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 Discharge := 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:
94 `A` becomes `(show A ?)` *)
95 definition show : ∀A.A→A ≝ λA:CProp.λa:A.a.
97 (* When something does not fit, this daemon is used *)
98 axiom cast: ∀A,B:CProp.B → A.
101 notation < "\infrule (t\atop ⋮) a ?" with precedence 19
102 for @{ 'leaf_ok $a $t }.
103 interpretation "leaf OK" 'leaf_ok a t = (show a t).
104 notation < "\infrule (t\atop ⋮) mstyle color #ff0000 (a) ?" with precedence 19
105 for @{ 'leaf_ko $a $t }.
106 interpretation "leaf KO" 'leaf_ko a t = (cast _ a (show _ t)).
108 (* begin a proof: draws the root *)
109 notation > "'prove' p" non associative with precedence 19
111 interpretation "prove KO" 'prove p = (cast _ _ (show p _)).
112 interpretation "prove OK" 'prove p = (show p _).
115 notation < "[ a ] \sup mstyle color #ff0000 (H)" with precedence 19
116 for @{ 'discharge_ko_1 $a $H }.
117 interpretation "discharge_ko_1" 'discharge_ko_1 a H =
118 (show a (cast _ _ (Discharge _ H))).
119 notation < "[ mstyle color #ff0000 (a) ] \sup mstyle color #ff0000 (H)" with precedence 19
120 for @{ 'discharge_ko_2 $a $H }.
121 interpretation "discharge_ko_2" 'discharge_ko_2 a H =
122 (cast _ _ (show a (cast _ _ (Discharge _ H)))).
124 notation < "[ a ] \sup H" with precedence 19
125 for @{ 'discharge_ok_1 $a $H }.
126 interpretation "discharge_ok_1" 'discharge_ok_1 a H =
127 (show a (Discharge _ H)).
128 notation < "[ mstyle color #ff0000 (a) ] \sup H" with precedence 19
129 for @{ 'discharge_ok_2 $a $H }.
130 interpretation "discharge_ok_2" 'discharge_ok_2 a H =
131 (cast _ _ (show a (Discharge _ H))).
133 notation > "'discharge' [H]" with precedence 19
134 for @{ 'discharge $H }.
135 interpretation "discharge KO" 'discharge H = (cast _ _ (Discharge _ H)).
136 interpretation "discharge OK" 'discharge H = (Discharge _ H).
139 notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
140 for @{ 'Imply_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
141 interpretation "Imply_intro_ko_1" 'Imply_intro_ko_1 ab \eta.b =
142 (show ab (cast _ _ (Imply_intro _ _ b))).
144 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (⇒\sub\i \emsp) ident H) " with precedence 19
145 for @{ 'Imply_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
146 interpretation "Imply_intro_ko_2" 'Imply_intro_ko_2 ab \eta.b =
147 (cast _ _ (show ab (cast _ _ (Imply_intro _ _ b)))).
149 notation < "\infrule hbox(\emsp b \emsp) ab (⇒\sub\i \emsp ident H) " with precedence 19
150 for @{ 'Imply_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
151 interpretation "Imply_intro_ok_1" 'Imply_intro_ok_1 ab \eta.b =
152 (show ab (Imply_intro _ _ b)).
154 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (⇒\sub\i \emsp ident H) " with precedence 19
155 for @{ 'Imply_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
156 interpretation "Imply_intro_ok_2" 'Imply_intro_ok_2 ab \eta.b =
157 (cast _ _ (show ab (Imply_intro _ _ b))).
159 notation > "⇒_'i' [ident H] term 90 b" with precedence 19
160 for @{ 'Imply_intro $b (λ${ident H}.show $b ?) }.
161 interpretation "Imply_intro KO" 'Imply_intro b pb =
162 (cast _ (Imply unit b) (Imply_intro _ b pb)).
163 interpretation "Imply_intro OK" 'Imply_intro b pb =
164 (Imply_intro _ b pb).
167 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (⇒\sub\e) " with precedence 19
168 for @{ 'Imply_elim_ko_1 $ab $a $b }.
169 interpretation "Imply_elim_ko_1" 'Imply_elim_ko_1 ab a b =
170 (show b (cast _ _ (Imply_elim _ _ (cast _ _ ab) (cast _ _ a)))).
172 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (⇒\sub\e) " with precedence 19
173 for @{ 'Imply_elim_ko_2 $ab $a $b }.
174 interpretation "Imply_elim_ko_2" 'Imply_elim_ko_2 ab a b =
175 (cast _ _ (show b (cast _ _ (Imply_elim _ _ (cast _ _ ab) (cast _ _ a))))).
177 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (⇒\sub\e) " with precedence 19
178 for @{ 'Imply_elim_ok_1 $ab $a $b }.
179 interpretation "Imply_elim_ok_1" 'Imply_elim_ok_1 ab a b =
180 (show b (Imply_elim _ _ ab a)).
182 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (⇒\sub\e) " with precedence 19
183 for @{ 'Imply_elim_ok_2 $ab $a $b }.
184 interpretation "Imply_elim_ok_2" 'Imply_elim_ok_2 ab a b =
185 (cast _ _ (show b (Imply_elim _ _ ab a))).
187 notation > "⇒_'e' term 90 ab term 90 a" with precedence 19
188 for @{ 'Imply_elim (show $ab ?) (show $a ?) }.
189 interpretation "Imply_elim KO" 'Imply_elim ab a =
190 (cast _ _ (Imply_elim _ _ (cast (Imply unit unit) _ ab) (cast unit _ a))).
191 interpretation "Imply_elim OK" 'Imply_elim ab a =
192 (Imply_elim _ _ ab a).
195 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab mstyle color #ff0000 (∧\sub\i)" with precedence 19
196 for @{ 'And_intro_ko_1 $a $b $ab }.
197 interpretation "And_intro_ko_1" 'And_intro_ko_1 a b ab =
198 (show ab (cast _ _ (And_intro _ _ a b))).
200 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∧\sub\i)" with precedence 19
201 for @{ 'And_intro_ko_2 $a $b $ab }.
202 interpretation "And_intro_ko_2" 'And_intro_ko_2 a b ab =
203 (cast _ _ (show ab (cast _ _ (And_intro _ _ a b)))).
205 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) ab (∧\sub\i)" with precedence 19
206 for @{ 'And_intro_ok_1 $a $b $ab }.
207 interpretation "And_intro_ok_1" 'And_intro_ok_1 a b ab =
208 (show ab (And_intro _ _ a b)).
210 notation < "\infrule hbox(\emsp a \emsp\emsp\emsp b \emsp) mstyle color #ff0000 (ab) (∧\sub\i)" with precedence 19
211 for @{ 'And_intro_ok_2 $a $b $ab }.
212 interpretation "And_intro_ok_2" 'And_intro_ok_2 a b ab =
213 (cast _ _ (show ab (And_intro _ _ a b))).
215 notation > "∧_'i' term 90 a term 90 b" with precedence 19
216 for @{ 'And_intro (show $a ?) (show $b ?) }.
217 interpretation "And_intro KO" 'And_intro a b = (cast _ _ (And_intro _ _ a b)).
218 interpretation "And_intro OK" 'And_intro a b = (And_intro _ _ a b).
221 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
222 for @{ 'And_elim_l_ko_1 $ab $a }.
223 interpretation "And_elim_l_ko_1" 'And_elim_l_ko_1 ab a =
224 (show a (cast _ _ (And_elim_l _ _ (cast _ _ ab)))).
226 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\l))" with precedence 19
227 for @{ 'And_elim_l_ko_2 $ab $a }.
228 interpretation "And_elim_l_ko_2" 'And_elim_l_ko_2 ab a =
229 (cast _ _ (show a (cast _ _ (And_elim_l _ _ (cast _ _ ab))))).
231 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\l))" with precedence 19
232 for @{ 'And_elim_l_ok_1 $ab $a }.
233 interpretation "And_elim_l_ok_1" 'And_elim_l_ok_1 ab a =
234 (show a (And_elim_l _ _ ab)).
236 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\l))" with precedence 19
237 for @{ 'And_elim_l_ok_2 $ab $a }.
238 interpretation "And_elim_l_ok_2" 'And_elim_l_ok_2 ab a =
239 (cast _ _ (show a (And_elim_l _ _ ab))).
241 notation > "∧_'e_l' term 90 ab" with precedence 19
242 for @{ 'And_elim_l (show $ab ?) }.
243 interpretation "And_elim_l KO" 'And_elim_l a = (cast _ _ (And_elim_l _ _ (cast (And unit unit) _ a))).
244 interpretation "And_elim_l OK" 'And_elim_l a = (And_elim_l _ _ a).
246 notation < "\infrule hbox(\emsp ab \emsp) a mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
247 for @{ 'And_elim_r_ko_1 $ab $a }.
248 interpretation "And_elim_r_ko_1" 'And_elim_r_ko_1 ab a =
249 (show a (cast _ _ (And_elim_r _ _ (cast _ _ ab)))).
251 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) mstyle color #ff0000 (∧\sub(\e_\r))" with precedence 19
252 for @{ 'And_elim_r_ko_2 $ab $a }.
253 interpretation "And_elim_r_ko_2" 'And_elim_r_ko_2 ab a =
254 (cast _ _ (show a (cast _ _ (And_elim_r _ _ (cast _ _ ab))))).
256 notation < "\infrule hbox(\emsp ab \emsp) a (∧\sub(\e_\r))" with precedence 19
257 for @{ 'And_elim_r_ok_1 $ab $a }.
258 interpretation "And_elim_r_ok_1" 'And_elim_r_ok_1 ab a =
259 (show a (And_elim_r _ _ ab)).
261 notation < "\infrule hbox(\emsp ab \emsp) mstyle color #ff0000 (a) (∧\sub(\e_\r))" with precedence 19
262 for @{ 'And_elim_r_ok_2 $ab $a }.
263 interpretation "And_elim_r_ok_2" 'And_elim_r_ok_2 ab a =
264 (cast _ _ (show a (And_elim_r _ _ ab))).
266 notation > "∧_'e_r' term 90 ab" with precedence 19
267 for @{ 'And_elim_r (show $ab ?) }.
268 interpretation "And_elim_r KO" 'And_elim_r a = (cast _ _ (And_elim_r _ _ (cast (And unit unit) _ a))).
269 interpretation "And_elim_r OK" 'And_elim_r a = (And_elim_r _ _ a).
272 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
273 for @{ 'Or_intro_l_ko_1 $a $ab }.
274 interpretation "Or_intro_l_ko_1" 'Or_intro_l_ko_1 a ab =
275 (show ab (cast _ _ (Or_intro_l _ _ a))).
277 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\l))" with precedence 19
278 for @{ 'Or_intro_l_ko_2 $a $ab }.
279 interpretation "Or_intro_l_ko_2" 'Or_intro_l_ko_2 a ab =
280 (cast _ _ (show ab (cast _ _ (Or_intro_l _ _ a)))).
282 notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\l))" with precedence 19
283 for @{ 'Or_intro_l_ok_1 $a $ab }.
284 interpretation "Or_intro_l_ok_1" 'Or_intro_l_ok_1 a ab =
285 (show ab (Or_intro_l _ _ a)).
287 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\l))" with precedence 19
288 for @{ 'Or_intro_l_ok_1 $a $ab }.
289 interpretation "Or_intro_l_ok_2" 'Or_intro_l_ok_2 a ab =
290 (cast _ _ (show ab (Or_intro_l _ _ a))).
292 notation > "∨_'i_l' term 90 a" with precedence 19
293 for @{ 'Or_intro_l (show $a ?) }.
294 interpretation "Or_intro_l KO" 'Or_intro_l a = (cast _ (Or _ unit) (Or_intro_l _ _ a)).
295 interpretation "Or_intro_l OK" 'Or_intro_l a = (Or_intro_l _ _ a).
297 notation < "\infrule hbox(\emsp a \emsp) ab mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
298 for @{ 'Or_intro_r_ko_1 $a $ab }.
299 interpretation "Or_intro_r_ko_1" 'Or_intro_r_ko_1 a ab =
300 (show ab (cast _ _ (Or_intro_r _ _ a))).
302 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) mstyle color #ff0000 (∨\sub(\i_\r))" with precedence 19
303 for @{ 'Or_intro_r_ko_2 $a $ab }.
304 interpretation "Or_intro_r_ko_2" 'Or_intro_r_ko_2 a ab =
305 (cast _ _ (show ab (cast _ _ (Or_intro_r _ _ a)))).
307 notation < "\infrule hbox(\emsp a \emsp) ab (∨\sub(\i_\r))" with precedence 19
308 for @{ 'Or_intro_r_ok_1 $a $ab }.
309 interpretation "Or_intro_r_ok_1" 'Or_intro_r_ok_1 a ab =
310 (show ab (Or_intro_r _ _ a)).
312 notation < "\infrule hbox(\emsp a \emsp) mstyle color #ff0000 (ab) (∨\sub(\i_\r))" with precedence 19
313 for @{ 'Or_intro_r_ok_1 $a $ab }.
314 interpretation "Or_intro_r_ok_2" 'Or_intro_r_ok_2 a ab =
315 (cast _ _ (show ab (Or_intro_r _ _ a))).
317 notation > "∨_'i_r' term 90 a" with precedence 19
318 for @{ 'Or_intro_r (show $a ?) }.
319 interpretation "Or_intro_r KO" 'Or_intro_r a = (cast _ (Or unit _) (Or_intro_r _ _ a)).
320 interpretation "Or_intro_r OK" 'Or_intro_r a = (Or_intro_r _ _ a).
323 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
324 for @{ 'Or_elim_ko_1 $ab $c (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) }.
325 interpretation "Or_elim_ko_1" 'Or_elim_ko_1 ab c \eta.ac \eta.bc =
326 (show c (cast _ _ (Or_elim _ _ _ (cast _ _ ab) (cast _ _ ac) (cast _ _ bc)))).
328 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
329 for @{ 'Or_elim_ko_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
330 interpretation "Or_elim_ko_2" 'Or_elim_ko_2 ab \eta.ac \eta.bc c =
331 (cast _ _ (show c (cast _ _ (Or_elim _ _ _ (cast _ _ ab) (cast _ _ ac) (cast _ _ bc))))).
333 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
334 for @{ 'Or_elim_ok_1 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
335 interpretation "Or_elim_ok_1" 'Or_elim_ok_1 ab \eta.ac \eta.bc c =
336 (show c (Or_elim _ _ _ ab ac bc)).
338 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
339 for @{ 'Or_elim_ok_2 $ab (λ${ident Ha}:$ta.$ac) (λ${ident Hb}:$tb.$bc) $c }.
340 interpretation "Or_elim_ok_2" 'Or_elim_ok_2 ab \eta.ac \eta.bc c =
341 (cast _ _ (show c (Or_elim _ _ _ ab ac bc))).
343 definition unit_to ≝ λx:CProp.unit → x.
345 notation > "∨_'e' term 90 ab [ident Ha] term 90 cl [ident Hb] term 90 cr" with precedence 19
346 for @{ 'Or_elim (show $ab ?) (λ${ident Ha}.show $cl ?) (λ${ident Hb}.show $cr ?) }.
347 interpretation "Or_elim KO" 'Or_elim ab ac bc =
348 (cast _ _ (Or_elim _ _ _
349 (cast (Or unit unit) _ ab)
350 (cast (unit_to unit) (unit_to _) ac)
351 (cast (unit_to unit) (unit_to _) bc))).
352 interpretation "Or_elim OK" 'Or_elim ab ac bc = (Or_elim _ _ _ ab ac bc).
355 notation < "\infrule \nbsp ⊤ mstyle color #ff0000 (⊤\sub\i)" with precedence 19
356 for @{'Top_intro_ko_1}.
357 interpretation "Top_intro_ko_1" 'Top_intro_ko_1 =
358 (show _ (cast _ _ Top_intro)).
360 notation < "\infrule \nbsp mstyle color #ff0000 (⊤) mstyle color #ff0000 (⊤\sub\i)" with precedence 19
361 for @{'Top_intro_ko_2}.
362 interpretation "Top_intro_ko_2" 'Top_intro_ko_2 =
363 (cast _ _ (show _ (cast _ _ Top_intro))).
365 notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19
366 for @{'Top_intro_ok_1}.
367 interpretation "Top_intro_ok_1" 'Top_intro_ok_1 = (show _ Top_intro).
369 notation < "\infrule \nbsp ⊤ (⊤\sub\i)" with precedence 19
370 for @{'Top_intro_ok_2 }.
371 interpretation "Top_intro_ok_2" 'Top_intro_ok_2 = (cast _ _ (show _ Top_intro)).
373 notation > "⊤_'i'" with precedence 19 for @{ 'Top_intro }.
374 interpretation "Top_intro KO" 'Top_intro = (cast _ _ Top_intro).
375 interpretation "Top_intro OK" 'Top_intro = Top_intro.
378 notation < "\infrule b a mstyle color #ff0000 (⊥\sub\e)" with precedence 19
379 for @{'Bot_elim_ko_1 $a $b}.
380 interpretation "Bot_elim_ko_1" 'Bot_elim_ko_1 a b =
381 (show a (Bot_elim _ (cast _ _ b))).
383 notation < "\infrule b mstyle color #ff0000 (a) mstyle color #ff0000 (⊥\sub\e)" with precedence 19
384 for @{'Bot_elim_ko_2 $a $b}.
385 interpretation "Bot_elim_ko_2" 'Bot_elim_ko_2 a b =
386 (cast _ _ (show a (Bot_elim _ (cast _ _ b)))).
388 notation < "\infrule b a (⊥\sub\e)" with precedence 19
389 for @{'Bot_elim_ok_1 $a $b}.
390 interpretation "Bot_elim_ok_1" 'Bot_elim_ok_1 a b =
391 (show a (Bot_elim _ b)).
393 notation < "\infrule b mstyle color #ff0000 (a) (⊥\sub\e)" with precedence 19
394 for @{'Bot_elim_ok_2 $a $b}.
395 interpretation "Bot_elim_ok_2" 'Bot_elim_ok_2 a b =
396 (cast _ _ (show a (Bot_elim _ b))).
398 notation > "⊥_'e' term 90 b" with precedence 19
399 for @{ 'Bot_elim (show $b ?) }.
400 interpretation "Bot_elim KO" 'Bot_elim a = (Bot_elim _ (cast _ _ a)).
401 interpretation "Bot_elim OK" 'Bot_elim a = (Bot_elim _ a).
404 notation < "\infrule hbox(\emsp b \emsp) ab (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
405 for @{ 'Not_intro_ko_1 $ab (λ${ident H}:$p.$b) }.
406 interpretation "Not_intro_ko_1" 'Not_intro_ko_1 ab \eta.b =
407 (show ab (cast _ _ (Not_intro _ (cast _ _ b)))).
409 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (mstyle color #ff0000 (\lnot\sub\i) \emsp ident H)" with precedence 19
410 for @{ 'Not_intro_ko_2 $ab (λ${ident H}:$p.$b) }.
411 interpretation "Not_intro_ko_2" 'Not_intro_ko_2 ab \eta.b =
412 (cast _ _ (show ab (cast _ _ (Not_intro _ (cast _ _ b))))).
414 notation < "\infrule hbox(\emsp b \emsp) ab (\lnot\sub\i \emsp ident H) " with precedence 19
415 for @{ 'Not_intro_ok_1 $ab (λ${ident H}:$p.$b) }.
416 interpretation "Not_intro_ok_1" 'Not_intro_ok_1 ab \eta.b =
417 (show ab (Not_intro _ b)).
419 notation < "\infrule hbox(\emsp b \emsp) mstyle color #ff0000 (ab) (\lnot\sub\i \emsp ident H) " with precedence 19
420 for @{ 'Not_intro_ok_2 $ab (λ${ident H}:$p.$b) }.
421 interpretation "Not_intro_ok_2" 'Not_intro_ok_2 ab \eta.b =
422 (cast _ _ (show ab (Not_intro _ b))).
424 notation > "¬_'i' [ident H] term 90 b" with precedence 19
425 for @{ 'Not_intro (λ${ident H}.show $b ?) }.
426 interpretation "Not_intro KO" 'Not_intro a = (cast _ _ (Not_intro _ (cast _ _ a))).
427 interpretation "Not_intro OK" 'Not_intro a = (Not_intro _ a).
430 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
431 for @{ 'Not_elim_ko_1 $ab $a $b }.
432 interpretation "Not_elim_ko_1" 'Not_elim_ko_1 ab a b =
433 (show b (cast _ _ (Not_elim _ (cast _ _ ab) a))).
435 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) mstyle color #ff0000 (\lnot\sub\e) " with precedence 19
436 for @{ 'Not_elim_ko_2 $ab $a $b }.
437 interpretation "Not_elim_ko_2" 'Not_elim_ko_2 ab a b =
438 (cast _ _ (show b (cast _ _ (Not_elim _ (cast _ _ ab) a)))).
440 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) b (\lnot\sub\e) " with precedence 19
441 for @{ 'Not_elim_ok_1 $ab $a $b }.
442 interpretation "Not_elim_ok_1" 'Not_elim_ok_1 ab a b =
443 (show b (Not_elim _ ab a)).
445 notation < "\infrule hbox(\emsp ab \emsp\emsp\emsp a\emsp) mstyle color #ff0000 (b) (\lnot\sub\e) " with precedence 19
446 for @{ 'Not_elim_ok_2 $ab $a $b }.
447 interpretation "Not_elim_ok_2" 'Not_elim_ok_2 ab a b =
448 (cast _ _ (show b (Not_elim _ ab a))).
450 notation > "¬_'e' term 90 ab term 90 a" with precedence 19
451 for @{ 'Not_elim (show $ab ?) (show $a ?) }.
452 interpretation "Not_elim KO" 'Not_elim ab a = (cast _ unit (Not_elim _ (cast _ _ ab) a)).
453 interpretation "Not_elim OK" 'Not_elim ab a = (Not_elim _ ab a).
456 notation < "\infrule hbox(\emsp Px \emsp) Pn (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
457 for @{ 'RAA_ko_1 (λ${ident x}:$tx.$Px) $Pn }.
458 interpretation "RAA_ko_1" 'RAA_ko_1 Px Pn =
459 (show Pn (cast _ _ (Raa _ (cast _ _ Px)))).
461 notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (mstyle color #ff0000 (\RAA) \emsp ident x)" with precedence 19
462 for @{ 'RAA_ko_2 (λ${ident x}:$tx.$Px) $Pn }.
463 interpretation "RAA_ko_2" 'RAA_ko_2 Px Pn =
464 (cast _ _ (show Pn (cast _ _ (Raa _ (cast _ _ Px))))).
466 notation < "\infrule hbox(\emsp Px \emsp) Pn (\RAA \emsp ident x)" with precedence 19
467 for @{ 'RAA_ok_1 (λ${ident x}:$tx.$Px) $Pn }.
468 interpretation "RAA_ok_1" 'RAA_ok_1 Px Pn =
469 (show Pn (Raa _ Px)).
471 notation < "\infrule hbox(\emsp Px \emsp) mstyle color #ff0000 (Pn) (\RAA \emsp ident x)" with precedence 19
472 for @{ 'RAA_ok_2 (λ${ident x}:$tx.$Px) $Pn }.
473 interpretation "RAA_ok_2" 'RAA_ok_2 Px Pn =
474 (cast _ _ (show Pn (Raa _ Px))).
476 notation > "'RAA' [ident H] term 90 b" with precedence 19
477 for @{ 'Raa (λ${ident H}.show $b ?) }.
478 interpretation "RAA KO" 'Raa p = (cast _ unit (Raa _ (cast _ (unit_to _) p))).
479 interpretation "RAA OK" 'Raa p = (Raa _ p).
484 apply rule (⇒_i […] (…)).
486 apply rule (∧_i (…) (…));
491 apply rule (∨_i_l (…)).
493 apply rule (∨_i_r (…)).
495 apply rule (¬_i […] (…)).
499 apply rule (⇒_e (…) (…));
504 apply rule (∧_e_l (…)).
506 apply rule (∧_e_r (…)).
508 apply rule (∨_e (…) […] (…) […] (…));
514 apply rule (¬_e (…) (…));
519 apply rule (⊥_e (…)).
521 apply rule (prove (…)).
523 apply rule (RAA […] (…)).
525 apply rule (discharge […]).