doc-src/TutorialI/Rules/Basic.thy

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/doc-src/TutorialI/Rules/Basic.thy	Mon Oct 23 16:25:04 2000 +0200
@@ -0,0 +1,290 @@
+theory Basic = Main:
+
+lemma conj_rule: "\<lbrakk> P; Q \<rbrakk> \<Longrightarrow> P \<and> (Q \<and> P)"
+apply (rule conjI)
+ apply assumption
+apply (rule conjI)
+ apply assumption
+apply assumption
+done
+    
+
+lemma disj_swap: "P | Q \<Longrightarrow> Q | P"
+apply (erule disjE)
+ apply (rule disjI2)
+ apply assumption
+apply (rule disjI1)
+apply assumption
+done
+
+lemma conj_swap: "P \<and> Q \<Longrightarrow> Q \<and> P"
+apply (rule conjI)
+ apply (drule conjunct2)
+ apply assumption
+apply (drule conjunct1)
+apply assumption
+done
+
+lemma imp_uncurry: "P \<longrightarrow> Q \<longrightarrow> R \<Longrightarrow> P \<and> Q \<longrightarrow> R"
+apply (rule impI)
+apply (erule conjE)
+apply (drule mp)
+ apply assumption
+apply (drule mp)
+  apply assumption
+ apply assumption
+done
+
+text {*
+substitution
+
+@{thm[display] ssubst}
+\rulename{ssubst}
+*};
+
+lemma "\<lbrakk> x = f x; P(f x) \<rbrakk> \<Longrightarrow> P x"
+apply (erule ssubst)
+apply assumption
+done
+
+text {*
+also provable by simp (re-orients)
+*};
+
+lemma "\<lbrakk> x = f x; P (f x) (f x) x \<rbrakk> \<Longrightarrow> P x x x"
+apply (erule ssubst)
+back
+back
+back
+back
+apply assumption
+done
+
+text {*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}x\ {\isacharequal}\ f\ x{\isacharsemicolon}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ x\ x\isanewline
+\ \isadigit{1}{\isachardot}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}
+
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}x\ {\isacharequal}\ f\ x{\isacharsemicolon}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ x\ x\isanewline
+\ \isadigit{1}{\isachardot}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isasymLongrightarrow}\ P\ x\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}
+
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}x\ {\isacharequal}\ f\ x{\isacharsemicolon}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ x\ x\isanewline
+\ \isadigit{1}{\isachardot}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isacharparenleft}f\ x{\isacharparenright}
+
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}x\ {\isacharequal}\ f\ x{\isacharsemicolon}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ x\ x\isanewline
+\ \isadigit{1}{\isachardot}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isasymLongrightarrow}\ P\ x\ x\ {\isacharparenleft}f\ x{\isacharparenright}
+
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ \isadigit{1}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}x\ {\isacharequal}\ f\ x{\isacharsemicolon}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ x\ x\ x\isanewline
+\ \isadigit{1}{\isachardot}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}\ {\isacharparenleft}f\ x{\isacharparenright}\ x
+*};
+
+lemma "\<lbrakk> x = f x; P (f x) (f x) x \<rbrakk> \<Longrightarrow> P x x x"
+apply (erule ssubst, assumption)
+done
+
+lemma "\<lbrakk> x = f x; P (f x) (f x) x \<rbrakk> \<Longrightarrow> P x x x"
+apply (erule_tac P="\<lambda>u. P u u x" in ssubst);
+apply assumption
+done
+
+
+text {*
+negation
+
+@{thm[display] notI}
+\rulename{notI}
+
+@{thm[display] notE}
+\rulename{notE}
+
+@{thm[display] classical}
+\rulename{classical}
+
+@{thm[display] contrapos_pp}
+\rulename{contrapos_pp}
+
+@{thm[display] contrapos_np}
+\rulename{contrapos_np}
+
+@{thm[display] contrapos_nn}
+\rulename{contrapos_nn}
+*};
+
+
+lemma "\<lbrakk>\<not>(P\<longrightarrow>Q); \<not>(R\<longrightarrow>Q)\<rbrakk> \<Longrightarrow> R"
+apply (erule_tac Q="R\<longrightarrow>Q" in contrapos_np)
+txt{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isacharsemicolon}\ {\isasymnot}\ {\isacharparenleft}R\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ R\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isacharsemicolon}\ {\isasymnot}\ R{\isasymrbrakk}\ {\isasymLongrightarrow}\ R\ {\isasymlongrightarrow}\ Q
+*}
+
+apply intro
+txt{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{3}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isacharsemicolon}\ {\isasymnot}\ {\isacharparenleft}R\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ R\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}P\ {\isasymlongrightarrow}\ Q{\isacharparenright}{\isacharsemicolon}\ {\isasymnot}\ R{\isacharsemicolon}\ R{\isasymrbrakk}\ {\isasymLongrightarrow}\ Q
+*}
+apply (erule notE, assumption)
+done
+
+
+lemma "(P \<or> Q) \<and> R \<Longrightarrow> P \<or> Q \<and> R"
+apply intro
+txt{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R{\isacharsemicolon}\ {\isasymnot}\ {\isacharparenleft}Q\ {\isasymand}\ R{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P
+*}
+
+apply (elim conjE disjE)
+ apply assumption
+
+txt{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{4}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isasymnot}\ {\isacharparenleft}Q\ {\isasymand}\ R{\isacharparenright}{\isacharsemicolon}\ R{\isacharsemicolon}\ Q{\isasymrbrakk}\ {\isasymLongrightarrow}\ P
+*}
+
+apply (erule contrapos_np, rule conjI)
+txt{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{6}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isacharparenleft}P\ {\isasymor}\ Q{\isacharparenright}\ {\isasymand}\ R\ {\isasymLongrightarrow}\ P\ {\isasymor}\ Q\ {\isasymand}\ R\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ Q\isanewline
+\ {\isadigit{2}}{\isachardot}\ {\isasymlbrakk}R{\isacharsemicolon}\ Q{\isacharsemicolon}\ {\isasymnot}\ P{\isasymrbrakk}\ {\isasymLongrightarrow}\ R
+*}
+
+  apply assumption
+ apply assumption
+done
+
+
+
+text{*Quantifiers*}
+
+lemma "\<forall>x. P x \<longrightarrow> P x"
+apply (rule allI)
+apply (rule impI)
+apply assumption
+done
+
+lemma "(\<forall>x. P \<longrightarrow> Q x) \<Longrightarrow> P \<longrightarrow> (\<forall>x. Q x)"
+apply (rule impI)
+apply (rule allI)
+apply (drule spec)
+apply (drule mp)
+  apply assumption
+ apply assumption
+done
+
+lemma "\<lbrakk>\<forall>x. P x \<longrightarrow> P (f x); P a\<rbrakk> \<Longrightarrow> P(f (f a))"
+apply (frule spec)
+apply (drule mp, assumption)
+apply (drule spec)
+apply (drule mp, assumption, assumption)
+done
+
+text
+{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{1}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}{\isasymforall}x{\isachardot}\ P\ x\ {\isasymlongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}{\isacharsemicolon}\ P\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ {\isacharparenleft}f\ a{\isacharparenright}{\isacharparenright}\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isasymforall}x{\isachardot}\ P\ x\ {\isasymlongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}{\isacharsemicolon}\ P\ a{\isacharsemicolon}\ P\ {\isacharquery}x\ {\isasymlongrightarrow}\ P\ {\isacharparenleft}f\ {\isacharquery}x{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ {\isacharparenleft}f\ a{\isacharparenright}{\isacharparenright}
+*}
+
+text{*
+proof\ {\isacharparenleft}prove{\isacharparenright}{\isacharcolon}\ step\ {\isadigit{3}}\isanewline
+\isanewline
+goal\ {\isacharparenleft}lemma{\isacharparenright}{\isacharcolon}\isanewline
+{\isasymlbrakk}{\isasymforall}x{\isachardot}\ P\ x\ {\isasymlongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}{\isacharsemicolon}\ P\ a{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ {\isacharparenleft}f\ a{\isacharparenright}{\isacharparenright}\isanewline
+\ {\isadigit{1}}{\isachardot}\ {\isasymlbrakk}{\isasymforall}x{\isachardot}\ P\ x\ {\isasymlongrightarrow}\ P\ {\isacharparenleft}f\ x{\isacharparenright}{\isacharsemicolon}\ P\ a{\isacharsemicolon}\ P\ {\isacharparenleft}f\ a{\isacharparenright}{\isasymrbrakk}\ {\isasymLongrightarrow}\ P\ {\isacharparenleft}f\ {\isacharparenleft}f\ a{\isacharparenright}{\isacharparenright}
+*}
+
+lemma "\<lbrakk>\<forall>x. P x \<longrightarrow> P (f x); P a\<rbrakk> \<Longrightarrow> P(f (f a))"
+by blast
+
+lemma "(\<exists>x. P x) \<or> (\<exists>x. Q x) \<Longrightarrow> \<exists>x. P x \<or> Q x"
+apply elim
+ apply intro
+ apply assumption
+apply (intro exI disjI2) (*or else we get disjCI*)
+apply assumption
+done
+
+lemma "(P\<longrightarrow>Q) \<or> (Q\<longrightarrow>P)"
+apply intro
+apply elim
+apply assumption
+done
+
+lemma "(P\<or>Q)\<and>(P\<or>R) \<Longrightarrow> P \<or> (Q\<and>R)"
+apply intro
+apply (elim conjE disjE)
+apply blast
+apply blast
+apply blast
+apply blast
+(*apply elim*)
+done
+
+lemma "(\<exists>x. P \<and> Q x) \<Longrightarrow> P \<and> (\<exists>x. Q x)"
+apply (erule exE)
+apply (erule conjE)
+apply (rule conjI)
+ apply assumption
+apply (rule exI)
+ apply assumption
+done
+
+lemma "(\<exists>x. P x) \<and> (\<exists>x. Q x) \<Longrightarrow> \<exists>x. P x \<and> Q x"
+apply (erule conjE)
+apply (erule exE)
+apply (erule exE)
+apply (rule exI)
+apply (rule conjI)
+ apply assumption
+oops
+
+lemma "\<forall> z. R z z \<Longrightarrow> \<exists>x. \<forall> y. R x y"
+apply (rule exI)
+apply (rule allI)
+apply (drule spec)
+oops
+
+lemma "\<forall>x. \<exists> y. x=y"
+apply (rule allI)
+apply (rule exI)
+apply (rule refl)
+done
+
+lemma "\<exists>x. \<forall> y. x=y"
+apply (rule exI)
+apply (rule allI)
+oops
+
+end