author | haftmann |
Fri, 23 Jan 2009 19:51:49 +0100 | |
changeset 29623 | 1219985d24b5 |
parent 29305 | 76af2a3c9d28 |
child 32740 | 9dd0a2f83429 |
permissions | -rw-r--r-- |
1477 | 1 |
(* Title: FOLP/IFOLP.thy |
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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 First-Order 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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setup PureThy.old_appl_syntax_setup -- theory Pure provides regular application syntax by default;
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parents:
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diff
changeset
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setup PureThy.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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parents:
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diff
changeset
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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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FOLP/IFOLP.thy: tightening precedences to eliminate syntactic ambiguities.
lcp
parents:
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diff
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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:
283
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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parents:
283
diff
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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):P|Q" |
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disjI2: "b:Q ==> inr(b):P|Q" |
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disjE: "[| a:P|Q; !!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:P|Q ==> when(a,%x. inl(x),%y. inr(y)) = a : P|Q" |
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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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(*** Sequent-style 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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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 => Term.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); |
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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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(*** If-and-only-if ***) |
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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)+ |
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done |
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(*Observe use of rewrite_rule to unfold "<->" in meta-assumptions (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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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))+ |
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apply (erule assms(2), assumption) |
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done |
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345 |
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(*** <-> congruence rules for simplification ***) |
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348 |
(*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) |
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*} |
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lemma conj_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 conjI | |
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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:(P|Q) <-> (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 *} | |
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tactic {* iff_tac @{thms assms} 1 *})+ |
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done |
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lemma iff_cong: |
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"[| p:P <-> P'; q:Q <-> Q' |] ==> ?p:(P<->Q) <-> (P'<->Q')" |
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apply (erule iffE | assumption | rule iffI | tactic {* mp_tac 1 *})+ |
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done |
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lemma not_cong: |
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"p:P <-> P' ==> ?p:~P <-> ~P'" |
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apply (assumption | rule iffI notI | tactic {* mp_tac 1 *} | erule iffE notE)+ |
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done |
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lemma all_cong: |
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assumes "!!x. f(x):P(x) <-> Q(x)" |
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shows "?p:(ALL x. P(x)) <-> (ALL x. Q(x))" |
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apply (assumption | rule iffI allI | tactic {* mp_tac 1 *} | erule allE | |
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tactic {* iff_tac @{thms assms} 1 *})+ |
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done |
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lemma ex_cong: |
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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 |
||
27152
192954a9a549
changed pred_congs: merely cover pred1_cong pred2_cong pred3_cong;
wenzelm
parents:
27150
diff
changeset
|
503 |
lemmas pred_congs = pred1_cong pred2_cong pred3_cong |
26322 | 504 |
|
505 |
(*special case for the equality predicate!*) |
|
506 |
lemmas eq_cong = pred2_cong [where P = "op =", standard] |
|
507 |
||
508 |
||
509 |
(*** Simplifications of assumed implications. |
|
510 |
Roy Dyckhoff has proved that conj_impE, disj_impE, and imp_impE |
|
511 |
used with mp_tac (restricted to atomic formulae) is COMPLETE for |
|
512 |
intuitionistic propositional logic. See |
|
513 |
R. Dyckhoff, Contraction-free sequent calculi for intuitionistic logic |
|
514 |
(preprint, University of St Andrews, 1991) ***) |
|
515 |
||
516 |
lemma conj_impE: |
|
517 |
assumes major: "p:(P&Q)-->S" |
|
518 |
and minor: "!!x. x:P-->(Q-->S) ==> q(x):R" |
|
519 |
shows "?p:R" |
|
520 |
apply (assumption | rule conjI impI major [THEN mp] minor)+ |
|
521 |
done |
|
522 |
||
523 |
lemma disj_impE: |
|
524 |
assumes major: "p:(P|Q)-->S" |
|
525 |
and minor: "!!x y.[| x:P-->S; y:Q-->S |] ==> q(x,y):R" |
|
526 |
shows "?p:R" |
|
527 |
apply (tactic {* DEPTH_SOLVE (atac 1 ORELSE |
|
528 |
resolve_tac [@{thm disjI1}, @{thm disjI2}, @{thm impI}, |
|
529 |
@{thm major} RS @{thm mp}, @{thm minor}] 1) *}) |
|
530 |
done |
|
531 |
||
532 |
(*Simplifies the implication. Classical version is stronger. |
|
533 |
Still UNSAFE since Q must be provable -- backtracking needed. *) |
|
534 |
lemma imp_impE: |
|
535 |
assumes major: "p:(P-->Q)-->S" |
|
536 |
and r1: "!!x y.[| x:P; y:Q-->S |] ==> q(x,y):Q" |
|
537 |
and r2: "!!x. x:S ==> r(x):R" |
|
538 |
shows "?p:R" |
|
539 |
apply (assumption | rule impI major [THEN mp] r1 r2)+ |
|
540 |
done |
|
541 |
||
542 |
(*Simplifies the implication. Classical version is stronger. |
|
543 |
Still UNSAFE since ~P must be provable -- backtracking needed. *) |
|
544 |
lemma not_impE: |
|
545 |
assumes major: "p:~P --> S" |
|
546 |
and r1: "!!y. y:P ==> q(y):False" |
|
547 |
and r2: "!!y. y:S ==> r(y):R" |
|
548 |
shows "?p:R" |
|
549 |
apply (assumption | rule notI impI major [THEN mp] r1 r2)+ |
|
550 |
done |
|
551 |
||
552 |
(*Simplifies the implication. UNSAFE. *) |
|
553 |
lemma iff_impE: |
|
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 y.[| x:Q; y:P-->S |] ==> r(x,y):P" |
|
557 |
and r3: "!!x. x:S ==> s(x):R" |
|
558 |
shows "?p:R" |
|
559 |
apply (assumption | rule iffI impI major [THEN mp] r1 r2 r3)+ |
|
560 |
done |
|
561 |
||
562 |
(*What if (ALL x.~~P(x)) --> ~~(ALL x.P(x)) is an assumption? UNSAFE*) |
|
563 |
lemma all_impE: |
|
564 |
assumes major: "p:(ALL x. P(x))-->S" |
|
565 |
and r1: "!!x. q:P(x)" |
|
566 |
and r2: "!!y. y:S ==> r(y):R" |
|
567 |
shows "?p:R" |
|
568 |
apply (assumption | rule allI impI major [THEN mp] r1 r2)+ |
|
569 |
done |
|
570 |
||
571 |
(*Unsafe: (EX x.P(x))-->S is equivalent to ALL x.P(x)-->S. *) |
|
572 |
lemma ex_impE: |
|
573 |
assumes major: "p:(EX x. P(x))-->S" |
|
574 |
and r: "!!y. y:P(a)-->S ==> q(y):R" |
|
575 |
shows "?p:R" |
|
576 |
apply (assumption | rule exI impI major [THEN mp] r)+ |
|
577 |
done |
|
578 |
||
579 |
||
580 |
lemma rev_cut_eq: |
|
581 |
assumes "p:a=b" |
|
582 |
and "!!x. x:a=b ==> f(x):R" |
|
583 |
shows "?p:R" |
|
584 |
apply (rule assms)+ |
|
585 |
done |
|
586 |
||
587 |
lemma thin_refl: "!!X. [|p:x=x; PROP W|] ==> PROP W" . |
|
588 |
||
589 |
use "hypsubst.ML" |
|
590 |
||
591 |
ML {* |
|
592 |
||
593 |
(*** Applying HypsubstFun to generate hyp_subst_tac ***) |
|
594 |
||
595 |
structure Hypsubst_Data = |
|
596 |
struct |
|
597 |
(*Take apart an equality judgement; otherwise raise Match!*) |
|
598 |
fun dest_eq (Const (@{const_name Proof}, _) $ |
|
599 |
(Const (@{const_name "op ="}, _) $ t $ u) $ _) = (t, u); |
|
600 |
||
601 |
val imp_intr = @{thm impI} |
|
602 |
||
603 |
(*etac rev_cut_eq moves an equality to be the last premise. *) |
|
604 |
val rev_cut_eq = @{thm rev_cut_eq} |
|
605 |
||
606 |
val rev_mp = @{thm rev_mp} |
|
607 |
val subst = @{thm subst} |
|
608 |
val sym = @{thm sym} |
|
609 |
val thin_refl = @{thm thin_refl} |
|
610 |
end; |
|
611 |
||
612 |
structure Hypsubst = HypsubstFun(Hypsubst_Data); |
|
613 |
open Hypsubst; |
|
614 |
*} |
|
615 |
||
616 |
use "intprover.ML" |
|
617 |
||
618 |
||
619 |
(*** Rewrite rules ***) |
|
620 |
||
621 |
lemma conj_rews: |
|
622 |
"?p1 : P & True <-> P" |
|
623 |
"?p2 : True & P <-> P" |
|
624 |
"?p3 : P & False <-> False" |
|
625 |
"?p4 : False & P <-> False" |
|
626 |
"?p5 : P & P <-> P" |
|
627 |
"?p6 : P & ~P <-> False" |
|
628 |
"?p7 : ~P & P <-> False" |
|
629 |
"?p8 : (P & Q) & R <-> P & (Q & R)" |
|
630 |
apply (tactic {* fn st => IntPr.fast_tac 1 st *})+ |
|
631 |
done |
|
632 |
||
633 |
lemma disj_rews: |
|
634 |
"?p1 : P | True <-> True" |
|
635 |
"?p2 : True | P <-> True" |
|
636 |
"?p3 : P | False <-> P" |
|
637 |
"?p4 : False | P <-> P" |
|
638 |
"?p5 : P | P <-> P" |
|
639 |
"?p6 : (P | Q) | R <-> P | (Q | R)" |
|
640 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
641 |
done |
|
642 |
||
643 |
lemma not_rews: |
|
644 |
"?p1 : ~ False <-> True" |
|
645 |
"?p2 : ~ True <-> False" |
|
646 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
647 |
done |
|
648 |
||
649 |
lemma imp_rews: |
|
650 |
"?p1 : (P --> False) <-> ~P" |
|
651 |
"?p2 : (P --> True) <-> True" |
|
652 |
"?p3 : (False --> P) <-> True" |
|
653 |
"?p4 : (True --> P) <-> P" |
|
654 |
"?p5 : (P --> P) <-> True" |
|
655 |
"?p6 : (P --> ~P) <-> ~P" |
|
656 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
657 |
done |
|
658 |
||
659 |
lemma iff_rews: |
|
660 |
"?p1 : (True <-> P) <-> P" |
|
661 |
"?p2 : (P <-> True) <-> P" |
|
662 |
"?p3 : (P <-> P) <-> True" |
|
663 |
"?p4 : (False <-> P) <-> ~P" |
|
664 |
"?p5 : (P <-> False) <-> ~P" |
|
665 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
666 |
done |
|
667 |
||
668 |
lemma quant_rews: |
|
669 |
"?p1 : (ALL x. P) <-> P" |
|
670 |
"?p2 : (EX x. P) <-> P" |
|
671 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
672 |
done |
|
673 |
||
674 |
(*These are NOT supplied by default!*) |
|
675 |
lemma distrib_rews1: |
|
676 |
"?p1 : ~(P|Q) <-> ~P & ~Q" |
|
677 |
"?p2 : P & (Q | R) <-> P&Q | P&R" |
|
678 |
"?p3 : (Q | R) & P <-> Q&P | R&P" |
|
679 |
"?p4 : (P | Q --> R) <-> (P --> R) & (Q --> R)" |
|
680 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
681 |
done |
|
682 |
||
683 |
lemma distrib_rews2: |
|
684 |
"?p1 : ~(EX x. NORM(P(x))) <-> (ALL x. ~NORM(P(x)))" |
|
685 |
"?p2 : ((EX x. NORM(P(x))) --> Q) <-> (ALL x. NORM(P(x)) --> Q)" |
|
686 |
"?p3 : (EX x. NORM(P(x))) & NORM(Q) <-> (EX x. NORM(P(x)) & NORM(Q))" |
|
687 |
"?p4 : NORM(Q) & (EX x. NORM(P(x))) <-> (EX x. NORM(Q) & NORM(P(x)))" |
|
688 |
apply (tactic {* IntPr.fast_tac 1 *})+ |
|
689 |
done |
|
690 |
||
691 |
lemmas distrib_rews = distrib_rews1 distrib_rews2 |
|
692 |
||
693 |
lemma P_Imp_P_iff_T: "p:P ==> ?p:(P <-> True)" |
|
694 |
apply (tactic {* IntPr.fast_tac 1 *}) |
|
695 |
done |
|
696 |
||
697 |
lemma not_P_imp_P_iff_F: "p:~P ==> ?p:(P <-> False)" |
|
698 |
apply (tactic {* IntPr.fast_tac 1 *}) |
|
699 |
done |
|
0 | 700 |
|
701 |
end |