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(* Title: FOLP/ex/Quantifiers_Int.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
First-Order Logic: quantifier examples (intuitionistic and classical)
Needs declarations of the theory "thy" and the tactic "tac".
*)
theory Quantifiers_Int
imports IFOLP
begin
schematic_goal "?p : (ALL x y. P(x,y)) --> (ALL y x. P(x,y))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (EX x y. P(x,y)) --> (EX y x. P(x,y))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*Converse is false*)
schematic_goal "?p : (ALL x. P(x)) | (ALL x. Q(x)) --> (ALL x. P(x) | Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (ALL x. P-->Q(x)) <-> (P--> (ALL x. Q(x)))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (ALL x. P(x)-->Q) <-> ((EX x. P(x)) --> Q)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Some harder ones"
schematic_goal "?p : (EX x. P(x) | Q(x)) <-> (EX x. P(x)) | (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*Converse is false*)
schematic_goal "?p : (EX x. P(x)&Q(x)) --> (EX x. P(x)) & (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Basic test of quantifier reasoning"
(*TRUE*)
schematic_goal "?p : (EX y. ALL x. Q(x,y)) --> (ALL x. EX y. Q(x,y))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (ALL x. Q(x)) --> (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "The following should fail, as they are false!"
schematic_goal "?p : (ALL x. EX y. Q(x,y)) --> (EX y. ALL x. Q(x,y))"
apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
oops
schematic_goal "?p : (EX x. Q(x)) --> (ALL x. Q(x))"
apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
oops
schematic_goal "?p : P(?a) --> (ALL x. P(x))"
apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
oops
schematic_goal "?p : (P(?a) --> (ALL x. Q(x))) --> (ALL x. P(x) --> Q(x))"
apply (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)?
oops
text "Back to things that are provable..."
schematic_goal "?p : (ALL x. P(x)-->Q(x)) & (EX x. P(x)) --> (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*An example of why exI should be delayed as long as possible*)
schematic_goal "?p : (P --> (EX x. Q(x))) & P --> (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (ALL x. P(x)-->Q(f(x))) & (ALL x. Q(x)-->R(g(x))) & P(d) --> R(?a)"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
schematic_goal "?p : (ALL x. Q(x)) --> (EX x. Q(x))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
text "Some slow ones"
(*Principia Mathematica *11.53 *)
schematic_goal "?p : (ALL x y. P(x) --> Q(y)) <-> ((EX x. P(x)) --> (ALL y. Q(y)))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*Principia Mathematica *11.55 *)
schematic_goal "?p : (EX x y. P(x) & Q(x,y)) <-> (EX x. P(x) & (EX y. Q(x,y)))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
(*Principia Mathematica *11.61 *)
schematic_goal "?p : (EX y. ALL x. P(x) --> Q(x,y)) --> (ALL x. P(x) --> (EX y. Q(x,y)))"
by (tactic \<open>IntPr.fast_tac \<^context> 1\<close>)
end
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