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src/HOL/TLA/Intensional.thy

author | wenzelm |

Fri, 17 Dec 2010 17:43:54 +0100 | |

changeset 41229 | d797baa3d57c |

parent 38786 | e46e7a9cb622 |

child 42018 | 878f33040280 |

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

replaced command 'nonterminals' by slightly modernized version 'nonterminal';

(* Title: HOL/TLA/Intensional.thy Author: Stephan Merz Copyright: 1998 University of Munich *) header {* A framework for "intensional" (possible-world based) logics on top of HOL, with lifting of constants and functions *} theory Intensional imports Main begin classes world classrel world < type (** abstract syntax **) types ('w,'a) expr = "'w => 'a" (* intention: 'w::world, 'a::type *) 'w form = "('w, bool) expr" consts Valid :: "('w::world) form => bool" const :: "'a => ('w::world, 'a) expr" lift :: "['a => 'b, ('w::world, 'a) expr] => ('w,'b) expr" lift2 :: "['a => 'b => 'c, ('w::world,'a) expr, ('w,'b) expr] => ('w,'c) expr" lift3 :: "['a => 'b => 'c => 'd, ('w::world,'a) expr, ('w,'b) expr, ('w,'c) expr] => ('w,'d) expr" (* "Rigid" quantification (logic level) *) RAll :: "('a => ('w::world) form) => 'w form" (binder "Rall " 10) REx :: "('a => ('w::world) form) => 'w form" (binder "Rex " 10) REx1 :: "('a => ('w::world) form) => 'w form" (binder "Rex! " 10) (** concrete syntax **) nonterminal lift and liftargs syntax "" :: "id => lift" ("_") "" :: "longid => lift" ("_") "" :: "var => lift" ("_") "_applC" :: "[lift, cargs] => lift" ("(1_/ _)" [1000, 1000] 999) "" :: "lift => lift" ("'(_')") "_lambda" :: "[idts, 'a] => lift" ("(3%_./ _)" [0, 3] 3) "_constrain" :: "[lift, type] => lift" ("(_::_)" [4, 0] 3) "" :: "lift => liftargs" ("_") "_liftargs" :: "[lift, liftargs] => liftargs" ("_,/ _") "_Valid" :: "lift => bool" ("(|- _)" 5) "_holdsAt" :: "['a, lift] => bool" ("(_ |= _)" [100,10] 10) (* Syntax for lifted expressions outside the scope of |- or |= *) "_LIFT" :: "lift => 'a" ("LIFT _") (* generic syntax for lifted constants and functions *) "_const" :: "'a => lift" ("(#_)" [1000] 999) "_lift" :: "['a, lift] => lift" ("(_<_>)" [1000] 999) "_lift2" :: "['a, lift, lift] => lift" ("(_<_,/ _>)" [1000] 999) "_lift3" :: "['a, lift, lift, lift] => lift" ("(_<_,/ _,/ _>)" [1000] 999) (* concrete syntax for common infix functions: reuse same symbol *) "_liftEqu" :: "[lift, lift] => lift" ("(_ =/ _)" [50,51] 50) "_liftNeq" :: "[lift, lift] => lift" ("(_ ~=/ _)" [50,51] 50) "_liftNot" :: "lift => lift" ("(~ _)" [40] 40) "_liftAnd" :: "[lift, lift] => lift" ("(_ &/ _)" [36,35] 35) "_liftOr" :: "[lift, lift] => lift" ("(_ |/ _)" [31,30] 30) "_liftImp" :: "[lift, lift] => lift" ("(_ -->/ _)" [26,25] 25) "_liftIf" :: "[lift, lift, lift] => lift" ("(if (_)/ then (_)/ else (_))" 10) "_liftPlus" :: "[lift, lift] => lift" ("(_ +/ _)" [66,65] 65) "_liftMinus" :: "[lift, lift] => lift" ("(_ -/ _)" [66,65] 65) "_liftTimes" :: "[lift, lift] => lift" ("(_ */ _)" [71,70] 70) "_liftDiv" :: "[lift, lift] => lift" ("(_ div _)" [71,70] 70) "_liftMod" :: "[lift, lift] => lift" ("(_ mod _)" [71,70] 70) "_liftLess" :: "[lift, lift] => lift" ("(_/ < _)" [50, 51] 50) "_liftLeq" :: "[lift, lift] => lift" ("(_/ <= _)" [50, 51] 50) "_liftMem" :: "[lift, lift] => lift" ("(_/ : _)" [50, 51] 50) "_liftNotMem" :: "[lift, lift] => lift" ("(_/ ~: _)" [50, 51] 50) "_liftFinset" :: "liftargs => lift" ("{(_)}") (** TODO: syntax for lifted collection / comprehension **) "_liftPair" :: "[lift,liftargs] => lift" ("(1'(_,/ _'))") (* infix syntax for list operations *) "_liftCons" :: "[lift, lift] => lift" ("(_ #/ _)" [65,66] 65) "_liftApp" :: "[lift, lift] => lift" ("(_ @/ _)" [65,66] 65) "_liftList" :: "liftargs => lift" ("[(_)]") (* Rigid quantification (syntax level) *) "_ARAll" :: "[idts, lift] => lift" ("(3! _./ _)" [0, 10] 10) "_AREx" :: "[idts, lift] => lift" ("(3? _./ _)" [0, 10] 10) "_AREx1" :: "[idts, lift] => lift" ("(3?! _./ _)" [0, 10] 10) "_RAll" :: "[idts, lift] => lift" ("(3ALL _./ _)" [0, 10] 10) "_REx" :: "[idts, lift] => lift" ("(3EX _./ _)" [0, 10] 10) "_REx1" :: "[idts, lift] => lift" ("(3EX! _./ _)" [0, 10] 10) translations "_const" == "CONST const" "_lift" == "CONST lift" "_lift2" == "CONST lift2" "_lift3" == "CONST lift3" "_Valid" == "CONST Valid" "_RAll x A" == "Rall x. A" "_REx x A" == "Rex x. A" "_REx1 x A" == "Rex! x. A" "_ARAll" => "_RAll" "_AREx" => "_REx" "_AREx1" => "_REx1" "w |= A" => "A w" "LIFT A" => "A::_=>_" "_liftEqu" == "_lift2 (op =)" "_liftNeq u v" == "_liftNot (_liftEqu u v)" "_liftNot" == "_lift (CONST Not)" "_liftAnd" == "_lift2 (op &)" "_liftOr" == "_lift2 (op | )" "_liftImp" == "_lift2 (op -->)" "_liftIf" == "_lift3 (CONST If)" "_liftPlus" == "_lift2 (op +)" "_liftMinus" == "_lift2 (op -)" "_liftTimes" == "_lift2 (op *)" "_liftDiv" == "_lift2 (op div)" "_liftMod" == "_lift2 (op mod)" "_liftLess" == "_lift2 (op <)" "_liftLeq" == "_lift2 (op <=)" "_liftMem" == "_lift2 (op :)" "_liftNotMem x xs" == "_liftNot (_liftMem x xs)" "_liftFinset (_liftargs x xs)" == "_lift2 (CONST insert) x (_liftFinset xs)" "_liftFinset x" == "_lift2 (CONST insert) x (_const {})" "_liftPair x (_liftargs y z)" == "_liftPair x (_liftPair y z)" "_liftPair" == "_lift2 (CONST Pair)" "_liftCons" == "CONST lift2 (CONST Cons)" "_liftApp" == "CONST lift2 (op @)" "_liftList (_liftargs x xs)" == "_liftCons x (_liftList xs)" "_liftList x" == "_liftCons x (_const [])" "w |= ~A" <= "_liftNot A w" "w |= A & B" <= "_liftAnd A B w" "w |= A | B" <= "_liftOr A B w" "w |= A --> B" <= "_liftImp A B w" "w |= u = v" <= "_liftEqu u v w" "w |= ALL x. A" <= "_RAll x A w" "w |= EX x. A" <= "_REx x A w" "w |= EX! x. A" <= "_REx1 x A w" syntax (xsymbols) "_Valid" :: "lift => bool" ("(\<turnstile> _)" 5) "_holdsAt" :: "['a, lift] => bool" ("(_ \<Turnstile> _)" [100,10] 10) "_liftNeq" :: "[lift, lift] => lift" (infixl "\<noteq>" 50) "_liftNot" :: "lift => lift" ("\<not> _" [40] 40) "_liftAnd" :: "[lift, lift] => lift" (infixr "\<and>" 35) "_liftOr" :: "[lift, lift] => lift" (infixr "\<or>" 30) "_liftImp" :: "[lift, lift] => lift" (infixr "\<longrightarrow>" 25) "_RAll" :: "[idts, lift] => lift" ("(3\<forall>_./ _)" [0, 10] 10) "_REx" :: "[idts, lift] => lift" ("(3\<exists>_./ _)" [0, 10] 10) "_REx1" :: "[idts, lift] => lift" ("(3\<exists>!_./ _)" [0, 10] 10) "_liftLeq" :: "[lift, lift] => lift" ("(_/ \<le> _)" [50, 51] 50) "_liftMem" :: "[lift, lift] => lift" ("(_/ \<in> _)" [50, 51] 50) "_liftNotMem" :: "[lift, lift] => lift" ("(_/ \<notin> _)" [50, 51] 50) syntax (HTML output) "_liftNeq" :: "[lift, lift] => lift" (infixl "\<noteq>" 50) "_liftNot" :: "lift => lift" ("\<not> _" [40] 40) "_liftAnd" :: "[lift, lift] => lift" (infixr "\<and>" 35) "_liftOr" :: "[lift, lift] => lift" (infixr "\<or>" 30) "_RAll" :: "[idts, lift] => lift" ("(3\<forall>_./ _)" [0, 10] 10) "_REx" :: "[idts, lift] => lift" ("(3\<exists>_./ _)" [0, 10] 10) "_REx1" :: "[idts, lift] => lift" ("(3\<exists>!_./ _)" [0, 10] 10) "_liftLeq" :: "[lift, lift] => lift" ("(_/ \<le> _)" [50, 51] 50) "_liftMem" :: "[lift, lift] => lift" ("(_/ \<in> _)" [50, 51] 50) "_liftNotMem" :: "[lift, lift] => lift" ("(_/ \<notin> _)" [50, 51] 50) defs Valid_def: "|- A == ALL w. w |= A" unl_con: "LIFT #c w == c" unl_lift: "lift f x w == f (x w)" unl_lift2: "LIFT f<x, y> w == f (x w) (y w)" unl_lift3: "LIFT f<x, y, z> w == f (x w) (y w) (z w)" unl_Rall: "w |= ALL x. A x == ALL x. (w |= A x)" unl_Rex: "w |= EX x. A x == EX x. (w |= A x)" unl_Rex1: "w |= EX! x. A x == EX! x. (w |= A x)" subsection {* Lemmas and tactics for "intensional" logics. *} lemmas intensional_rews [simp] = unl_con unl_lift unl_lift2 unl_lift3 unl_Rall unl_Rex unl_Rex1 lemma inteq_reflection: "|- x=y ==> (x==y)" apply (unfold Valid_def unl_lift2) apply (rule eq_reflection) apply (rule ext) apply (erule spec) done lemma intI [intro!]: "(!!w. w |= A) ==> |- A" apply (unfold Valid_def) apply (rule allI) apply (erule meta_spec) done lemma intD [dest]: "|- A ==> w |= A" apply (unfold Valid_def) apply (erule spec) done (** Lift usual HOL simplifications to "intensional" level. **) lemma int_simps: "|- (x=x) = #True" "|- (~#True) = #False" "|- (~#False) = #True" "|- (~~ P) = P" "|- ((~P) = P) = #False" "|- (P = (~P)) = #False" "|- (P ~= Q) = (P = (~Q))" "|- (#True=P) = P" "|- (P=#True) = P" "|- (#True --> P) = P" "|- (#False --> P) = #True" "|- (P --> #True) = #True" "|- (P --> P) = #True" "|- (P --> #False) = (~P)" "|- (P --> ~P) = (~P)" "|- (P & #True) = P" "|- (#True & P) = P" "|- (P & #False) = #False" "|- (#False & P) = #False" "|- (P & P) = P" "|- (P & ~P) = #False" "|- (~P & P) = #False" "|- (P | #True) = #True" "|- (#True | P) = #True" "|- (P | #False) = P" "|- (#False | P) = P" "|- (P | P) = P" "|- (P | ~P) = #True" "|- (~P | P) = #True" "|- (! x. P) = P" "|- (? x. P) = P" "|- (~Q --> ~P) = (P --> Q)" "|- (P|Q --> R) = ((P-->R)&(Q-->R))" apply (unfold Valid_def intensional_rews) apply blast+ done declare int_simps [THEN inteq_reflection, simp] lemma TrueW [simp]: "|- #True" by (simp add: Valid_def unl_con) (* ======== Functions to "unlift" intensional implications into HOL rules ====== *) ML {* (* Basic unlifting introduces a parameter "w" and applies basic rewrites, e.g. |- F = G becomes F w = G w |- F --> G becomes F w --> G w *) fun int_unlift th = rewrite_rule @{thms intensional_rews} (th RS @{thm intD} handle THM _ => th); (* Turn |- F = G into meta-level rewrite rule F == G *) fun int_rewrite th = zero_var_indexes (rewrite_rule @{thms intensional_rews} (th RS @{thm inteq_reflection})) (* flattening turns "-->" into "==>" and eliminates conjunctions in the antecedent. For example, P & Q --> (R | S --> T) becomes [| P; Q; R | S |] ==> T Flattening can be useful with "intensional" lemmas (after unlifting). Naive resolution with mp and conjI may run away because of higher-order unification, therefore the code is a little awkward. *) fun flatten t = let (* analogous to RS, but using matching instead of resolution *) fun matchres tha i thb = case Seq.chop 2 (Thm.biresolution true [(false,tha)] i thb) of ([th],_) => th | ([],_) => raise THM("matchres: no match", i, [tha,thb]) | _ => raise THM("matchres: multiple unifiers", i, [tha,thb]) (* match tha with some premise of thb *) fun matchsome tha thb = let fun hmatch 0 = raise THM("matchsome: no match", 0, [tha,thb]) | hmatch n = matchres tha n thb handle THM _ => hmatch (n-1) in hmatch (nprems_of thb) end fun hflatten t = case (concl_of t) of Const _ $ (Const (@{const_name HOL.implies}, _) $ _ $ _) => hflatten (t RS mp) | _ => (hflatten (matchsome conjI t)) handle THM _ => zero_var_indexes t in hflatten t end fun int_use th = case (concl_of th) of Const _ $ (Const ("Intensional.Valid", _) $ _) => (flatten (int_unlift th) handle THM _ => th) | _ => th *} attribute_setup int_unlift = {* Scan.succeed (Thm.rule_attribute (K int_unlift)) *} "" attribute_setup int_rewrite = {* Scan.succeed (Thm.rule_attribute (K int_rewrite)) *} "" attribute_setup flatten = {* Scan.succeed (Thm.rule_attribute (K flatten)) *} "" attribute_setup int_use = {* Scan.succeed (Thm.rule_attribute (K int_use)) *} "" lemma Not_Rall: "|- (~(! x. F x)) = (? x. ~F x)" by (simp add: Valid_def) lemma Not_Rex: "|- (~ (? x. F x)) = (! x. ~ F x)" by (simp add: Valid_def) end