author  paulson 
Thu, 14 Jun 2018 14:23:48 +0100  
changeset 68446  92ddca1edc43 
parent 67405  e9ab4ad7bd15 
child 69593  3dda49e08b9d 
permissions  rwrr 
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(* Title: FOLP/IFOLP.thy 
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Author: Martin D Coen, Cambridge University Computer Laboratory 

1142  3 
Copyright 1992 University of Cambridge 
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*) 

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section \<open>Intuitionistic FirstOrder Logic with Proofs\<close> 
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theory IFOLP 

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imports Pure 

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begin 

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ML_file "~~/src/Tools/misc_legacy.ML" 
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setup Pure_Thy.old_appl_syntax_setup 
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setup PureThy.old_appl_syntax_setup  theory Pure provides regular application syntax by default;
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class "term" 
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default_sort "term" 
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typedecl p 
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typedecl o 

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consts 
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(*** Judgements ***) 
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Proof :: "[o,p]=>prop" 
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EqProof :: "[p,p,o]=>prop" ("(3_ /= _ :/ _)" [10,10,10] 5) 
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(*** Logical Connectives  Type Formers ***) 
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eq :: "['a,'a] => o" (infixl "=" 50) 
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True :: "o" 
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False :: "o" 

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conj :: "[o,o] => o" (infixr "&" 35) 
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disj :: "[o,o] => o" (infixr "" 30) 

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imp :: "[o,o] => o" (infixr ">" 25) 

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(*Quantifiers*) 
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All :: "('a => o) => o" (binder "ALL " 10) 
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Ex :: "('a => o) => o" (binder "EX " 10) 

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(*Rewriting gadgets*) 
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NORM :: "o => o" 
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norm :: "'a => 'a" 

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(*** Proof Term Formers: precedence must exceed 50 ***) 
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tt :: "p" 
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contr :: "p=>p" 

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fst :: "p=>p" 
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snd :: "p=>p" 

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pair :: "[p,p]=>p" ("(1<_,/_>)") 
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split :: "[p, [p,p]=>p] =>p" 

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inl :: "p=>p" 
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inr :: "p=>p" 

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support 'when' statement, which corresponds to 'presume';
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"when" :: "[p, p=>p, p=>p]=>p" 
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lambda :: "(p => p) => p" (binder "lam " 55) 
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App :: "[p,p]=>p" (infixl "`" 60) 
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alll :: "['a=>p]=>p" (binder "all " 55) 
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app :: "[p,'a]=>p" (infixl "^" 55) 
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exists :: "['a,p]=>p" ("(1[_,/_])") 
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xsplit :: "[p,['a,p]=>p]=>p" 
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ideq :: "'a=>p" 

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idpeel :: "[p,'a=>p]=>p" 

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nrm :: p 
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NRM :: p 

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syntax "_Proof" :: "[p,o]=>prop" ("(_ /: _)" [51, 10] 5) 
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parse_translation \<open> 
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let fun proof_tr [p, P] = Const (@{const_syntax Proof}, dummyT) $ P $ p 
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in [(@{syntax_const "_Proof"}, K proof_tr)] end 
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\<close> 
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(*show_proofs = true displays the proof terms  they are ENORMOUS*) 
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ML \<open>val show_proofs = Attrib.setup_config_bool @{binding show_proofs} (K false)\<close> 
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print_translation \<open> 
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let 
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fun proof_tr' ctxt [P, p] = 

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if Config.get ctxt show_proofs then Const (@{syntax_const "_Proof"}, dummyT) $ p $ P 

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else P 

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in [(@{const_syntax Proof}, proof_tr')] end 

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\<close> 
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(**** Propositional logic ****) 

82 

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(*Equality*) 

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(* Like Intensional Equality in MLTT  but proofs distinct from terms *) 

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axiomatization where 
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ieqI: "ideq(a) : a=a" and 

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ieqE: "[ p : a=b; !!x. f(x) : P(x,x) ] ==> idpeel(p,f) : P(a,b)" 
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(* Truth and Falsity *) 

91 

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axiomatization where 
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TrueI: "tt : True" and 

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FalseE: "a:False ==> contr(a):P" 
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(* Conjunction *) 

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axiomatization where 
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conjI: "[ a:P; b:Q ] ==> <a,b> : P&Q" and 

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conjunct1: "p:P&Q ==> fst(p):P" and 

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conjunct2: "p:P&Q ==> snd(p):Q" 
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(* Disjunction *) 

104 

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axiomatization where 
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disjI1: "a:P ==> inl(a):PQ" and 

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disjI2: "b:Q ==> inr(b):PQ" and 

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disjE: "[ a:PQ; !!x. x:P ==> f(x):R; !!x. x:Q ==> g(x):R 
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] ==> when(a,f,g):R" 

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(* Implication *) 

112 

51306  113 
axiomatization where 
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impI: "\<And>P Q f. (!!x. x:P ==> f(x):Q) ==> lam x. f(x):P>Q" and 

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mp: "\<And>P Q f. [ f:P>Q; a:P ] ==> f`a:Q" 

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(*Quantifiers*) 

118 

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axiomatization where 
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allI: "\<And>P. (!!x. f(x) : P(x)) ==> all x. f(x) : ALL x. P(x)" and 

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spec: "\<And>P f. (f:ALL x. P(x)) ==> f^x : P(x)" 

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51306  123 
axiomatization where 
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exI: "p : P(x) ==> [x,p] : EX x. P(x)" and 

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exE: "[ p: EX x. P(x); !!x u. u:P(x) ==> f(x,u) : R ] ==> xsplit(p,f):R" 
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(**** Equality between proofs ****) 

128 

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axiomatization where 
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prefl: "a : P ==> a = a : P" and 

131 
psym: "a = b : P ==> b = a : P" and 

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ptrans: "[ a = b : P; b = c : P ] ==> a = c : P" 
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51306  134 
axiomatization where 
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idpeelB: "[ !!x. f(x) : P(x,x) ] ==> idpeel(ideq(a),f) = f(a) : P(a,a)" 
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51306  137 
axiomatization where 
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fstB: "a:P ==> fst(<a,b>) = a : P" and 

139 
sndB: "b:Q ==> snd(<a,b>) = b : Q" and 

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pairEC: "p:P&Q ==> p = <fst(p),snd(p)> : P&Q" 
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51306  142 
axiomatization where 
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whenBinl: "[ a:P; !!x. x:P ==> f(x) : Q ] ==> when(inl(a),f,g) = f(a) : Q" and 

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whenBinr: "[ b:P; !!x. x:P ==> g(x) : Q ] ==> when(inr(b),f,g) = g(b) : Q" and 

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plusEC: "a:PQ ==> when(a,%x. inl(x),%y. inr(y)) = a : PQ" 
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axiomatization where 
148 
applyB: "[ a:P; !!x. x:P ==> b(x) : Q ] ==> (lam x. b(x)) ` a = b(a) : Q" and 

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funEC: "f:P ==> f = lam x. f`x : P" 
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axiomatization where 
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specB: "[ !!x. f(x) : P(x) ] ==> (all x. f(x)) ^ a = f(a) : P(a)" 
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154 

155 
(**** Definitions ****) 

156 

62147  157 
definition Not :: "o => o" ("~ _" [40] 40) 
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where not_def: "~P == P>False" 

159 

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definition iff :: "[o,o] => o" (infixr "<>" 25) 

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where "P<>Q == (P>Q) & (Q>P)" 

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(*Unique existence*) 

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definition Ex1 :: "('a => o) => o" (binder "EX! " 10) 
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where ex1_def: "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) > y=x)" 

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(*Rewriting  special constants to flag normalized terms and formulae*) 

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axiomatization where 
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norm_eq: "nrm : norm(x) = x" and 

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NORM_iff: "NRM : NORM(P) <> P" 
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(*** Sequentstyle elimination rules for & > and ALL ***) 
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schematic_goal conjE: 
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assumes "p:P&Q" 
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and "!!x y.[ x:P; y:Q ] ==> f(x,y):R" 

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shows "?a:R" 

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apply (rule assms(2)) 

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apply (rule conjunct1 [OF assms(1)]) 

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apply (rule conjunct2 [OF assms(1)]) 

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done 

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schematic_goal impE: 
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assumes "p:P>Q" 
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and "q:P" 

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and "!!x. x:Q ==> r(x):R" 

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shows "?p:R" 

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apply (rule assms mp)+ 

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done 

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schematic_goal allE: 
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assumes "p:ALL x. P(x)" 
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and "!!y. y:P(x) ==> q(y):R" 

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shows "?p:R" 

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apply (rule assms spec)+ 

196 
done 

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(*Duplicates the quantifier; for use with eresolve_tac*) 

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schematic_goal all_dupE: 
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assumes "p:ALL x. P(x)" 
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and "!!y z.[ y:P(x); z:ALL x. P(x) ] ==> q(y,z):R" 

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shows "?p:R" 

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apply (rule assms spec)+ 

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done 

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(*** Negation rules, which translate between ~P and P>False ***) 

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schematic_goal notI: 
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assumes "!!x. x:P ==> q(x):False" 
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shows "?p:~P" 

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unfolding not_def 

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apply (assumption  rule assms impI)+ 

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done 

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schematic_goal notE: "p:~P \<Longrightarrow> q:P \<Longrightarrow> ?p:R" 
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unfolding not_def 
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apply (drule (1) mp) 

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apply (erule FalseE) 

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done 

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(*This is useful with the special implication rules for each kind of P. *) 

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schematic_goal not_to_imp: 
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assumes "p:~P" 
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and "!!x. x:(P>False) ==> q(x):Q" 

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shows "?p:Q" 

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apply (assumption  rule assms impI notE)+ 

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done 

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(* For substitution int an assumption P, reduce Q to P>Q, substitute into 

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this implication, then apply impI to move P back into the assumptions.*) 
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schematic_goal rev_mp: "[ p:P; q:P > Q ] ==> ?p:Q" 
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apply (assumption  rule mp)+ 
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done 

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(*Contrapositive of an inference rule*) 

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schematic_goal contrapos: 
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assumes major: "p:~Q" 
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and minor: "!!y. y:P==>q(y):Q" 

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shows "?a:~P" 

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apply (rule major [THEN notE, THEN notI]) 

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apply (erule minor) 

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done 

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(** Unique assumption tactic. 

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Ignores proof objects. 

248 
Fails unless one assumption is equal and exactly one is unifiable 

249 
**) 

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ML \<open> 
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local 
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fun discard_proof (Const (@{const_name Proof}, _) $ P $ _) = P; 

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in 

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fun uniq_assume_tac ctxt = 
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SUBGOAL 
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(fn (prem,i) => 

258 
let val hyps = map discard_proof (Logic.strip_assums_hyp prem) 

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and concl = discard_proof (Logic.strip_assums_concl prem) 

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in 

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if exists (fn hyp => hyp aconv concl) hyps 

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then case distinct (op =) (filter (fn hyp => Term.could_unify (hyp, concl)) hyps) of 
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[_] => assume_tac ctxt i 
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 _ => no_tac 
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else no_tac 

266 
end); 

267 
end; 

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\<close> 
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(*** Modus Ponens Tactics ***) 

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273 
(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

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ML \<open> 
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fun mp_tac ctxt i = 
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eresolve_tac ctxt [@{thm notE}, make_elim @{thm mp}] i THEN assume_tac ctxt i 
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\<close> 
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method_setup mp = \<open>Scan.succeed (SIMPLE_METHOD' o mp_tac)\<close> 
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(*Like mp_tac but instantiates no variables*) 

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ML \<open> 
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fun int_uniq_mp_tac ctxt i = 
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eresolve_tac ctxt [@{thm notE}, @{thm impE}] i THEN uniq_assume_tac ctxt i 
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\<close> 
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287 
(*** Ifandonlyif ***) 

288 

61337  289 
schematic_goal iffI: 
26322  290 
assumes "!!x. x:P ==> q(x):Q" 
291 
and "!!x. x:Q ==> r(x):P" 

292 
shows "?p:P<>Q" 

293 
unfolding iff_def 

294 
apply (assumption  rule assms conjI impI)+ 

295 
done 

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schematic_goal iffE: 
26322  299 
assumes "p:P <> Q" 
300 
and "!!x y.[ x:P>Q; y:Q>P ] ==> q(x,y):R" 

301 
shows "?p:R" 

302 
apply (rule conjE) 

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apply (rule assms(1) [unfolded iff_def]) 

304 
apply (rule assms(2)) 

305 
apply assumption+ 

306 
done 

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308 
(* Destruct rules for <> similar to Modus Ponens *) 

309 

61337  310 
schematic_goal iffD1: "[ p:P <> Q; q:P ] ==> ?p:Q" 
26322  311 
unfolding iff_def 
312 
apply (rule conjunct1 [THEN mp], assumption+) 

313 
done 

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61337  315 
schematic_goal iffD2: "[ p:P <> Q; q:Q ] ==> ?p:P" 
26322  316 
unfolding iff_def 
317 
apply (rule conjunct2 [THEN mp], assumption+) 

318 
done 

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schematic_goal iff_refl: "?p:P <> P" 
26322  321 
apply (rule iffI) 
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apply assumption+ 

323 
done 

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61337  325 
schematic_goal iff_sym: "p:Q <> P ==> ?p:P <> Q" 
26322  326 
apply (erule iffE) 
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apply (rule iffI) 

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apply (erule (1) mp)+ 

329 
done 

330 

61337  331 
schematic_goal iff_trans: "[ p:P <> Q; q:Q<> R ] ==> ?p:P <> R" 
26322  332 
apply (rule iffI) 
333 
apply (assumption  erule iffE  erule (1) impE)+ 

334 
done 

335 

336 
(*** Unique existence. NOTE THAT the following 2 quantifications 

337 
EX!x such that [EX!y such that P(x,y)] (sequential) 

338 
EX!x,y such that P(x,y) (simultaneous) 

339 
do NOT mean the same thing. The parser treats EX!x y.P(x,y) as sequential. 

340 
***) 

341 

61337  342 
schematic_goal ex1I: 
26322  343 
assumes "p:P(a)" 
344 
and "!!x u. u:P(x) ==> f(u) : x=a" 

345 
shows "?p:EX! x. P(x)" 

346 
unfolding ex1_def 

347 
apply (assumption  rule assms exI conjI allI impI)+ 

348 
done 

349 

61337  350 
schematic_goal ex1E: 
26322  351 
assumes "p:EX! x. P(x)" 
352 
and "!!x u v. [ u:P(x); v:ALL y. P(y) > y=x ] ==> f(x,u,v):R" 

353 
shows "?a : R" 

354 
apply (insert assms(1) [unfolded ex1_def]) 

355 
apply (erule exE conjE  assumption  rule assms(1))+ 

29305  356 
apply (erule assms(2), assumption) 
26322  357 
done 
358 

359 

360 
(*** <> congruence rules for simplification ***) 

361 

362 
(*Use iffE on a premise. For conj_cong, imp_cong, all_cong, ex_cong*) 

60770  363 
ML \<open> 
59529  364 
fun iff_tac ctxt prems i = 
365 
resolve_tac ctxt (prems RL [@{thm iffE}]) i THEN 

366 
REPEAT1 (eresolve_tac ctxt [asm_rl, @{thm mp}] i) 

60770  367 
\<close> 
26322  368 

59529  369 
method_setup iff = 
370 
\<open>Attrib.thms >> (fn prems => fn ctxt => SIMPLE_METHOD' (iff_tac ctxt prems))\<close> 

371 

61337  372 
schematic_goal conj_cong: 
26322  373 
assumes "p:P <> P'" 
374 
and "!!x. x:P' ==> q(x):Q <> Q'" 

375 
shows "?p:(P&Q) <> (P'&Q')" 

376 
apply (insert assms(1)) 

59529  377 
apply (assumption  rule iffI conjI  erule iffE conjE mp  iff assms)+ 
26322  378 
done 
379 

61337  380 
schematic_goal disj_cong: 
26322  381 
"[ p:P <> P'; q:Q <> Q' ] ==> ?p:(PQ) <> (P'Q')" 
59529  382 
apply (erule iffE disjE disjI1 disjI2  assumption  rule iffI  mp)+ 
26322  383 
done 
384 

61337  385 
schematic_goal imp_cong: 
26322  386 
assumes "p:P <> P'" 
387 
and "!!x. x:P' ==> q(x):Q <> Q'" 

388 
shows "?p:(P>Q) <> (P'>Q')" 

389 
apply (insert assms(1)) 

59529  390 
apply (assumption  rule iffI impI  erule iffE  mp  iff assms)+ 
26322  391 
done 
392 

61337  393 
schematic_goal iff_cong: 
26322  394 
"[ p:P <> P'; q:Q <> Q' ] ==> ?p:(P<>Q) <> (P'<>Q')" 
59529  395 
apply (erule iffE  assumption  rule iffI  mp)+ 
26322  396 
done 
397 

61337  398 
schematic_goal not_cong: 
26322  399 
"p:P <> P' ==> ?p:~P <> ~P'" 
59529  400 
apply (assumption  rule iffI notI  mp  erule iffE notE)+ 
26322  401 
done 
402 

61337  403 
schematic_goal all_cong: 
26322  404 
assumes "!!x. f(x):P(x) <> Q(x)" 
405 
shows "?p:(ALL x. P(x)) <> (ALL x. Q(x))" 

59529  406 
apply (assumption  rule iffI allI  mp  erule allE  iff assms)+ 
26322  407 
done 
408 

61337  409 
schematic_goal ex_cong: 
26322  410 
assumes "!!x. f(x):P(x) <> Q(x)" 
411 
shows "?p:(EX x. P(x)) <> (EX x. Q(x))" 

59529  412 
apply (erule exE  assumption  rule iffI exI  mp  iff assms)+ 
26322  413 
done 
414 

415 
(*NOT PROVED 

56199  416 
ML_Thms.bind_thm ("ex1_cong", prove_goal (the_context ()) 
26322  417 
"(!!x.f(x):P(x) <> Q(x)) ==> ?p:(EX! x.P(x)) <> (EX! x.Q(x))" 
418 
(fn prems => 

419 
[ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1 

420 
ORELSE mp_tac 1 

421 
ORELSE iff_tac prems 1)) ])) 

422 
*) 

423 

424 
(*** Equality rules ***) 

425 

426 
lemmas refl = ieqI 

427 

61337  428 
schematic_goal subst: 
26322  429 
assumes prem1: "p:a=b" 
430 
and prem2: "q:P(a)" 

431 
shows "?p : P(b)" 

432 
apply (rule prem2 [THEN rev_mp]) 

433 
apply (rule prem1 [THEN ieqE]) 

434 
apply (rule impI) 

435 
apply assumption 

436 
done 

437 

61337  438 
schematic_goal sym: "q:a=b ==> ?c:b=a" 
26322  439 
apply (erule subst) 
440 
apply (rule refl) 

441 
done 

442 

61337  443 
schematic_goal trans: "[ p:a=b; q:b=c ] ==> ?d:a=c" 
26322  444 
apply (erule (1) subst) 
445 
done 

446 

447 
(** ~ b=a ==> ~ a=b **) 

61337  448 
schematic_goal not_sym: "p:~ b=a ==> ?q:~ a=b" 
26322  449 
apply (erule contrapos) 
450 
apply (erule sym) 

451 
done 

452 

61337  453 
schematic_goal ssubst: "p:b=a \<Longrightarrow> q:P(a) \<Longrightarrow> ?p:P(b)" 
45594  454 
apply (drule sym) 
455 
apply (erule subst) 

456 
apply assumption 

457 
done 

26322  458 

459 
(*A special case of ex1E that would otherwise need quantifier expansion*) 

61337  460 
schematic_goal ex1_equalsE: "[ p:EX! x. P(x); q:P(a); r:P(b) ] ==> ?d:a=b" 
26322  461 
apply (erule ex1E) 
462 
apply (rule trans) 

463 
apply (rule_tac [2] sym) 

464 
apply (assumption  erule spec [THEN mp])+ 

465 
done 

466 

467 
(** Polymorphic congruence rules **) 

468 

61337  469 
schematic_goal subst_context: "[ p:a=b ] ==> ?d:t(a)=t(b)" 
26322  470 
apply (erule ssubst) 
471 
apply (rule refl) 

472 
done 

473 

61337  474 
schematic_goal subst_context2: "[ p:a=b; q:c=d ] ==> ?p:t(a,c)=t(b,d)" 
26322  475 
apply (erule ssubst)+ 
476 
apply (rule refl) 

477 
done 

478 

61337  479 
schematic_goal subst_context3: "[ p:a=b; q:c=d; r:e=f ] ==> ?p:t(a,c,e)=t(b,d,f)" 
26322  480 
apply (erule ssubst)+ 
481 
apply (rule refl) 

482 
done 

483 

484 
(*Useful with eresolve_tac for proving equalties from known equalities. 

485 
a = b 

486 
  

487 
c = d *) 

61337  488 
schematic_goal box_equals: "[ p:a=b; q:a=c; r:b=d ] ==> ?p:c=d" 
26322  489 
apply (rule trans) 
490 
apply (rule trans) 

491 
apply (rule sym) 

492 
apply assumption+ 

493 
done 

494 

495 
(*Dual of box_equals: for proving equalities backwards*) 

61337  496 
schematic_goal simp_equals: "[ p:a=c; q:b=d; r:c=d ] ==> ?p:a=b" 
26322  497 
apply (rule trans) 
498 
apply (rule trans) 

499 
apply (assumption  rule sym)+ 

500 
done 

501 

502 
(** Congruence rules for predicate letters **) 

503 

61337  504 
schematic_goal pred1_cong: "p:a=a' ==> ?p:P(a) <> P(a')" 
26322  505 
apply (rule iffI) 
60770  506 
apply (tactic \<open> 
507 
DEPTH_SOLVE (assume_tac @{context} 1 ORELSE eresolve_tac @{context} [@{thm subst}, @{thm ssubst}] 1)\<close>) 

26322  508 
done 
509 

61337  510 
schematic_goal pred2_cong: "[ p:a=a'; q:b=b' ] ==> ?p:P(a,b) <> P(a',b')" 
26322  511 
apply (rule iffI) 
60770  512 
apply (tactic \<open> 
513 
DEPTH_SOLVE (assume_tac @{context} 1 ORELSE eresolve_tac @{context} [@{thm subst}, @{thm ssubst}] 1)\<close>) 

26322  514 
done 
515 

61337  516 
schematic_goal pred3_cong: "[ p:a=a'; q:b=b'; r:c=c' ] ==> ?p:P(a,b,c) <> P(a',b',c')" 
26322  517 
apply (rule iffI) 
60770  518 
apply (tactic \<open> 
519 
DEPTH_SOLVE (assume_tac @{context} 1 ORELSE eresolve_tac @{context} [@{thm subst}, @{thm ssubst}] 1)\<close>) 

26322  520 
done 
521 

27152
192954a9a549
changed pred_congs: merely cover pred1_cong pred2_cong pred3_cong;
wenzelm
parents:
27150
diff
changeset

522 
lemmas pred_congs = pred1_cong pred2_cong pred3_cong 
26322  523 

524 
(*special case for the equality predicate!*) 

67399  525 
lemmas eq_cong = pred2_cong [where P = "(=)"] 
26322  526 

527 

528 
(*** Simplifications of assumed implications. 

529 
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE 

530 
used with mp_tac (restricted to atomic formulae) is COMPLETE for 

531 
intuitionistic propositional logic. See 

532 
R. Dyckhoff, Contractionfree sequent calculi for intuitionistic logic 

533 
(preprint, University of St Andrews, 1991) ***) 

534 

61337  535 
schematic_goal conj_impE: 
26322  536 
assumes major: "p:(P&Q)>S" 
537 
and minor: "!!x. x:P>(Q>S) ==> q(x):R" 

538 
shows "?p:R" 

539 
apply (assumption  rule conjI impI major [THEN mp] minor)+ 

540 
done 

541 

61337  542 
schematic_goal disj_impE: 
26322  543 
assumes major: "p:(PQ)>S" 
544 
and minor: "!!x y.[ x:P>S; y:Q>S ] ==> q(x,y):R" 

545 
shows "?p:R" 

60770  546 
apply (tactic \<open>DEPTH_SOLVE (assume_tac @{context} 1 ORELSE 
59498
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
wenzelm
parents:
58963
diff
changeset

547 
resolve_tac @{context} [@{thm disjI1}, @{thm disjI2}, @{thm impI}, 
60770  548 
@{thm major} RS @{thm mp}, @{thm minor}] 1)\<close>) 
26322  549 
done 
550 

551 
(*Simplifies the implication. Classical version is stronger. 

552 
Still UNSAFE since Q must be provable  backtracking needed. *) 

61337  553 
schematic_goal imp_impE: 
26322  554 
assumes major: "p:(P>Q)>S" 
555 
and r1: "!!x y.[ x:P; y:Q>S ] ==> q(x,y):Q" 

556 
and r2: "!!x. x:S ==> r(x):R" 

557 
shows "?p:R" 

558 
apply (assumption  rule impI major [THEN mp] r1 r2)+ 

559 
done 

560 

561 
(*Simplifies the implication. Classical version is stronger. 

562 
Still UNSAFE since ~P must be provable  backtracking needed. *) 

61337  563 
schematic_goal not_impE: 
26322  564 
assumes major: "p:~P > S" 
565 
and r1: "!!y. y:P ==> q(y):False" 

566 
and r2: "!!y. y:S ==> r(y):R" 

567 
shows "?p:R" 

568 
apply (assumption  rule notI impI major [THEN mp] r1 r2)+ 

569 
done 

570 

571 
(*Simplifies the implication. UNSAFE. *) 

61337  572 
schematic_goal iff_impE: 
26322  573 
assumes major: "p:(P<>Q)>S" 
574 
and r1: "!!x y.[ x:P; y:Q>S ] ==> q(x,y):Q" 

575 
and r2: "!!x y.[ x:Q; y:P>S ] ==> r(x,y):P" 

576 
and r3: "!!x. x:S ==> s(x):R" 

577 
shows "?p:R" 

578 
apply (assumption  rule iffI impI major [THEN mp] r1 r2 r3)+ 

579 
done 

580 

581 
(*What if (ALL x.~~P(x)) > ~~(ALL x.P(x)) is an assumption? UNSAFE*) 

61337  582 
schematic_goal all_impE: 
26322  583 
assumes major: "p:(ALL x. P(x))>S" 
584 
and r1: "!!x. q:P(x)" 

585 
and r2: "!!y. y:S ==> r(y):R" 

586 
shows "?p:R" 

587 
apply (assumption  rule allI impI major [THEN mp] r1 r2)+ 

588 
done 

589 

590 
(*Unsafe: (EX x.P(x))>S is equivalent to ALL x.P(x)>S. *) 

61337  591 
schematic_goal ex_impE: 
26322  592 
assumes major: "p:(EX x. P(x))>S" 
593 
and r: "!!y. y:P(a)>S ==> q(y):R" 

594 
shows "?p:R" 

595 
apply (assumption  rule exI impI major [THEN mp] r)+ 

596 
done 

597 

598 

61337  599 
schematic_goal rev_cut_eq: 
26322  600 
assumes "p:a=b" 
601 
and "!!x. x:a=b ==> f(x):R" 

602 
shows "?p:R" 

603 
apply (rule assms)+ 

604 
done 

605 

606 
lemma thin_refl: "!!X. [p:x=x; PROP W] ==> PROP W" . 

607 

48891  608 
ML_file "hypsubst.ML" 
26322  609 

60770  610 
ML \<open> 
42799  611 
structure Hypsubst = Hypsubst 
612 
( 

26322  613 
(*Take apart an equality judgement; otherwise raise Match!*) 
614 
fun dest_eq (Const (@{const_name Proof}, _) $ 

41310  615 
(Const (@{const_name eq}, _) $ t $ u) $ _) = (t, u); 
26322  616 

617 
val imp_intr = @{thm impI} 

618 

619 
(*etac rev_cut_eq moves an equality to be the last premise. *) 

620 
val rev_cut_eq = @{thm rev_cut_eq} 

621 

622 
val rev_mp = @{thm rev_mp} 

623 
val subst = @{thm subst} 

624 
val sym = @{thm sym} 

625 
val thin_refl = @{thm thin_refl} 

42799  626 
); 
26322  627 
open Hypsubst; 
60770  628 
\<close> 
26322  629 

48891  630 
ML_file "intprover.ML" 
26322  631 

632 

633 
(*** Rewrite rules ***) 

634 

61337  635 
schematic_goal conj_rews: 
26322  636 
"?p1 : P & True <> P" 
637 
"?p2 : True & P <> P" 

638 
"?p3 : P & False <> False" 

639 
"?p4 : False & P <> False" 

640 
"?p5 : P & P <> P" 

641 
"?p6 : P & ~P <> False" 

642 
"?p7 : ~P & P <> False" 

643 
"?p8 : (P & Q) & R <> P & (Q & R)" 

60770  644 
apply (tactic \<open>fn st => IntPr.fast_tac @{context} 1 st\<close>)+ 
26322  645 
done 
646 

61337  647 
schematic_goal disj_rews: 
26322  648 
"?p1 : P  True <> True" 
649 
"?p2 : True  P <> True" 

650 
"?p3 : P  False <> P" 

651 
"?p4 : False  P <> P" 

652 
"?p5 : P  P <> P" 

653 
"?p6 : (P  Q)  R <> P  (Q  R)" 

60770  654 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  655 
done 
656 

61337  657 
schematic_goal not_rews: 
26322  658 
"?p1 : ~ False <> True" 
659 
"?p2 : ~ True <> False" 

60770  660 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  661 
done 
662 

61337  663 
schematic_goal imp_rews: 
26322  664 
"?p1 : (P > False) <> ~P" 
665 
"?p2 : (P > True) <> True" 

666 
"?p3 : (False > P) <> True" 

667 
"?p4 : (True > P) <> P" 

668 
"?p5 : (P > P) <> True" 

669 
"?p6 : (P > ~P) <> ~P" 

60770  670 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  671 
done 
672 

61337  673 
schematic_goal iff_rews: 
26322  674 
"?p1 : (True <> P) <> P" 
675 
"?p2 : (P <> True) <> P" 

676 
"?p3 : (P <> P) <> True" 

677 
"?p4 : (False <> P) <> ~P" 

678 
"?p5 : (P <> False) <> ~P" 

60770  679 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  680 
done 
681 

61337  682 
schematic_goal quant_rews: 
26322  683 
"?p1 : (ALL x. P) <> P" 
684 
"?p2 : (EX x. P) <> P" 

60770  685 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  686 
done 
687 

688 
(*These are NOT supplied by default!*) 

61337  689 
schematic_goal distrib_rews1: 
26322  690 
"?p1 : ~(PQ) <> ~P & ~Q" 
691 
"?p2 : P & (Q  R) <> P&Q  P&R" 

692 
"?p3 : (Q  R) & P <> Q&P  R&P" 

693 
"?p4 : (P  Q > R) <> (P > R) & (Q > R)" 

60770  694 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  695 
done 
696 

61337  697 
schematic_goal distrib_rews2: 
26322  698 
"?p1 : ~(EX x. NORM(P(x))) <> (ALL x. ~NORM(P(x)))" 
699 
"?p2 : ((EX x. NORM(P(x))) > Q) <> (ALL x. NORM(P(x)) > Q)" 

700 
"?p3 : (EX x. NORM(P(x))) & NORM(Q) <> (EX x. NORM(P(x)) & NORM(Q))" 

701 
"?p4 : NORM(Q) & (EX x. NORM(P(x))) <> (EX x. NORM(Q) & NORM(P(x)))" 

60770  702 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>)+ 
26322  703 
done 
704 

705 
lemmas distrib_rews = distrib_rews1 distrib_rews2 

706 

61337  707 
schematic_goal P_Imp_P_iff_T: "p:P ==> ?p:(P <> True)" 
60770  708 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>) 
26322  709 
done 
710 

61337  711 
schematic_goal not_P_imp_P_iff_F: "p:~P ==> ?p:(P <> False)" 
60770  712 
apply (tactic \<open>IntPr.fast_tac @{context} 1\<close>) 
26322  713 
done 
0  714 

715 
end 