X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdidactic%2Fsupport%2Fnatural_deduction.ma;h=f5799b90e3a9435a90253876ed4903e42296fc52;hb=5755ebe4016316a474ad8ab8d33b1a1a9187cc9e;hp=3969197e98dddc6f2ea2cba42891408d1354e464;hpb=3895bfa2b76c88af1d6ac41605ff17e2626d353a;p=helm.git

diff --git a/helm/software/matita/library/didactic/support/natural_deduction.ma b/helm/software/matita/library/didactic/support/natural_deduction.ma
index 3969197e9..f5799b90e 100644
--- a/helm/software/matita/library/didactic/support/natural_deduction.ma
+++ b/helm/software/matita/library/didactic/support/natural_deduction.ma
@@ -40,11 +40,26 @@ definition Not_intro : ∀A.(A → Bot) → Not A ≝  λA.
 definition Not_elim : ∀A.Not A → A → Bot ≝ λA. 
   Imply_elim ? Bot.  
 
-definition assumpt := λA:CProp.λa:A.
+definition Discharge := λA:CProp.λa:A.
   a.
 
 axiom Raa : ∀A.(Not A → Bot) → A.
 
+axiom sort : Type.
+
+inductive Exists (A:Type) (P:A→CProp) : CProp ≝
+  Exists_intro: ∀w:A. P w → Exists A P.
+
+definition Exists_elim ≝
+  λA:Type.λP:A→CProp.λC:CProp.λc:Exists A P.λH:(Πx.P x → C).
+   match c with [ Exists_intro w p ⇒ H w p ].
+
+inductive Forall (A:Type) (P:A→CProp) : CProp ≝
+ Forall_intro: (∀n:A. P n) → Forall A P.
+
+definition Forall_elim ≝
+ λA:Type.λP:A→CProp.λn:A.λf:Forall A P.match f with [ Forall_intro g ⇒ g n ].
+
 (* Dummy proposition *)
 axiom unit : CProp.
 
@@ -61,7 +76,17 @@ interpretation "Bot" 'Bot = Bot.
 interpretation "Not" 'not a = (Not a).
 notation "✶" non associative with precedence 90 for @{'unit}.
 interpretation "dummy prop" 'unit = unit.
-
+notation > "\exists list1 ident x sep , . term 19 Px" with precedence 20
+for ${ fold right @{$Px} rec acc @{'myexists (λ${ident x}.$acc)} }.
+notation < "hvbox(\exists ident i break . p)" with precedence 20
+for @{ 'myexists (\lambda ${ident i} : $ty. $p) }.
+interpretation "constructive ex" 'myexists \eta.x = (Exists sort x).
+notation > "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}.$Px) }.
+notation < "\forall ident x.break term 19 Px" with precedence 20
+for @{ 'Forall (λ${ident x}:$tx.$Px) }.
+interpretation "Forall" 'Forall \eta.Px = (Forall _ Px).
+ 
 (* Variables *)
 axiom A : CProp.
 axiom B : CProp.
@@ -78,48 +103,60 @@ axiom L : CProp.
 axiom M : CProp.
 axiom N : CProp.
 axiom O : CProp.
-axiom P : CProp.
-axiom Q : CProp.
-axiom R : CProp.
-axiom S : CProp.
-axiom T : CProp.
-axiom U : CProp.
-axiom V : CProp.
-axiom W : CProp.
-axiom X : CProp.
-axiom Y : CProp.
-axiom Z : CProp.
-
-(* Every formula user provided annotates its proof A becomes (show A ?) *)
-definition show : ∀A.A→A ≝ λA:CProp.λa:A.a.
+axiom P : sort →CProp.
+axiom Q : sort →CProp.
+axiom R : sort →sort →CProp.
+axiom S : sort →sort →CProp.
+axiom f : sort → sort.
+axiom g : sort → sort.
+axiom h : sort → sort → sort.
+
+(* Every formula user provided annotates its proof:
+   `A` becomes `(show A ?)` *)
+definition show : ΠA.A→A ≝ λA:CProp.λa:A.a.
 
 (* When something does not fit, this daemon is used *)
-axiom cast: ∀A,B:CProp.B → A.
-
+axiom cast: ΠA,B:CProp.B → A.
 
+(* begin a proof: draws the root *)
 notation > "'prove' p" non associative with precedence 19 
 for @{ 'prove $p }.
 interpretation "prove KO" 'prove p = (cast _ _ (show p _)).
 interpretation "prove OK" 'prove p = (show p _).
 
-notation < "\infrule (t\atop ⋮) a ?" with precedence 19 for @{ 'leaf_ok $a $t }.
+(* Leaves *)
+notation < "\infrule (t\atop ⋮) a ?" with precedence 19
+for @{ 'leaf_ok $a $t }.
 interpretation "leaf OK" 'leaf_ok a t = (show a t).
-notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (a) ?" with precedence 19 for @{ 'leaf_ko $a $t }.
-interpretation "leaf KO" 'leaf_ko a t = (cast _ a (show _ t)).
-
-notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19 for @{ 'assumpt_ko $a $H }.
-interpretation "assumption_ko 1" 'assumpt_ko a H = (show a (cast _ _ (assumpt _ H))).
-interpretation "assumption_ko 2" 'assumpt_ko a H = (cast _ _ (show a (cast _ _ (assumpt _ H)))).
-
-notation < "[ a ] \sup H" with precedence 19 for @{ 'assumpt_ok $a $H }.
-interpretation "assumption_ok 1" 'assumpt_ok a H = (show a (assumpt a H)).
-notation < "[ mstyle color #ff0000 (a) ] \sup H" with precedence 19 for @{ 'assumpt_ok_2 $a $H }.
-interpretation "assumption_ok 2" 'assumpt_ok_2 a H = (cast _ _ (show a (assumpt a H))).
-
-notation > "[H]" with precedence 90 for @{ 'assumpt $H }.
-interpretation "assumpt KO" 'assumpt H = (cast _ _ (assumpt _ H)).
-interpretation "assumpt OK" 'assumpt H = (assumpt _ H).
-
+notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (a) ?" with precedence 19 
+for @{ 'leaf_ko $a $t }.
+interpretation "leaf KO" 'leaf_ko a t = (cast _ _ (show a t)).
+
+(* discharging *)
+notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19 
+for @{ 'discharge_ko_1 $a $H }.
+interpretation "discharge_ko_1" 'discharge_ko_1 a H = 
+  (show a (cast _ _ (Discharge _ H))).
+notation < "[ mstyle color #ff0000 (a) ] \sup mstyle color #ff0000 (H)" with precedence 19 
+for @{ 'discharge_ko_2 $a $H }.
+interpretation "discharge_ko_2" 'discharge_ko_2 a H = 
+  (cast _ _ (show a (cast _ _ (Discharge _ H)))).
+
+notation < "[ a ] \sup H" with precedence 19 
+for @{ 'discharge_ok_1 $a $H }.
+interpretation "discharge_ok_1" 'discharge_ok_1 a H = 
+  (show a (Discharge _ H)).
+notation < "[ mstyle color #ff0000 (a) ] \sup H" with precedence 19 
+for @{ 'discharge_ok_2 $a $H }.
+interpretation "discharge_ok_2" 'discharge_ok_2 a H = 
+  (cast _ _ (show a (Discharge _ H))).
+
+notation > "'discharge' [H]" with precedence 19 
+for @{ 'discharge $H }.
+interpretation "discharge KO" 'discharge H = (cast _ _ (Discharge _ H)).
+interpretation "discharge OK" 'discharge H = (Discharge _ H).
+
+(* ⇒ introduction *)
 notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
 for @{ 'Imply_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
 interpretation "Imply_intro_ko_1" 'Imply_intro_ko_1 ab \eta.b = 
@@ -127,7 +164,7 @@ interpretation "Imply_intro_ko_1" 'Imply_intro_ko_1 ab \eta.b =
 
 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
 for @{ 'Imply_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
-interpretation "Imply_intro_ko_1" 'Imply_intro_ko_2 ab \eta.b = 
+interpretation "Imply_intro_ko_2" 'Imply_intro_ko_2 ab \eta.b = 
   (cast _ _ (show ab (cast _ _ (Imply_intro _ _ b)))).
 
 notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) " with precedence 19
@@ -142,15 +179,21 @@ interpretation "Imply_intro_ok_2" 'Imply_intro_ok_2 ab \eta.b =
 
 notation > "⇒_'i' [ident H] term 90 b" with precedence 19
 for @{ 'Imply_intro $b (λ${ident H}.show $b ?) }.
-interpretation "Imply_intro KO" 'Imply_intro b pb = (cast _ (Imply unit b) (Imply_intro _ b pb)).
-interpretation "Imply_intro OK" 'Imply_intro b pb = (Imply_intro _ b pb).
-
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (⇒\sub\e) " with precedence 19 for @{ 'Imply_elim_ko_1 $ab $a $b }.
+interpretation "Imply_intro KO" 'Imply_intro b pb = 
+  (cast _ (Imply unit b) (Imply_intro _ b pb)).
+interpretation "Imply_intro OK" 'Imply_intro b pb = 
+  (Imply_intro _ b pb).
+
+(* ⇒ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (⇒\sub\e) " with precedence 19
+for @{ 'Imply_elim_ko_1 $ab $a $b }.
 interpretation "Imply_elim_ko_1" 'Imply_elim_ko_1 ab a b = 
-  (show b (cast _ _ (Imply_elim _ _ ab a))).
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (⇒\sub\e) " with precedence 19 for @{ 'Imply_elim_ko_2 $ab $a $b }.
+  (show b (cast _ _ (Imply_elim _ _ (cast _ _ ab) (cast _ _ a)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (⇒\sub\e) " with precedence 19 
+for @{ 'Imply_elim_ko_2 $ab $a $b }.
 interpretation "Imply_elim_ko_2" 'Imply_elim_ko_2 ab a b = 
-  (cast _ _ (show b (cast _ _ (Imply_elim _ _ ab a)))).
+  (cast _ _ (show b (cast _ _ (Imply_elim _ _ (cast _ _ ab) (cast _ _ a))))).
 
 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19 
 for @{ 'Imply_elim_ok_1 $ab $a $b }.
@@ -162,14 +205,21 @@ for @{ 'Imply_elim_ok_2 $ab $a $b }.
 interpretation "Imply_elim_ok_2" 'Imply_elim_ok_2 ab a b = 
   (cast _ _ (show b (Imply_elim _ _ ab a))).
 
-notation > "⇒_'e' term 90 ab term 90 a" with precedence 19 for @{ 'Imply_elim (show $ab ?) (show $a ?) }.
-interpretation "Imply_elim KO" 'Imply_elim ab a = (cast _ _ (Imply_elim _ _ (cast (Imply unit unit) _ ab) (cast unit _ a))).
-interpretation "Imply_elim OK" 'Imply_elim ab a = (Imply_elim _ _ ab a). 
+notation > "⇒_'e' term 90 ab term 90 a" with precedence 19 
+for @{ 'Imply_elim (show $ab ?) (show $a ?) }.
+interpretation "Imply_elim KO" 'Imply_elim ab a = 
+  (cast _ _ (Imply_elim _ _ (cast (Imply unit unit) _ ab) (cast unit _ a))).
+interpretation "Imply_elim OK" 'Imply_elim ab a = 
+  (Imply_elim _ _ ab a). 
 
-notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab mstyle color #ff0000 (∧\sub\i)" with precedence 19 for @{ 'And_intro_ko_1 $a $b $ab }.
+(* ∧ introduction *)
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab mstyle color #ff0000 (∧\sub\i)" with precedence 19 
+for @{ 'And_intro_ko_1 $a $b $ab }.
 interpretation "And_intro_ko_1" 'And_intro_ko_1 a b ab = 
   (show ab (cast _ _ (And_intro _ _ a b))).
-notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∧\sub\i)" with precedence 19 for @{ 'And_intro_ko_2 $a $b $ab }.
+
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∧\sub\i)" with precedence 19 
+for @{ 'And_intro_ko_2 $a $b $ab }.
 interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab = 
   (cast _ _ (show ab (cast _ _ (And_intro _ _ a b)))).
 
@@ -183,19 +233,21 @@ for @{ 'And_intro_ok_2 $a $b $ab }.
 interpretation "And_intro_ok_2" 'And_intro_ok_2 a b ab = 
   (cast _ _ (show ab (And_intro _ _ a b))).
 
-notation > "∧_'i' term 90 a term 90 b" with precedence 19 for @{ 'And_intro (show $a ?) (show $b ?) }.
+notation > "∧_'i' term 90 a term 90 b" with precedence 19 
+for @{ 'And_intro (show $a ?) (show $b ?) }.
 interpretation "And_intro KO" 'And_intro a b = (cast _ _ (And_intro _ _ a b)).
 interpretation "And_intro OK" 'And_intro a b = (And_intro _ _ a b).
 
+(* ∧ elimination *)
 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19 
 for @{ 'And_elim_l_ko_1 $ab $a }.
 interpretation "And_elim_l_ko_1" 'And_elim_l_ko_1 ab a = 
-  (show a (cast _ _ (And_elim_l _ _ ab))).
+  (show a (cast _ _ (And_elim_l _ _ (cast _ _ ab)))).
 
 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19 
 for @{ 'And_elim_l_ko_2 $ab $a }.
 interpretation "And_elim_l_ko_2" 'And_elim_l_ko_2 ab a = 
-  (cast _ _ (show a (cast _ _ (And_elim_l _ _ ab)))).
+  (cast _ _ (show a (cast _ _ (And_elim_l _ _ (cast _ _ ab))))).
 
 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))" with precedence 19 
 for @{ 'And_elim_l_ok_1 $ab $a }.
@@ -209,18 +261,18 @@ interpretation "And_elim_l_ok_2" 'And_elim_l_ok_2 ab a =
 
 notation > "∧_'e_l' term 90 ab" with precedence 19
 for @{ 'And_elim_l (show $ab ?) }.
-interpretation "And_elim_l KO" 'And_elim_l a = (And_elim_l _ _ (cast (And _ unit) _ a)).
+interpretation "And_elim_l KO" 'And_elim_l a = (cast _ _ (And_elim_l _ _ (cast (And unit unit) _ a))).
 interpretation "And_elim_l OK" 'And_elim_l a = (And_elim_l _ _ a).
 
 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19 
 for @{ 'And_elim_r_ko_1 $ab $a }.
 interpretation "And_elim_r_ko_1" 'And_elim_r_ko_1 ab a = 
-  (show a (cast _ _ (And_elim_r _ _ ab))).
+  (show a (cast _ _ (And_elim_r _ _ (cast _ _ ab)))).
 
 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19 
 for @{ 'And_elim_r_ko_2 $ab $a }.
 interpretation "And_elim_r_ko_2" 'And_elim_r_ko_2 ab a = 
-  (cast _ _ (show a (cast _ _ (And_elim_r _ _ ab)))).
+  (cast _ _ (show a (cast _ _ (And_elim_r _ _ (cast _ _ ab))))).
 
 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))" with precedence 19 
 for @{ 'And_elim_r_ok_1 $ab $a }.
@@ -234,19 +286,10 @@ interpretation "And_elim_r_ok_2" 'And_elim_r_ok_2 ab a =
 
 notation > "∧_'e_r' term 90 ab" with precedence 19
 for @{ 'And_elim_r (show $ab ?) }.
-interpretation "And_elim_r KO" 'And_elim_r a = (And_elim_r _ _ (cast (And unit _) _ a)).
+interpretation "And_elim_r KO" 'And_elim_r a = (cast _ _ (And_elim_r _ _ (cast (And unit unit) _ a))).
 interpretation "And_elim_r OK" 'And_elim_r a = (And_elim_r _ _ a).
 
-notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))" with precedence 19 
-for @{ 'Or_intro_l_ok_1 $a $ab }.
-interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab = 
-  (show ab (Or_intro_l _ _ a)).
-
-notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19 
-for @{ 'Or_intro_l_ok_1 $a $ab }.
-interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab = 
-  (cast _ _ (show ab (Or_intro_l _ _ a))).
-
+(* ∨ introduction *)
 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19 
 for @{ 'Or_intro_l_ko_1 $a $ab }.
 interpretation "Or_intro_l_ko_1" 'Or_intro_l_ko_1 a ab = 
@@ -257,21 +300,21 @@ for @{ 'Or_intro_l_ko_2 $a $ab }.
 interpretation "Or_intro_l_ko_2" 'Or_intro_l_ko_2 a ab = 
   (cast _ _ (show ab (cast _ _ (Or_intro_l _ _ a)))).
 
+notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))" with precedence 19 
+for @{ 'Or_intro_l_ok_1 $a $ab }.
+interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab = 
+  (show ab (Or_intro_l _ _ a)).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19 
+for @{ 'Or_intro_l_ok_2 $a $ab }.
+interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab = 
+  (cast _ _ (show ab (Or_intro_l _ _ a))).
+
 notation > "∨_'i_l' term 90 a" with precedence 19
 for @{ 'Or_intro_l (show $a ?) }.
 interpretation "Or_intro_l KO" 'Or_intro_l a = (cast _ (Or _ unit) (Or_intro_l _ _ a)).
 interpretation "Or_intro_l OK" 'Or_intro_l a = (Or_intro_l _ _ a). 
   
-notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))" with precedence 19 
-for @{ 'Or_intro_r_ok_1 $a $ab }.
-interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab = 
-  (show ab (Or_intro_r _ _ a)).
-
-notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19 
-for @{ 'Or_intro_r_ok_1 $a $ab }.
-interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab = 
-  (cast _ _ (show ab (Or_intro_r _ _ a))).
-
 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19 
 for @{ 'Or_intro_r_ko_1 $a $ab }.
 interpretation "Or_intro_r_ko_1" 'Or_intro_r_ko_1 a ab = 
@@ -282,11 +325,32 @@ for @{ 'Or_intro_r_ko_2 $a $ab }.
 interpretation "Or_intro_r_ko_2" 'Or_intro_r_ko_2 a ab = 
   (cast _ _ (show ab (cast _ _ (Or_intro_r _ _ a)))).
 
+notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))" with precedence 19 
+for @{ 'Or_intro_r_ok_1 $a $ab }.
+interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab = 
+  (show ab (Or_intro_r _ _ a)).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19 
+for @{ 'Or_intro_r_ok_2 $a $ab }.
+interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab = 
+  (cast _ _ (show ab (Or_intro_r _ _ a))).
+
 notation > "∨_'i_r' term 90 a" with precedence 19
 for @{ 'Or_intro_r (show $a ?) }.
 interpretation "Or_intro_r KO" 'Or_intro_r a = (cast _ (Or unit _) (Or_intro_r _ _ a)).
 interpretation "Or_intro_r OK" 'Or_intro_r a = (Or_intro_r _ _ a). 
 
+(* ∨ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (mstyle color #ff0000 (∨\sub\e \emsp) ident Ha \emsp ident Hb)" with precedence 19
+for @{ 'Or_elim_ko_1 $ab $c (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) }.
+interpretation "Or_elim_ko_1" 'Or_elim_ko_1 ab c \eta.ac \eta.bc = 
+  (show c (cast _ _ (Or_elim _ _ _ (cast _ _ ab) (cast _ _ ac) (cast _ _ bc)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∨\sub\e) \emsp ident Ha \emsp ident Hb)" with precedence 19
+for @{ 'Or_elim_ko_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
+interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c = 
+  (cast _ _ (show c (cast _ _ (Or_elim _ _ _ (cast _ _ ab) (cast _ _ ac) (cast _ _ bc))))).
+
 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e \emsp ident Ha \emsp ident Hb)" with precedence 19
 for @{ 'Or_elim_ok_1 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
 interpretation "Or_elim_ok_1" 'Or_elim_ok_1 ab \eta.ac \eta.bc c = 
@@ -297,65 +361,83 @@ for @{ 'Or_elim_ok_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
 interpretation "Or_elim_ok_2" 'Or_elim_ok_2 ab \eta.ac \eta.bc c = 
   (cast _ _ (show c (Or_elim _ _ _ ab ac bc))).
 
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (mstyle color #ff0000 (∨\sub\e \emsp) ident Ha \emsp ident Hb)" with precedence 19
-for @{ 'Or_elim_ko_1 $ab $c (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) }.
-interpretation "Or_elim_ko_1" 'Or_elim_ko_1 ab c \eta.ac \eta.bc = 
-  (show c (cast _ _ (Or_elim _ _ _ ab (cast _ _ ac) (cast _ _ bc)))).
-
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∨\sub\e) \emsp ident Ha \emsp ident Hb)" with precedence 19
-for @{ 'Or_elim_ko_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
-interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c = 
-  (cast _ _ (show c (cast _ _ (Or_elim _ _ _ ab (cast _ _ ac) (cast _ _ bc))))).
-
 definition unit_to ≝ λx:CProp.unit → x.
 
 notation > "∨_'e' term 90 ab [ident Ha] term 90 cl [ident Hb] term 90 cr" with precedence 19
-for @{ 'Or_elim (show $ab ?) (λ${ident Ha}.show $cl ?) (λ${ident Hb}.show $cr ?) $cl $cr }.
-interpretation "Or_elim KO" 'Or_elim ab ac bc c1 c2 = 
-  (cast _ _ (Or_elim _ _ _ (cast (Or unit unit) _ ab) (cast (unit_to unit) (unit_to _) ac) (cast (unit_to unit) (unit_to _) bc))).
-interpretation "Or_elim OK" 'Or_elim ab ac bc c1 c2 = (Or_elim _ _ _ ab ac bc).
-
+for @{ 'Or_elim (show $ab ?) (λ${ident Ha}.show $cl ?) (λ${ident Hb}.show $cr ?) }.
+interpretation "Or_elim KO" 'Or_elim ab ac bc = 
+  (cast _ _ (Or_elim _ _ _ 
+    (cast (Or unit unit) _ ab) 
+    (cast (unit_to unit) (unit_to _) ac) 
+    (cast (unit_to unit) (unit_to _) bc))).
+interpretation "Or_elim OK" 'Or_elim ab ac bc = (Or_elim _ _ _ ab ac bc).
+
+(* ⊤ introduction *)
 notation < "\infrule \nbsp ⊤ mstyle color #ff0000 (⊤\sub\i)" with precedence 19 
 for @{'Top_intro_ko_1}.
-interpretation "Top_intro_ko_1" 'Top_intro_ko_1 = (show _ (cast _ _ Top_intro)).
+interpretation "Top_intro_ko_1" 'Top_intro_ko_1 = 
+  (show _ (cast _ _ Top_intro)).
+
+notation < "\infrule \nbsp mstyle color #ff0000 (⊤) mstyle color #ff0000 (⊤\sub\i)" with precedence 19 
+for @{'Top_intro_ko_2}.
+interpretation "Top_intro_ko_2" 'Top_intro_ko_2 = 
+  (cast _ _ (show _ (cast _ _ Top_intro))).
 
 notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19 
 for @{'Top_intro_ok_1}.
 interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show _ Top_intro).
 
+notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19 
+for @{'Top_intro_ok_2 }.
+interpretation "Top_intro_ok_2" 'Top_intro_ok_2 = (cast _ _ (show _ Top_intro)).
+
 notation > "⊤_'i'" with precedence 19 for @{ 'Top_intro }.
 interpretation "Top_intro KO" 'Top_intro = (cast _ _ Top_intro).
 interpretation "Top_intro OK" 'Top_intro = Top_intro.
 
-notation < "\infrule b a (⊥\sub\e)" with precedence 19 for @{'Bot_elim_ok_1 $a $b}.
+(* ⊥ introduction *)
+notation < "\infrule b a mstyle color #ff0000 (⊥\sub\e)" with precedence 19 
+for @{'Bot_elim_ko_1 $a $b}.
+interpretation "Bot_elim_ko_1" 'Bot_elim_ko_1 a b = 
+  (show a (Bot_elim _ (cast _ _ b))).
+
+notation < "\infrule b mstyle color #ff0000 (a) mstyle color #ff0000 (⊥\sub\e)" with precedence 19 
+for @{'Bot_elim_ko_2 $a $b}.
+interpretation "Bot_elim_ko_2" 'Bot_elim_ko_2 a b = 
+  (cast _ _ (show a (Bot_elim _ (cast _ _ b)))).
+
+notation < "\infrule b a (⊥\sub\e)" with precedence 19 
+for @{'Bot_elim_ok_1 $a $b}.
 interpretation "Bot_elim_ok_1" 'Bot_elim_ok_1 a b = 
-  (show a (Bot_elim a b)).
+  (show a (Bot_elim _ b)).
 
-notation < "\infrule b a mstyle color #ff0000 (⊥\sub\e)" with precedence 19 for @{'Bot_elim_ko_1 $a $b}.
-interpretation "Bot_elim_ko_1" 'Bot_elim_ko_1 a b = 
-  (show a (Bot_elim a (cast _ _ b))).
+notation < "\infrule b mstyle color #ff0000 (a) (⊥\sub\e)" with precedence 19 
+for @{'Bot_elim_ok_2 $a $b}.
+interpretation "Bot_elim_ok_2" 'Bot_elim_ok_2 a b = 
+  (cast _ _ (show a (Bot_elim _ b))).
 
 notation > "⊥_'e' term 90 b" with precedence 19 
 for @{ 'Bot_elim (show $b ?) }.
 interpretation "Bot_elim KO" 'Bot_elim a = (Bot_elim _ (cast _ _ a)).  
 interpretation "Bot_elim OK" 'Bot_elim a = (Bot_elim _ a).
 
-notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
+(* ¬ introduction *)
+notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
 for @{ 'Not_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
 interpretation "Not_intro_ko_1" 'Not_intro_ko_1 ab \eta.b = 
   (show ab (cast _ _ (Not_intro _ (cast _ _ b)))).
 
-notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub(\emsp\i)) \emsp ident H)" with precedence 19
 for @{ 'Not_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
 interpretation "Not_intro_ko_2" 'Not_intro_ko_2 ab \eta.b = 
   (cast _ _ (show ab (cast _ _ (Not_intro _ (cast _ _ b))))).
 
-notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub\i \emsp ident H) " with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub(\emsp\i) \emsp ident H) " with precedence 19
 for @{ 'Not_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
 interpretation "Not_intro_ok_1" 'Not_intro_ok_1 ab \eta.b = 
   (show ab (Not_intro _ b)).
 
-notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub\i \emsp ident H) " with precedence 19
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub(\emsp\i) \emsp ident H) " with precedence 19
 for @{ 'Not_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
 interpretation "Not_intro_ok_2" 'Not_intro_ok_2 ab \eta.b = 
   (cast _ _ (show ab (Not_intro _ b))).
@@ -365,30 +447,44 @@ for @{ 'Not_intro (λ${ident H}.show $b ?) }.
 interpretation "Not_intro KO" 'Not_intro a = (cast _ _ (Not_intro _ (cast _ _ a))).
 interpretation "Not_intro OK" 'Not_intro a = (Not_intro _ a).
 
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub\e) " with precedence 19 
+(* ¬ elimination *)
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19 
+for @{ 'Not_elim_ko_1 $ab $a $b }.
+interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b = 
+  (show b (cast _ _ (Not_elim _ (cast _ _ ab) (cast _ _ a)))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub(\emsp\e)) " with precedence 19 
+for @{ 'Not_elim_ko_2 $ab $a $b }.
+interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b = 
+  (cast _ _ (show b (cast _ _ (Not_elim _ (cast _ _ ab) (cast _ _ a))))).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub(\emsp\e)) " with precedence 19 
 for @{ 'Not_elim_ok_1 $ab $a $b }.
 interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b = 
   (show b (Not_elim _ ab a)).
 
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub\e) " with precedence 19 
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub(\emsp\e)) " with precedence 19 
 for @{ 'Not_elim_ok_2 $ab $a $b }.
 interpretation "Not_elim_ok_2" 'Not_elim_ok_2 ab a b = 
   (cast _ _ (show b (Not_elim _ ab a))).
 
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub\e) " with precedence 19 
-for @{ 'Not_elim_ko_1 $ab $a $b }.
-interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b = 
-  (show b (Not_elim _ (cast _ _ ab) a)).
-
-notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub\e) " with precedence 19 
-for @{ 'Not_elim_ko_2 $ab $a $b }.
-interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b = 
-  (cast _ _ (show b (Not_elim _ (cast _ _ ab) a))).
-
 notation > "¬_'e' term 90 ab term 90 a" with precedence 19
 for @{ 'Not_elim (show $ab ?) (show $a ?) }.
-interpretation "Not_elim KO" 'Not_elim ab a = (Not_elim _ (cast _ _ ab) a).
-interpretation "Not_elim OK" 'Not_elim ab a = (Not_elim _ ab a).
+interpretation "Not_elim KO" 'Not_elim ab a = 
+  (cast _ _ (Not_elim unit (cast _ _ ab) (cast _ _ a))).
+interpretation "Not_elim OK" 'Not_elim ab a = 
+  (Not_elim _ ab a).
+
+(* RAA *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
+for @{ 'RAA_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ko_1" 'RAA_ko_1 Px Pn = 
+  (show Pn (cast _ _ (Raa _ (cast _ _ Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
+for @{ 'RAA_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn = 
+  (cast _ _ (show Pn (cast _ _ (Raa _ (cast _ _ Px))))).
 
 notation < "\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)" with precedence 19
 for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
@@ -400,39 +496,351 @@ for @{ 'RAA_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
 interpretation "RAA_ok_2" 'RAA_ok_2 Px Pn = 
   (cast _ _ (show Pn (Raa _ Px))).
 
-notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
-for @{ 'RAA_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
-interpretation "RAA_ko_1" 'RAA_ko_1 Px Pn = 
-  (show Pn (Raa _ (cast _ _ Px))).
-
-notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
-for @{ 'RAA_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
-interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn = 
-  (cast _ _ (show Pn (Raa _ (cast _ _ Px)))).
-
 notation > "'RAA' [ident H] term 90 b" with precedence 19 
 for @{ 'Raa (λ${ident H}.show $b ?) }. 
-interpretation "RAA KO" 'Raa p = (Raa _ (cast _ (unit_to _) p)).
+interpretation "RAA KO" 'Raa p = (cast _ unit (Raa _ (cast _ (unit_to _) p))).
 interpretation "RAA OK" 'Raa p = (Raa _ p).
 
-(*DOCBEGIN
-Templates for rules
-⇒_i […] (…)
-∧_i (…) (…)
-∨_i_l (…)
-∨_i_r (…)
-¬_i […] (…)
-⊤_i
-⇒_e (…) (…)
-∧_e_l (…)
-∧_e_r (…)
-∨_e (…) […] (…) […] (…)
-¬_e (…) (…)
-⊥_e (…)
-prove (…)
-RAA […] (…)
-DOCEND*)
-
-
-
-
+(* ∃ introduction *)
+notation < "\infrule hbox(\emsp Pn \emsp) Px mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_1 $Pn $Px }.
+interpretation "Exists_intro_ko_1" 'Exists_intro_ko_1 Pn Px = 
+  (show Px (cast _ _ (Exists_intro _ _ _ (cast _ _ Pn)))).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) mstyle color #ff0000 (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ko_2 $Pn $Px }.
+interpretation "Exists_intro_ko_2" 'Exists_intro_ko_2 Pn Px = 
+  (cast _ _ (show Px (cast _ _ (Exists_intro _ _ _ (cast _ _ Pn))))).
+
+notation < "\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ok_1 $Pn $Px }.
+interpretation "Exists_intro_ok_1" 'Exists_intro_ok_1 Pn Px = 
+  (show Px (Exists_intro _ _ _ Pn)).
+
+notation < "\infrule hbox(\emsp Pn \emsp) mstyle color #ff0000 (Px) (∃\sub\i)" with precedence 19
+for @{ 'Exists_intro_ok_2 $Pn $Px }.
+interpretation "Exists_intro_ok_2" 'Exists_intro_ok_2 Pn Px = 
+  (cast _ _ (show Px (Exists_intro _ _ _ Pn))).
+
+notation >"∃_'i' {term 90 t} term 90 Pt" non associative with precedence 19
+for @{'Exists_intro $t (λ_.?) (show $Pt ?)}. 
+interpretation "Exists_intro KO" 'Exists_intro t P Pt =
+ (cast _ _ (Exists_intro sort P t (cast _ _ Pt))).
+interpretation "Exists_intro OK" 'Exists_intro t P Pt =
+ (Exists_intro sort P t Pt).
+ 
+(* ∃ elimination *)
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_1" 'Exists_elim_ko_1 ExPx Pc c =
+    (show c (cast _ _ (Exists_elim _ _ _ (cast _ _ ExPx) (cast _ _ Pc)))).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (mstyle color #ff0000 (∃\sub\e) \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ko_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ko_2" 'Exists_elim_ko_2 ExPx Pc c =
+    (cast _ _ (show c (cast _ _ (Exists_elim _ _ _ (cast _ _ ExPx) (cast _ _ Pc))))).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ok_1 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_1" 'Exists_elim_ok_1 ExPx Pc c =
+    (show c (Exists_elim _ _ _ ExPx Pc)).
+
+notation < "\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) mstyle color #ff0000 (c) (∃\sub\e \emsp ident n \emsp ident HPn)" with precedence 19
+for @{ 'Exists_elim_ok_2 $ExPx (λ${ident n}:$tn.λ${ident HPn}:$Pn.$Pc) $c }.
+interpretation "Exists_elim_ok_2" 'Exists_elim_ok_2 ExPx Pc c =
+    (cast _ _ (show c (Exists_elim _ _ _ ExPx Pc))).
+
+definition ex_concl := λx:CProp.sort → unit → x.
+definition fake_pred := λx:sort.unit.
+
+notation >"∃_'e' term 90 ExPt {ident t} [ident H] term 90 c" non associative with precedence 19
+for @{'Exists_elim (λ_.?) (show $ExPt ?) (λ${ident t}:sort.λ${ident H}.show $c ?)}. 
+interpretation "Exists_elim KO" 'Exists_elim P ExPt c =
+ (cast _ _ (Exists_elim sort P _ 
+   (cast (Exists _ P)  _ ExPt) (cast (ex_concl unit) (ex_concl _) c))).
+interpretation "Exists_elim OK" 'Exists_elim P ExPt c =
+ (Exists_elim sort P _ ExPt c).
+
+(* ∀ introduction *)
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_1" 'Forall_intro_ko_1 Px Pn = 
+  (show Pn (cast _ _ (Forall_intro _ _ (cast _ _ Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ko_2" 'Forall_intro_ko_2 Px Pn = 
+  (cast _ _ (show Pn (cast _ _ (Forall_intro _ _ (cast _ _ Px))))).
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_1" 'Forall_intro_ok_1 Px Pn = 
+  (show Pn (Forall_intro _ _ Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\i \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "Forall_intro_ok_2" 'Forall_intro_ok_2 Px Pn = 
+  (cast _ _ (show Pn (Forall_intro _ _ Px))).
+
+notation > "∀_'i' {ident y} term 90 Px" non associative with precedence 19
+for @{ 'Forall_intro (λ_.?) (λ${ident y}.show $Px ?) }. 
+interpretation "Forall_intro KO" 'Forall_intro P Px =
+  (cast _ _ (Forall_intro sort P (cast _ _ Px))). 
+interpretation "Forall_intro OK" 'Forall_intro P Px =
+  (Forall_intro sort P Px). 
+
+(* ∀ elimination *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_1 $Px $Pn }.
+interpretation "Forall_elim_ko_1" 'Forall_elim_ko_1 Px Pn = 
+  (show Pn (cast _ _ (Forall_elim _ _ _ (cast _ _ Px)))).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000(Pn) (mstyle color #ff0000 (∀\sub\e))" with precedence 19
+for @{ 'Forall_elim_ko_2 $Px $Pn }.
+interpretation "Forall_elim_ko_2" 'Forall_elim_ko_2 Px Pn = 
+  (cast _ _ (show Pn (cast _ _ (Forall_elim _ _ _ (cast _ _ Px))))).
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (∀\sub\e)" with precedence 19
+for @{ 'Forall_elim_ok_1 $Px $Pn }.
+interpretation "Forall_elim_ok_1" 'Forall_elim_ok_1 Px Pn = 
+  (show Pn (Forall_elim _ _ _ Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (∀\sub\e)" with precedence 19
+for @{ 'Forall_elim_ok_2 $Px $Pn }.
+interpretation "Forall_elim_ok_2" 'Forall_elim_ok_2 Px Pn = 
+  (cast _ _ (show Pn (Forall_elim _ _ _ Px))).
+
+notation > "∀_'e' {term 90 t} term 90 Pn" non associative with precedence 19
+for @{ 'Forall_elim (λ_.?) $t (show $Pn ?) }. 
+interpretation "Forall_elim KO" 'Forall_elim P t Px =
+  (cast _ unit (Forall_elim sort P t (cast _ _ Px))). 
+interpretation "Forall_elim OK" 'Forall_elim P t Px =
+  (Forall_elim sort P t Px). 
+
+(* already proved lemma *)
+definition hide_args : ∀A:Type.∀a:A.A := λA:Type.λa:A.a.
+notation < "t" non associative with precedence 90 for @{'hide_args $t}.
+interpretation "hide 0 args"  'hide_args t = (hide_args _ t).
+interpretation "hide 1 args"  'hide_args t = (hide_args _ t _).
+interpretation "hide 2 args"  'hide_args t = (hide_args _ t _ _).
+interpretation "hide 3 args"  'hide_args t = (hide_args _ t _ _ _).
+interpretation "hide 4 args"  'hide_args t = (hide_args _ t _ _ _ _). 
+interpretation "hide 5 args"  'hide_args t = (hide_args _ t _ _ _ _ _).
+interpretation "hide 6 args"  'hide_args t = (hide_args _ t _ _ _ _ _ _).
+interpretation "hide 7 args"  'hide_args t = (hide_args _ t _ _ _ _ _ _ _).
+
+(* more args crashes the pattern matcher *)
+
+(* already proved lemma, 0 assumptions *)
+definition Lemma : ΠA.A→A ≝ λA:CProp.λa:A.a.
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma_ko_1 $p ($H : $_) }.
+interpretation "lemma_ko_1" 'lemma_ko_1 p H = 
+  (show p (cast _ _ (Lemma _ (cast _ _ H)))). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma_ko_2 $p ($H : $_) }.
+interpretation "lemma_ko_2" 'lemma_ko_2 p H = 
+  (cast _ _ (show p (cast _ _ 
+    (Lemma _ (cast _ _ H))))). 
+
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma_ok_1 $p ($H : $_) }.
+interpretation "lemma_ok_1" 'lemma_ok_1 p H = 
+  (show p (Lemma _ H)). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma_ok_2 $p ($H : $_) }.
+interpretation "lemma_ok_2" 'lemma_ok_2 p H = 
+  (cast _ _ (show p (Lemma _ H))). 
+
+notation > "'lem' 0 term 90 l" non associative with precedence 19
+for @{ 'Lemma (hide_args ? $l : ?) }.
+interpretation "lemma KO" 'Lemma l = 
+  (cast _ _ (Lemma unit (cast unit _ l))). 
+interpretation "lemma OK" 'Lemma l = (Lemma _ l).
+
+
+(* already proved lemma, 1 assumption *)
+definition Lemma1 : ΠA,B. (A ⇒ B) → A → B ≝ 
+ λA,B:CProp.λf:A⇒B.λa:A.
+  Imply_elim A B f a.
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma1_ko_1 $a $p ($H : $_) }.
+interpretation "lemma1_ko_1" 'lemma1_ko_1 a p H = 
+  (show p (cast _ _ (Lemma1 _ _ (cast _ _ H) (cast _ _ a)))). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma1_ko_2 $a $p ($H : $_) }.
+interpretation "lemma1_ko_2" 'lemma1_ko_2 a p H = 
+  (cast _ _ (show p (cast _ _ 
+    (Lemma1 _ _ (cast _ _ H) (cast _ _ a))))). 
+
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma1_ok_1 $a $p ($H : $_) }.
+interpretation "lemma1_ok_1" 'lemma1_ok_1 a p H = 
+  (show p (Lemma1 _ _ H a)). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma1_ok_2 $a $p ($H : $_) }.
+interpretation "lemma1_ok_2" 'lemma1_ok_2 a p H = 
+  (cast _ _ (show p (Lemma1 _ _ H a))). 
+
+
+notation > "'lem' 1 term 90 l term 90 p" non associative with precedence 19
+for @{ 'Lemma1 (hide_args ? $l : ?) (show $p ?) }.
+interpretation "lemma 1 KO" 'Lemma1 l p = 
+  (cast _ _ (Lemma1 unit unit (cast (Imply unit unit) _ l) (cast unit _ p))). 
+interpretation "lemma 1 OK" 'Lemma1 l p = (Lemma1 _ _ l p).
+
+(* already proved lemma, 2 assumptions *)
+definition Lemma2 : ΠA,B,C. (A ⇒ B ⇒ C) → A → B → C ≝ 
+ λA,B,C:CProp.λf:A⇒B⇒C.λa:A.λb:B.
+  Imply_elim B C (Imply_elim A (B⇒C) f a) b.
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma2_ko_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_1" 'lemma2_ko_1 a b p H = 
+  (show p (cast _ _ (Lemma2 _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b)))). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma2_ko_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ko_2" 'lemma2_ko_2 a b p H = 
+  (cast _ _ (show p (cast _ _ 
+    (Lemma2 _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b))))). 
+
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma2_ok_1 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_1" 'lemma2_ok_1 a b p H = 
+  (show p (Lemma2 _ _ _ H a b)). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma2_ok_2 $a $b $p ($H : $_) }.
+interpretation "lemma2_ok_2" 'lemma2_ok_2 a b p H = 
+  (cast _ _ (show p (Lemma2 _ _ _ H a b))). 
+
+notation > "'lem' 2 term 90 l term 90 p term 90 q" non associative with precedence 19
+for @{ 'Lemma2 (hide_args ? $l : ?) (show $p ?) (show $q ?) }.
+interpretation "lemma 2 KO" 'Lemma2 l p q = 
+  (cast _ _ (Lemma2 unit unit unit (cast (Imply unit (Imply unit unit)) _ l) (cast unit _ p) (cast unit _ q))). 
+interpretation "lemma 2 OK" 'Lemma2 l p q = (Lemma2 _ _ _ l p q).
+
+(* already proved lemma, 3 assumptions *)
+definition Lemma3 : ΠA,B,C,D. (A ⇒ B ⇒ C ⇒ D) → A → B → C → D ≝ 
+ λA,B,C,D:CProp.λf:A⇒B⇒C⇒D.λa:A.λb:B.λc:C.
+  Imply_elim C D (Imply_elim B (C⇒D) (Imply_elim A (B⇒C⇒D) f a) b) c.
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma3_ko_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_1" 'lemma3_ko_1 a b c p H = 
+  (show p (cast _ _ 
+   (Lemma3 _ _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b) (cast _ _ c)))). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp) 
+             (╲ mstyle mathsize normal (mstyle color #ff0000 (H)) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma3_ko_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ko_2" 'lemma3_ko_2 a b c p H = 
+  (cast _ _ (show p (cast _ _ 
+    (Lemma3 _ _ _ _ (cast _ _ H) (cast _ _ a) (cast _ _ b) (cast _ _ c))))). 
+
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         p \nbsp" 
+non associative with precedence 19
+for @{ 'lemma3_ok_1 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_1" 'lemma3_ok_1 a b c p H = 
+  (show p (Lemma3 _ _ _ _ H a b c)). 
+
+notation < "\infrule 
+         (\infrule 
+             (\emsp a \emsp\emsp\emsp b \emsp\emsp\emsp c \emsp) 
+             (╲ mstyle mathsize normal (H) ╱) \nbsp) 
+         mstyle color #ff0000 (p) \nbsp" 
+non associative with precedence 19
+for @{ 'lemma3_ok_2 $a $b $c $p ($H : $_) }.
+interpretation "lemma3_ok_2" 'lemma3_ok_2 a b c p H = 
+  (cast _ _ (show p (Lemma3 _ _ _ _ H a b c))). 
+
+notation > "'lem' 3 term 90 l term 90 p term 90 q term 90 r" non associative with precedence 19
+for @{ 'Lemma3 (hide_args ? $l : ?) (show $p ?) (show $q ?) (show $r ?) }.
+interpretation "lemma 3 KO" 'Lemma3 l p q r = 
+  (cast _ _ (Lemma3 unit unit unit unit (cast (Imply unit (Imply unit (Imply unit unit))) _ l) (cast unit _ p) (cast unit _ q) (cast unit _ r))). 
+interpretation "lemma 3 OK" 'Lemma3 l p q r = (Lemma3 _ _ _ _ l p q r).