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src/FOL/IFOL.thy

author | blanchet |

Tue, 15 Nov 2011 22:13:39 +0100 | |

changeset 45509 | 624872fc47bf |

parent 44121 | 44adaa6db327 |

child 46972 | ef6fc1a0884d |

permissions | -rw-r--r-- |

use consts, not frees, for lambda-lifting

(* Title: FOL/IFOL.thy Author: Lawrence C Paulson and Markus Wenzel *) header {* Intuitionistic first-order logic *} theory IFOL imports Pure uses "~~/src/Tools/misc_legacy.ML" "~~/src/Provers/splitter.ML" "~~/src/Provers/hypsubst.ML" "~~/src/Tools/IsaPlanner/zipper.ML" "~~/src/Tools/IsaPlanner/isand.ML" "~~/src/Tools/IsaPlanner/rw_tools.ML" "~~/src/Tools/IsaPlanner/rw_inst.ML" "~~/src/Tools/eqsubst.ML" "~~/src/Provers/quantifier1.ML" "~~/src/Tools/intuitionistic.ML" "~~/src/Tools/project_rule.ML" "~~/src/Tools/atomize_elim.ML" ("fologic.ML") ("intprover.ML") begin subsection {* Syntax and axiomatic basis *} setup Pure_Thy.old_appl_syntax_setup classes "term" default_sort "term" typedecl o judgment Trueprop :: "o => prop" ("(_)" 5) consts True :: o False :: o (* Connectives *) eq :: "['a, 'a] => o" (infixl "=" 50) Not :: "o => o" ("~ _" [40] 40) conj :: "[o, o] => o" (infixr "&" 35) disj :: "[o, o] => o" (infixr "|" 30) imp :: "[o, o] => o" (infixr "-->" 25) iff :: "[o, o] => o" (infixr "<->" 25) (* Quantifiers *) All :: "('a => o) => o" (binder "ALL " 10) Ex :: "('a => o) => o" (binder "EX " 10) Ex1 :: "('a => o) => o" (binder "EX! " 10) abbreviation not_equal :: "['a, 'a] => o" (infixl "~=" 50) where "x ~= y == ~ (x = y)" notation (xsymbols) not_equal (infixl "\<noteq>" 50) notation (HTML output) not_equal (infixl "\<noteq>" 50) notation (xsymbols) Not ("\<not> _" [40] 40) and conj (infixr "\<and>" 35) and disj (infixr "\<or>" 30) and All (binder "\<forall>" 10) and Ex (binder "\<exists>" 10) and Ex1 (binder "\<exists>!" 10) and imp (infixr "\<longrightarrow>" 25) and iff (infixr "\<longleftrightarrow>" 25) notation (HTML output) Not ("\<not> _" [40] 40) and conj (infixr "\<and>" 35) and disj (infixr "\<or>" 30) and All (binder "\<forall>" 10) and Ex (binder "\<exists>" 10) and Ex1 (binder "\<exists>!" 10) finalconsts False All Ex eq conj disj imp axiomatization where (* Equality *) refl: "a=a" and subst: "a=b \<Longrightarrow> P(a) \<Longrightarrow> P(b)" axiomatization where (* Propositional logic *) 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" axiomatization where (* Quantifiers *) allI: "(!!x. P(x)) ==> (ALL x. P(x))" and spec: "(ALL x. P(x)) ==> P(x)" and exI: "P(x) ==> (EX x. P(x))" and exE: "[| EX x. P(x); !!x. P(x) ==> R |] ==> R" axiomatization where (* Reflection, admissible *) eq_reflection: "(x=y) ==> (x==y)" and iff_reflection: "(P<->Q) ==> (P==Q)" lemmas strip = impI allI defs (* Definitions *) True_def: "True == False-->False" not_def: "~P == P-->False" iff_def: "P<->Q == (P-->Q) & (Q-->P)" (* Unique existence *) ex1_def: "Ex1(P) == EX x. P(x) & (ALL y. P(y) --> y=x)" subsection {* Lemmas and proof tools *} lemma TrueI: True unfolding True_def by (rule impI) (*** Sequent-style elimination rules for & --> and ALL ***) 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: "ALL x. P(x)" and r: "P(x) ==> R" shows R apply (rule r) apply (rule major [THEN spec]) done (*Duplicates the quantifier; for use with eresolve_tac*) lemma all_dupE: assumes major: "ALL x. P(x)" and r: "[| P(x); ALL x. P(x) |] ==> R" shows R apply (rule r) apply (rule major [THEN spec]) apply (rule major) done (*** 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) (*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 (* 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) (*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 (*** Modus Ponens Tactics ***) (*Finds P-->Q and P in the assumptions, replaces implication by Q *) ML {* fun mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i THEN assume_tac i fun eq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i THEN eq_assume_tac i *} (*** 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 (*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (prems) *) 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 (* 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 (*** Unique existence. NOTE THAT the following 2 quantifications EX!x such that [EX!y such that P(x,y)] (sequential) 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) \<Longrightarrow> (!!x. P(x) ==> x=a) \<Longrightarrow> EX! x. P(x)" apply (unfold ex1_def) apply (assumption | rule exI conjI allI impI)+ done (*Sometimes easier to use: the premises have no shared variables. Safe!*) lemma ex_ex1I: "EX x. P(x) \<Longrightarrow> (!!x y. [| P(x); P(y) |] ==> x=y) \<Longrightarrow> EX! x. P(x)" apply (erule exE) apply (rule ex1I) apply assumption apply assumption done lemma ex1E: "EX! x. P(x) \<Longrightarrow> (!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R) \<Longrightarrow> R" apply (unfold ex1_def) apply (assumption | erule exE conjE)+ done (*** <-> congruence rules for simplification ***) (*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*) ML {* fun iff_tac prems i = resolve_tac (prems RL @{thms iffE}) i THEN REPEAT1 (eresolve_tac [@{thm asm_rl}, @{thm mp}] i) *} 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 | tactic {* iff_tac @{thms assms} 1 *})+ done (*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 | tactic {* iff_tac @{thms assms} 1 *})+ 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 | tactic {* iff_tac @{thms assms} 1 *})+ 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 "(ALL x. P(x)) <-> (ALL x. Q(x))" apply (assumption | rule iffI allI | erule (1) notE impE | erule allE | tactic {* iff_tac @{thms assms} 1 *})+ done lemma ex_cong: assumes "!!x. P(x) <-> Q(x)" shows "(EX x. P(x)) <-> (EX x. Q(x))" apply (erule exE | assumption | rule iffI exI | erule (1) notE impE | tactic {* iff_tac @{thms assms} 1 *})+ done lemma ex1_cong: assumes "!!x. P(x) <-> Q(x)" shows "(EX! x. P(x)) <-> (EX! x. Q(x))" apply (erule ex1E spec [THEN mp] | assumption | rule iffI ex1I | erule (1) notE impE | tactic {* iff_tac @{thms assms} 1 *})+ done (*** 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 (* Two theorms 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) (*substitution*) lemma ssubst: "[| b = a; P(a) |] ==> P(b)" apply (drule sym) apply (erule (1) subst) done (*A special case of ex1E that would otherwise need quantifier expansion*) lemma ex1_equalsE: "[| EX! 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 (** 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 (*Useful with eresolve_tac for proving equalties 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 (*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 (** 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 (*special case for the equality predicate!*) lemma eq_cong: "[| a = a'; b = b' |] ==> a = b <-> a' = b'" apply (erule (1) pred2_cong) done (*** Simplifications of assumed implications. Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE used with 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)+ (*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)+ (*Simplifies the implication. Classical version is stronger. Still UNSAFE since ~P must be provable -- backtracking needed. *) lemma not_impE: "~P --> S \<Longrightarrow> (P ==> False) \<Longrightarrow> (S ==> R) \<Longrightarrow> R" apply (drule mp) apply (rule notI) apply assumption apply assumption done (*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 (*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*) lemma all_impE: assumes major: "(ALL 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 (*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *) lemma ex_impE: assumes major: "(EX x. P(x))-->S" and r: "P(x)-->S ==> R" shows R apply (assumption | rule exI impI major [THEN mp] r)+ done (*** Courtesy of Krzysztof Grabczewski ***) lemma disj_imp_disj: "P|Q \<Longrightarrow> (P==>R) \<Longrightarrow> (Q==>S) \<Longrightarrow> 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} ) *} use "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; *} setup hypsubst_setup use "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: "ALL x. P(x)" and 2: "P(x) ==> ALL 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 tac => hyp_subst_tac 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 (ALL x. P(x))" proof assume "!!x. P(x)" then show "ALL x. P(x)" .. next assume "ALL 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 *} setup AtomizeElim.setup lemma atomize_exL[atomize_elim]: "(!!x. P(x) ==> Q) == ((EX 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+ (*The x=t versions are needed for the simplification procedures*) lemma quant_simps: "!!P. (ALL x. P) <-> P" "(ALL x. x=t --> P(x)) <-> P(t)" "(ALL x. t=x --> P(x)) <-> P(t)" "!!P. (EX x. P) <-> P" "EX x. x=t" "EX x. t=x" "(EX x. x=t & P(x)) <-> P(t)" "(EX x. t=x & P(x)) <-> P(t)" by iprover+ (*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+ text {* 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]) text {* 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: "(~ (EX x. P(x))) <-> (ALL x.~P(x))" by iprover lemma imp_ex: "((EX x. P(x)) --> Q) <-> (ALL x. P(x) --> Q)" by iprover lemma ex_disj_distrib: "(EX x. P(x) | Q(x)) <-> ((EX x. P(x)) | (EX x. Q(x)))" by iprover lemma all_conj_distrib: "(ALL x. P(x) & Q(x)) <-> ((ALL x. P(x)) & (ALL x. Q(x)))" by iprover end