(* Title: FOL/IFOL.thy
Author: Lawrence C Paulson and Markus Wenzel
*)
section \<open>Intuitionistic first-order logic\<close>
theory IFOL
imports Pure
begin
ML_file \<open>~~/src/Tools/misc_legacy.ML\<close>
ML_file \<open>~~/src/Provers/splitter.ML\<close>
ML_file \<open>~~/src/Provers/hypsubst.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/zipper.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/isand.ML\<close>
ML_file \<open>~~/src/Tools/IsaPlanner/rw_inst.ML\<close>
ML_file \<open>~~/src/Provers/quantifier1.ML\<close>
ML_file \<open>~~/src/Tools/intuitionistic.ML\<close>
ML_file \<open>~~/src/Tools/project_rule.ML\<close>
ML_file \<open>~~/src/Tools/atomize_elim.ML\<close>
subsection \<open>Syntax and axiomatic basis\<close>
setup Pure_Thy.old_appl_syntax_setup
setup \<open>Proofterm.set_preproc (Proof_Rewrite_Rules.standard_preproc [])\<close>
class "term"
default_sort \<open>term\<close>
typedecl o
judgment
Trueprop :: \<open>o \<Rightarrow> prop\<close> (\<open>(_)\<close> 5)
subsubsection \<open>Equality\<close>
axiomatization
eq :: \<open>['a, 'a] \<Rightarrow> o\<close> (infixl \<open>=\<close> 50)
where
refl: \<open>a = a\<close> and
subst: \<open>a = b \<Longrightarrow> P(a) \<Longrightarrow> P(b)\<close>
subsubsection \<open>Propositional logic\<close>
axiomatization
False :: \<open>o\<close> and
conj :: \<open>[o, o] => o\<close> (infixr \<open>\<and>\<close> 35) and
disj :: \<open>[o, o] => o\<close> (infixr \<open>\<or>\<close> 30) and
imp :: \<open>[o, o] => o\<close> (infixr \<open>\<longrightarrow>\<close> 25)
where
conjI: \<open>\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> P \<and> Q\<close> and
conjunct1: \<open>P \<and> Q \<Longrightarrow> P\<close> and
conjunct2: \<open>P \<and> Q \<Longrightarrow> Q\<close> and
disjI1: \<open>P \<Longrightarrow> P \<or> Q\<close> and
disjI2: \<open>Q \<Longrightarrow> P \<or> Q\<close> and
disjE: \<open>\<lbrakk>P \<or> Q; P \<Longrightarrow> R; Q \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R\<close> and
impI: \<open>(P \<Longrightarrow> Q) \<Longrightarrow> P \<longrightarrow> Q\<close> and
mp: \<open>\<lbrakk>P \<longrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q\<close> and
FalseE: \<open>False \<Longrightarrow> P\<close>
subsubsection \<open>Quantifiers\<close>
axiomatization
All :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close> (binder \<open>\<forall>\<close> 10) and
Ex :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close> (binder \<open>\<exists>\<close> 10)
where
allI: \<open>(\<And>x. P(x)) \<Longrightarrow> (\<forall>x. P(x))\<close> and
spec: \<open>(\<forall>x. P(x)) \<Longrightarrow> P(x)\<close> and
exI: \<open>P(x) \<Longrightarrow> (\<exists>x. P(x))\<close> and
exE: \<open>\<lbrakk>\<exists>x. P(x); \<And>x. P(x) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R\<close>
subsubsection \<open>Definitions\<close>
definition \<open>True \<equiv> False \<longrightarrow> False\<close>
definition Not (\<open>\<not> _\<close> [40] 40)
where not_def: \<open>\<not> P \<equiv> P \<longrightarrow> False\<close>
definition iff (infixr \<open>\<longleftrightarrow>\<close> 25)
where \<open>P \<longleftrightarrow> Q \<equiv> (P \<longrightarrow> Q) \<and> (Q \<longrightarrow> P)\<close>
definition Ex1 :: \<open>('a \<Rightarrow> o) \<Rightarrow> o\<close> (binder \<open>\<exists>!\<close> 10)
where ex1_def: \<open>\<exists>!x. P(x) \<equiv> \<exists>x. P(x) \<and> (\<forall>y. P(y) \<longrightarrow> y = x)\<close>
axiomatization where \<comment> \<open>Reflection, admissible\<close>
eq_reflection: \<open>(x = y) \<Longrightarrow> (x \<equiv> y)\<close> and
iff_reflection: \<open>(P \<longleftrightarrow> Q) \<Longrightarrow> (P \<equiv> Q)\<close>
abbreviation not_equal :: \<open>['a, 'a] \<Rightarrow> o\<close> (infixl \<open>\<noteq>\<close> 50)
where \<open>x \<noteq> y \<equiv> \<not> (x = y)\<close>
subsubsection \<open>Old-style ASCII syntax\<close>
notation (ASCII)
not_equal (infixl \<open>~=\<close> 50) and
Not (\<open>~ _\<close> [40] 40) and
conj (infixr \<open>&\<close> 35) and
disj (infixr \<open>|\<close> 30) and
All (binder \<open>ALL \<close> 10) and
Ex (binder \<open>EX \<close> 10) and
Ex1 (binder \<open>EX! \<close> 10) and
imp (infixr \<open>-->\<close> 25) and
iff (infixr \<open><->\<close> 25)
subsection \<open>Lemmas and proof tools\<close>
lemmas strip = impI allI
lemma TrueI: \<open>True\<close>
unfolding True_def by (rule impI)
subsubsection \<open>Sequent-style elimination rules for \<open>\<and>\<close> \<open>\<longrightarrow>\<close> and \<open>\<forall>\<close>\<close>
lemma conjE:
assumes major: \<open>P \<and> Q\<close>
and r: \<open>\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule r)
apply (rule major [THEN conjunct1])
apply (rule major [THEN conjunct2])
done
lemma impE:
assumes major: \<open>P \<longrightarrow> Q\<close>
and \<open>P\<close>
and r: \<open>Q \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule r)
apply (rule major [THEN mp])
apply (rule \<open>P\<close>)
done
lemma allE:
assumes major: \<open>\<forall>x. P(x)\<close>
and r: \<open>P(x) \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule r)
apply (rule major [THEN spec])
done
text \<open>Duplicates the quantifier; for use with \<^ML>\<open>eresolve_tac\<close>.\<close>
lemma all_dupE:
assumes major: \<open>\<forall>x. P(x)\<close>
and r: \<open>\<lbrakk>P(x); \<forall>x. P(x)\<rbrakk> \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule r)
apply (rule major [THEN spec])
apply (rule major)
done
subsubsection \<open>Negation rules, which translate between \<open>\<not> P\<close> and \<open>P \<longrightarrow> False\<close>\<close>
lemma notI: \<open>(P \<Longrightarrow> False) \<Longrightarrow> \<not> P\<close>
unfolding not_def by (erule impI)
lemma notE: \<open>\<lbrakk>\<not> P; P\<rbrakk> \<Longrightarrow> R\<close>
unfolding not_def by (erule mp [THEN FalseE])
lemma rev_notE: \<open>\<lbrakk>P; \<not> P\<rbrakk> \<Longrightarrow> R\<close>
by (erule notE)
text \<open>This is useful with the special implication rules for each kind of \<open>P\<close>.\<close>
lemma not_to_imp:
assumes \<open>\<not> P\<close>
and r: \<open>P \<longrightarrow> False \<Longrightarrow> Q\<close>
shows \<open>Q\<close>
apply (rule r)
apply (rule impI)
apply (erule notE [OF \<open>\<not> P\<close>])
done
text \<open>
For substitution into an assumption \<open>P\<close>, reduce \<open>Q\<close> to \<open>P \<longrightarrow> Q\<close>, substitute into this implication, then apply \<open>impI\<close> to
move \<open>P\<close> back into the assumptions.
\<close>
lemma rev_mp: \<open>\<lbrakk>P; P \<longrightarrow> Q\<rbrakk> \<Longrightarrow> Q\<close>
by (erule mp)
text \<open>Contrapositive of an inference rule.\<close>
lemma contrapos:
assumes major: \<open>\<not> Q\<close>
and minor: \<open>P \<Longrightarrow> Q\<close>
shows \<open>\<not> P\<close>
apply (rule major [THEN notE, THEN notI])
apply (erule minor)
done
subsubsection \<open>Modus Ponens Tactics\<close>
text \<open>
Finds \<open>P \<longrightarrow> Q\<close> and P in the assumptions, replaces implication by
\<open>Q\<close>.
\<close>
ML \<open>
fun mp_tac ctxt i =
eresolve_tac ctxt @{thms notE impE} i THEN assume_tac ctxt i;
fun eq_mp_tac ctxt i =
eresolve_tac ctxt @{thms notE impE} i THEN eq_assume_tac i;
\<close>
subsection \<open>If-and-only-if\<close>
lemma iffI: \<open>\<lbrakk>P \<Longrightarrow> Q; Q \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P \<longleftrightarrow> Q\<close>
apply (unfold iff_def)
apply (rule conjI)
apply (erule impI)
apply (erule impI)
done
lemma iffE:
assumes major: \<open>P \<longleftrightarrow> Q\<close>
and r: \<open>P \<longrightarrow> Q \<Longrightarrow> Q \<longrightarrow> P \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (insert major, unfold iff_def)
apply (erule conjE)
apply (erule r)
apply assumption
done
subsubsection \<open>Destruct rules for \<open>\<longleftrightarrow>\<close> similar to Modus Ponens\<close>
lemma iffD1: \<open>\<lbrakk>P \<longleftrightarrow> Q; P\<rbrakk> \<Longrightarrow> Q\<close>
apply (unfold iff_def)
apply (erule conjunct1 [THEN mp])
apply assumption
done
lemma iffD2: \<open>\<lbrakk>P \<longleftrightarrow> Q; Q\<rbrakk> \<Longrightarrow> P\<close>
apply (unfold iff_def)
apply (erule conjunct2 [THEN mp])
apply assumption
done
lemma rev_iffD1: \<open>\<lbrakk>P; P \<longleftrightarrow> Q\<rbrakk> \<Longrightarrow> Q\<close>
apply (erule iffD1)
apply assumption
done
lemma rev_iffD2: \<open>\<lbrakk>Q; P \<longleftrightarrow> Q\<rbrakk> \<Longrightarrow> P\<close>
apply (erule iffD2)
apply assumption
done
lemma iff_refl: \<open>P \<longleftrightarrow> P\<close>
by (rule iffI)
lemma iff_sym: \<open>Q \<longleftrightarrow> P \<Longrightarrow> P \<longleftrightarrow> Q\<close>
apply (erule iffE)
apply (rule iffI)
apply (assumption | erule mp)+
done
lemma iff_trans: \<open>\<lbrakk>P \<longleftrightarrow> Q; Q \<longleftrightarrow> R\<rbrakk> \<Longrightarrow> P \<longleftrightarrow> R\<close>
apply (rule iffI)
apply (assumption | erule iffE | erule (1) notE impE)+
done
subsection \<open>Unique existence\<close>
text \<open>
NOTE THAT the following 2 quantifications:
\<^item> \<open>\<exists>!x\<close> such that [\<open>\<exists>!y\<close> such that P(x,y)] (sequential)
\<^item> \<open>\<exists>!x,y\<close> such that P(x,y) (simultaneous)
do NOT mean the same thing. The parser treats \<open>\<exists>!x y.P(x,y)\<close> as sequential.
\<close>
lemma ex1I: \<open>P(a) \<Longrightarrow> (\<And>x. P(x) \<Longrightarrow> x = a) \<Longrightarrow> \<exists>!x. P(x)\<close>
apply (unfold ex1_def)
apply (assumption | rule exI conjI allI impI)+
done
text \<open>Sometimes easier to use: the premises have no shared variables. Safe!\<close>
lemma ex_ex1I: \<open>\<exists>x. P(x) \<Longrightarrow> (\<And>x y. \<lbrakk>P(x); P(y)\<rbrakk> \<Longrightarrow> x = y) \<Longrightarrow> \<exists>!x. P(x)\<close>
apply (erule exE)
apply (rule ex1I)
apply assumption
apply assumption
done
lemma ex1E: \<open>\<exists>! x. P(x) \<Longrightarrow> (\<And>x. \<lbrakk>P(x); \<forall>y. P(y) \<longrightarrow> y = x\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R\<close>
apply (unfold ex1_def)
apply (assumption | erule exE conjE)+
done
subsubsection \<open>\<open>\<longleftrightarrow>\<close> congruence rules for simplification\<close>
text \<open>Use \<open>iffE\<close> on a premise. For \<open>conj_cong\<close>, \<open>imp_cong\<close>, \<open>all_cong\<close>, \<open>ex_cong\<close>.\<close>
ML \<open>
fun iff_tac ctxt prems i =
resolve_tac ctxt (prems RL @{thms iffE}) i THEN
REPEAT1 (eresolve_tac ctxt @{thms asm_rl mp} i);
\<close>
method_setup iff =
\<open>Attrib.thms >>
(fn prems => fn ctxt => SIMPLE_METHOD' (iff_tac ctxt prems))\<close>
lemma conj_cong:
assumes \<open>P \<longleftrightarrow> P'\<close>
and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
shows \<open>(P \<and> Q) \<longleftrightarrow> (P' \<and> Q')\<close>
apply (insert assms)
apply (assumption | rule iffI conjI | erule iffE conjE mp | iff assms)+
done
text \<open>Reversed congruence rule! Used in ZF/Order.\<close>
lemma conj_cong2:
assumes \<open>P \<longleftrightarrow> P'\<close>
and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
shows \<open>(Q \<and> P) \<longleftrightarrow> (Q' \<and> P')\<close>
apply (insert assms)
apply (assumption | rule iffI conjI | erule iffE conjE mp | iff assms)+
done
lemma disj_cong:
assumes \<open>P \<longleftrightarrow> P'\<close> and \<open>Q \<longleftrightarrow> Q'\<close>
shows \<open>(P \<or> Q) \<longleftrightarrow> (P' \<or> Q')\<close>
apply (insert assms)
apply (erule iffE disjE disjI1 disjI2 |
assumption | rule iffI | erule (1) notE impE)+
done
lemma imp_cong:
assumes \<open>P \<longleftrightarrow> P'\<close>
and \<open>P' \<Longrightarrow> Q \<longleftrightarrow> Q'\<close>
shows \<open>(P \<longrightarrow> Q) \<longleftrightarrow> (P' \<longrightarrow> Q')\<close>
apply (insert assms)
apply (assumption | rule iffI impI | erule iffE | erule (1) notE impE | iff assms)+
done
lemma iff_cong: \<open>\<lbrakk>P \<longleftrightarrow> P'; Q \<longleftrightarrow> Q'\<rbrakk> \<Longrightarrow> (P \<longleftrightarrow> Q) \<longleftrightarrow> (P' \<longleftrightarrow> Q')\<close>
apply (erule iffE | assumption | rule iffI | erule (1) notE impE)+
done
lemma not_cong: \<open>P \<longleftrightarrow> P' \<Longrightarrow> \<not> P \<longleftrightarrow> \<not> P'\<close>
apply (assumption | rule iffI notI | erule (1) notE impE | erule iffE notE)+
done
lemma all_cong:
assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
shows \<open>(\<forall>x. P(x)) \<longleftrightarrow> (\<forall>x. Q(x))\<close>
apply (assumption | rule iffI allI | erule (1) notE impE | erule allE | iff assms)+
done
lemma ex_cong:
assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
shows \<open>(\<exists>x. P(x)) \<longleftrightarrow> (\<exists>x. Q(x))\<close>
apply (erule exE | assumption | rule iffI exI | erule (1) notE impE | iff assms)+
done
lemma ex1_cong:
assumes \<open>\<And>x. P(x) \<longleftrightarrow> Q(x)\<close>
shows \<open>(\<exists>!x. P(x)) \<longleftrightarrow> (\<exists>!x. Q(x))\<close>
apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE | iff assms)+
done
subsection \<open>Equality rules\<close>
lemma sym: \<open>a = b \<Longrightarrow> b = a\<close>
apply (erule subst)
apply (rule refl)
done
lemma trans: \<open>\<lbrakk>a = b; b = c\<rbrakk> \<Longrightarrow> a = c\<close>
apply (erule subst, assumption)
done
lemma not_sym: \<open>b \<noteq> a \<Longrightarrow> a \<noteq> b\<close>
apply (erule contrapos)
apply (erule sym)
done
text \<open>
Two theorems for rewriting only one instance of a definition:
the first for definitions of formulae and the second for terms.
\<close>
lemma def_imp_iff: \<open>(A \<equiv> B) \<Longrightarrow> A \<longleftrightarrow> B\<close>
apply unfold
apply (rule iff_refl)
done
lemma meta_eq_to_obj_eq: \<open>(A \<equiv> B) \<Longrightarrow> A = B\<close>
apply unfold
apply (rule refl)
done
lemma meta_eq_to_iff: \<open>x \<equiv> y \<Longrightarrow> x \<longleftrightarrow> y\<close>
by unfold (rule iff_refl)
text \<open>Substitution.\<close>
lemma ssubst: \<open>\<lbrakk>b = a; P(a)\<rbrakk> \<Longrightarrow> P(b)\<close>
apply (drule sym)
apply (erule (1) subst)
done
text \<open>A special case of \<open>ex1E\<close> that would otherwise need quantifier
expansion.\<close>
lemma ex1_equalsE: \<open>\<lbrakk>\<exists>!x. P(x); P(a); P(b)\<rbrakk> \<Longrightarrow> a = b\<close>
apply (erule ex1E)
apply (rule trans)
apply (rule_tac [2] sym)
apply (assumption | erule spec [THEN mp])+
done
subsubsection \<open>Polymorphic congruence rules\<close>
lemma subst_context: \<open>a = b \<Longrightarrow> t(a) = t(b)\<close>
apply (erule ssubst)
apply (rule refl)
done
lemma subst_context2: \<open>\<lbrakk>a = b; c = d\<rbrakk> \<Longrightarrow> t(a,c) = t(b,d)\<close>
apply (erule ssubst)+
apply (rule refl)
done
lemma subst_context3: \<open>\<lbrakk>a = b; c = d; e = f\<rbrakk> \<Longrightarrow> t(a,c,e) = t(b,d,f)\<close>
apply (erule ssubst)+
apply (rule refl)
done
text \<open>
Useful with \<^ML>\<open>eresolve_tac\<close> for proving equalities from known
equalities.
a = b
| |
c = d
\<close>
lemma box_equals: \<open>\<lbrakk>a = b; a = c; b = d\<rbrakk> \<Longrightarrow> c = d\<close>
apply (rule trans)
apply (rule trans)
apply (rule sym)
apply assumption+
done
text \<open>Dual of \<open>box_equals\<close>: for proving equalities backwards.\<close>
lemma simp_equals: \<open>\<lbrakk>a = c; b = d; c = d\<rbrakk> \<Longrightarrow> a = b\<close>
apply (rule trans)
apply (rule trans)
apply assumption+
apply (erule sym)
done
subsubsection \<open>Congruence rules for predicate letters\<close>
lemma pred1_cong: \<open>a = a' \<Longrightarrow> P(a) \<longleftrightarrow> P(a')\<close>
apply (rule iffI)
apply (erule (1) subst)
apply (erule (1) ssubst)
done
lemma pred2_cong: \<open>\<lbrakk>a = a'; b = b'\<rbrakk> \<Longrightarrow> P(a,b) \<longleftrightarrow> P(a',b')\<close>
apply (rule iffI)
apply (erule subst)+
apply assumption
apply (erule ssubst)+
apply assumption
done
lemma pred3_cong: \<open>\<lbrakk>a = a'; b = b'; c = c'\<rbrakk> \<Longrightarrow> P(a,b,c) \<longleftrightarrow> P(a',b',c')\<close>
apply (rule iffI)
apply (erule subst)+
apply assumption
apply (erule ssubst)+
apply assumption
done
text \<open>Special case for the equality predicate!\<close>
lemma eq_cong: \<open>\<lbrakk>a = a'; b = b'\<rbrakk> \<Longrightarrow> a = b \<longleftrightarrow> a' = b'\<close>
apply (erule (1) pred2_cong)
done
subsection \<open>Simplifications of assumed implications\<close>
text \<open>
Roy Dyckhoff has proved that \<open>conj_impE\<close>, \<open>disj_impE\<close>, and
\<open>imp_impE\<close> used with \<^ML>\<open>mp_tac\<close> (restricted to atomic formulae) is
COMPLETE for intuitionistic propositional logic.
See R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic
(preprint, University of St Andrews, 1991).
\<close>
lemma conj_impE:
assumes major: \<open>(P \<and> Q) \<longrightarrow> S\<close>
and r: \<open>P \<longrightarrow> (Q \<longrightarrow> S) \<Longrightarrow> R\<close>
shows \<open>R\<close>
by (assumption | rule conjI impI major [THEN mp] r)+
lemma disj_impE:
assumes major: \<open>(P \<or> Q) \<longrightarrow> S\<close>
and r: \<open>\<lbrakk>P \<longrightarrow> S; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> R\<close>
shows \<open>R\<close>
by (assumption | rule disjI1 disjI2 impI major [THEN mp] r)+
text \<open>Simplifies the implication. Classical version is stronger.
Still UNSAFE since Q must be provable -- backtracking needed.\<close>
lemma imp_impE:
assumes major: \<open>(P \<longrightarrow> Q) \<longrightarrow> S\<close>
and r1: \<open>\<lbrakk>P; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> Q\<close>
and r2: \<open>S \<Longrightarrow> R\<close>
shows \<open>R\<close>
by (assumption | rule impI major [THEN mp] r1 r2)+
text \<open>Simplifies the implication. Classical version is stronger.
Still UNSAFE since ~P must be provable -- backtracking needed.\<close>
lemma not_impE: \<open>\<not> P \<longrightarrow> S \<Longrightarrow> (P \<Longrightarrow> False) \<Longrightarrow> (S \<Longrightarrow> R) \<Longrightarrow> R\<close>
apply (drule mp)
apply (rule notI)
apply assumption
apply assumption
done
text \<open>Simplifies the implication. UNSAFE.\<close>
lemma iff_impE:
assumes major: \<open>(P \<longleftrightarrow> Q) \<longrightarrow> S\<close>
and r1: \<open>\<lbrakk>P; Q \<longrightarrow> S\<rbrakk> \<Longrightarrow> Q\<close>
and r2: \<open>\<lbrakk>Q; P \<longrightarrow> S\<rbrakk> \<Longrightarrow> P\<close>
and r3: \<open>S \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+
done
text \<open>What if \<open>(\<forall>x. \<not> \<not> P(x)) \<longrightarrow> \<not> \<not> (\<forall>x. P(x))\<close> is an assumption?
UNSAFE.\<close>
lemma all_impE:
assumes major: \<open>(\<forall>x. P(x)) \<longrightarrow> S\<close>
and r1: \<open>\<And>x. P(x)\<close>
and r2: \<open>S \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (rule allI impI major [THEN mp] r1 r2)+
done
text \<open>
Unsafe: \<open>\<exists>x. P(x)) \<longrightarrow> S\<close> is equivalent
to \<open>\<forall>x. P(x) \<longrightarrow> S\<close>.\<close>
lemma ex_impE:
assumes major: \<open>(\<exists>x. P(x)) \<longrightarrow> S\<close>
and r: \<open>P(x) \<longrightarrow> S \<Longrightarrow> R\<close>
shows \<open>R\<close>
apply (assumption | rule exI impI major [THEN mp] r)+
done
text \<open>Courtesy of Krzysztof Grabczewski.\<close>
lemma disj_imp_disj: \<open>P \<or> Q \<Longrightarrow> (P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> S) \<Longrightarrow> R \<or> S\<close>
apply (erule disjE)
apply (rule disjI1) apply assumption
apply (rule disjI2) apply assumption
done
ML \<open>
structure Project_Rule = Project_Rule
(
val conjunct1 = @{thm conjunct1}
val conjunct2 = @{thm conjunct2}
val mp = @{thm mp}
)
\<close>
ML_file \<open>fologic.ML\<close>
lemma thin_refl: \<open>\<lbrakk>x = x; PROP W\<rbrakk> \<Longrightarrow> PROP W\<close> .
ML \<open>
structure Hypsubst = Hypsubst
(
val dest_eq = FOLogic.dest_eq
val dest_Trueprop = FOLogic.dest_Trueprop
val dest_imp = FOLogic.dest_imp
val eq_reflection = @{thm eq_reflection}
val rev_eq_reflection = @{thm meta_eq_to_obj_eq}
val imp_intr = @{thm impI}
val rev_mp = @{thm rev_mp}
val subst = @{thm subst}
val sym = @{thm sym}
val thin_refl = @{thm thin_refl}
);
open Hypsubst;
\<close>
ML_file \<open>intprover.ML\<close>
subsection \<open>Intuitionistic Reasoning\<close>
setup \<open>Intuitionistic.method_setup \<^binding>\<open>iprover\<close>\<close>
lemma impE':
assumes 1: \<open>P \<longrightarrow> Q\<close>
and 2: \<open>Q \<Longrightarrow> R\<close>
and 3: \<open>P \<longrightarrow> Q \<Longrightarrow> P\<close>
shows \<open>R\<close>
proof -
from 3 and 1 have \<open>P\<close> .
with 1 have \<open>Q\<close> by (rule impE)
with 2 show \<open>R\<close> .
qed
lemma allE':
assumes 1: \<open>\<forall>x. P(x)\<close>
and 2: \<open>P(x) \<Longrightarrow> \<forall>x. P(x) \<Longrightarrow> Q\<close>
shows \<open>Q\<close>
proof -
from 1 have \<open>P(x)\<close> by (rule spec)
from this and 1 show \<open>Q\<close> by (rule 2)
qed
lemma notE':
assumes 1: \<open>\<not> P\<close>
and 2: \<open>\<not> P \<Longrightarrow> P\<close>
shows \<open>R\<close>
proof -
from 2 and 1 have \<open>P\<close> .
with 1 show \<open>R\<close> by (rule notE)
qed
lemmas [Pure.elim!] = disjE iffE FalseE conjE exE
and [Pure.intro!] = iffI conjI impI TrueI notI allI refl
and [Pure.elim 2] = allE notE' impE'
and [Pure.intro] = exI disjI2 disjI1
setup \<open>
Context_Rules.addSWrapper
(fn ctxt => fn tac => hyp_subst_tac ctxt ORELSE' tac)
\<close>
lemma iff_not_sym: \<open>\<not> (Q \<longleftrightarrow> P) \<Longrightarrow> \<not> (P \<longleftrightarrow> Q)\<close>
by iprover
lemmas [sym] = sym iff_sym not_sym iff_not_sym
and [Pure.elim?] = iffD1 iffD2 impE
lemma eq_commute: \<open>a = b \<longleftrightarrow> b = a\<close>
apply (rule iffI)
apply (erule sym)+
done
subsection \<open>Atomizing meta-level rules\<close>
lemma atomize_all [atomize]: \<open>(\<And>x. P(x)) \<equiv> Trueprop (\<forall>x. P(x))\<close>
proof
assume \<open>\<And>x. P(x)\<close>
then show \<open>\<forall>x. P(x)\<close> ..
next
assume \<open>\<forall>x. P(x)\<close>
then show \<open>\<And>x. P(x)\<close> ..
qed
lemma atomize_imp [atomize]: \<open>(A \<Longrightarrow> B) \<equiv> Trueprop (A \<longrightarrow> B)\<close>
proof
assume \<open>A \<Longrightarrow> B\<close>
then show \<open>A \<longrightarrow> B\<close> ..
next
assume \<open>A \<longrightarrow> B\<close> and \<open>A\<close>
then show \<open>B\<close> by (rule mp)
qed
lemma atomize_eq [atomize]: \<open>(x \<equiv> y) \<equiv> Trueprop (x = y)\<close>
proof
assume \<open>x \<equiv> y\<close>
show \<open>x = y\<close> unfolding \<open>x \<equiv> y\<close> by (rule refl)
next
assume \<open>x = y\<close>
then show \<open>x \<equiv> y\<close> by (rule eq_reflection)
qed
lemma atomize_iff [atomize]: \<open>(A \<equiv> B) \<equiv> Trueprop (A \<longleftrightarrow> B)\<close>
proof
assume \<open>A \<equiv> B\<close>
show \<open>A \<longleftrightarrow> B\<close> unfolding \<open>A \<equiv> B\<close> by (rule iff_refl)
next
assume \<open>A \<longleftrightarrow> B\<close>
then show \<open>A \<equiv> B\<close> by (rule iff_reflection)
qed
lemma atomize_conj [atomize]: \<open>(A &&& B) \<equiv> Trueprop (A \<and> B)\<close>
proof
assume conj: \<open>A &&& B\<close>
show \<open>A \<and> B\<close>
proof (rule conjI)
from conj show \<open>A\<close> by (rule conjunctionD1)
from conj show \<open>B\<close> by (rule conjunctionD2)
qed
next
assume conj: \<open>A \<and> B\<close>
show \<open>A &&& B\<close>
proof -
from conj show \<open>A\<close> ..
from conj show \<open>B\<close> ..
qed
qed
lemmas [symmetric, rulify] = atomize_all atomize_imp
and [symmetric, defn] = atomize_all atomize_imp atomize_eq atomize_iff
subsection \<open>Atomizing elimination rules\<close>
lemma atomize_exL[atomize_elim]: \<open>(\<And>x. P(x) \<Longrightarrow> Q) \<equiv> ((\<exists>x. P(x)) \<Longrightarrow> Q)\<close>
by rule iprover+
lemma atomize_conjL[atomize_elim]: \<open>(A \<Longrightarrow> B \<Longrightarrow> C) \<equiv> (A \<and> B \<Longrightarrow> C)\<close>
by rule iprover+
lemma atomize_disjL[atomize_elim]: \<open>((A \<Longrightarrow> C) \<Longrightarrow> (B \<Longrightarrow> C) \<Longrightarrow> C) \<equiv> ((A \<or> B \<Longrightarrow> C) \<Longrightarrow> C)\<close>
by rule iprover+
lemma atomize_elimL[atomize_elim]: \<open>(\<And>B. (A \<Longrightarrow> B) \<Longrightarrow> B) \<equiv> Trueprop(A)\<close> ..
subsection \<open>Calculational rules\<close>
lemma forw_subst: \<open>a = b \<Longrightarrow> P(b) \<Longrightarrow> P(a)\<close>
by (rule ssubst)
lemma back_subst: \<open>P(a) \<Longrightarrow> a = b \<Longrightarrow> P(b)\<close>
by (rule subst)
text \<open>
Note that this list of rules is in reverse order of priorities.
\<close>
lemmas basic_trans_rules [trans] =
forw_subst
back_subst
rev_mp
mp
trans
subsection \<open>``Let'' declarations\<close>
nonterminal letbinds and letbind
definition Let :: \<open>['a::{}, 'a => 'b] \<Rightarrow> ('b::{})\<close>
where \<open>Let(s, f) \<equiv> f(s)\<close>
syntax
"_bind" :: \<open>[pttrn, 'a] => letbind\<close> (\<open>(2_ =/ _)\<close> 10)
"" :: \<open>letbind => letbinds\<close> (\<open>_\<close>)
"_binds" :: \<open>[letbind, letbinds] => letbinds\<close> (\<open>_;/ _\<close>)
"_Let" :: \<open>[letbinds, 'a] => 'a\<close> (\<open>(let (_)/ in (_))\<close> 10)
translations
"_Let(_binds(b, bs), e)" == "_Let(b, _Let(bs, e))"
"let x = a in e" == "CONST Let(a, \<lambda>x. e)"
lemma LetI:
assumes \<open>\<And>x. x = t \<Longrightarrow> P(u(x))\<close>
shows \<open>P(let x = t in u(x))\<close>
apply (unfold Let_def)
apply (rule refl [THEN assms])
done
subsection \<open>Intuitionistic simplification rules\<close>
lemma conj_simps:
\<open>P \<and> True \<longleftrightarrow> P\<close>
\<open>True \<and> P \<longleftrightarrow> P\<close>
\<open>P \<and> False \<longleftrightarrow> False\<close>
\<open>False \<and> P \<longleftrightarrow> False\<close>
\<open>P \<and> P \<longleftrightarrow> P\<close>
\<open>P \<and> P \<and> Q \<longleftrightarrow> P \<and> Q\<close>
\<open>P \<and> \<not> P \<longleftrightarrow> False\<close>
\<open>\<not> P \<and> P \<longleftrightarrow> False\<close>
\<open>(P \<and> Q) \<and> R \<longleftrightarrow> P \<and> (Q \<and> R)\<close>
by iprover+
lemma disj_simps:
\<open>P \<or> True \<longleftrightarrow> True\<close>
\<open>True \<or> P \<longleftrightarrow> True\<close>
\<open>P \<or> False \<longleftrightarrow> P\<close>
\<open>False \<or> P \<longleftrightarrow> P\<close>
\<open>P \<or> P \<longleftrightarrow> P\<close>
\<open>P \<or> P \<or> Q \<longleftrightarrow> P \<or> Q\<close>
\<open>(P \<or> Q) \<or> R \<longleftrightarrow> P \<or> (Q \<or> R)\<close>
by iprover+
lemma not_simps:
\<open>\<not> (P \<or> Q) \<longleftrightarrow> \<not> P \<and> \<not> Q\<close>
\<open>\<not> False \<longleftrightarrow> True\<close>
\<open>\<not> True \<longleftrightarrow> False\<close>
by iprover+
lemma imp_simps:
\<open>(P \<longrightarrow> False) \<longleftrightarrow> \<not> P\<close>
\<open>(P \<longrightarrow> True) \<longleftrightarrow> True\<close>
\<open>(False \<longrightarrow> P) \<longleftrightarrow> True\<close>
\<open>(True \<longrightarrow> P) \<longleftrightarrow> P\<close>
\<open>(P \<longrightarrow> P) \<longleftrightarrow> True\<close>
\<open>(P \<longrightarrow> \<not> P) \<longleftrightarrow> \<not> P\<close>
by iprover+
lemma iff_simps:
\<open>(True \<longleftrightarrow> P) \<longleftrightarrow> P\<close>
\<open>(P \<longleftrightarrow> True) \<longleftrightarrow> P\<close>
\<open>(P \<longleftrightarrow> P) \<longleftrightarrow> True\<close>
\<open>(False \<longleftrightarrow> P) \<longleftrightarrow> \<not> P\<close>
\<open>(P \<longleftrightarrow> False) \<longleftrightarrow> \<not> P\<close>
by iprover+
text \<open>The \<open>x = t\<close> versions are needed for the simplification
procedures.\<close>
lemma quant_simps:
\<open>\<And>P. (\<forall>x. P) \<longleftrightarrow> P\<close>
\<open>(\<forall>x. x = t \<longrightarrow> P(x)) \<longleftrightarrow> P(t)\<close>
\<open>(\<forall>x. t = x \<longrightarrow> P(x)) \<longleftrightarrow> P(t)\<close>
\<open>\<And>P. (\<exists>x. P) \<longleftrightarrow> P\<close>
\<open>\<exists>x. x = t\<close>
\<open>\<exists>x. t = x\<close>
\<open>(\<exists>x. x = t \<and> P(x)) \<longleftrightarrow> P(t)\<close>
\<open>(\<exists>x. t = x \<and> P(x)) \<longleftrightarrow> P(t)\<close>
by iprover+
text \<open>These are NOT supplied by default!\<close>
lemma distrib_simps:
\<open>P \<and> (Q \<or> R) \<longleftrightarrow> P \<and> Q \<or> P \<and> R\<close>
\<open>(Q \<or> R) \<and> P \<longleftrightarrow> Q \<and> P \<or> R \<and> P\<close>
\<open>(P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)\<close>
by iprover+
subsubsection \<open>Conversion into rewrite rules\<close>
lemma P_iff_F: \<open>\<not> P \<Longrightarrow> (P \<longleftrightarrow> False)\<close>
by iprover
lemma iff_reflection_F: \<open>\<not> P \<Longrightarrow> (P \<equiv> False)\<close>
by (rule P_iff_F [THEN iff_reflection])
lemma P_iff_T: \<open>P \<Longrightarrow> (P \<longleftrightarrow> True)\<close>
by iprover
lemma iff_reflection_T: \<open>P \<Longrightarrow> (P \<equiv> True)\<close>
by (rule P_iff_T [THEN iff_reflection])
subsubsection \<open>More rewrite rules\<close>
lemma conj_commute: \<open>P \<and> Q \<longleftrightarrow> Q \<and> P\<close> by iprover
lemma conj_left_commute: \<open>P \<and> (Q \<and> R) \<longleftrightarrow> Q \<and> (P \<and> R)\<close> by iprover
lemmas conj_comms = conj_commute conj_left_commute
lemma disj_commute: \<open>P \<or> Q \<longleftrightarrow> Q \<or> P\<close> by iprover
lemma disj_left_commute: \<open>P \<or> (Q \<or> R) \<longleftrightarrow> Q \<or> (P \<or> R)\<close> by iprover
lemmas disj_comms = disj_commute disj_left_commute
lemma conj_disj_distribL: \<open>P \<and> (Q \<or> R) \<longleftrightarrow> (P \<and> Q \<or> P \<and> R)\<close> by iprover
lemma conj_disj_distribR: \<open>(P \<or> Q) \<and> R \<longleftrightarrow> (P \<and> R \<or> Q \<and> R)\<close> by iprover
lemma disj_conj_distribL: \<open>P \<or> (Q \<and> R) \<longleftrightarrow> (P \<or> Q) \<and> (P \<or> R)\<close> by iprover
lemma disj_conj_distribR: \<open>(P \<and> Q) \<or> R \<longleftrightarrow> (P \<or> R) \<and> (Q \<or> R)\<close> by iprover
lemma imp_conj_distrib: \<open>(P \<longrightarrow> (Q \<and> R)) \<longleftrightarrow> (P \<longrightarrow> Q) \<and> (P \<longrightarrow> R)\<close> by iprover
lemma imp_conj: \<open>((P \<and> Q) \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> (Q \<longrightarrow> R))\<close> by iprover
lemma imp_disj: \<open>(P \<or> Q \<longrightarrow> R) \<longleftrightarrow> (P \<longrightarrow> R) \<and> (Q \<longrightarrow> R)\<close> by iprover
lemma de_Morgan_disj: \<open>(\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)\<close> by iprover
lemma not_ex: \<open>(\<not> (\<exists>x. P(x))) \<longleftrightarrow> (\<forall>x. \<not> P(x))\<close> by iprover
lemma imp_ex: \<open>((\<exists>x. P(x)) \<longrightarrow> Q) \<longleftrightarrow> (\<forall>x. P(x) \<longrightarrow> Q)\<close> by iprover
lemma ex_disj_distrib: \<open>(\<exists>x. P(x) \<or> Q(x)) \<longleftrightarrow> ((\<exists>x. P(x)) \<or> (\<exists>x. Q(x)))\<close>
by iprover
lemma all_conj_distrib: \<open>(\<forall>x. P(x) \<and> Q(x)) \<longleftrightarrow> ((\<forall>x. P(x)) \<and> (\<forall>x. Q(x)))\<close>
by iprover
end