X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Flib%2Fdidactic%2Fnatural_deduction.ma;fp=matita%2Fmatita%2Flib%2Fdidactic%2Fnatural_deduction.ma;h=45ce24f9180708629bd1c9a9e4cf30caeccd4908;hb=600fba840c748f67593838673a6eb40eab9b68e5;hp=0000000000000000000000000000000000000000;hpb=b4f76b0d8fa0e5365fb48e91474febe200b647a7;p=helm.git

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+(**************************************************************************)
+(*       ___	                                                          *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* Logic system *)
+
+include "basics/pts.ma".
+include "hints_declaration.ma".
+
+inductive Imply (A,B:Prop) : Prop ≝
+| Imply_intro: (A → B) → Imply A B.
+ 
+definition Imply_elim ≝ λA,B:Prop.λf:Imply A B. λa:A.
+  match f with [ Imply_intro g ⇒ g a].
+
+inductive And (A,B:Prop) : Prop ≝
+| And_intro: A → B → And A B.
+
+definition And_elim_l ≝ λA,B.λc:And A B.
+  match c with [ And_intro a b ⇒ a ].
+
+definition And_elim_r ≝ λA,B.λc:And A B.
+  match c with [ And_intro a b ⇒ b ].
+
+inductive Or (A,B:Prop) : Prop ≝
+| Or_intro_l: A → Or A B
+| Or_intro_r: B → Or A B. 
+ 
+definition Or_elim ≝ λA,B,C:Prop.λc:Or A B.λfa: A → C.λfb: B → C.
+  match c with 
+  [ Or_intro_l a ⇒ fa a 
+  | Or_intro_r b ⇒ fb b].
+
+inductive Top : Prop := 
+| Top_intro : Top.
+
+inductive Bot : Prop := .
+
+definition Bot_elim ≝ λP:Prop.λx:Bot.
+  match x in Bot return λx.P with []. 
+
+definition Not := λA:Prop.Imply A Bot.
+
+definition Not_intro : ∀A.(A → Bot) → Not A ≝  λA. 
+  Imply_intro A Bot.
+
+definition Not_elim : ∀A.Not A → A → Bot ≝ λA. 
+  Imply_elim ? Bot.  
+
+definition Discharge := λA:Prop.λa:A.
+  a.
+
+axiom Raa : ∀A.(Not A → Bot) → A.
+
+axiom sort : Type[0].
+
+inductive Exists (A:Type[0]) (P:A→Prop) : Prop ≝
+  Exists_intro: ∀w:A. P w → Exists A P.
+
+definition Exists_elim ≝
+  λA:Type[0].λP:A→Prop.λC:Prop.λc:Exists A P.λH:(Πx.P x → C).
+   match c with [ Exists_intro w p ⇒ H w p ].
+
+inductive Forall (A:Type[0]) (P:A→Prop) : Prop ≝
+ Forall_intro: (∀n:A. P n) → Forall A P.
+
+definition Forall_elim ≝
+ λA:Type[0].λP:A→Prop.λn:A.λf:Forall A P.match f with [ Forall_intro g ⇒ g n ].
+
+(* Dummy proposition *)
+axiom unit : Prop.
+
+(* Notations *)
+notation "hbox(a break ⇒ b)" right associative with precedence 20 
+for @{ 'Imply $a $b }.
+interpretation "Imply" 'Imply a b = (Imply a b).
+interpretation "constructive or" 'or x y = (Or x y).
+interpretation "constructive and" 'and x y = (And x y).
+notation "⊤" non associative with precedence 90 for @{'Top}.
+interpretation "Top" 'Top = Top.
+notation "⊥" non associative with precedence 90 for @{'Bot}.
+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 : Prop.
+axiom B : Prop.
+axiom C : Prop.
+axiom D : Prop.
+axiom E : Prop.
+axiom F : Prop.
+axiom G : Prop.
+axiom H : Prop.
+axiom I : Prop.
+axiom J : Prop.
+axiom K : Prop.
+axiom L : Prop.
+axiom M : Prop.
+axiom N : Prop.
+axiom O : Prop.
+axiom  x: sort.
+axiom  y: sort.
+axiom  z: sort.
+axiom  w: sort.
+
+(* Every formula user provided annotates its proof:
+   `A` becomes `(show A ?)` *)
+definition show : ΠA.A→A ≝ λA:Prop.λa:A.a.
+
+(* When something does not fit, this daemon is used *)
+axiom cast: ΠA,B:Prop.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 ?).
+
+(* 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 ? ? (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 = 
+  (show ab (cast ? ? (Imply_intro ? ? 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_2" 'Imply_intro_ko_2 ab \eta.b = 
+  (cast ? ? (show ab (cast ? ? (Imply_intro ? ? b)))).
+
+notation < "maction (\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) ) (\vdots)" with precedence 19
+for @{ 'Imply_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ok_1" 'Imply_intro_ok_1 ab \eta.b = 
+  (show ab (Imply_intro ? ? b)).
+
+notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (⇒\sub\i \emsp ident H) " with precedence 19
+for @{ 'Imply_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
+interpretation "Imply_intro_ok_2" 'Imply_intro_ok_2 ab \eta.b = 
+  (cast ? ? (show ab (Imply_intro ? ? 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).
+
+(* ⇒ 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 ? ? (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 ? ? (cast ? ? ab) (cast ? ? a))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) ) (\vdots)" with precedence 19 
+for @{ 'Imply_elim_ok_1 $ab $a $b }.
+interpretation "Imply_elim_ok_1" 'Imply_elim_ok_1 ab a b = 
+  (show b (Imply_elim ? ? ab a)).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (⇒\sub\e) " with precedence 19 
+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). 
+
+(* ∧ 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 }.
+interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab = 
+  (cast ? ? (show ab (cast ? ? (And_intro ? ? a b)))).
+
+notation < "maction (\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)) (\vdots)" with precedence 19 
+for @{ 'And_intro_ok_1 $a $b $ab }.
+interpretation "And_intro_ok_1" 'And_intro_ok_1 a b ab = 
+  (show ab (And_intro ? ? a b)).
+
+notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) (∧\sub\i)" with precedence 19 
+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 ?) }.
+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 ? ? (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 ? ? (cast ? ? ab))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))) (\vdots)" with precedence 19 
+for @{ 'And_elim_l_ok_1 $ab $a }.
+interpretation "And_elim_l_ok_1" 'And_elim_l_ok_1 ab a = 
+  (show a (And_elim_l ? ? ab)).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\l))" with precedence 19 
+for @{ 'And_elim_l_ok_2 $ab $a }.
+interpretation "And_elim_l_ok_2" 'And_elim_l_ok_2 ab a = 
+  (cast ? ? (show a (And_elim_l ? ? ab))).
+
+notation > "∧#'e_l' term 90 ab" with precedence 19
+for @{ 'And_elim_l (show $ab ?) }.
+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 ? ? (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 ? ? (cast ? ? ab))))).
+
+notation < "maction (\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))) (\vdots)" with precedence 19 
+for @{ 'And_elim_r_ok_1 $ab $a }.
+interpretation "And_elim_r_ok_1" 'And_elim_r_ok_1 ab a = 
+  (show a (And_elim_r ? ? ab)).
+
+notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\r))" with precedence 19 
+for @{ 'And_elim_r_ok_2 $ab $a }.
+interpretation "And_elim_r_ok_2" 'And_elim_r_ok_2 ab a = 
+  (cast ? ? (show a (And_elim_r ? ? ab))).
+
+notation > "∧#'e_r' term 90 ab" with precedence 19
+for @{ 'And_elim_r (show $ab ?) }.
+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).
+
+(* ∨ 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 = 
+  (show ab (cast ? ? (Or_intro_l ? ? a))).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19 
+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 < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))) (\vdots)" 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 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 = 
+  (show ab (cast ? ? (Or_intro_r ? ? a))).
+
+notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19 
+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 < "maction (\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))) (\vdots)" 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 < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) c (∨\sub\e \emsp ident Ha \emsp ident Hb)) (\vdots)" 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 = 
+  (show c (Or_elim ? ? ? ab ac bc)).
+
+notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp ac \emsp\emsp\emsp bc \emsp) mstyle color #ff0000 (c) (∨\sub\e \emsp ident Ha \emsp ident Hb)" with precedence 19
+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))).
+
+definition unit_to ≝ λx:Prop.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 ?) }.
+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)).
+
+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 < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" with precedence 19 
+for @{'Top_intro_ok_1}.
+interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show ? Top_intro).
+
+notation < "maction (\infrule \nbsp ⊤ (⊤\sub\i)) (\vdots)" 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.
+
+(* ⊥ 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 < "maction (\infrule b a (⊥\sub\e)) (\vdots)" 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 ? 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).
+
+(* ¬ 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(\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 < "maction (\infrule hbox(\emsp b \emsp) ab (\lnot\sub(\emsp\i) \emsp ident H) ) (\vdots)" 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(\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))).
+
+notation > "¬#'i' [ident H] term 90 b" with precedence 19
+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).
+
+(* ¬ 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 < "maction (\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub(\emsp\e)) ) (\vdots)" 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(\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 > "¬#'e' term 90 ab term 90 a" with precedence 19
+for @{ 'Not_elim (show $ab ?) (show $a ?) }.
+interpretation "Not_elim KO" 'Not_elim ab a = 
+  (cast ? ? (Not_elim unit (cast ? ? ab) (cast ? ? a))).
+interpretation "Not_elim OK" 'Not_elim ab a = 
+  (Not_elim ? ab a).
+
+(* RAA *)
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
+for @{ 'RAA_ko_1 (λ${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) $Pn }.
+interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn = 
+  (cast ? ? (show Pn (cast ? ? (Raa ? (cast ? ? Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)) (\vdots)" with precedence 19
+for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ok_1" 'RAA_ok_1 Px Pn = 
+  (show Pn (Raa ? Px)).
+
+notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (\RAA \emsp ident x)" with precedence 19
+for @{ 'RAA_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
+interpretation "RAA_ok_2" 'RAA_ok_2 Px Pn = 
+  (cast ? ? (show Pn (Raa ? Px))).
+
+notation > "'RAA' [ident H] term 90 b" with precedence 19 
+for @{ 'Raa (λ${ident H}.show $b ?) }. 
+interpretation "RAA KO" 'Raa p = (cast ? unit (Raa ? (cast ? (unit_to ?) p))).
+interpretation "RAA OK" 'Raa p = (Raa ? p).
+
+(* ∃ 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 < "maction (\infrule hbox(\emsp Pn \emsp) Px (∃\sub\i)) (\vdots)" 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 (λw.? w) (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 < "maction (\infrule hbox(\emsp ExPx \emsp\emsp\emsp Pc \emsp) c (∃\sub\e \emsp ident n \emsp ident HPn)) (\vdots)" 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:sort → Prop.Πy:sort.unit → x y.
+definition ex_concl_dummy := Πy:sort.unit → unit.
+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 (λx.? x) (show $ExPt ?) (λ${ident t}:sort.λ${ident H}.show $c ?)}. 
+interpretation "Exists_elim KO" 'Exists_elim P ExPt c =
+ (cast ? ? (Exists_elim sort P ? 
+   (cast (Exists ? P)  ? ExPt) 
+   (cast ex_concl_dummy (ex_concl ?) c))).
+interpretation "Exists_elim OK" 'Exists_elim P ExPt c =
+ (Exists_elim sort P ? ExPt c).
+
+(* ∀ introduction *)
+
+notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (∀\sub\i) \emsp ident x)" with precedence 19
+for @{ 'Forall_intro_ko_1 (λ${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) $Pn }.
+interpretation "Forall_intro_ko_2" 'Forall_intro_ko_2 Px Pn = 
+  (cast ? ? (show Pn (cast ? ? (Forall_intro ? ? (cast ? ? Px))))).
+
+notation < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\i \emsp ident x)) (\vdots)" with precedence 19
+for @{ 'Forall_intro_ok_1 (λ${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 < "maction (\infrule hbox(\emsp Px \emsp) Pn (∀\sub\e)) (\vdots)" 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 : ΠA:Type[0].A→A := λA:Type[0].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:Prop.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:Prop.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:Prop.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:Prop.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).