author  wenzelm 
Tue, 18 Mar 2008 22:19:18 +0100  
changeset 26322  eaf634e975fa 
parent 17480  fd19f77dcf60 
child 26480  544cef16045b 
permissions  rwrr 
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(* Title: FOLP/IFOLP.thy 
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ID: $Id$ 
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Author: Martin D Coen, Cambridge University Computer Laboratory 
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Copyright 1992 University of Cambridge 
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*) 

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header {* Intuitionistic FirstOrder Logic with Proofs *} 
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theory IFOLP 

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imports Pure 

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uses ("hypsubst.ML") ("intprover.ML") 
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begin 
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global 
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classes "term" 
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defaultsort "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" :: "[p,o]=>prop" ("(_ /: _)" [51,10] 5) 
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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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"=" :: "['a,'a] => o" (infixl 50) 
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True :: "o" 
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False :: "o" 

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Not :: "o => o" ("~ _" [40] 40) 
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"&" :: "[o,o] => o" (infixr 35) 
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"" :: "[o,o] => o" (infixr 30) 

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

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"<>" :: "[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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Ex1 :: "('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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when :: "[p, p=>p, p=>p]=>p" 
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lambda :: "(p => p) => p" (binder "lam " 55) 

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"`" :: "[p,p]=>p" (infixl 60) 

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FOLP/IFOLP.thy: tightening precedences to eliminate syntactic ambiguities.
lcp
parents:
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diff
changeset

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alll :: "['a=>p]=>p" (binder "all " 55) 
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FOLP/IFOLP.thy: tightening precedences to eliminate syntactic ambiguities.
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"^" :: "[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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local 
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ML {* 
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(*show_proofs:=true displays the proof terms  they are ENORMOUS*) 

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val show_proofs = ref false; 

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fun proof_tr [p,P] = Const (@{const_name Proof}, dummyT) $ P $ p; 
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fun proof_tr' [P,p] = 

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if !show_proofs then Const("@Proof",dummyT) $ p $ P 

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else P (*this case discards the proof term*); 

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*} 

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parse_translation {* [("@Proof", proof_tr)] *} 

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print_translation {* [("Proof", proof_tr')] *} 

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axioms 

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(**** Propositional logic ****) 

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(*Equality*) 

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

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ieqI: "ideq(a) : a=a" 
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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 *) 

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TrueI: "tt : True" 
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FalseE: "a:False ==> contr(a):P" 

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(* Conjunction *) 

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conjI: "[ a:P; b:Q ] ==> <a,b> : P&Q" 
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conjunct1: "p:P&Q ==> fst(p):P" 

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

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(* Disjunction *) 

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disjI1: "a:P ==> inl(a):PQ" 
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disjI2: "b:Q ==> inr(b):PQ" 

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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 *) 

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impI: "(!!x. x:P ==> f(x):Q) ==> lam x. f(x):P>Q" 
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mp: "[ f:P>Q; a:P ] ==> f`a:Q" 

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(*Quantifiers*) 

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allI: "(!!x. f(x) : P(x)) ==> all x. f(x) : ALL x. P(x)" 
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spec: "(f:ALL x. P(x)) ==> f^x : P(x)" 

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exI: "p : P(x) ==> [x,p] : EX x. P(x)" 
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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 ****) 

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prefl: "a : P ==> a = a : P" 
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psym: "a = b : P ==> b = a : P" 

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ptrans: "[ a = b : P; b = c : P ] ==> a = c : P" 

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idpeelB: "[ !!x. f(x) : P(x,x) ] ==> idpeel(ideq(a),f) = f(a) : P(a,a)" 
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fstB: "a:P ==> fst(<a,b>) = a : P" 
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sndB: "b:Q ==> snd(<a,b>) = b : Q" 

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pairEC: "p:P&Q ==> p = <fst(p),snd(p)> : P&Q" 

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

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plusEC: "a:PQ ==> when(a,%x. inl(x),%y. inr(y)) = a : PQ" 

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applyB: "[ a:P; !!x. x:P ==> b(x) : Q ] ==> (lam x. b(x)) ` a = b(a) : Q" 
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funEC: "f:P ==> f = lam x. f`x : P" 

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specB: "[ !!x. f(x) : P(x) ] ==> (all x. f(x)) ^ a = f(a) : P(a)" 
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(**** Definitions ****) 

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not_def: "~P == P>False" 
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iff_def: "P<>Q == (P>Q) & (Q>P)" 

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(*Unique existence*) 

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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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norm_eq: "nrm : norm(x) = x" 
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NORM_iff: "NRM : NORM(P) <> P" 

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(*** Sequentstyle elimination rules for & > and ALL ***) 
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lemma 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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lemma 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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lemma 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)+ 

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done 

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

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lemma 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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lemma 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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lemma 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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lemma 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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To specify P use something like 

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eres_inst_tac [ ("P","ALL y. ?S(x,y)") ] rev_mp 1 *) 

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lemma 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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lemma 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. 

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Fails unless one assumption is equal and exactly one is unifiable 

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**) 

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ML {* 

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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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val uniq_assume_tac = 

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SUBGOAL 

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(fn (prem,i) => 

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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 => could_unify (hyp, concl)) hyps) of 

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[_] => assume_tac i 

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 _ => no_tac 

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else no_tac 

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end); 

255 
end; 

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*} 

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(*** Modus Ponens Tactics ***) 

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

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ML {* 

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fun mp_tac i = eresolve_tac [@{thm notE}, make_elim @{thm mp}] i THEN assume_tac i 

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*} 

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(*Like mp_tac but instantiates no variables*) 

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ML {* 

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fun int_uniq_mp_tac i = eresolve_tac [@{thm notE}, @{thm impE}] i THEN uniq_assume_tac i 

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*} 

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(*** Ifandonlyif ***) 

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lemma iffI: 

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assumes "!!x. x:P ==> q(x):Q" 

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

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shows "?p:P<>Q" 

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unfolding iff_def 

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apply (assumption  rule assms conjI impI)+ 

280 
done 

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(*Observe use of rewrite_rule to unfold "<>" in metaassumptions (prems) *) 

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lemma iffE: 

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assumes "p:P <> Q" 

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and "!!x y.[ x:P>Q; y:Q>P ] ==> q(x,y):R" 

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shows "?p:R" 

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apply (rule conjE) 

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

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apply (rule assms(2)) 

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apply assumption+ 

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done 

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

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lemma iffD1: "[ p:P <> Q; q:P ] ==> ?p:Q" 

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unfolding iff_def 

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apply (rule conjunct1 [THEN mp], assumption+) 

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done 

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lemma iffD2: "[ p:P <> Q; q:Q ] ==> ?p:P" 

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unfolding iff_def 

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apply (rule conjunct2 [THEN mp], assumption+) 

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done 

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307 
lemma iff_refl: "?p:P <> P" 

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apply (rule iffI) 

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apply assumption+ 

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done 

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lemma iff_sym: "p:Q <> P ==> ?p:P <> Q" 

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apply (erule iffE) 

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apply (rule iffI) 

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apply (erule (1) mp)+ 

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done 

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lemma iff_trans: "[ p:P <> Q; q:Q<> R ] ==> ?p:P <> R" 

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apply (rule iffI) 

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apply (assumption  erule iffE  erule (1) impE)+ 

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done 

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

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

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

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

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***) 

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lemma ex1I: 

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assumes "p:P(a)" 

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and "!!x u. u:P(x) ==> f(u) : x=a" 

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

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unfolding ex1_def 

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apply (assumption  rule assms exI conjI allI impI)+ 

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done 

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lemma ex1E: 

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assumes "p:EX! x. P(x)" 

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and "!!x u v. [ u:P(x); v:ALL y. P(y) > y=x ] ==> f(x,u,v):R" 

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shows "?a : R" 

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

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

343 
done 

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345 

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

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

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ML {* 

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fun iff_tac prems i = 

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resolve_tac (prems RL [@{thm iffE}]) i THEN 

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

353 
*} 

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lemma conj_cong: 

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assumes "p:P <> P'" 

357 
and "!!x. x:P' ==> q(x):Q <> Q'" 

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

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apply (insert assms(1)) 

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apply (assumption  rule iffI conjI  

361 
erule iffE conjE mp  tactic {* iff_tac @{thms assms} 1 *})+ 

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done 

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lemma disj_cong: 

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"[ p:P <> P'; q:Q <> Q' ] ==> ?p:(PQ) <> (P'Q')" 

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apply (erule iffE disjE disjI1 disjI2  assumption  rule iffI  tactic {* mp_tac 1 *})+ 

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done 

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lemma imp_cong: 

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assumes "p:P <> P'" 

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and "!!x. x:P' ==> q(x):Q <> Q'" 

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

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apply (insert assms(1)) 

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apply (assumption  rule iffI impI  erule iffE  tactic {* mp_tac 1 *}  

375 
tactic {* iff_tac @{thms assms} 1 *})+ 

376 
done 

377 

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lemma iff_cong: 

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"[ p:P <> P'; q:Q <> Q' ] ==> ?p:(P<>Q) <> (P'<>Q')" 

380 
apply (erule iffE  assumption  rule iffI  tactic {* mp_tac 1 *})+ 

381 
done 

382 

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lemma not_cong: 

384 
"p:P <> P' ==> ?p:~P <> ~P'" 

385 
apply (assumption  rule iffI notI  tactic {* mp_tac 1 *}  erule iffE notE)+ 

386 
done 

387 

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lemma all_cong: 

389 
assumes "!!x. f(x):P(x) <> Q(x)" 

390 
shows "?p:(ALL x. P(x)) <> (ALL x. Q(x))" 

391 
apply (assumption  rule iffI allI  tactic {* mp_tac 1 *}  erule allE  

392 
tactic {* iff_tac @{thms assms} 1 *})+ 

393 
done 

394 

395 
lemma ex_cong: 

396 
assumes "!!x. f(x):P(x) <> Q(x)" 

397 
shows "?p:(EX x. P(x)) <> (EX x. Q(x))" 

398 
apply (erule exE  assumption  rule iffI exI  tactic {* mp_tac 1 *}  

399 
tactic {* iff_tac @{thms assms} 1 *})+ 

400 
done 

401 

402 
(*NOT PROVED 

403 
bind_thm ("ex1_cong", prove_goal (the_context ()) 

404 
"(!!x.f(x):P(x) <> Q(x)) ==> ?p:(EX! x.P(x)) <> (EX! x.Q(x))" 

405 
(fn prems => 

406 
[ (REPEAT (eresolve_tac [ex1E, spec RS mp] 1 ORELSE ares_tac [iffI,ex1I] 1 

407 
ORELSE mp_tac 1 

408 
ORELSE iff_tac prems 1)) ])) 

409 
*) 

410 

411 
(*** Equality rules ***) 

412 

413 
lemmas refl = ieqI 

414 

415 
lemma subst: 

416 
assumes prem1: "p:a=b" 

417 
and prem2: "q:P(a)" 

418 
shows "?p : P(b)" 

419 
apply (rule prem2 [THEN rev_mp]) 

420 
apply (rule prem1 [THEN ieqE]) 

421 
apply (rule impI) 

422 
apply assumption 

423 
done 

424 

425 
lemma sym: "q:a=b ==> ?c:b=a" 

426 
apply (erule subst) 

427 
apply (rule refl) 

428 
done 

429 

430 
lemma trans: "[ p:a=b; q:b=c ] ==> ?d:a=c" 

431 
apply (erule (1) subst) 

432 
done 

433 

434 
(** ~ b=a ==> ~ a=b **) 

435 
lemma not_sym: "p:~ b=a ==> ?q:~ a=b" 

436 
apply (erule contrapos) 

437 
apply (erule sym) 

438 
done 

439 

440 
(*calling "standard" reduces maxidx to 0*) 

441 
lemmas ssubst = sym [THEN subst, standard] 

442 

443 
(*A special case of ex1E that would otherwise need quantifier expansion*) 

444 
lemma ex1_equalsE: "[ p:EX! x. P(x); q:P(a); r:P(b) ] ==> ?d:a=b" 

445 
apply (erule ex1E) 

446 
apply (rule trans) 

447 
apply (rule_tac [2] sym) 

448 
apply (assumption  erule spec [THEN mp])+ 

449 
done 

450 

451 
(** Polymorphic congruence rules **) 

452 

453 
lemma subst_context: "[ p:a=b ] ==> ?d:t(a)=t(b)" 

454 
apply (erule ssubst) 

455 
apply (rule refl) 

456 
done 

457 

458 
lemma subst_context2: "[ p:a=b; q:c=d ] ==> ?p:t(a,c)=t(b,d)" 

459 
apply (erule ssubst)+ 

460 
apply (rule refl) 

461 
done 

462 

463 
lemma subst_context3: "[ p:a=b; q:c=d; r:e=f ] ==> ?p:t(a,c,e)=t(b,d,f)" 

464 
apply (erule ssubst)+ 

465 
apply (rule refl) 

466 
done 

467 

468 
(*Useful with eresolve_tac for proving equalties from known equalities. 

469 
a = b 

470 
  

471 
c = d *) 

472 
lemma box_equals: "[ p:a=b; q:a=c; r:b=d ] ==> ?p:c=d" 

473 
apply (rule trans) 

474 
apply (rule trans) 

475 
apply (rule sym) 

476 
apply assumption+ 

477 
done 

478 

479 
(*Dual of box_equals: for proving equalities backwards*) 

480 
lemma simp_equals: "[ p:a=c; q:b=d; r:c=d ] ==> ?p:a=b" 

481 
apply (rule trans) 

482 
apply (rule trans) 

483 
apply (assumption  rule sym)+ 

484 
done 

485 

486 
(** Congruence rules for predicate letters **) 

487 

488 
lemma pred1_cong: "p:a=a' ==> ?p:P(a) <> P(a')" 

489 
apply (rule iffI) 

490 
apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *}) 

491 
done 

492 

493 
lemma pred2_cong: "[ p:a=a'; q:b=b' ] ==> ?p:P(a,b) <> P(a',b')" 

494 
apply (rule iffI) 

495 
apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *}) 

496 
done 

497 

498 
lemma pred3_cong: "[ p:a=a'; q:b=b'; r:c=c' ] ==> ?p:P(a,b,c) <> P(a',b',c')" 

499 
apply (rule iffI) 

500 
apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE eresolve_tac [@{thm subst}, @{thm ssubst}] 1) *}) 

501 
done 

502 

503 
(*special cases for free variables P, Q, R, S  up to 3 arguments*) 

504 

505 
ML_setup {* 

506 
bind_thms ("pred_congs", 

507 
flat (map (fn c => 

508 
map (fn th => read_instantiate [("P",c)] th) 

509 
[@{thm pred1_cong}, @{thm pred2_cong}, @{thm pred3_cong}]) 

510 
(explode"PQRS"))) 

511 
*} 

512 

513 
(*special case for the equality predicate!*) 

514 
lemmas eq_cong = pred2_cong [where P = "op =", standard] 

515 

516 

517 
(*** Simplifications of assumed implications. 

518 
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE 

519 
used with mp_tac (restricted to atomic formulae) is COMPLETE for 

520 
intuitionistic propositional logic. See 

521 
R. Dyckhoff, Contractionfree sequent calculi for intuitionistic logic 

522 
(preprint, University of St Andrews, 1991) ***) 

523 

524 
lemma conj_impE: 

525 
assumes major: "p:(P&Q)>S" 

526 
and minor: "!!x. x:P>(Q>S) ==> q(x):R" 

527 
shows "?p:R" 

528 
apply (assumption  rule conjI impI major [THEN mp] minor)+ 

529 
done 

530 

531 
lemma disj_impE: 

532 
assumes major: "p:(PQ)>S" 

533 
and minor: "!!x y.[ x:P>S; y:Q>S ] ==> q(x,y):R" 

534 
shows "?p:R" 

535 
apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE 

536 
resolve_tac [@{thm disjI1}, @{thm disjI2}, @{thm impI}, 

537 
@{thm major} RS @{thm mp}, @{thm minor}] 1) *}) 

538 
done 

539 

540 
(*Simplifies the implication. Classical version is stronger. 

541 
Still UNSAFE since Q must be provable  backtracking needed. *) 

542 
lemma imp_impE: 

543 
assumes major: "p:(P>Q)>S" 

544 
and r1: "!!x y.[ x:P; y:Q>S ] ==> q(x,y):Q" 

545 
and r2: "!!x. x:S ==> r(x):R" 

546 
shows "?p:R" 

547 
apply (assumption  rule impI major [THEN mp] r1 r2)+ 

548 
done 

549 

550 
(*Simplifies the implication. Classical version is stronger. 

551 
Still UNSAFE since ~P must be provable  backtracking needed. *) 

552 
lemma not_impE: 

553 
assumes major: "p:~P > S" 

554 
and r1: "!!y. y:P ==> q(y):False" 

555 
and r2: "!!y. y:S ==> r(y):R" 

556 
shows "?p:R" 

557 
apply (assumption  rule notI impI major [THEN mp] r1 r2)+ 

558 
done 

559 

560 
(*Simplifies the implication. UNSAFE. *) 

561 
lemma iff_impE: 

562 
assumes major: "p:(P<>Q)>S" 

563 
and r1: "!!x y.[ x:P; y:Q>S ] ==> q(x,y):Q" 

564 
and r2: "!!x y.[ x:Q; y:P>S ] ==> r(x,y):P" 

565 
and r3: "!!x. x:S ==> s(x):R" 

566 
shows "?p:R" 

567 
apply (assumption  rule iffI impI major [THEN mp] r1 r2 r3)+ 

568 
done 

569 

570 
(*What if (ALL x.~~P(x)) > ~~(ALL x.P(x)) is an assumption? UNSAFE*) 

571 
lemma all_impE: 

572 
assumes major: "p:(ALL x. P(x))>S" 

573 
and r1: "!!x. q:P(x)" 

574 
and r2: "!!y. y:S ==> r(y):R" 

575 
shows "?p:R" 

576 
apply (assumption  rule allI impI major [THEN mp] r1 r2)+ 

577 
done 

578 

579 
(*Unsafe: (EX x.P(x))>S is equivalent to ALL x.P(x)>S. *) 

580 
lemma ex_impE: 

581 
assumes major: "p:(EX x. P(x))>S" 

582 
and r: "!!y. y:P(a)>S ==> q(y):R" 

583 
shows "?p:R" 

584 
apply (assumption  rule exI impI major [THEN mp] r)+ 

585 
done 

586 

587 

588 
lemma rev_cut_eq: 

589 
assumes "p:a=b" 

590 
and "!!x. x:a=b ==> f(x):R" 

591 
shows "?p:R" 

592 
apply (rule assms)+ 

593 
done 

594 

595 
lemma thin_refl: "!!X. [p:x=x; PROP W] ==> PROP W" . 

596 

597 
use "hypsubst.ML" 

598 

599 
ML {* 

600 

601 
(*** Applying HypsubstFun to generate hyp_subst_tac ***) 

602 

603 
structure Hypsubst_Data = 

604 
struct 

605 
(*Take apart an equality judgement; otherwise raise Match!*) 

606 
fun dest_eq (Const (@{const_name Proof}, _) $ 

607 
(Const (@{const_name "op ="}, _) $ t $ u) $ _) = (t, u); 

608 

609 
val imp_intr = @{thm impI} 

610 

611 
(*etac rev_cut_eq moves an equality to be the last premise. *) 

612 
val rev_cut_eq = @{thm rev_cut_eq} 

613 

614 
val rev_mp = @{thm rev_mp} 

615 
val subst = @{thm subst} 

616 
val sym = @{thm sym} 

617 
val thin_refl = @{thm thin_refl} 

618 
end; 

619 

620 
structure Hypsubst = HypsubstFun(Hypsubst_Data); 

621 
open Hypsubst; 

622 
*} 

623 

624 
use "intprover.ML" 

625 

626 

627 
(*** Rewrite rules ***) 

628 

629 
lemma conj_rews: 

630 
"?p1 : P & True <> P" 

631 
"?p2 : True & P <> P" 

632 
"?p3 : P & False <> False" 

633 
"?p4 : False & P <> False" 

634 
"?p5 : P & P <> P" 

635 
"?p6 : P & ~P <> False" 

636 
"?p7 : ~P & P <> False" 

637 
"?p8 : (P & Q) & R <> P & (Q & R)" 

638 
apply (tactic {* fn st => IntPr.fast_tac 1 st *})+ 

639 
done 

640 

641 
lemma disj_rews: 

642 
"?p1 : P  True <> True" 

643 
"?p2 : True  P <> True" 

644 
"?p3 : P  False <> P" 

645 
"?p4 : False  P <> P" 

646 
"?p5 : P  P <> P" 

647 
"?p6 : (P  Q)  R <> P  (Q  R)" 

648 
apply (tactic {* IntPr.fast_tac 1 *})+ 

649 
done 

650 

651 
lemma not_rews: 

652 
"?p1 : ~ False <> True" 

653 
"?p2 : ~ True <> False" 

654 
apply (tactic {* IntPr.fast_tac 1 *})+ 

655 
done 

656 

657 
lemma imp_rews: 

658 
"?p1 : (P > False) <> ~P" 

659 
"?p2 : (P > True) <> True" 

660 
"?p3 : (False > P) <> True" 

661 
"?p4 : (True > P) <> P" 

662 
"?p5 : (P > P) <> True" 

663 
"?p6 : (P > ~P) <> ~P" 

664 
apply (tactic {* IntPr.fast_tac 1 *})+ 

665 
done 

666 

667 
lemma iff_rews: 

668 
"?p1 : (True <> P) <> P" 

669 
"?p2 : (P <> True) <> P" 

670 
"?p3 : (P <> P) <> True" 

671 
"?p4 : (False <> P) <> ~P" 

672 
"?p5 : (P <> False) <> ~P" 

673 
apply (tactic {* IntPr.fast_tac 1 *})+ 

674 
done 

675 

676 
lemma quant_rews: 

677 
"?p1 : (ALL x. P) <> P" 

678 
"?p2 : (EX x. P) <> P" 

679 
apply (tactic {* IntPr.fast_tac 1 *})+ 

680 
done 

681 

682 
(*These are NOT supplied by default!*) 

683 
lemma distrib_rews1: 

684 
"?p1 : ~(PQ) <> ~P & ~Q" 

685 
"?p2 : P & (Q  R) <> P&Q  P&R" 

686 
"?p3 : (Q  R) & P <> Q&P  R&P" 

687 
"?p4 : (P  Q > R) <> (P > R) & (Q > R)" 

688 
apply (tactic {* IntPr.fast_tac 1 *})+ 

689 
done 

690 

691 
lemma distrib_rews2: 

692 
"?p1 : ~(EX x. NORM(P(x))) <> (ALL x. ~NORM(P(x)))" 

693 
"?p2 : ((EX x. NORM(P(x))) > Q) <> (ALL x. NORM(P(x)) > Q)" 

694 
"?p3 : (EX x. NORM(P(x))) & NORM(Q) <> (EX x. NORM(P(x)) & NORM(Q))" 

695 
"?p4 : NORM(Q) & (EX x. NORM(P(x))) <> (EX x. NORM(Q) & NORM(P(x)))" 

696 
apply (tactic {* IntPr.fast_tac 1 *})+ 

697 
done 

698 

699 
lemmas distrib_rews = distrib_rews1 distrib_rews2 

700 

701 
lemma P_Imp_P_iff_T: "p:P ==> ?p:(P <> True)" 

702 
apply (tactic {* IntPr.fast_tac 1 *}) 

703 
done 

704 

705 
lemma not_P_imp_P_iff_F: "p:~P ==> ?p:(P <> False)" 

706 
apply (tactic {* IntPr.fast_tac 1 *}) 

707 
done 

0  708 

709 
end 