3 inductive Imply (A,B:CProp) : CProp ≝
4 | Imply_intro: (A → B) → Imply A B.
6 definition Imply_elim ≝ λA,B.λf:Imply A B. λa:A.
7 match f with [ Imply_intro g ⇒ g a].
9 inductive And (A,B:CProp) : CProp ≝
10 | And_intro: A → B → And A B.
12 definition And_elim_l ≝ λA,B.λc:And A B.
13 match c with [ And_intro a b ⇒ a ].
15 definition And_elim_r ≝ λA,B.λc:And A B.
16 match c with [ And_intro a b ⇒ b ].
18 inductive Or (A,B:CProp) : CProp ≝
19 | Or_intro_l: A → Or A B
20 | Or_intro_r: B → Or A B.
22 definition Or_elim ≝ λA,B,C:CProp.λc:Or A B.λfa: A → C.λfb: B → C.
25 | Or_intro_r b ⇒ fb b].
27 inductive Top : CProp :=
30 inductive Bot : CProp := .
32 definition Bot_elim ≝ λP:CProp.λx:Bot.
33 match x in Bot return λx.P with [].
35 definition Not := λA:CProp.Imply A Bot.
37 definition Not_intro : ∀A.(A → Bot) → Not A ≝ λA.
40 definition Not_elim : ∀A.Not A → A → Bot ≝ λA.
43 definition assumpt := λA:CProp.λa:A.
46 axiom Raa : ∀A.(Not A → Bot) → A.
48 (* Dummy proposition *)
52 notation "hbox(a break ⇒ b)" right associative with precedence 20
53 for @{ 'Imply $a $b }.
54 interpretation "Imply" 'Imply a b = (Imply a b).
55 interpretation "constructive or" 'or x y = (Or x y).
56 interpretation "constructive and" 'and x y = (And x y).
57 notation "⊤" non associative with precedence 90 for @{'Top}.
58 interpretation "Top" 'Top = Top.
59 notation "⊥" non associative with precedence 90 for @{'Bot}.
60 interpretation "Bot" 'Bot = Bot.
61 interpretation "Not" 'not a = (Not a).
62 notation "✶" non associative with precedence 90 for @{'unit}.
63 interpretation "dummy prop" 'unit = unit.
93 (* Every formula user provided annotates its proof A becomes (show A ?) *)
94 definition show : ∀A.A→A ≝ λA:CProp.λa:A.a.
96 (* When something does not fit, this daemon is used *)
97 axiom cast: ∀A,B:CProp.B → A.
100 notation > "'prove' p" non associative with precedence 19
102 interpretation "prove KO" 'prove p = (cast _ _ (show p _)).
103 interpretation "prove OK" 'prove p = (show p _).
105 notation < "\infrule (t\atop ⋮) a ?" with precedence 19 for @{ 'leaf_ok $a $t }.
106 interpretation "leaf OK" 'leaf_ok a t = (show a t).
107 notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (a) ?" with precedence 19 for @{ 'leaf_ko $a $t }.
108 interpretation "leaf KO" 'leaf_ko a t = (cast _ a (show _ t)).
110 notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19 for @{ 'assumpt_ko $a $H }.
111 interpretation "assumption_ko 1" 'assumpt_ko a H = (show a (cast _ _ (assumpt _ H))).
112 interpretation "assumption_ko 2" 'assumpt_ko a H = (cast _ _ (show a (cast _ _ (assumpt _ H)))).
114 notation < "[ a ] \sup H" with precedence 19 for @{ 'assumpt_ok $a $H }.
115 interpretation "assumption_ok 1" 'assumpt_ok a H = (show a (assumpt a H)).
116 notation < "[ mstyle color #ff0000 (a) ] \sup H" with precedence 19 for @{ 'assumpt_ok_2 $a $H }.
117 interpretation "assumption_ok 2" 'assumpt_ok_2 a H = (cast _ _ (show a (assumpt a H))).
119 notation > "[H]" with precedence 90 for @{ 'assumpt $H }.
120 interpretation "assumpt KO" 'assumpt H = (cast _ _ (assumpt _ H)).
121 interpretation "assumpt OK" 'assumpt H = (assumpt _ H).
123 notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
124 for @{ 'Imply_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
125 interpretation "Imply_intro_ko_1" 'Imply_intro_ko_1 ab \eta.b =
126 (show ab (cast _ _ (Imply_intro _ _ b))).
128 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
129 for @{ 'Imply_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
130 interpretation "Imply_intro_ko_1" 'Imply_intro_ko_2 ab \eta.b =
131 (cast _ _ (show ab (cast _ _ (Imply_intro _ _ b)))).
133 notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) " with precedence 19
134 for @{ 'Imply_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
135 interpretation "Imply_intro_ok_1" 'Imply_intro_ok_1 ab \eta.b =
136 (show ab (Imply_intro _ _ b)).
138 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (⇒\sub\i \emsp ident H) " with precedence 19
139 for @{ 'Imply_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
140 interpretation "Imply_intro_ok_2" 'Imply_intro_ok_2 ab \eta.b =
141 (cast _ _ (show ab (Imply_intro _ _ b))).
143 notation > "⇒_'i' [ident H] term 90 b" with precedence 19
144 for @{ 'Imply_intro $b (λ${ident H}.show $b ?) }.
145 interpretation "Imply_intro KO" 'Imply_intro b pb = (cast _ (Imply unit b) (Imply_intro _ b pb)).
146 interpretation "Imply_intro OK" 'Imply_intro b pb = (Imply_intro _ b pb).
148 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 }.
149 interpretation "Imply_elim_ko_1" 'Imply_elim_ko_1 ab a b =
150 (show b (cast _ _ (Imply_elim _ _ ab a))).
151 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 }.
152 interpretation "Imply_elim_ko_2" 'Imply_elim_ko_2 ab a b =
153 (cast _ _ (show b (cast _ _ (Imply_elim _ _ ab a)))).
155 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19
156 for @{ 'Imply_elim_ok_1 $ab $a $b }.
157 interpretation "Imply_elim_ok_1" 'Imply_elim_ok_1 ab a b =
158 (show b (Imply_elim _ _ ab a)).
160 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (⇒\sub\e) " with precedence 19
161 for @{ 'Imply_elim_ok_2 $ab $a $b }.
162 interpretation "Imply_elim_ok_2" 'Imply_elim_ok_2 ab a b =
163 (cast _ _ (show b (Imply_elim _ _ ab a))).
165 notation > "⇒_'e' term 90 ab term 90 a" with precedence 19 for @{ 'Imply_elim (show $ab ?) (show $a ?) }.
166 interpretation "Imply_elim KO" 'Imply_elim ab a = (cast _ _ (Imply_elim _ _ (cast (Imply unit unit) _ ab) (cast unit _ a))).
167 interpretation "Imply_elim OK" 'Imply_elim ab a = (Imply_elim _ _ ab a).
169 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 }.
170 interpretation "And_intro_ko_1" 'And_intro_ko_1 a b ab =
171 (show ab (cast _ _ (And_intro _ _ a b))).
172 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 }.
173 interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab =
174 (cast _ _ (show ab (cast _ _ (And_intro _ _ a b)))).
176 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)" with precedence 19
177 for @{ 'And_intro_ok_1 $a $b $ab }.
178 interpretation "And_intro_ok_1" 'And_intro_ok_1 a b ab =
179 (show ab (And_intro _ _ a b)).
181 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) (∧\sub\i)" with precedence 19
182 for @{ 'And_intro_ok_2 $a $b $ab }.
183 interpretation "And_intro_ok_2" 'And_intro_ok_2 a b ab =
184 (cast _ _ (show ab (And_intro _ _ a b))).
186 notation > "∧_'i' term 90 a term 90 b" with precedence 19 for @{ 'And_intro (show $a ?) (show $b ?) }.
187 interpretation "And_intro KO" 'And_intro a b = (cast _ _ (And_intro _ _ a b)).
188 interpretation "And_intro OK" 'And_intro a b = (And_intro _ _ a b).
190 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
191 for @{ 'And_elim_l_ko_1 $ab $a }.
192 interpretation "And_elim_l_ko_1" 'And_elim_l_ko_1 ab a =
193 (show a (cast _ _ (And_elim_l _ _ ab))).
195 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
196 for @{ 'And_elim_l_ko_2 $ab $a }.
197 interpretation "And_elim_l_ko_2" 'And_elim_l_ko_2 ab a =
198 (cast _ _ (show a (cast _ _ (And_elim_l _ _ ab)))).
200 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))" with precedence 19
201 for @{ 'And_elim_l_ok_1 $ab $a }.
202 interpretation "And_elim_l_ok_1" 'And_elim_l_ok_1 ab a =
203 (show a (And_elim_l _ _ ab)).
205 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\l))" with precedence 19
206 for @{ 'And_elim_l_ok_2 $ab $a }.
207 interpretation "And_elim_l_ok_2" 'And_elim_l_ok_2 ab a =
208 (cast _ _ (show a (And_elim_l _ _ ab))).
210 notation > "∧_'e_l' term 90 ab" with precedence 19
211 for @{ 'And_elim_l (show $ab ?) }.
212 interpretation "And_elim_l KO" 'And_elim_l a = (And_elim_l _ _ (cast (And _ unit) _ a)).
213 interpretation "And_elim_l OK" 'And_elim_l a = (And_elim_l _ _ a).
215 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
216 for @{ 'And_elim_r_ko_1 $ab $a }.
217 interpretation "And_elim_r_ko_1" 'And_elim_r_ko_1 ab a =
218 (show a (cast _ _ (And_elim_r _ _ ab))).
220 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
221 for @{ 'And_elim_r_ko_2 $ab $a }.
222 interpretation "And_elim_r_ko_2" 'And_elim_r_ko_2 ab a =
223 (cast _ _ (show a (cast _ _ (And_elim_r _ _ ab)))).
225 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))" with precedence 19
226 for @{ 'And_elim_r_ok_1 $ab $a }.
227 interpretation "And_elim_r_ok_1" 'And_elim_r_ok_1 ab a =
228 (show a (And_elim_r _ _ ab)).
230 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\r))" with precedence 19
231 for @{ 'And_elim_r_ok_2 $ab $a }.
232 interpretation "And_elim_r_ok_2" 'And_elim_r_ok_2 ab a =
233 (cast _ _ (show a (And_elim_r _ _ ab))).
235 notation > "∧_'e_r' term 90 ab" with precedence 19
236 for @{ 'And_elim_r (show $ab ?) }.
237 interpretation "And_elim_r KO" 'And_elim_r a = (And_elim_r _ _ (cast (And unit _) _ a)).
238 interpretation "And_elim_r OK" 'And_elim_r a = (And_elim_r _ _ a).
240 notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))" with precedence 19
241 for @{ 'Or_intro_l_ok_1 $a $ab }.
242 interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab =
243 (show ab (Or_intro_l _ _ a)).
245 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19
246 for @{ 'Or_intro_l_ok_1 $a $ab }.
247 interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab =
248 (cast _ _ (show ab (Or_intro_l _ _ a))).
250 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
251 for @{ 'Or_intro_l_ko_1 $a $ab }.
252 interpretation "Or_intro_l_ko_1" 'Or_intro_l_ko_1 a ab =
253 (show ab (cast _ _ (Or_intro_l _ _ a))).
255 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
256 for @{ 'Or_intro_l_ko_2 $a $ab }.
257 interpretation "Or_intro_l_ko_2" 'Or_intro_l_ko_2 a ab =
258 (cast _ _ (show ab (cast _ _ (Or_intro_l _ _ a)))).
260 notation > "∨_'i_l' term 90 a" with precedence 19
261 for @{ 'Or_intro_l (show $a ?) }.
262 interpretation "Or_intro_l KO" 'Or_intro_l a = (cast _ (Or _ unit) (Or_intro_l _ _ a)).
263 interpretation "Or_intro_l OK" 'Or_intro_l a = (Or_intro_l _ _ a).
265 notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))" with precedence 19
266 for @{ 'Or_intro_r_ok_1 $a $ab }.
267 interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab =
268 (show ab (Or_intro_r _ _ a)).
270 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19
271 for @{ 'Or_intro_r_ok_1 $a $ab }.
272 interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab =
273 (cast _ _ (show ab (Or_intro_r _ _ a))).
275 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
276 for @{ 'Or_intro_r_ko_1 $a $ab }.
277 interpretation "Or_intro_r_ko_1" 'Or_intro_r_ko_1 a ab =
278 (show ab (cast _ _ (Or_intro_r _ _ a))).
280 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
281 for @{ 'Or_intro_r_ko_2 $a $ab }.
282 interpretation "Or_intro_r_ko_2" 'Or_intro_r_ko_2 a ab =
283 (cast _ _ (show ab (cast _ _ (Or_intro_r _ _ a)))).
285 notation > "∨_'i_r' term 90 a" with precedence 19
286 for @{ 'Or_intro_r (show $a ?) }.
287 interpretation "Or_intro_r KO" 'Or_intro_r a = (cast _ (Or unit _) (Or_intro_r _ _ a)).
288 interpretation "Or_intro_r OK" 'Or_intro_r a = (Or_intro_r _ _ a).
290 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
291 for @{ 'Or_elim_ok_1 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
292 interpretation "Or_elim_ok_1" 'Or_elim_ok_1 ab \eta.ac \eta.bc c =
293 (show c (Or_elim _ _ _ ab ac bc)).
295 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
296 for @{ 'Or_elim_ok_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
297 interpretation "Or_elim_ok_2" 'Or_elim_ok_2 ab \eta.ac \eta.bc c =
298 (cast _ _ (show c (Or_elim _ _ _ ab ac bc))).
300 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
301 for @{ 'Or_elim_ko_1 $ab $c (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) }.
302 interpretation "Or_elim_ko_1" 'Or_elim_ko_1 ab c \eta.ac \eta.bc =
303 (show c (cast _ _ (Or_elim _ _ _ ab (cast _ _ ac) (cast _ _ bc)))).
305 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
306 for @{ 'Or_elim_ko_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
307 interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c =
308 (cast _ _ (show c (cast _ _ (Or_elim _ _ _ ab (cast _ _ ac) (cast _ _ bc))))).
310 definition unit_to ≝ λx:CProp.unit → x.
312 notation > "∨_'e' term 90 ab [ident Ha] term 90 cl [ident Hb] term 90 cr" with precedence 19
313 for @{ 'Or_elim (show $ab ?) (λ${ident Ha}.show $cl ?) (λ${ident Hb}.show $cr ?) $cl $cr }.
314 interpretation "Or_elim KO" 'Or_elim ab ac bc c1 c2 =
315 (cast _ _ (Or_elim _ _ _ (cast (Or unit unit) _ ab) (cast (unit_to unit) (unit_to _) ac) (cast (unit_to unit) (unit_to _) bc))).
316 interpretation "Or_elim OK" 'Or_elim ab ac bc c1 c2 = (Or_elim _ _ _ ab ac bc).
318 notation < "\infrule \nbsp ⊤ mstyle color #ff0000 (⊤\sub\i)" with precedence 19
319 for @{'Top_intro_ko_1}.
320 interpretation "Top_intro_ko_1" 'Top_intro_ko_1 = (show _ (cast _ _ Top_intro)).
322 notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19
323 for @{'Top_intro_ok_1}.
324 interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show _ Top_intro).
326 notation > "⊤_'i'" with precedence 19 for @{ 'Top_intro }.
327 interpretation "Top_intro KO" 'Top_intro = (cast _ _ Top_intro).
328 interpretation "Top_intro OK" 'Top_intro = Top_intro.
330 notation < "\infrule b a (⊥\sub\e)" with precedence 19 for @{'Bot_elim_ok_1 $a $b}.
331 interpretation "Bot_elim_ok_1" 'Bot_elim_ok_1 a b =
332 (show a (Bot_elim a b)).
334 notation < "\infrule b a mstyle color #ff0000 (⊥\sub\e)" with precedence 19 for @{'Bot_elim_ko_1 $a $b}.
335 interpretation "Bot_elim_ko_1" 'Bot_elim_ko_1 a b =
336 (show a (Bot_elim a (cast _ _ b))).
338 notation > "⊥_'e' term 90 b" with precedence 19
339 for @{ 'Bot_elim (show $b ?) }.
340 interpretation "Bot_elim KO" 'Bot_elim a = (Bot_elim _ (cast _ _ a)).
341 interpretation "Bot_elim OK" 'Bot_elim a = (Bot_elim _ a).
343 notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
344 for @{ 'Not_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
345 interpretation "Not_intro_ko_1" 'Not_intro_ko_1 ab \eta.b =
346 (show ab (cast _ _ (Not_intro _ (cast _ _ b)))).
348 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
349 for @{ 'Not_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
350 interpretation "Not_intro_ko_2" 'Not_intro_ko_2 ab \eta.b =
351 (cast _ _ (show ab (cast _ _ (Not_intro _ (cast _ _ b))))).
353 notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub\i \emsp ident H) " with precedence 19
354 for @{ 'Not_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
355 interpretation "Not_intro_ok_1" 'Not_intro_ok_1 ab \eta.b =
356 (show ab (Not_intro _ b)).
358 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub\i \emsp ident H) " with precedence 19
359 for @{ 'Not_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
360 interpretation "Not_intro_ok_2" 'Not_intro_ok_2 ab \eta.b =
361 (cast _ _ (show ab (Not_intro _ b))).
363 notation > "¬_'i' [ident H] term 90 b" with precedence 19
364 for @{ 'Not_intro (λ${ident H}.show $b ?) }.
365 interpretation "Not_intro KO" 'Not_intro a = (cast _ _ (Not_intro _ (cast _ _ a))).
366 interpretation "Not_intro OK" 'Not_intro a = (Not_intro _ a).
368 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub\e) " with precedence 19
369 for @{ 'Not_elim_ok_1 $ab $a $b }.
370 interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b =
371 (show b (Not_elim _ ab a)).
373 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub\e) " with precedence 19
374 for @{ 'Not_elim_ok_2 $ab $a $b }.
375 interpretation "Not_elim_ok_2" 'Not_elim_ok_2 ab a b =
376 (cast _ _ (show b (Not_elim _ ab a))).
378 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
379 for @{ 'Not_elim_ko_1 $ab $a $b }.
380 interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b =
381 (show b (Not_elim _ (cast _ _ ab) a)).
383 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
384 for @{ 'Not_elim_ko_2 $ab $a $b }.
385 interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b =
386 (cast _ _ (show b (Not_elim _ (cast _ _ ab) a))).
388 notation > "¬_'e' term 90 ab term 90 a" with precedence 19
389 for @{ 'Not_elim (show $ab ?) (show $a ?) }.
390 interpretation "Not_elim KO" 'Not_elim ab a = (Not_elim _ (cast _ _ ab) a).
391 interpretation "Not_elim OK" 'Not_elim ab a = (Not_elim _ ab a).
393 notation < "\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)" with precedence 19
394 for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
395 interpretation "RAA_ok_1" 'RAA_ok_1 Px Pn =
396 (show Pn (Raa _ Px)).
398 notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (\RAA \emsp ident x)" with precedence 19
399 for @{ 'RAA_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
400 interpretation "RAA_ok_2" 'RAA_ok_2 Px Pn =
401 (cast _ _ (show Pn (Raa _ Px))).
403 notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
404 for @{ 'RAA_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
405 interpretation "RAA_ko_1" 'RAA_ko_1 Px Pn =
406 (show Pn (Raa _ (cast _ _ Px))).
408 notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
409 for @{ 'RAA_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
410 interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn =
411 (cast _ _ (show Pn (Raa _ (cast _ _ Px)))).
413 notation > "'RAA' [ident H] term 90 b" with precedence 19
414 for @{ 'Raa (λ${ident H}.show $b ?) }.
415 interpretation "RAA KO" 'Raa p = (Raa _ (cast _ (unit_to _) p)).
416 interpretation "RAA OK" 'Raa p = (Raa _ p).
429 ∨_e (…) […] (…) […] (…)