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