(* Title: HOL/Eisbach/Examples_FOL.thy
Author: Daniel Matichuk, NICTA/UNSW
*)
section \Basic Eisbach examples (in FOL)\
theory Examples_FOL
imports FOL Eisbach_Old_Appl_Syntax
begin
subsection \Basic methods\
method my_intros = (rule conjI | rule impI)
lemma "P \ Q \ Z \ X"
apply my_intros+
oops
method my_intros' uses intros = (rule conjI | rule impI | rule intros)
lemma "P \ Q \ Z \ X"
apply (my_intros' intros: disjI1)+
oops
method my_spec for x :: 'a = (drule spec[where x="x"])
lemma "\x. P(x) \ P(x)"
apply (my_spec x)
apply assumption
done
subsection \Demo\
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 \ B) \ (A \ C) \ (B \ C)) \ C"
apply prop_solver
done
method guess_all =
(match premises in U[thin]:"\x. P (x :: 'a)" for P \
\match premises in "?H (y :: 'a)" for y \
\rule allE[where P = P and x = y, OF U]\
| match conclusion in "?H (y :: 'a)" for y \
\rule allE[where P = P and x = y, OF U]\\)
lemma "(\x. P(x) \ Q(x)) \ P(y) \ Q(y)"
apply guess_all
apply prop_solver
done
lemma "(\x. P(x) \ Q(x)) \ P(z) \ P(y) \ Q(y)"
apply (solves \guess_all, prop_solver\) \ \Try it without solve\
done
method guess_ex =
(match conclusion in
"\x. P (x :: 'a)" for P \
\match premises in "?H (x :: 'a)" for x \
\rule exI[where x=x]\\)
lemma "P(x) \ \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 "(\x. P(x)) \ (\x. Q(x)) \ (\x. P(x) \ Q(x))"
and "\x. P(x) \ (\x. P(x))"
and "(\x. \y. R(x, y)) \ (\y. \x. R(x, y))"
by fol_solver+
end