# Theory Intro

theory Intro
imports FOL
```(*  Title:      FOL/ex/Intro.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

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

section ‹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

text ‹Correct version, delaying use of ‹spec› until last.›
lemma "(∀x y. P(x,y)) ⟶ (∀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››

lemma "(∃y. ∀x. J(y,x) ⟷ ¬ J(x,x)) ⟶ ¬ (∀x. ∃y. ∀z. J(z,y) ⟷ ¬ J(z,x))"
apply fast
done

lemma "∀x. P(x,f(x)) ⟷ (∃y. (∀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 assms)
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
```