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