session-qualified theory imports: isabelle imports -U -i -d '~~/src/Benchmarks' -a;
(* Title: HOL/Eisbach/Examples_FOL.thy
Author: Daniel Matichuk, NICTA/UNSW
*)
section \<open>Basic Eisbach examples (in FOL)\<close>
theory Examples_FOL
imports FOL Eisbach_Old_Appl_Syntax
begin
subsection \<open>Basic methods\<close>
method my_intros = (rule conjI | rule impI)
lemma "P \<and> Q \<longrightarrow> Z \<and> X"
apply my_intros+
oops
method my_intros' uses intros = (rule conjI | rule impI | rule intros)
lemma "P \<and> Q \<longrightarrow> Z \<or> X"
apply (my_intros' intros: disjI1)+
oops
method my_spec for x :: 'a = (drule spec[where x="x"])
lemma "\<forall>x. P(x) \<Longrightarrow> P(x)"
apply (my_spec x)
apply assumption
done
subsection \<open>Demo\<close>
named_theorems intros and elims and subst
method prop_solver declares intros elims subst =
(assumption |
rule intros | erule elims |
subst subst | subst (asm) subst |
(erule notE; solves prop_solver))+
lemmas [intros] =
conjI
impI
disjCI
iffI
notI
lemmas [elims] =
impCE
conjE
disjE
lemma "((A \<or> B) \<and> (A \<longrightarrow> C) \<and> (B \<longrightarrow> C)) \<longrightarrow> C"
apply prop_solver
done
method guess_all =
(match premises in U[thin]:"\<forall>x. P (x :: 'a)" for P \<Rightarrow>
\<open>match premises in "?H (y :: 'a)" for y \<Rightarrow>
\<open>rule allE[where P = P and x = y, OF U]\<close>
| match conclusion in "?H (y :: 'a)" for y \<Rightarrow>
\<open>rule allE[where P = P and x = y, OF U]\<close>\<close>)
lemma "(\<forall>x. P(x) \<longrightarrow> Q(x)) \<Longrightarrow> P(y) \<Longrightarrow> Q(y)"
apply guess_all
apply prop_solver
done
lemma "(\<forall>x. P(x) \<longrightarrow> Q(x)) \<Longrightarrow> P(z) \<Longrightarrow> P(y) \<Longrightarrow> Q(y)"
apply (solves \<open>guess_all, prop_solver\<close>) \<comment> \<open>Try it without solve\<close>
done
method guess_ex =
(match conclusion in
"\<exists>x. P (x :: 'a)" for P \<Rightarrow>
\<open>match premises in "?H (x :: 'a)" for x \<Rightarrow>
\<open>rule exI[where x=x]\<close>\<close>)
lemma "P(x) \<Longrightarrow> \<exists>x. P(x)"
apply guess_ex
apply prop_solver
done
method fol_solver =
((guess_ex | guess_all | prop_solver); solves fol_solver)
declare
allI [intros]
exE [elims]
ex_simps [subst]
all_simps [subst]
lemma "(\<forall>x. P(x)) \<and> (\<forall>x. Q(x)) \<Longrightarrow> (\<forall>x. P(x) \<and> Q(x))"
and "\<exists>x. P(x) \<longrightarrow> (\<forall>x. P(x))"
and "(\<exists>x. \<forall>y. R(x, y)) \<longrightarrow> (\<forall>y. \<exists>x. R(x, y))"
by fol_solver+
end