn Px = (cast Px (Exists_intro _ _ _ Pn)).
81 definition Exists_elim ≝
82 λA:Type.λP:A→CProp.λC:CProp.λc:∃x:A.P x.λH:(∀x.P x → C).
83 match c with [ Exists_intro w p ⇒ H w p ].
85 notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e)" with precedence 19 for @{ 'Exists_elim $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
86 interpretation "Exists_elim" 'Exists_elim ExPx Pc c = (cast c (Exists_elim _ _ _ ExPx Pc)).
88 inductive Forall (A:Type) (P:A→CProp) : CProp ≝
89 Forall_intro: (∀n:A. P n) → Forall A P.
91 notation "\forall ident x:A.break term 19 Px" with precedence 20 for @{ 'Forall (λ${ident x}:$A.$Px) }.
92 interpretation "Forall" 'Forall \eta.Px = (Forall _ Px).
94 notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i)" with precedence 19 for @{ 'Forall_intro (λ${ident x}:$tx.$Px) $Pn }.
95 interpretation "Forall_intro" 'Forall_intro Px Pn = (cast Pn (Forall_intro _ _ Px)).
97 definition Forall_elim ≝
98 λA:Type.λP:A→CProp.λn:A.λf:∀x:A.P x.match f with [ Forall_intro g ⇒ g n ].
100 notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i)" with precedence 19 for @{ 'Forall_elim $Px $Pn }.
101 interpretation "Forall_elim" 'Forall_elim Px Pn = (cast Pn (Forall_elim _ _ _ Px)).
109 lemma ex1 : (A ⇒ E) ∨ B ⇒ A ∧ C ⇒ (E ∧ C) ∨ B.
110 repeat (apply cast; constructor 1; intro);
111 apply cast; apply (Or_elim (A ⇒ E) B (E∧C∨B)); try intro;
112 [ apply cast; assumption
113 | apply cast; apply Or_intro_l;
114 apply cast; constructor 1;
115 [ apply cast; apply (Imply_elim A E);
116 [ apply cast; assumption
117 | apply cast; apply (And_elim_l A C);
118 apply cast; assumption
120 | apply cast; apply (And_elim_r A C);
121 apply cast; assumption
123 | apply cast; apply Or_intro_r;
124 apply cast; assumption
129 axiom R: N → N → CProp.
131 lemma ex2: (∀a:N.∀b:N.R a b ⇒ R b a) ⇒ ∀z:N.(∃x.R x z) ⇒ ∃y. R z y.
132 apply cast; apply Imply_intro; intro;
133 apply cast; apply Forall_intro; intro z;
134 apply cast; apply Imply_intro; intro;
135 apply cast; apply (Exists_elim N (λy.R y z)); try intros (n);
136 [ apply cast; assumption
137 | apply cast; apply (Exists_intro ? ? n);
138 apply cast; apply (Imply_elim (R n z) (R z n));
139 [ apply cast; apply (Forall_elim N (λb:N.R n b ⇒ R b n) z);
140 apply cast; apply (Forall_elim N (λa:N.∀b:N.R a b ⇒ R b a) n);
141 apply cast; assumption
142 | apply cast; assumption