theory Demo3 = Main:

section{* Propositional logic *}

subsection{* Basic rules *}

text{* \<and> *}
thm conjI conjE 

text{* \<or> *}
thm disjI1 disjI2 disjE

text{* \<longrightarrow> *}
thm impI impE


subsection{* Examples *}

text{* a simple backward step: *}
lemma "A \<and> B"
apply(rule conjI)
oops

text{* a simple backward proof: *}
lemma "B \<and> A \<longrightarrow> A \<and> B"
apply(rule impI)
apply(erule conjE)
apply(rule conjI)
 apply(assumption)
apply(assumption)
done

lemma "A \<or> B \<longrightarrow> B \<or> A"
oops

lemma "\<lbrakk> A \<longrightarrow> B; B \<longrightarrow> C \<rbrakk> \<Longrightarrow> A \<longrightarrow> C"
oops

thm notI notE 

lemma "\<not>A \<or> \<not>B \<Longrightarrow> \<not>(A \<and> B)"
oops


text {* Case distinctions *}

thm FalseE

lemma "(\<not>P \<longrightarrow> False) \<longrightarrow> P"
  apply (cases "P")
  oops


text{* Explicit backtracking: *}
lemma "\<lbrakk> P \<and> Q; A \<and> B \<rbrakk> \<Longrightarrow> A"
apply(erule conjE)
back
apply(assumption)
done

text{* UGLY! EVIL! AVOID! *}


text{* Implict backtracking: chaining with , *}
lemma "\<lbrakk> P \<and> Q; A \<and> B \<rbrakk> \<Longrightarrow> A"
apply(erule conjE, assumption)
done


text {* more rules *}
text{* \<and> *}
thm conjunct1 conjunct2

text{* \<or> *}
thm disjCI  

text{* \<longrightarrow> *}
thm mp

text{* = (iff) *}
thm iffI iffE iffD1 iffD2

text{* Equality *}
thm refl sym trans

text{* \<not> *}
thm notI notE

text{* Contradictions *}
thm FalseE ccontr classical excluded_middle


end