src/HOL/Isar_examples/BasicLogic.thy

tuned comments;

(*  Title:      HOL/Isar_examples/BasicLogic.thy
    ID:         $Id$
    Author:     Markus Wenzel, TU Muenchen

Basic propositional and quantifier reasoning.
*)

theory BasicLogic = Main:;


text {* Just a few tiny examples to get an idea of how Isabelle/Isar
  proof documents may look like. *};

lemma I: "A --> A";
proof;
  assume A;
  show A; .;
qed;

lemma K: "A --> B --> A";
proof;
  assume A;
  show "B --> A";
  proof;
    show A; .;
  qed;
qed;

lemma K': "A --> B --> A";
proof single+   -- {* better use sufficient-to-show here \dots *};
  assume A;
  show A; .;
qed;

lemma S: "(A --> B --> C) --> (A --> B) --> A --> C";
proof;
  assume "A --> B --> C";
  show "(A --> B) --> A --> C";
  proof;
    assume "A --> B";
    show "A --> C";
    proof;
      assume A;
      show C;
      proof (rule mp);
	show "B --> C"; by (rule mp);
        show B; by (rule mp);
      qed;
    qed;
  qed;
qed;


text {* Variations of backward vs.\ forward reasonong \dots *};

lemma "A & B --> B & A";
proof;
  assume "A & B";
  show "B & A";
  proof;
    show B; by (rule conjunct2);
    show A; by (rule conjunct1);
  qed;
qed;

lemma "A & B --> B & A";
proof;
  assume "A & B";
  then; show "B & A";
  proof;
    assume A B;
    show ??thesis; ..;
  qed;
qed;

lemma "A & B --> B & A";
proof;
  assume ab: "A & B";
  from ab; have a: A; ..;
  from ab; have b: B; ..;
  from b a; show "B & A"; ..;
qed;


section {* Examples from 'Introduction to Isabelle' *};

text {* 'Propositional proof' *};

lemma "P | P --> P";
proof;
  assume "P | P";
  then; show P;
  proof;
    assume P;
    show P; .;
    show P; .;
  qed;
qed;

lemma "P | P --> P";
proof;
  assume "P | P";
  then; show P; ..;
qed;


text {* 'Quantifier proof' *};

lemma "(EX x. P(f(x))) --> (EX x. P(x))";
proof;
  assume "EX x. P(f(x))";
  then; show "EX x. P(x)";
  proof (rule exE);
    fix a;
    assume "P(f(a))" (is "P(??witness)");
    show ??thesis; by (rule exI [of P ??witness]);
  qed;
qed;

lemma "(EX x. P(f(x))) --> (EX x. P(x))";
proof;
  assume "EX x. P(f(x))";
  then; show "EX x. P(x)";
  proof;
    fix a;
    assume "P(f(a))";
    show ??thesis; ..;
  qed;
qed;

lemma "(EX x. P(f(x))) --> (EX x. P(x))";
  by blast;


subsection {* 'Deriving rules in Isabelle' *};

text {* We derive the conjunction elimination rule from the
 projections.  The proof below follows the basic reasoning of the
 script given in the Isabelle Intro Manual closely.  Although, the
 actual underlying operations on rules and proof states are quite
 different: Isabelle/Isar supports non-atomic goals and assumptions
 fully transparently, while the original Isabelle classic script
 depends on the primitive goal command to decompose the rule into
 premises and conclusion, with the result emerging by discharging the
 context again later. *};

theorem conjE: "A & B ==> (A ==> B ==> C) ==> C";
proof same;
  assume ab: "A & B";
  assume ab_c: "A ==> B ==> C";
  show C;
  proof (rule ab_c);
    from ab; show A; ..;
    from ab; show B; ..;
  qed;
qed;


end;