--- a/src/FOL/ex/If.thy Mon Oct 19 20:31:13 2015 +0200
+++ b/src/FOL/ex/If.thy Mon Oct 19 23:00:07 2015 +0200
@@ -5,57 +5,56 @@
section \<open>First-Order Logic: the 'if' example\<close>
-theory If imports FOL begin
-
-definition "if" :: "[o,o,o]=>o" where
- "if(P,Q,R) == P&Q | ~P&R"
+theory If
+imports FOL
+begin
-lemma ifI:
- "[| P ==> Q; ~P ==> R |] ==> if(P,Q,R)"
-apply (simp add: if_def, blast)
-done
+definition "if" :: "[o,o,o]=>o"
+ where "if(P,Q,R) \<equiv> P \<and> Q \<or> \<not> P \<and> R"
+
+lemma ifI: "\<lbrakk>P \<Longrightarrow> Q; \<not> P \<Longrightarrow> R\<rbrakk> \<Longrightarrow> if(P,Q,R)"
+ unfolding if_def by blast
-lemma ifE:
- "[| if(P,Q,R); [| P; Q |] ==> S; [| ~P; R |] ==> S |] ==> S"
-apply (simp add: if_def, blast)
-done
+lemma ifE: "\<lbrakk>if(P,Q,R); \<lbrakk>P; Q\<rbrakk> \<Longrightarrow> S; \<lbrakk>\<not> P; R\<rbrakk> \<Longrightarrow> S\<rbrakk> \<Longrightarrow> S"
+ unfolding if_def by blast
-lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))"
-apply (rule iffI)
-apply (erule ifE)
-apply (erule ifE)
-apply (rule ifI)
-apply (rule ifI)
-oops
+lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) \<longleftrightarrow> if(Q, if(P,A,C), if(P,B,D))"
+ apply (rule iffI)
+ apply (erule ifE)
+ apply (erule ifE)
+ apply (rule ifI)
+ apply (rule ifI)
+ oops
text\<open>Trying again from the beginning in order to use @{text blast}\<close>
declare ifI [intro!]
declare ifE [elim!]
-lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) <-> if(Q, if(P,A,C), if(P,B,D))"
-by blast
+lemma if_commute: "if(P, if(Q,A,B), if(Q,C,D)) \<longleftrightarrow> if(Q, if(P,A,C), if(P,B,D))"
+ by blast
-lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))"
-by blast
+lemma "if(if(P,Q,R), A, B) \<longleftrightarrow> if(P, if(Q,A,B), if(R,A,B))"
+ by blast
text\<open>Trying again from the beginning in order to prove from the definitions\<close>
-lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,A,B))"
-by (simp add: if_def, blast)
+lemma "if(if(P,Q,R), A, B) \<longleftrightarrow> if(P, if(Q,A,B), if(R,A,B))"
+ unfolding if_def by blast
-text\<open>An invalid formula. High-level rules permit a simpler diagnosis\<close>
-lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))"
-apply auto
- -- \<open>The next step will fail unless subgoals remain\<close>
-apply (tactic all_tac)
-oops
+text \<open>An invalid formula. High-level rules permit a simpler diagnosis.\<close>
+lemma "if(if(P,Q,R), A, B) \<longleftrightarrow> if(P, if(Q,A,B), if(R,B,A))"
+ apply auto
+ -- \<open>The next step will fail unless subgoals remain\<close>
+ apply (tactic all_tac)
+ oops
-text\<open>Trying again from the beginning in order to prove from the definitions\<close>
-lemma "if(if(P,Q,R), A, B) <-> if(P, if(Q,A,B), if(R,B,A))"
-apply (simp add: if_def, auto)
- -- \<open>The next step will fail unless subgoals remain\<close>
-apply (tactic all_tac)
-oops
+text \<open>Trying again from the beginning in order to prove from the definitions.\<close>
+lemma "if(if(P,Q,R), A, B) \<longleftrightarrow> if(P, if(Q,A,B), if(R,B,A))"
+ unfolding if_def
+ apply auto
+ -- \<open>The next step will fail unless subgoals remain\<close>
+ apply (tactic all_tac)
+ oops
end