Theory Propositional_Int

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theory Propositional_Int
imports IFOLP
begin

(*  Title:      FOLP/ex/Propositional_Int.thy
    ID:         $Id: Propositional_Int.thy,v 1.1 2008/03/26 21:38:55 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

header {* First-Order Logic: propositional examples *}

theory Propositional_Int
imports IFOLP
begin


text "commutative laws of & and | "
lemma "?p : P & Q  -->  Q & P"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : P | Q  -->  Q | P"
  by (tactic {* IntPr.fast_tac 1 *})


text "associative laws of & and | "
lemma "?p : (P & Q) & R  -->  P & (Q & R)"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P | Q) | R  -->  P | (Q | R)"
  by (tactic {* IntPr.fast_tac 1 *})


text "distributive laws of & and | "
lemma "?p : (P & Q) | R  --> (P | R) & (Q | R)"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P | R) & (Q | R)  --> (P & Q) | R"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P | Q) & R  --> (P & R) | (Q & R)"
  by (tactic {* IntPr.fast_tac 1 *})


lemma "?p : (P & R) | (Q & R)  --> (P | Q) & R"
  by (tactic {* IntPr.fast_tac 1 *})


text "Laws involving implication"

lemma "?p : (P-->R) & (Q-->R) <-> (P|Q --> R)"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P & Q --> R) <-> (P--> (Q-->R))"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : ((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : ~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P --> Q & R) <-> (P-->Q)  &  (P-->R)"
  by (tactic {* IntPr.fast_tac 1 *})


text "Propositions-as-types"

(*The combinator K*)
lemma "?p : P --> (Q --> P)"
  by (tactic {* IntPr.fast_tac 1 *})

(*The combinator S*)
lemma "?p : (P-->Q-->R)  --> (P-->Q) --> (P-->R)"
  by (tactic {* IntPr.fast_tac 1 *})


(*Converse is classical*)
lemma "?p : (P-->Q) | (P-->R)  -->  (P --> Q | R)"
  by (tactic {* IntPr.fast_tac 1 *})

lemma "?p : (P-->Q)  -->  (~Q --> ~P)"
  by (tactic {* IntPr.fast_tac 1 *})


text "Schwichtenberg's examples (via T. Nipkow)"

lemma stab_imp: "?p : (((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q"
  by (tactic {* IntPr.fast_tac 1 *})

lemma stab_to_peirce: "?p : (((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q)  
              --> ((P --> Q) --> P) --> P"
  by (tactic {* IntPr.fast_tac 1 *})

lemma peirce_imp1: "?p : (((Q --> R) --> Q) --> Q)  
               --> (((P --> Q) --> R) --> P --> Q) --> P --> Q"
  by (tactic {* IntPr.fast_tac 1 *})
  
lemma peirce_imp2: "?p : (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P"
  by (tactic {* IntPr.fast_tac 1 *})

lemma mints: "?p : ((((P --> Q) --> P) --> P) --> Q) --> Q"
  by (tactic {* IntPr.fast_tac 1 *})

lemma mints_solovev: "?p : (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R"
  by (tactic {* IntPr.fast_tac 1 *})

lemma tatsuta: "?p : (((P7 --> P1) --> P10) --> P4 --> P5)  
          --> (((P8 --> P2) --> P9) --> P3 --> P10)  
          --> (P1 --> P8) --> P6 --> P7  
          --> (((P3 --> P2) --> P9) --> P4)  
          --> (P1 --> P3) --> (((P6 --> P1) --> P2) --> P9) --> P5"
  by (tactic {* IntPr.fast_tac 1 *})

lemma tatsuta1: "?p : (((P8 --> P2) --> P9) --> P3 --> P10)  
     --> (((P3 --> P2) --> P9) --> P4)  
     --> (((P6 --> P1) --> P2) --> P9)  
     --> (((P7 --> P1) --> P10) --> P4 --> P5)  
     --> (P1 --> P3) --> (P1 --> P8) --> P6 --> P7 --> P5"
  by (tactic {* IntPr.fast_tac 1 *})

end

lemma

  P & Q --> Q & P

lemma

  P | Q --> Q | P

lemma

  (P & Q) & R --> P & Q & R

lemma

  (P | Q) | R --> P | Q | R

lemma

  P & Q | R --> (P | R) & (Q | R)

lemma

  (P | R) & (Q | R) --> P & Q | R

lemma

  (P | Q) & R --> P & R | Q & R

lemma

  P & R | Q & R --> (P | Q) & R

lemma

  (P --> R) & (Q --> R) <-> P | Q --> R

lemma

  (P & Q --> R) <-> P --> Q --> R

lemma

  ((P --> R) --> R) --> ((Q --> R) --> R) --> (P & Q --> R) --> R

lemma

  ~ (P --> R) --> ~ (Q --> R) --> ~ (P & Q --> R)

lemma

  (P --> Q & R) <-> (P --> Q) & (P --> R)

lemma

  P --> Q --> P

lemma

  (P --> Q --> R) --> (P --> Q) --> P --> R

lemma

  (P --> Q) | (P --> R) --> P --> Q | R

lemma

  (P --> Q) --> ~ Q --> ~ P

lemma stab_imp:

  (((Q --> R) --> R) --> Q) --> (((P --> Q) --> R) --> R) --> P --> Q

lemma stab_to_peirce:

  (((P --> R) --> R) --> P) -->
  (((Q --> R) --> R) --> Q) --> ((P --> Q) --> P) --> P

lemma peirce_imp1:

  (((Q --> R) --> Q) --> Q) --> (((P --> Q) --> R) --> P --> Q) --> P --> Q

lemma peirce_imp2:

  (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P

lemma mints:

  ((((P --> Q) --> P) --> P) --> Q) --> Q

lemma mints_solovev:

  (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R

lemma tatsuta:

  (((P7.0 --> P1.0) --> P10.0) --> P4.0 --> P5.0) -->
  (((P8.0 --> P2.0) --> P9.0) --> P3.0 --> P10.0) -->
  (P1.0 --> P8.0) -->
  P6.0 -->
  P7.0 -->
  (((P3.0 --> P2.0) --> P9.0) --> P4.0) -->
  (P1.0 --> P3.0) --> (((P6.0 --> P1.0) --> P2.0) --> P9.0) --> P5.0

lemma tatsuta1:

  (((P8.0 --> P2.0) --> P9.0) --> P3.0 --> P10.0) -->
  (((P3.0 --> P2.0) --> P9.0) --> P4.0) -->
  (((P6.0 --> P1.0) --> P2.0) --> P9.0) -->
  (((P7.0 --> P1.0) --> P10.0) --> P4.0 --> P5.0) -->
  (P1.0 --> P3.0) --> (P1.0 --> P8.0) --> P6.0 --> P7.0 --> P5.0