Anforderungen  |   Konzepte  |   Entwurf  |   Entwicklung  |   Qualitätssicherung  |   Lebenszyklus  |   Steuerung
 
 
 
 


Quellcode-Bibliothek

© Kompilation durch diese Firma

[Weder Korrektheit noch Funktionsfähigkeit der Software werden zugesichert.]

Datei: Intuitionistic.thy   Sprache: Isabelle

Original von: Isabelle©

(*  Title:      FOLP/ex/Intuitionistic.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

Intuitionistic First-Order Logic.

Single-step commands:
by (IntPr.step_tac 1)
by (biresolve_tac safe_brls 1);
by (biresolve_tac haz_brls 1);
by (assume_tac 1);
by (IntPr.safe_tac 1);
by (IntPr.mp_tac 1);
by (IntPr.fast_tac 1);
*)


(*Note: for PROPOSITIONAL formulae...
  ~A is classically provable iff it is intuitionistically provable.  
  Therefore A is classically provable iff ~~A is intuitionistically provable.

Let Q be the conjuction of the propositions A|~A, one for each atom A in
P.  If P is provable classically, then clearly P&Q is provable
intuitionistically, so ~~(P&Q) is also provable intuitionistically.
The latter is intuitionistically equivalent to ~~P&~~Q, hence to ~~P,
since ~~Q is intuitionistically provable.  Finally, if P is a negation then
~~P is intuitionstically equivalent to P.  [Andy Pitts]
*)


theory Intuitionistic
imports IFOLP
begin

schematic_goal "?p : ~~(P&Q) <-> ~~P & ~~Q"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ~~~P <-> ~P"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ~~((P --> Q | R) --> (P-->Q) | (P-->R))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : (P<->Q) <-> (Q<->P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)


subsection \<open>Lemmas for the propositional double-negation translation\<close>

schematic_goal "?p : P --> ~~P"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ~~(~~P --> P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ~~P & ~~(P --> Q) --> ~~Q"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)


subsection \<open>The following are classically but not constructively valid\<close>

(*The attempt to prove them terminates quickly!*)
schematic_goal "?p : ((P-->Q) --> P) --> P"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops

schematic_goal "?p : (P&Q-->R) --> (P-->R) | (Q-->R)"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops


subsection \<open>Intuitionistic FOL: propositional problems based on Pelletier\<close>

text "Problem ~~1"
schematic_goal "?p : ~~((P-->Q) <-> (~Q --> ~P))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~2"
schematic_goal "?p : ~~(~~P <-> P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 3"
schematic_goal "?p : ~(P-->Q) --> (Q-->P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~4"
schematic_goal "?p : ~~((~P-->Q) <-> (~Q --> P))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~5"
schematic_goal "?p : ~~((P|Q-->P|R) --> P|(Q-->R))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~6"
schematic_goal "?p : ~~(P | ~P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~7"
schematic_goal "?p : ~~(P | ~~~P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~8. Peirce's law"
schematic_goal "?p : ~~(((P-->Q) --> P) --> P)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 9"
schematic_goal "?p : ((P|Q) & (~P|Q) & (P| ~Q)) --> ~ (~P | ~Q)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 10"
schematic_goal "?p : (Q-->R) --> (R-->P&Q) --> (P-->(Q|R)) --> (P<->Q)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "11. Proved in each direction (incorrectly, says Pelletier!!) "
schematic_goal "?p : P<->P"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~12. Dijkstra's law "
schematic_goal "?p : ~~(((P <-> Q) <-> R) <-> (P <-> (Q <-> R)))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ((P <-> Q) <-> R) --> ~~(P <-> (Q <-> R))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 13. Distributive law"
schematic_goal "?p : P | (Q & R) <-> (P | Q) & (P | R)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~14"
schematic_goal "?p : ~~((P <-> Q) <-> ((Q | ~P) & (~Q|P)))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~15"
schematic_goal "?p : ~~((P --> Q) <-> (~P | Q))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~16"
schematic_goal "?p : ~~((P-->Q) | (Q-->P))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~17"
schematic_goal "?p : ~~(((P & (Q-->R))-->S) <-> ((~P | Q | S) & (~P | ~R | S)))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)  \<comment> \<open>slow\<close>


subsection \<open>Examples with quantifiers\<close>

text "The converse is classical in the following implications..."

schematic_goal "?p : (EX x. P(x)-->Q) --> (ALL x. P(x)) --> Q"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ((ALL x. P(x))-->Q) --> ~ (ALL x. P(x) & ~Q)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : ((ALL x. ~P(x))-->Q) --> ~ (ALL x. ~ (P(x)|Q))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : (ALL x. P(x)) | Q --> (ALL x. P(x) | Q)"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

schematic_goal "?p : (EX x. P --> Q(x)) --> (P --> (EX x. Q(x)))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)


text "The following are not constructively valid!"
text "The attempt to prove them terminates quickly!"

schematic_goal "?p : ((ALL x. P(x))-->Q) --> (EX x. P(x)-->Q)"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops

schematic_goal "?p : (P --> (EX x. Q(x))) --> (EX x. P-->Q(x))"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops

schematic_goal "?p : (ALL x. P(x) | Q) --> ((ALL x. P(x)) | Q)"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops

schematic_goal "?p : (ALL x. ~~P(x)) --> ~~(ALL x. P(x))"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops

(*Classically but not intuitionistically valid.  Proved by a bug in 1986!*)
schematic_goal "?p : EX x. Q(x) --> (ALL x. Q(x))"
  apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
  oops


subsection "Hard examples with quantifiers"

text \<open>
  The ones that have not been proved are not known to be valid!
  Some will require quantifier duplication -- not currently available.
\<close>

text "Problem ~~18"
schematic_goal "?p : ~~(EX y. ALL x. P(y)-->P(x))" oops
(*NOT PROVED*)

text "Problem ~~19"
schematic_goal "?p : ~~(EX x. ALL y z. (P(y)-->Q(z)) --> (P(x)-->Q(x)))" oops
(*NOT PROVED*)

text "Problem 20"
schematic_goal "?p : (ALL x y. EX z. ALL w. (P(x)&Q(y)-->R(z)&S(w)))
    --> (EX x y. P(x) & Q(y)) --> (EX z. R(z))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 21"
schematic_goal "?p : (EX x. P-->Q(x)) & (EX x. Q(x)-->P) --> ~~(EX x. P<->Q(x))" oops
(*NOT PROVED*)

text "Problem 22"
schematic_goal "?p : (ALL x. P <-> Q(x)) --> (P <-> (ALL x. Q(x)))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem ~~23"
schematic_goal "?p : ~~ ((ALL x. P | Q(x)) <-> (P | (ALL x. Q(x))))"
  by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)

text "Problem 24"
schematic_goal "?p : ~(EX x. S(x)&Q(x)) & (ALL x. P(x) --> Q(x)|R(x)) &
     (~(EX x. P(x)) --> (EX x. Q(x))) & (ALL x. Q(x)|R(x) --> S(x))   
    --> ~~(EX x. P(x)&R(x))"
(*Not clear why fast_tac, best_tac, ASTAR and ITER_DEEPEN all take forever*)
  apply (tactic "IntPr.safe_tac \<^context>")
  apply (erule impE)
   apply (tactic "IntPr.fast_tac \<^context> 1")
  apply (tactic "IntPr.fast_tac \<^context> 1")
  done

text "Problem 25"
schematic_goal "?p : (EX x. P(x)) &
        (ALL x. L(x) --> ~ (M(x) & R(x))) &   
        (ALL x. P(x) --> (M(x) & L(x))) &    
        ((ALL x. P(x)-->Q(x)) | (EX x. P(x)&R(x)))   
    --> (EX x. Q(x)&P(x))"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 29. Essentially the same as Principia Mathematica *11.71"
schematic_goal "?p : (EX x. P(x)) & (EX y. Q(y))
    --> ((ALL x. P(x)-->R(x)) & (ALL y. Q(y)-->S(y))   <->      
         (ALL x y. P(x) & Q(y) --> R(x) & S(y)))"
  by (tactic "IntPr.fast_tac \<^context> 1")

text "Problem ~~30"
schematic_goal "?p : (ALL x. (P(x) | Q(x)) --> ~ R(x)) &
        (ALL x. (Q(x) --> ~ S(x)) --> P(x) & R(x))   
    --> (ALL x. ~~S(x))"
  by (tactic "IntPr.fast_tac \<^context> 1")

text "Problem 31"
schematic_goal "?p : ~(EX x. P(x) & (Q(x) | R(x))) &
        (EX x. L(x) & P(x)) &  
        (ALL x. ~ R(x) --> M(x))   
    --> (EX x. L(x) & M(x))"
  by (tactic "IntPr.fast_tac \<^context> 1")

text "Problem 32"
schematic_goal "?p : (ALL x. P(x) & (Q(x)|R(x))-->S(x)) &
        (ALL x. S(x) & R(x) --> L(x)) &  
        (ALL x. M(x) --> R(x))   
    --> (ALL x. P(x) & M(x) --> L(x))"
  by (tactic "IntPr.best_tac \<^context> 1") \ \slow\

text "Problem 39"
schematic_goal "?p : ~ (EX x. ALL y. F(y,x) <-> ~F(y,y))"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 40. AMENDED"
schematic_goal "?p : (EX y. ALL x. F(x,y) <-> F(x,x)) -->
              ~(ALL x. EX y. ALL z. F(z,y) <-> ~ F(z,x))"
  by (tactic "IntPr.best_tac \<^context> 1") \ \slow\

text "Problem 44"
schematic_goal "?p : (ALL x. f(x) -->
              (EX y. g(y) & h(x,y) & (EX y. g(y) & ~ h(x,y))))  &        
              (EX x. j(x) & (ALL y. g(y) --> h(x,y)))                    
              --> (EX x. j(x) & ~f(x))"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 48"
schematic_goal "?p : (a=b | c=d) & (a=c | b=d) --> a=d | b=c"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 51"
schematic_goal
    "?p : (EX z w. ALL x y. P(x,y) <-> (x=z & y=w)) -->
     (EX z. ALL x. EX w. (ALL y. P(x,y) <-> y=w) <-> x=z)"
  by (tactic "IntPr.best_tac \<^context> 1") \ \60 seconds\

text "Problem 56"
schematic_goal "?p : (ALL x. (EX y. P(y) & x=f(y)) --> P(x)) <-> (ALL x. P(x) --> P(f(x)))"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 57"
schematic_goal
    "?p : P(f(a,b), f(b,c)) & P(f(b,c), f(a,c)) &
     (ALL x y z. P(x,y) & P(y,z) --> P(x,z))    -->   P(f(a,b), f(a,c))"
  by (tactic "IntPr.best_tac \<^context> 1")

text "Problem 60"
schematic_goal "?p : ALL x. P(x,f(x)) <-> (EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))">
  by (tactic "IntPr.best_tac \<^context> 1")

end

¤ Dauer der Verarbeitung: 0.0 Sekunden  (vorverarbeitet)  ¤





Download des
Quellennavigators
Download des
sprechenden Kalenders

in der Quellcodebibliothek suchen




Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.


Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.


Bot Zugriff



                                                                                                                                                                                                                                                                                                                                                                                                     


Neuigkeiten

     Aktuelles
     Motto des Tages

Software

     Produkte
     Quellcodebibliothek

Aktivitäten

     Artikel über Sicherheit
     Anleitung zur Aktivierung von SSL

Muße

     Gedichte
     Musik
     Bilder

Jenseits des Üblichen ....
    

Besucherstatistik

Besucherstatistik