Theory Propositional_Cla

Up to index of Isabelle/FOL/ex

theory Propositional_Cla
imports FOL
begin

(*  Title:      FOL/ex/Propositional_Cla.thy
    ID:         $Id: Propositional_Cla.thy,v 1.1 2007/07/22 20:01:30 wenzelm Exp $
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge
*)

header {* First-Order Logic: propositional examples (classical version) *}

theory Propositional_Cla
imports FOL
begin

text {* commutative laws of @{text "&"} and @{text "|"} *}

lemma "P & Q  -->  Q & P"
  by (tactic "IntPr.fast_tac 1")

lemma "P | Q  -->  Q | P"
  by fast


text {* associative laws of @{text "&"} and @{text "|"} *}
lemma "(P & Q) & R  -->  P & (Q & R)"
  by fast

lemma "(P | Q) | R  -->  P | (Q | R)"
  by fast


text {* distributive laws of @{text "&"} and @{text "|"} *}
lemma "(P & Q) | R  --> (P | R) & (Q | R)"
  by fast

lemma "(P | R) & (Q | R)  --> (P & Q) | R"
  by fast

lemma "(P | Q) & R  --> (P & R) | (Q & R)"
  by fast

lemma "(P & R) | (Q & R)  --> (P | Q) & R"
  by fast


text {* Laws involving implication *}

lemma "(P-->R) & (Q-->R) <-> (P|Q --> R)"
  by fast

lemma "(P & Q --> R) <-> (P--> (Q-->R))"
  by fast

lemma "((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R"
  by fast

lemma "~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)"
  by fast

lemma "(P --> Q & R) <-> (P-->Q)  &  (P-->R)"
  by fast


text {* Propositions-as-types *}

-- {* The combinator K *}
lemma "P --> (Q --> P)"
  by fast

-- {* The combinator S *}
lemma "(P-->Q-->R)  --> (P-->Q) --> (P-->R)"
  by fast


-- {* Converse is classical *}
lemma "(P-->Q) | (P-->R)  -->  (P --> Q | R)"
  by fast

lemma "(P-->Q)  -->  (~Q --> ~P)"
  by fast


text {* Schwichtenberg's examples (via T. Nipkow) *}

lemma stab_imp: "(((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q"
  by fast

lemma stab_to_peirce:
  "(((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q)  
                              --> ((P --> Q) --> P) --> P"
  by fast

lemma peirce_imp1: "(((Q --> R) --> Q) --> Q)  
                --> (((P --> Q) --> R) --> P --> Q) --> P --> Q"
  by fast
  
lemma peirce_imp2: "(((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P"
  by fast

lemma mints: "((((P --> Q) --> P) --> P) --> Q) --> Q"
  by fast

lemma mints_solovev: "(P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R"
  by fast

lemma tatsuta: "(((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 fast

lemma tatsuta1: "(((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 fast

end

lemma

  PQ --> QP

lemma

  PQ --> QP

lemma

  (PQ) ∧ R --> PQR

lemma

  (PQ) ∨ R --> PQR

lemma

  PQR --> (PR) ∧ (QR)

lemma

  (PR) ∧ (QR) --> PQR

lemma

  (PQ) ∧ R --> PRQR

lemma

  PRQR --> (PQ) ∧ R

lemma

  (P --> R) ∧ (Q --> R) <-> PQ --> R

lemma

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

lemma

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

lemma

  ¬ (P --> R) --> ¬ (Q --> R) --> ¬ (PQ --> R)

lemma

  (P --> QR) <-> (P --> Q) ∧ (P --> R)

lemma

  P --> Q --> P

lemma

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

lemma

  (P --> Q) ∨ (P --> R) --> P --> QR

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