header{*Examples of Intuitionistic Reasoning*}
theory IFOL_examples imports IFOL begin
text{*Quantifier example from the book Logic and Computation*}
lemma "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule impI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule allI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule exI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule exE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule allE)
--{* @{subgoals[display,indent=0,margin=65]} *}
txt{*Now @{text "apply assumption"} fails*}
oops
text{*Trying again, with the same first two steps*}
lemma "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule impI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule allI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule exE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule exI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule allE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply assumption
--{* @{subgoals[display,indent=0,margin=65]} *}
done
lemma "(EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"
by (tactic {*IntPr.fast_tac 1*})
text{*Example of Dyckhoff's method*}
lemma "~ ~ ((P-->Q) | (Q-->P))"
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (unfold not_def)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (rule impI)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule disj_impE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule imp_impE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply (erule imp_impE)
--{* @{subgoals[display,indent=0,margin=65]} *}
apply assumption
apply (erule FalseE)+
done
end