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src/FOL/IFOL.thy

author | wenzelm |

Sun, 26 Nov 2006 23:43:53 +0100 | |

changeset 21539 | c5cf9243ad62 |

parent 21524 | 7843e2fd14a9 |

child 22139 | 539a63b98f76 |

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

converted legacy ML scripts;

(* Title: FOL/IFOL.thy ID: $Id$ Author: Lawrence C Paulson and Markus Wenzel *) header {* Intuitionistic first-order logic *} theory IFOL imports Pure uses ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML") begin subsection {* Syntax and axiomatic basis *} global classes "term" defaultsort "term" typedecl o judgment Trueprop :: "o => prop" ("(_)" 5) consts True :: o False :: o (* Connectives *) "op =" :: "['a, 'a] => o" (infixl "=" 50) Not :: "o => o" ("~ _" [40] 40) "op &" :: "[o, o] => o" (infixr "&" 35) "op |" :: "[o, o] => o" (infixr "|" 30) "op -->" :: "[o, o] => o" (infixr "-->" 25) "op <->" :: "[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 "op &" (infixr "\<and>" 35) and "op |" (infixr "\<or>" 30) and All (binder "\<forall>" 10) and Ex (binder "\<exists>" 10) and Ex1 (binder "\<exists>!" 10) and "op -->" (infixr "\<longrightarrow>" 25) and "op <->" (infixr "\<longleftrightarrow>" 25) notation (HTML output) Not ("\<not> _" [40] 40) and "op &" (infixr "\<and>" 35) and "op |" (infixr "\<or>" 30) and All (binder "\<forall>" 10) and Ex (binder "\<exists>" 10) and Ex1 (binder "\<exists>!" 10) local finalconsts False All Ex "op =" "op &" "op |" "op -->" axioms (* Equality *) refl: "a=a" (* Propositional logic *) conjI: "[| P; Q |] ==> P&Q" conjunct1: "P&Q ==> P" conjunct2: "P&Q ==> Q" disjI1: "P ==> P|Q" disjI2: "Q ==> P|Q" disjE: "[| P|Q; P ==> R; Q ==> R |] ==> R" impI: "(P ==> Q) ==> P-->Q" mp: "[| P-->Q; P |] ==> Q" FalseE: "False ==> P" (* Quantifiers *) allI: "(!!x. P(x)) ==> (ALL x. P(x))" spec: "(ALL x. P(x)) ==> P(x)" exI: "P(x) ==> (EX x. P(x))" exE: "[| EX x. P(x); !!x. P(x) ==> R |] ==> R" (* Reflection *) eq_reflection: "(x=y) ==> (x==y)" iff_reflection: "(P<->Q) ==> (P==Q)" lemmas strip = impI allI text{*Thanks to Stephan Merz*} theorem subst: assumes eq: "a = b" and p: "P(a)" shows "P(b)" proof - from eq have meta: "a \<equiv> b" by (rule eq_reflection) from p show ?thesis by (unfold meta) qed 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. To specify P use something like eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *) 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 {* local val notE = thm "notE" val impE = thm "impE" in fun mp_tac i = eresolve_tac [notE,impE] i THEN assume_tac i fun eq_mp_tac i = eresolve_tac [notE,impE] i THEN eq_assume_tac i end *} (*** 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: assumes "P(a)" and "!!x. P(x) ==> x=a" shows "EX! x. P(x)" apply (unfold ex1_def) apply (assumption | rule assms exI conjI allI impI)+ done (*Sometimes easier to use: the premises have no shared variables. Safe!*) lemma ex_ex1I: assumes ex: "EX x. P(x)" and eq: "!!x y. [| P(x); P(y) |] ==> x=y" shows "EX! x. P(x)" apply (rule ex [THEN exE]) apply (assumption | rule ex1I eq)+ done lemma ex1E: assumes ex1: "EX! x. P(x)" and r: "!!x. [| P(x); ALL y. P(y) --> y=x |] ==> R" shows R apply (insert ex1, 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 {* local val iffE = thm "iffE" val mp = thm "mp" in fun iff_tac prems i = resolve_tac (prems RL [iffE]) i THEN REPEAT1 (eresolve_tac [asm_rl, mp] i) end *} 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 cases for free variables P, Q, R, S -- up to 3 arguments*) ML {* bind_thms ("pred_congs", List.concat (map (fn c => map (fn th => read_instantiate [("P",c)] th) [thm "pred1_cong", thm "pred2_cong", thm "pred3_cong"]) (explode"PQRS"))) *} (*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: assumes major: "~P --> S" and r1: "P ==> False" and r2: "S ==> R" shows R apply (assumption | rule notI impI major [THEN mp] r1 r2)+ 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 (assumption | 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: assumes major: "P|Q" and "P==>R" and "Q==>S" shows "R|S" apply (rule disjE [OF major]) apply (rule disjI1) apply assumption apply (rule disjI2) apply assumption done ML {* structure ProjectRule = ProjectRuleFun (struct val conjunct1 = thm "conjunct1"; val conjunct2 = thm "conjunct2"; val mp = thm "mp"; end) *} use "fologic.ML" lemma thin_refl: "!!X. [|x=x; PROP W|] ==> PROP W" . use "hypsubstdata.ML" setup hypsubst_setup use "intprover.ML" subsection {* Intuitionistic Reasoning *} 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 {* ContextRules.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)" show "ALL x. P(x)" .. next assume "ALL x. P(x)" thus "!!x. P(x)" .. qed lemma atomize_imp [atomize]: "(A ==> B) == Trueprop (A --> B)" proof assume "A ==> B" thus "A --> B" .. next assume "A --> B" and A thus B by (rule mp) qed lemma atomize_eq [atomize]: "(x == y) == Trueprop (x = y)" proof assume "x == y" show "x = y" by (unfold prems) (rule refl) next assume "x = y" thus "x == y" by (rule eq_reflection) qed lemma atomize_iff [atomize]: "(A == B) == Trueprop (A <-> B)" proof assume "A == B" show "A <-> B" by (unfold prems) (rule iff_refl) next assume "A <-> B" thus "A == B" by (rule iff_reflection) qed lemma atomize_conj [atomize]: includes meta_conjunction_syntax shows "(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 {* 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 *} nonterminals letbinds letbind constdefs Let :: "['a::{}, 'a => 'b] => ('b::{})" "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" == "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 {* ML bindings *} ML {* val refl = thm "refl" val trans = thm "trans" val sym = thm "sym" val subst = thm "subst" val ssubst = thm "ssubst" val conjI = thm "conjI" val conjE = thm "conjE" val conjunct1 = thm "conjunct1" val conjunct2 = thm "conjunct2" val disjI1 = thm "disjI1" val disjI2 = thm "disjI2" val disjE = thm "disjE" val impI = thm "impI" val impE = thm "impE" val mp = thm "mp" val rev_mp = thm "rev_mp" val TrueI = thm "TrueI" val FalseE = thm "FalseE" val iff_refl = thm "iff_refl" val iff_trans = thm "iff_trans" val iffI = thm "iffI" val iffE = thm "iffE" val iffD1 = thm "iffD1" val iffD2 = thm "iffD2" val notI = thm "notI" val notE = thm "notE" val allI = thm "allI" val allE = thm "allE" val spec = thm "spec" val exI = thm "exI" val exE = thm "exE" val eq_reflection = thm "eq_reflection" val iff_reflection = thm "iff_reflection" val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq" val meta_eq_to_iff = thm "meta_eq_to_iff" *} end