(* Title: HOL/prod
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
For prod.thy. Ordered Pairs, the Cartesian product type, the unit type
*)
open Prod;
(*This counts as a non-emptiness result for admitting 'a * 'b as a type*)
goalw Prod.thy [Prod_def] "Pair_Rep a b : Prod";
by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]);
qed "ProdI";
val [major] = goalw Prod.thy [Pair_Rep_def]
"Pair_Rep a b = Pair_Rep a' b' ==> a=a' & b=b'";
by (EVERY1 [rtac (major RS fun_cong RS fun_cong RS subst),
rtac conjI, rtac refl, rtac refl]);
qed "Pair_Rep_inject";
goal Prod.thy "inj_onto Abs_Prod Prod";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Prod_inverse 1);
qed "inj_onto_Abs_Prod";
val prems = goalw Prod.thy [Pair_def]
"[| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R";
by (rtac (inj_onto_Abs_Prod RS inj_ontoD RS Pair_Rep_inject RS conjE) 1);
by (REPEAT (ares_tac (prems@[ProdI]) 1));
qed "Pair_inject";
goal Prod.thy "((a,b) = (a',b')) = (a=a' & b=b')";
by (fast_tac (set_cs addIs [Pair_inject]) 1);
qed "Pair_eq";
goalw Prod.thy [fst_def] "fst((a,b)) = a";
by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1);
qed "fst_conv";
goalw Prod.thy [snd_def] "snd((a,b)) = b";
by (fast_tac (set_cs addIs [select_equality] addSEs [Pair_inject]) 1);
qed "snd_conv";
goalw Prod.thy [Pair_def] "? x y. p = (x,y)";
by (rtac (rewrite_rule [Prod_def] Rep_Prod RS CollectE) 1);
by (EVERY1[etac exE, etac exE, rtac exI, rtac exI,
rtac (Rep_Prod_inverse RS sym RS trans), etac arg_cong]);
qed "PairE_lemma";
val [prem] = goal Prod.thy "[| !!x y. p = (x,y) ==> Q |] ==> Q";
by (rtac (PairE_lemma RS exE) 1);
by (REPEAT (eresolve_tac [prem,exE] 1));
qed "PairE";
goalw Prod.thy [split_def] "split c (a,b) = c a b";
by (sstac [fst_conv, snd_conv] 1);
by (rtac refl 1);
qed "split";
Addsimps [fst_conv, snd_conv, split, Pair_eq];
goal Prod.thy "(s=t) = (fst(s)=fst(t) & snd(s)=snd(t))";
by (res_inst_tac[("p","s")] PairE 1);
by (res_inst_tac[("p","t")] PairE 1);
by (Asm_simp_tac 1);
qed "Pair_fst_snd_eq";
(*Prevents simplification of c: much faster*)
qed_goal "split_weak_cong" Prod.thy
"p=q ==> split c p = split c q"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
(* Do not add as rewrite rule: invalidates some proofs in IMP *)
goal Prod.thy "p = (fst(p),snd(p))";
by (res_inst_tac [("p","p")] PairE 1);
by (Asm_simp_tac 1);
qed "surjective_pairing";
goal Prod.thy "p = split (%x y.(x,y)) p";
by (res_inst_tac [("p","p")] PairE 1);
by (Asm_simp_tac 1);
qed "surjective_pairing2";
(*For use with split_tac and the simplifier*)
goal Prod.thy "R(split c p) = (! x y. p = (x,y) --> R(c x y))";
by (stac surjective_pairing 1);
by (stac split 1);
by (fast_tac (HOL_cs addSEs [Pair_inject]) 1);
qed "expand_split";
(** split used as a logical connective or set former **)
(*These rules are for use with fast_tac.
Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*)
goal Prod.thy "!!a b c. c a b ==> split c (a,b)";
by (Asm_simp_tac 1);
qed "splitI";
val prems = goalw Prod.thy [split_def]
"[| split c p; !!x y. [| p = (x,y); c x y |] ==> Q |] ==> Q";
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
qed "splitE";
goal Prod.thy "!!R a b. split R (a,b) ==> R a b";
by (etac (split RS iffD1) 1);
qed "splitD";
goal Prod.thy "!!a b c. z: c a b ==> z: split c (a,b)";
by (Asm_simp_tac 1);
qed "mem_splitI";
val prems = goalw Prod.thy [split_def]
"[| z: split c p; !!x y. [| p = (x,y); z: c x y |] ==> Q |] ==> Q";
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
qed "mem_splitE";
(*** prod_fun -- action of the product functor upon functions ***)
goalw Prod.thy [prod_fun_def] "prod_fun f g (a,b) = (f(a),g(b))";
by (rtac split 1);
qed "prod_fun";
goal Prod.thy
"prod_fun (f1 o f2) (g1 o g2) = ((prod_fun f1 g1) o (prod_fun f2 g2))";
by (rtac ext 1);
by (res_inst_tac [("p","x")] PairE 1);
by (asm_simp_tac (!simpset addsimps [prod_fun,o_def]) 1);
qed "prod_fun_compose";
goal Prod.thy "prod_fun (%x.x) (%y.y) = (%z.z)";
by (rtac ext 1);
by (res_inst_tac [("p","z")] PairE 1);
by (asm_simp_tac (!simpset addsimps [prod_fun]) 1);
qed "prod_fun_ident";
val prems = goal Prod.thy "(a,b):r ==> (f(a),g(b)) : (prod_fun f g)``r";
by (rtac image_eqI 1);
by (rtac (prod_fun RS sym) 1);
by (resolve_tac prems 1);
qed "prod_fun_imageI";
val major::prems = goal Prod.thy
"[| c: (prod_fun f g)``r; !!x y. [| c=(f(x),g(y)); (x,y):r |] ==> P \
\ |] ==> P";
by (rtac (major RS imageE) 1);
by (res_inst_tac [("p","x")] PairE 1);
by (resolve_tac prems 1);
by (fast_tac HOL_cs 2);
by (fast_tac (HOL_cs addIs [prod_fun]) 1);
qed "prod_fun_imageE";
(*** Disjoint union of a family of sets - Sigma ***)
qed_goalw "SigmaI" Prod.thy [Sigma_def]
"[| a:A; b:B(a) |] ==> (a,b) : Sigma A B"
(fn prems=> [ (REPEAT (resolve_tac (prems@[singletonI,UN_I]) 1)) ]);
(*The general elimination rule*)
qed_goalw "SigmaE" Prod.thy [Sigma_def]
"[| c: Sigma A B; \
\ !!x y.[| x:A; y:B(x); c=(x,y) |] ==> P \
\ |] ==> P"
(fn major::prems=>
[ (cut_facts_tac [major] 1),
(REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ]);
(** Elimination of (a,b):A*B -- introduces no eigenvariables **)
qed_goal "SigmaD1" Prod.thy "(a,b) : Sigma A B ==> a : A"
(fn [major]=>
[ (rtac (major RS SigmaE) 1),
(REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
qed_goal "SigmaD2" Prod.thy "(a,b) : Sigma A B ==> b : B(a)"
(fn [major]=>
[ (rtac (major RS SigmaE) 1),
(REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ]);
qed_goal "SigmaE2" Prod.thy
"[| (a,b) : Sigma A B; \
\ [| a:A; b:B(a) |] ==> P \
\ |] ==> P"
(fn [major,minor]=>
[ (rtac minor 1),
(rtac (major RS SigmaD1) 1),
(rtac (major RS SigmaD2) 1) ]);
(*** Domain of a relation ***)
val prems = goalw Prod.thy [image_def] "(a,b) : r ==> a : fst``r";
by (rtac CollectI 1);
by (rtac bexI 1);
by (rtac (fst_conv RS sym) 1);
by (resolve_tac prems 1);
qed "fst_imageI";
val major::prems = goal Prod.thy
"[| a : fst``r; !!y.[| (a,y) : r |] ==> P |] ==> P";
by (rtac (major RS imageE) 1);
by (resolve_tac prems 1);
by (etac ssubst 1);
by (rtac (surjective_pairing RS subst) 1);
by (assume_tac 1);
qed "fst_imageE";
(*** Range of a relation ***)
val prems = goalw Prod.thy [image_def] "(a,b) : r ==> b : snd``r";
by (rtac CollectI 1);
by (rtac bexI 1);
by (rtac (snd_conv RS sym) 1);
by (resolve_tac prems 1);
qed "snd_imageI";
val major::prems = goal Prod.thy
"[| a : snd``r; !!y.[| (y,a) : r |] ==> P |] ==> P";
by (rtac (major RS imageE) 1);
by (resolve_tac prems 1);
by (etac ssubst 1);
by (rtac (surjective_pairing RS subst) 1);
by (assume_tac 1);
qed "snd_imageE";
(** Exhaustion rule for unit -- a degenerate form of induction **)
goalw Prod.thy [Unity_def]
"u = ()";
by (stac (rewrite_rule [Unit_def] Rep_Unit RS CollectD RS sym) 1);
by (rtac (Rep_Unit_inverse RS sym) 1);
qed "unit_eq";
val prod_cs = set_cs addSIs [SigmaI, mem_splitI]
addIs [fst_imageI, snd_imageI, prod_fun_imageI]
addSEs [SigmaE2, SigmaE, mem_splitE,
fst_imageE, snd_imageE, prod_fun_imageE,
Pair_inject];