# Theory Intro

```(*  Title:      FOLP/ex/Intro.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1992  University of Cambridge

Derives some inference rules, illustrating the use of definitions.
*)

section ‹Examples for the manual ``Introduction to Isabelle''›

theory Intro
imports FOLP
begin

subsubsection ‹Some simple backward proofs›

schematic_goal mythm: "?p : P|P --> P"
apply (rule impI)
apply (rule disjE)
prefer 3 apply (assumption)
prefer 2 apply (assumption)
apply assumption
done

schematic_goal "?p : (P & Q) | R --> (P | R)"
apply (rule impI)
apply (erule disjE)
apply (drule conjunct1)
apply (rule disjI1)
apply (rule_tac  disjI2)
apply assumption+
done

(*Correct version, delaying use of "spec" until last*)
schematic_goal "?p : (ALL x y. P(x,y)) --> (ALL z w. P(w,z))"
apply (rule impI)
apply (rule allI)
apply (rule allI)
apply (drule spec)
apply (drule spec)
apply assumption
done

subsubsection ‹Demonstration of ‹fast››

schematic_goal "?p : (EX y. ALL x. J(y,x) <-> ~J(x,x))
-->  ~ (ALL x. EX y. ALL z. J(z,y) <-> ~ J(z,x))"
apply (tactic ‹fast_tac \<^context> FOLP_cs 1›)
done

schematic_goal "?p : ALL x. P(x,f(x)) <->
(EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
apply (tactic ‹fast_tac \<^context> FOLP_cs 1›)
done

subsubsection ‹Derivation of conjunction elimination rule›

schematic_goal
assumes major: "p : P&Q"
and minor: "!!x y. [| x : P; y : Q |] ==> f(x, y) : R"
shows "?p : R"
apply (rule minor)
apply (rule major [THEN conjunct1])
apply (rule major [THEN conjunct2])
done

subsection ‹Derived rules involving definitions›

text ‹Derivation of negation introduction›

schematic_goal
assumes "!!x. x : P ==> f(x) : False"
shows "?p : ~ P"
apply (unfold not_def)
apply (rule impI)
apply (rule assms)
apply assumption
done

schematic_goal
assumes major: "p : ~P"
and minor: "q : P"
shows "?p : R"
apply (rule FalseE)
apply (rule mp)
apply (rule major [unfolded not_def])
apply (rule minor)
done

text ‹Alternative proof of the result above›
schematic_goal
assumes major: "p : ~P"
and minor: "q : P"
shows "?p : R"
apply (rule minor [THEN major [unfolded not_def, THEN mp, THEN FalseE]])
done

end
```