--- a/CONTRIBUTORS Sun Oct 22 14:39:41 2017 +0200
+++ b/CONTRIBUTORS Sun Oct 22 16:35:24 2017 +0200
@@ -6,6 +6,8 @@
Contributions to this Isabelle version
--------------------------------------
+* October 2017: Alexander Maletzky
+ Derivation of axiom "iff" in HOL.thy from the other axioms.
Contributions to Isabelle2017
-----------------------------
--- a/src/HOL/HOL.thy Sun Oct 22 14:39:41 2017 +0200
+++ b/src/HOL/HOL.thy Sun Oct 22 16:35:24 2017 +0200
@@ -55,6 +55,15 @@
subsection \<open>Primitive logic\<close>
+text \<open>
+The definition of the logic is based on Mike Gordon's technical report \cite{Gordon-TR68} that
+describes the first implementation of HOL. However, there are a number of differences.
+In particular, we start with the definite description operator and introduce Hilbert's \<open>\<epsilon>\<close> operator
+only much later. Moreover, axiom \<open>(P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P) \<longrightarrow> (P = Q)\<close> is derived from the other
+axioms. The fact that this axiom is derivable was first noticed by Bruno Barras (for Mike Gordon's
+line of HOL systems) and later independently by Alexander Maletzky (for Isabelle/HOL).
+\<close>
+
subsubsection \<open>Core syntax\<close>
setup \<open>Axclass.class_axiomatization (@{binding type}, [])\<close>
@@ -195,7 +204,6 @@
impI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q" and
mp: "\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q" and
- iff: "(P \<longrightarrow> Q) \<longrightarrow> (Q \<longrightarrow> P) \<longrightarrow> (P = Q)" and
True_or_False: "(P = True) \<or> (P = False)"
definition If :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(if (_)/ then (_)/ else (_))" [0, 0, 10] 10)
@@ -283,9 +291,6 @@
subsubsection \<open>Equality of booleans -- iff\<close>
-lemma iffI: assumes "P \<Longrightarrow> Q" and "Q \<Longrightarrow> P" shows "P = Q"
- by (iprover intro: iff [THEN mp, THEN mp] impI assms)
-
lemma iffD2: "\<lbrakk>P = Q; Q\<rbrakk> \<Longrightarrow> P"
by (erule ssubst)
@@ -305,24 +310,16 @@
by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
-subsubsection \<open>True\<close>
+subsubsection \<open>True (1)\<close>
lemma TrueI: True
unfolding True_def by (rule refl)
-lemma eqTrueI: "P \<Longrightarrow> P = True"
- by (iprover intro: iffI TrueI)
-
lemma eqTrueE: "P = True \<Longrightarrow> P"
by (erule iffD2) (rule TrueI)
-subsubsection \<open>Universal quantifier\<close>
-
-lemma allI:
- assumes "\<And>x::'a. P x"
- shows "\<forall>x. P x"
- unfolding All_def by (iprover intro: ext eqTrueI assms)
+subsubsection \<open>Universal quantifier (1)\<close>
lemma spec: "\<forall>x::'a. P x \<Longrightarrow> P x"
apply (unfold All_def)
@@ -420,6 +417,70 @@
by (erule subst, erule ssubst, assumption)
+subsubsection \<open>Disjunction (1)\<close>
+
+lemma disjE:
+ assumes major: "P \<or> Q"
+ and minorP: "P \<Longrightarrow> R"
+ and minorQ: "Q \<Longrightarrow> R"
+ shows R
+ by (iprover intro: minorP minorQ impI
+ major [unfolded or_def, THEN spec, THEN mp, THEN mp])
+
+
+subsubsection \<open>Derivation of \<open>iffI\<close>\<close>
+
+text \<open>In an intuitionistic version of HOL \<open>iffI\<close> needs to be an axiom.\<close>
+
+lemma iffI:
+ assumes "P \<Longrightarrow> Q" and "Q \<Longrightarrow> P"
+ shows "P = Q"
+proof (rule disjE[OF True_or_False[of P]])
+ assume 1: "P = True"
+ note Q = assms(1)[OF eqTrueE[OF this]]
+ from 1 show ?thesis
+ proof (rule ssubst)
+ from True_or_False[of Q] show "True = Q"
+ proof (rule disjE)
+ assume "Q = True"
+ thus ?thesis by(rule sym)
+ next
+ assume "Q = False"
+ with Q have False by (rule rev_iffD1)
+ thus ?thesis by (rule FalseE)
+ qed
+ qed
+next
+ assume 2: "P = False"
+ thus ?thesis
+ proof (rule ssubst)
+ from True_or_False[of Q] show "False = Q"
+ proof (rule disjE)
+ assume "Q = True"
+ from 2 assms(2)[OF eqTrueE[OF this]] have False by (rule iffD1)
+ thus ?thesis by (rule FalseE)
+ next
+ assume "Q = False"
+ thus ?thesis by(rule sym)
+ qed
+ qed
+qed
+
+
+subsubsection \<open>True (2)\<close>
+
+lemma eqTrueI: "P \<Longrightarrow> P = True"
+ by (iprover intro: iffI TrueI)
+
+
+subsubsection \<open>Universal quantifier (2)\<close>
+
+lemma allI:
+ assumes "\<And>x::'a. P x"
+ shows "\<forall>x. P x"
+ unfolding All_def by (iprover intro: ext eqTrueI assms)
+
+
subsubsection \<open>Existential quantifier\<close>
lemma exI: "P x \<Longrightarrow> \<exists>x::'a. P x"
@@ -458,7 +519,7 @@
by (iprover intro: conjI assms)
-subsubsection \<open>Disjunction\<close>
+subsubsection \<open>Disjunction (2)\<close>
lemma disjI1: "P \<Longrightarrow> P \<or> Q"
unfolding or_def by (iprover intro: allI impI mp)
@@ -466,14 +527,6 @@
lemma disjI2: "Q \<Longrightarrow> P \<or> Q"
unfolding or_def by (iprover intro: allI impI mp)
-lemma disjE:
- assumes major: "P \<or> Q"
- and minorP: "P \<Longrightarrow> R"
- and minorQ: "Q \<Longrightarrow> R"
- shows R
- by (iprover intro: minorP minorQ impI
- major [unfolded or_def, THEN spec, THEN mp, THEN mp])
-
subsubsection \<open>Classical logic\<close>
--- a/src/HOL/ZF/Games.thy Sun Oct 22 14:39:41 2017 +0200
+++ b/src/HOL/ZF/Games.thy Sun Oct 22 16:35:24 2017 +0200
@@ -153,7 +153,7 @@
}
note games = this
show ?thesis
- apply (rule iff[rule_format])
+ apply (rule iffI)
apply (erule games)
apply (simp add: games_lfp_unfold[symmetric])
done
--- a/src/HOL/document/root.bib Sun Oct 22 14:39:41 2017 +0200
+++ b/src/HOL/document/root.bib Sun Oct 22 16:35:24 2017 +0200
@@ -1,3 +1,25 @@
+@book{Birkhoff79,
+ author = {Garret Birkhoff},
+ title = {Lattice Theory},
+ publisher = {American Mathematical Society},
+ year=1979
+}
+
+@Book{davenport92,
+ author = {H. Davenport},
+ title = {The Higher Arithmetic},
+ publisher = {Cambridge University Press},
+ year = 1992
+}
+
+@techreport{Gordon-TR68,
+ author = "Mike Gordon",
+ title = "{HOL}: {A} Machine Oriented Formulation of Higher-Order Logic",
+ institution = "University of Cambridge, Computer Laboratory",
+ number = 68,
+ year = 1985
+}
+
@book{card-book,
title = {Introduction to {C}ardinal {A}rithmetic},
author = {M. Holz and K. Steffens and E. Weitz},
@@ -5,11 +27,13 @@
year = 1999,
}
-@Book{davenport92,
- author = {H. Davenport},
- title = {The Higher Arithmetic},
- publisher = {Cambridge University Press},
- year = 1992
+@book{Nipkow-et-al:2002:tutorial,
+ author = {T. Nipkow and L. C. Paulson and M. Wenzel},
+ title = {{Isabelle/HOL} --- A Proof Assistant for Higher-Order Logic},
+ series = {LNCS},
+ volume = 2283,
+ year = 2002,
+ publisher = {Springer-Verlag}
}
@InProceedings{paulin-tlca,
@@ -26,19 +50,3 @@
year = 1993,
publisher = {Springer},
series = {LNCS 664}}
-
-@book{Birkhoff79,
- author = {Garret Birkhoff},
- title = {Lattice Theory},
- publisher = {American Mathematical Society},
- year=1979
-}
-
-@book{Nipkow-et-al:2002:tutorial,
- author = {T. Nipkow and L. C. Paulson and M. Wenzel},
- title = {{Isabelle/HOL} --- A Proof Assistant for Higher-Order Logic},
- series = {LNCS},
- volume = 2283,
- year = 2002,
- publisher = {Springer-Verlag}
-}
\ No newline at end of file