src/FOL/ex/Foundation.thy
 author haftmann Tue, 10 Jul 2007 17:30:50 +0200 changeset 23709 fd31da8f752a parent 19819 14de4d05d275 child 26682 310c3b1a4157 permissions -rw-r--r--
moved lfp_induct2 here
```
(*  Title:      FOL/ex/Foundation.ML
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright   1991  University of Cambridge
*)

header "Intuitionistic FOL: Examples from The Foundation of a Generic Theorem Prover"

theory Foundation
imports IFOL
begin

lemma "A&B  --> (C-->A&C)"
apply (rule impI)
apply (rule impI)
apply (rule conjI)
prefer 2 apply assumption
apply (rule conjunct1)
apply assumption
done

text {*A form of conj-elimination*}
lemma
assumes "A & B"
and "A ==> B ==> C"
shows C
apply (rule prems)
apply (rule conjunct1)
apply (rule prems)
apply (rule conjunct2)
apply (rule prems)
done

lemma
assumes "!!A. ~ ~A ==> A"
shows "B | ~B"
apply (rule prems)
apply (rule notI)
apply (rule_tac P = "~B" in notE)
apply (rule_tac [2] notI)
apply (rule_tac [2] P = "B | ~B" in notE)
prefer 2 apply assumption
apply (rule_tac [2] disjI1)
prefer 2 apply assumption
apply (rule notI)
apply (rule_tac P = "B | ~B" in notE)
apply assumption
apply (rule disjI2)
apply assumption
done

lemma
assumes "!!A. ~ ~A ==> A"
shows "B | ~B"
apply (rule prems)
apply (rule notI)
apply (rule notE)
apply (rule_tac [2] notI)
apply (erule_tac [2] notE)
apply (erule_tac [2] disjI1)
apply (rule notI)
apply (erule notE)
apply (erule disjI2)
done

lemma
assumes "A | ~A"
and "~ ~A"
shows A
apply (rule disjE)
apply (rule prems)
apply assumption
apply (rule FalseE)
apply (rule_tac P = "~A" in notE)
apply (rule prems)
apply assumption
done

subsection "Examples with quantifiers"

lemma
assumes "ALL z. G(z)"
shows "ALL z. G(z)|H(z)"
apply (rule allI)
apply (rule disjI1)
apply (rule prems [THEN spec])
done

lemma "ALL x. EX y. x=y"
apply (rule allI)
apply (rule exI)
apply (rule refl)
done

lemma "EX y. ALL x. x=y"
apply (rule exI)
apply (rule allI)
apply (rule refl)?
oops

text {* Parallel lifting example. *}
lemma "EX u. ALL x. EX v. ALL y. EX w. P(u,x,v,y,w)"
apply (rule exI allI)
apply (rule exI allI)
apply (rule exI allI)
apply (rule exI allI)
apply (rule exI allI)
oops

lemma
assumes "(EX z. F(z)) & B"
shows "EX z. F(z) & B"
apply (rule conjE)
apply (rule prems)
apply (rule exE)
apply assumption
apply (rule exI)
apply (rule conjI)
apply assumption
apply assumption
done

text {* A bigger demonstration of quantifiers -- not in the paper. *}
lemma "(EX y. ALL x. Q(x,y)) -->  (ALL x. EX y. Q(x,y))"
apply (rule impI)
apply (rule allI)
apply (rule exE, assumption)
apply (rule exI)
apply (rule allE, assumption)
apply assumption
done

end
```