src/HOL/Eisbach/Examples_FOL.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Eisbach/Examples_FOL.thy	Wed Jan 13 16:41:32 2016 +0100
@@ -0,0 +1,100 @@
+(*  Title:      HOL/Eisbach/Examples.thy
+    Author:     Daniel Matichuk, NICTA/UNSW
+*)
+
+section \<open>Basic Eisbach examples (in FOL)\<close>
+
+theory Examples_FOL
+imports "~~/src/FOL/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