(* Title: FOL/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.
*)
header {* Examples for the manual ``Introduction to Isabelle'' *}
theory Intro
imports FOL
begin
subsubsection {* Some simple backward proofs *}
lemma mythm: "P|P --> P"
apply (rule impI)
apply (rule disjE)
prefer 3 apply (assumption)
prefer 2 apply (assumption)
apply assumption
done
lemma "(P & Q) | R --> (P | R)"
apply (rule impI)
apply (erule disjE)
apply (drule conjunct1)
apply (rule disjI1)
apply (rule_tac [2] disjI2)
apply assumption+
done
(*Correct version, delaying use of "spec" until last*)
lemma "(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 @{text "fast"} *}
lemma "(EX y. ALL x. J(y,x) <-> ~J(x,x))
--> ~ (ALL x. EX y. ALL z. J(z,y) <-> ~ J(z,x))"
apply fast
done
lemma "ALL x. P(x,f(x)) <->
(EX y. (ALL z. P(z,y) --> P(z,f(x))) & P(x,y))"
apply fast
done
subsubsection {* Derivation of conjunction elimination rule *}
lemma
assumes major: "P&Q"
and minor: "[| P; Q |] ==> R"
shows 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 *}
lemma
assumes "P ==> False"
shows "~ P"
apply (unfold not_def)
apply (rule impI)
apply (rule prems)
apply assumption
done
lemma
assumes major: "~P"
and minor: P
shows R
apply (rule FalseE)
apply (rule mp)
apply (rule major [unfolded not_def])
apply (rule minor)
done
text {* Alternative proof of the result above *}
lemma
assumes major: "~P"
and minor: P
shows R
apply (rule minor [THEN major [unfolded not_def, THEN mp, THEN FalseE]])
done
end