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

section ‹First-Order Logic: propositional examples›

theory Propositional_Cla
imports FOLP
begin


text "commutative laws of & and | "
schematic_goal "?p : P & Q --> Q & P"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : P | Q --> Q | P"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


text "associative laws of & and | "
schematic_goal "?p : (P & Q) & R --> P & (Q & R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P | Q) | R --> P | (Q | R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


text "distributive laws of & and | "
schematic_goal "?p : (P & Q) | R --> (P | R) & (Q | R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P | R) & (Q | R) --> (P & Q) | R"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P | Q) & R --> (P & R) | (Q & R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


schematic_goal "?p : (P & R) | (Q & R) --> (P | Q) & R"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


text "Laws involving implication"

schematic_goal "?p : (P-->R) & (Q-->R) <-> (P|Q --> R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P & Q --> R) <-> (P--> (Q-->R))"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : ((P-->R)-->R) --> ((Q-->R)-->R) --> (P&Q-->R) --> R"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : ~(P-->R) --> ~(Q-->R) --> ~(P&Q-->R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P --> Q & R) <-> (P-->Q) & (P-->R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


text "Propositions-as-types"

(*The combinator K*)
schematic_goal "?p : P --> (Q --> P)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

(*The combinator S*)
schematic_goal "?p : (P-->Q-->R) --> (P-->Q) --> (P-->R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


(*Converse is classical*)
schematic_goal "?p : (P-->Q) | (P-->R) --> (P --> Q | R)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal "?p : (P-->Q) --> (~Q --> ~P)"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)


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

schematic_goal stab_imp: "?p : (((Q-->R)-->R)-->Q) --> (((P-->Q)-->R)-->R)-->P-->Q"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal stab_to_peirce: "?p : (((P --> R) --> R) --> P) --> (((Q --> R) --> R) --> Q)
              --> ((P --> Q) --> P) --> P"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal peirce_imp1: "?p : (((Q --> R) --> Q) --> Q)
               --> (((P --> Q) --> R) --> P --> Q) --> P --> Q"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)
  
schematic_goal peirce_imp2: "?p : (((P --> R) --> P) --> P) --> ((P --> Q --> R) --> P) --> P"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal mints: "?p : ((((P --> Q) --> P) --> P) --> Q) --> Q"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal mints_solovev: "?p : (P --> (Q --> R) --> Q) --> ((P --> Q) --> R) --> R"
  by (tactic ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal 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 ‹Cla.fast_tac context FOLP_cs 1›)

schematic_goal 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 ‹Cla.fast_tac context FOLP_cs 1›)

end