Added example file for intuitionistic logic (taken from FOL).
authorberghofe
Mon Dec 10 15:36:05 2001 +0100 (2001-12-10)
changeset 124501162b280700a
parent 12449 95fb2e206dc7
child 12451 0224f472be71
Added example file for intuitionistic logic (taken from FOL).
src/HOL/ex/Intuitionistic.thy
src/HOL/ex/ROOT.ML
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/ex/Intuitionistic.thy	Mon Dec 10 15:36:05 2001 +0100
     1.3 @@ -0,0 +1,329 @@
     1.4 +(*  Title:      HOL/ex/Intuitionistic.thy
     1.5 +    ID:         $Id$
     1.6 +    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     1.7 +    Copyright   1991  University of Cambridge
     1.8 +
     1.9 +Higher-Order Logic: Intuitionistic predicate calculus problems
    1.10 +
    1.11 +Taken from FOL/ex/int.ML
    1.12 +*)
    1.13 +
    1.14 +theory Intuitionistic = Main:
    1.15 +
    1.16 +
    1.17 +(*Metatheorem (for PROPOSITIONAL formulae...):
    1.18 +  P is classically provable iff ~~P is intuitionistically provable.
    1.19 +  Therefore ~P is classically provable iff it is intuitionistically provable.  
    1.20 +
    1.21 +Proof: Let Q be the conjuction of the propositions A|~A, one for each atom A
    1.22 +in P.  Now ~~Q is intuitionistically provable because ~~(A|~A) is and because
    1.23 +~~ distributes over &.  If P is provable classically, then clearly Q-->P is
    1.24 +provable intuitionistically, so ~~(Q-->P) is also provable intuitionistically.
    1.25 +The latter is intuitionistically equivalent to ~~Q-->~~P, hence to ~~P, since
    1.26 +~~Q is intuitionistically provable.  Finally, if P is a negation then ~~P is
    1.27 +intuitionstically equivalent to P.  [Andy Pitts] *)
    1.28 +
    1.29 +lemma "(~~(P&Q)) = ((~~P) & (~~Q))"
    1.30 +  by rules
    1.31 +
    1.32 +lemma "~~ ((~P --> Q) --> (~P --> ~Q) --> P)"
    1.33 +  by rules
    1.34 +
    1.35 +(* ~~ does NOT distribute over | *)
    1.36 +
    1.37 +lemma "(~~(P-->Q))  = (~~P --> ~~Q)"
    1.38 +  by rules
    1.39 +
    1.40 +lemma "(~~~P) = (~P)"
    1.41 +  by rules
    1.42 +
    1.43 +lemma "~~((P --> Q | R)  -->  (P-->Q) | (P-->R))"
    1.44 +  by rules
    1.45 +
    1.46 +lemma "(P=Q) = (Q=P)"
    1.47 +  by rules
    1.48 +
    1.49 +lemma "((P --> (Q | (Q-->R))) --> R) --> R"
    1.50 +  by rules
    1.51 +
    1.52 +lemma "(((G-->A) --> J) --> D --> E) --> (((H-->B)-->I)-->C-->J)
    1.53 +      --> (A-->H) --> F --> G --> (((C-->B)-->I)-->D)-->(A-->C)
    1.54 +      --> (((F-->A)-->B) --> I) --> E"
    1.55 +  by rules
    1.56 +
    1.57 +
    1.58 +(* Lemmas for the propositional double-negation translation *)
    1.59 +
    1.60 +lemma "P --> ~~P"
    1.61 +  by rules
    1.62 +
    1.63 +lemma "~~(~~P --> P)"
    1.64 +  by rules
    1.65 +
    1.66 +lemma "~~P & ~~(P --> Q) --> ~~Q"
    1.67 +  by rules
    1.68 +
    1.69 +
    1.70 +(* de Bruijn formulae *)
    1.71 +
    1.72 +(*de Bruijn formula with three predicates*)
    1.73 +lemma "((P=Q) --> P&Q&R) &
    1.74 +       ((Q=R) --> P&Q&R) &
    1.75 +       ((R=P) --> P&Q&R) --> P&Q&R"
    1.76 +  by rules
    1.77 +
    1.78 +(*de Bruijn formula with five predicates*)
    1.79 +lemma "((P=Q) --> P&Q&R&S&T) &
    1.80 +       ((Q=R) --> P&Q&R&S&T) &
    1.81 +       ((R=S) --> P&Q&R&S&T) &
    1.82 +       ((S=T) --> P&Q&R&S&T) &
    1.83 +       ((T=P) --> P&Q&R&S&T) --> P&Q&R&S&T"
    1.84 +  by rules
    1.85 +
    1.86 +
    1.87 +(*** Problems from Sahlin, Franzen and Haridi, 
    1.88 +     An Intuitionistic Predicate Logic Theorem Prover.
    1.89 +     J. Logic and Comp. 2 (5), October 1992, 619-656.
    1.90 +***)
    1.91 +
    1.92 +(*Problem 1.1*)
    1.93 +lemma "(ALL x. EX y. ALL z. p(x) & q(y) & r(z)) =
    1.94 +       (ALL z. EX y. ALL x. p(x) & q(y) & r(z))"
    1.95 +  by (rules del: allE elim 2: allE')
    1.96 +
    1.97 +(*Problem 3.1*)
    1.98 +lemma "~ (EX x. ALL y. p y x = (~ p x x))"
    1.99 +  by rules
   1.100 +
   1.101 +
   1.102 +(* Intuitionistic FOL: propositional problems based on Pelletier. *)
   1.103 +
   1.104 +(* Problem ~~1 *)
   1.105 +lemma "~~((P-->Q)  =  (~Q --> ~P))"
   1.106 +  by rules
   1.107 +
   1.108 +(* Problem ~~2 *)
   1.109 +lemma "~~(~~P  =  P)"
   1.110 +  by rules
   1.111 +
   1.112 +(* Problem 3 *)
   1.113 +lemma "~(P-->Q) --> (Q-->P)"
   1.114 +  by rules
   1.115 +
   1.116 +(* Problem ~~4 *)
   1.117 +lemma "~~((~P-->Q)  =  (~Q --> P))"
   1.118 +  by rules
   1.119 +
   1.120 +(* Problem ~~5 *)
   1.121 +lemma "~~((P|Q-->P|R) --> P|(Q-->R))"
   1.122 +  by rules
   1.123 +
   1.124 +(* Problem ~~6 *)
   1.125 +lemma "~~(P | ~P)"
   1.126 +  by rules
   1.127 +
   1.128 +(* Problem ~~7 *)
   1.129 +lemma "~~(P | ~~~P)"
   1.130 +  by rules
   1.131 +
   1.132 +(* Problem ~~8.  Peirce's law *)
   1.133 +lemma "~~(((P-->Q) --> P)  -->  P)"
   1.134 +  by rules
   1.135 +
   1.136 +(* Problem 9 *)
   1.137 +lemma "((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
   1.138 +  by rules
   1.139 +
   1.140 +(* Problem 10 *)
   1.141 +lemma "(Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P=Q)"
   1.142 +  by rules
   1.143 +
   1.144 +(* 11.  Proved in each direction (incorrectly, says Pelletier!!) *)
   1.145 +lemma "P=P"
   1.146 +  by rules
   1.147 +
   1.148 +(* Problem ~~12.  Dijkstra's law *)
   1.149 +lemma "~~(((P = Q) = R)  =  (P = (Q = R)))"
   1.150 +  by rules
   1.151 +
   1.152 +lemma "((P = Q) = R)  -->  ~~(P = (Q = R))"
   1.153 +  by rules
   1.154 +
   1.155 +(* Problem 13.  Distributive law *)
   1.156 +lemma "(P | (Q & R))  = ((P | Q) & (P | R))"
   1.157 +  by rules
   1.158 +
   1.159 +(* Problem ~~14 *)
   1.160 +lemma "~~((P = Q) = ((Q | ~P) & (~Q|P)))"
   1.161 +  by rules
   1.162 +
   1.163 +(* Problem ~~15 *)
   1.164 +lemma "~~((P --> Q) = (~P | Q))"
   1.165 +  by rules
   1.166 +
   1.167 +(* Problem ~~16 *)
   1.168 +lemma "~~((P-->Q) | (Q-->P))"
   1.169 +by rules
   1.170 +
   1.171 +(* Problem ~~17 *)
   1.172 +lemma "~~(((P & (Q-->R))-->S) = ((~P | Q | S) & (~P | ~R | S)))"
   1.173 +  oops
   1.174 +
   1.175 +(*Dijkstra's "Golden Rule"*)
   1.176 +lemma "(P&Q) = (P = (Q = (P|Q)))"
   1.177 +  by rules
   1.178 +
   1.179 +
   1.180 +(****Examples with quantifiers****)
   1.181 +
   1.182 +(* The converse is classical in the following implications... *)
   1.183 +
   1.184 +lemma "(EX x. P(x)-->Q)  -->  (ALL x. P(x)) --> Q"
   1.185 +  by rules
   1.186 +
   1.187 +lemma "((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
   1.188 +  by rules
   1.189 +
   1.190 +lemma "((ALL x. ~P(x))-->Q)  -->  ~ (ALL x. ~ (P(x)|Q))"
   1.191 +  by rules
   1.192 +
   1.193 +lemma "(ALL x. P(x)) | Q  -->  (ALL x. P(x) | Q)"
   1.194 +  by rules 
   1.195 +
   1.196 +lemma "(EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
   1.197 +  by rules
   1.198 +
   1.199 +
   1.200 +(* Hard examples with quantifiers *)
   1.201 +
   1.202 +(*The ones that have not been proved are not known to be valid!
   1.203 +  Some will require quantifier duplication -- not currently available*)
   1.204 +
   1.205 +(* Problem ~~19 *)
   1.206 +lemma "~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))"
   1.207 +  by rules
   1.208 +
   1.209 +(* Problem 20 *)
   1.210 +lemma "(ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))
   1.211 +    --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
   1.212 +  by rules
   1.213 +
   1.214 +(* Problem 21 *)
   1.215 +lemma "(EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P=Q(x))"
   1.216 +  by rules
   1.217 +
   1.218 +(* Problem 22 *)
   1.219 +lemma "(ALL x. P = Q(x))  -->  (P = (ALL x. Q(x)))"
   1.220 +  by rules
   1.221 +
   1.222 +(* Problem ~~23 *)
   1.223 +lemma "~~ ((ALL x. P | Q(x))  =  (P | (ALL x. Q(x))))"
   1.224 +  by rules
   1.225 +
   1.226 +(* Problem 25 *)
   1.227 +lemma "(EX x. P(x)) &
   1.228 +       (ALL x. L(x) --> ~ (M(x) & R(x))) &
   1.229 +       (ALL x. P(x) --> (M(x) & L(x))) &
   1.230 +       ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))
   1.231 +   --> (EX x. Q(x)&P(x))"
   1.232 +  by rules
   1.233 +
   1.234 +(* Problem 27 *)
   1.235 +lemma "(EX x. P(x) & ~Q(x)) &
   1.236 +             (ALL x. P(x) --> R(x)) &
   1.237 +             (ALL x. M(x) & L(x) --> P(x)) &
   1.238 +             ((EX x. R(x) & ~ Q(x)) --> (ALL x. L(x) --> ~ R(x)))
   1.239 +         --> (ALL x. M(x) --> ~L(x))"
   1.240 +  by rules
   1.241 +
   1.242 +(* Problem ~~28.  AMENDED *)
   1.243 +lemma "(ALL x. P(x) --> (ALL x. Q(x))) &
   1.244 +       (~~(ALL x. Q(x)|R(x)) --> (EX x. Q(x)&S(x))) &
   1.245 +       (~~(EX x. S(x)) --> (ALL x. L(x) --> M(x)))
   1.246 +   --> (ALL x. P(x) & L(x) --> M(x))"
   1.247 +  by rules
   1.248 +
   1.249 +(* Problem 29.  Essentially the same as Principia Mathematica *11.71 *)
   1.250 +lemma "(((EX x. P(x)) & (EX y. Q(y))) -->
   1.251 +   (((ALL x. (P(x) --> R(x))) & (ALL y. (Q(y) --> S(y)))) =
   1.252 +    (ALL x y. ((P(x) & Q(y)) --> (R(x) & S(y))))))"
   1.253 +  by rules
   1.254 +
   1.255 +(* Problem ~~30 *)
   1.256 +lemma "(ALL x. (P(x) | Q(x)) --> ~ R(x)) &
   1.257 +       (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))
   1.258 +   --> (ALL x. ~~S(x))"
   1.259 +  by rules
   1.260 +
   1.261 +(* Problem 31 *)
   1.262 +lemma "~(EX x. P(x) & (Q(x) | R(x))) & 
   1.263 +        (EX x. L(x) & P(x)) &
   1.264 +        (ALL x. ~ R(x) --> M(x))
   1.265 +    --> (EX x. L(x) & M(x))"
   1.266 +  by rules
   1.267 +
   1.268 +(* Problem 32 *)
   1.269 +lemma "(ALL x. P(x) & (Q(x)|R(x))-->S(x)) &
   1.270 +       (ALL x. S(x) & R(x) --> L(x)) &
   1.271 +       (ALL x. M(x) --> R(x))
   1.272 +   --> (ALL x. P(x) & M(x) --> L(x))"
   1.273 +  by rules
   1.274 +
   1.275 +(* Problem ~~33 *)
   1.276 +lemma "(ALL x. ~~(P(a) & (P(x)-->P(b))-->P(c)))  =
   1.277 +       (ALL x. ~~((~P(a) | P(x) | P(c)) & (~P(a) | ~P(b) | P(c))))"
   1.278 +  oops
   1.279 +
   1.280 +(* Problem 36 *)
   1.281 +lemma
   1.282 +     "(ALL x. EX y. J x y) &
   1.283 +      (ALL x. EX y. G x y) &
   1.284 +      (ALL x y. J x y | G x y --> (ALL z. J y z | G y z --> H x z))
   1.285 +  --> (ALL x. EX y. H x y)"
   1.286 +  by rules
   1.287 +
   1.288 +(* Problem 39 *)
   1.289 +lemma "~ (EX x. ALL y. F y x = (~F y y))"
   1.290 +  by rules
   1.291 +
   1.292 +(* Problem 40.  AMENDED *)
   1.293 +lemma "(EX y. ALL x. F x y = F x x) -->
   1.294 +             ~(ALL x. EX y. ALL z. F z y = (~ F z x))"
   1.295 +  by rules
   1.296 +
   1.297 +(* Problem 44 *)
   1.298 +lemma "(ALL x. f(x) -->
   1.299 +             (EX y. g(y) & h x y & (EX y. g(y) & ~ h x y)))  &
   1.300 +             (EX x. j(x) & (ALL y. g(y) --> h x y))
   1.301 +             --> (EX x. j(x) & ~f(x))"
   1.302 +  by rules
   1.303 +
   1.304 +(* Problem 48 *)
   1.305 +lemma "(a=b | c=d) & (a=c | b=d) --> a=d | b=c"
   1.306 +  by rules
   1.307 +
   1.308 +(* Problem 51 *)
   1.309 +lemma "((EX z w. (ALL x y. (P x y = ((x = z) & (y = w))))) -->
   1.310 +  (EX z. (ALL x. (EX w. ((ALL y. (P x y = (y = w))) = (x = z))))))"
   1.311 +  by rules
   1.312 +
   1.313 +(* Problem 52 *)
   1.314 +(*Almost the same as 51. *)
   1.315 +lemma "((EX z w. (ALL x y. (P x y = ((x = z) & (y = w))))) -->
   1.316 +   (EX w. (ALL y. (EX z. ((ALL x. (P x y = (x = z))) = (y = w))))))"
   1.317 +  by rules
   1.318 +
   1.319 +(* Problem 56 *)
   1.320 +lemma "(ALL x. (EX y. P(y) & x=f(y)) --> P(x)) = (ALL x. P(x) --> P(f(x)))"
   1.321 +  by rules
   1.322 +
   1.323 +(* Problem 57 *)
   1.324 +lemma "P (f a b) (f b c) & P (f b c) (f a c) &
   1.325 +     (ALL x y z. P x y & P y z --> P x z) --> P (f a b) (f a c)"
   1.326 +  by rules
   1.327 +
   1.328 +(* Problem 60 *)
   1.329 +lemma "ALL x. P x (f x) = (EX y. (ALL z. P z y --> P z (f x)) & P x y)"
   1.330 +  by rules
   1.331 +
   1.332 +end
     2.1 --- a/src/HOL/ex/ROOT.ML	Mon Dec 10 15:35:03 2001 +0100
     2.2 +++ b/src/HOL/ex/ROOT.ML	Mon Dec 10 15:36:05 2001 +0100
     2.3 @@ -19,6 +19,7 @@
     2.4  time_use_thy "Tuple";
     2.5  
     2.6  time_use_thy "NatSum";
     2.7 +time_use_thy "Intuitionistic";
     2.8  time_use     "cla.ML";
     2.9  time_use     "mesontest.ML";
    2.10  time_use_thy "mesontest2";