(* Title: HOL/prod
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
Ordered Pairs, the Cartesian product type, the unit type
*)
(*This counts as a non-emptiness result for admitting 'a * 'b as a type*)
Goalw [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 "inj_on Abs_Prod Prod";
by (rtac inj_on_inverseI 1);
by (etac Abs_Prod_inverse 1);
qed "inj_on_Abs_Prod";
val prems = Goalw [Pair_def]
"[| (a, b) = (a',b'); [| a=a'; b=b' |] ==> R |] ==> R";
by (rtac (inj_on_Abs_Prod RS inj_onD RS Pair_Rep_inject RS conjE) 1);
by (REPEAT (ares_tac (prems@[ProdI]) 1));
qed "Pair_inject";
Goal "((a,b) = (a',b')) = (a=a' & b=b')";
by (blast_tac (claset() addSEs [Pair_inject]) 1);
qed "Pair_eq";
AddIffs [Pair_eq];
Goalw [fst_def] "fst (a,b) = a";
by (Blast_tac 1);
qed "fst_conv";
Goalw [snd_def] "snd (a,b) = b";
by (Blast_tac 1);
qed "snd_conv";
Addsimps [fst_conv, snd_conv];
Goal "fst (x, y) = a ==> x = a";
by (Asm_full_simp_tac 1);
qed "fst_eqD";
Goal "snd (x, y) = a ==> y = a";
by (Asm_full_simp_tac 1);
qed "snd_eqD";
Goalw [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 "[| !!x y. p = (x,y) ==> Q |] ==> Q";
by (rtac (PairE_lemma RS exE) 1);
by (REPEAT (eresolve_tac [prem,exE] 1));
qed "PairE";
fun pair_tac s = EVERY' [res_inst_tac [("p",s)] PairE, hyp_subst_tac,
K prune_params_tac];
(* Do not add as rewrite rule: invalidates some proofs in IMP *)
Goal "p = (fst(p),snd(p))";
by (pair_tac "p" 1);
by (Asm_simp_tac 1);
qed "surjective_pairing";
Addsimps [surjective_pairing RS sym];
Goal "? x y. z = (x, y)";
by (rtac exI 1);
by (rtac exI 1);
by (rtac surjective_pairing 1);
qed "surj_pair";
Addsimps [surj_pair];
bind_thm ("split_paired_all",
SplitPairedAll.rule (standard (surjective_pairing RS eq_reflection)));
(*
Addsimps [split_paired_all] does not work with simplifier
because it also affects premises in congrence rules,
where is can lead to premises of the form !!a b. ... = ?P(a,b)
which cannot be solved by reflexivity.
*)
(* replace parameters of product type by individual component parameters *)
local
fun is_pair (_,Type("*",_)) = true
| is_pair _ = false;
fun exists_paired_all prem = exists is_pair (Logic.strip_params prem);
val split_tac = full_simp_tac (HOL_basic_ss addsimps [split_paired_all]);
in
val split_all_tac = SUBGOAL (fn (prem,i) =>
if exists_paired_all prem then split_tac i else no_tac);
end;
claset_ref() := claset() addSWrapper ("split_all_tac",
fn tac2 => split_all_tac ORELSE' tac2);
Goal "(!x. P x) = (!a b. P(a,b))";
by (Fast_tac 1);
qed "split_paired_All";
Addsimps [split_paired_All];
(* AddIffs is not a good idea because it makes Blast_tac loop *)
bind_thm ("prod_induct",
allI RS (allI RS (split_paired_All RS iffD2)) RS spec);
Goal "(? x. P x) = (? a b. P(a,b))";
by (Fast_tac 1);
qed "split_paired_Ex";
Addsimps [split_paired_Ex];
Goalw [split_def] "split c (a,b) = c a b";
by (Simp_tac 1);
qed "split";
Addsimps [split];
(*Subsumes the old split_Pair when f is the identity function*)
Goal "split (%x y. f(x,y)) = f";
by (rtac ext 1);
by (pair_tac "x" 1);
by (Simp_tac 1);
qed "split_Pair_apply";
(*Can't be added to simpset: loops!*)
Goal "(SOME x. P x) = (SOME (a,b). P(a,b))";
by (simp_tac (simpset() addsimps [split_Pair_apply]) 1);
qed "split_paired_Eps";
Goal "!!s t. (s=t) = (fst(s)=fst(t) & snd(s)=snd(t))";
by (split_all_tac 1);
by (Asm_simp_tac 1);
qed "Pair_fst_snd_eq";
(*Prevents simplification of c: much faster*)
val [prem] = goal Prod.thy
"p=q ==> split c p = split c q";
by (rtac (prem RS arg_cong) 1);
qed "split_weak_cong";
Goal "(%(x,y). f(x,y)) = f";
by (rtac ext 1);
by (split_all_tac 1);
by (rtac split 1);
qed "split_eta";
val prems = Goal "(!!x y. f x y = g(x,y)) ==> (%(x,y). f x y) = g";
by (asm_simp_tac (simpset() addsimps prems@[split_eta]) 1);
qed "cond_split_eta";
(*simplification procedure for cond_split_eta.
using split_eta a rewrite rule is not general enough, and using
cond_split_eta directly would render some existing proofs very inefficient.
similarly for split_beta. *)
local
fun Pair_pat k 0 (Bound m) = (m = k)
| Pair_pat k i (Const ("Pair", _) $ Bound m $ t) = i > 0 andalso
m = k+i andalso Pair_pat k (i-1) t
| Pair_pat _ _ _ = false;
fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
| no_args k i (t $ u) = no_args k i t andalso no_args k i u
| no_args k i (Bound m) = m < k orelse m > k+i
| no_args _ _ _ = true;
fun split_pat tp i (Abs (_,_,t)) = if tp 0 i t then Some (i,t) else None
| split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
| split_pat tp i _ = None;
fun metaeq sg lhs rhs = mk_meta_eq (prove_goalw_cterm []
(cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))))
(K [simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1]));
fun simproc name patstr = Simplifier.mk_simproc name
[Thm.read_cterm (sign_of Prod.thy) (patstr, HOLogic.termT)];
val beta_patstr = "split f z";
val eta_patstr = "split f";
fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
| beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
(beta_term_pat k i t andalso beta_term_pat k i u)
| beta_term_pat k i t = no_args k i t;
fun eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
| eta_term_pat _ _ _ = false;
fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
| subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
else (subst arg k i t $ subst arg k i u)
| subst arg k i t = t;
fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
(case split_pat beta_term_pat 1 t of
Some (i,f) => Some (metaeq sg s (subst arg 0 i f))
| None => None)
| beta_proc _ _ _ = None;
fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) =
(case split_pat eta_term_pat 1 t of
Some (_,ft) => Some (metaeq sg s (let val (f $ arg) = ft in f end))
| None => None)
| eta_proc _ _ _ = None;
in
val split_beta_proc = simproc "split_beta" beta_patstr beta_proc;
val split_eta_proc = simproc "split_eta" eta_patstr eta_proc;
end;
Addsimprocs [split_beta_proc,split_eta_proc];
Goal "(%(x,y). P x y) z = P (fst z) (snd z)";
by (stac surjective_pairing 1 THEN rtac split 1);
qed "split_beta";
(*For use with split_tac and the simplifier*)
Goal "R (split c p) = (! x y. p = (x,y) --> R (c x y))";
by (stac surjective_pairing 1);
by (stac split 1);
by (Blast_tac 1);
qed "split_split";
(* could be done after split_tac has been speeded up significantly:
simpset_ref() := simpset() addsplits [split_split];
precompute the constants involved and don't do anything unless
the current goal contains one of those constants
*)
Goal "R (split c p) = (~(? x y. p = (x,y) & (~R (c x y))))";
by (stac split_split 1);
by (Simp_tac 1);
qed "expand_split_asm";
(** split used as a logical connective or set former **)
(*These rules are for use with blast_tac.
Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*)
Goal "!!p. [| !!a b. p=(a,b) ==> c a b |] ==> split c p";
by (split_all_tac 1);
by (Asm_simp_tac 1);
qed "splitI2";
Goal "!!p. [| !!a b. (a,b)=p ==> c a b x |] ==> split c p x";
by (split_all_tac 1);
by (Asm_simp_tac 1);
qed "splitI2'";
Goal "c a b ==> split c (a,b)";
by (Asm_simp_tac 1);
qed "splitI";
val prems = Goalw [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";
val prems = Goalw [split_def]
"[| split c p z; !!x y. [| p = (x,y); c x y z |] ==> Q |] ==> Q";
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
qed "splitE'";
val major::prems = goal Prod.thy
"[| Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R \
\ |] ==> R";
by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
by (rtac (split_beta RS subst) 1 THEN rtac major 1);
qed "splitE2";
Goal "split R (a,b) ==> R a b";
by (etac (split RS iffD1) 1);
qed "splitD";
Goal "z: c a b ==> z: split c (a,b)";
by (Asm_simp_tac 1);
qed "mem_splitI";
Goal "!!p. [| !!a b. p=(a,b) ==> z: c a b |] ==> z: split c p";
by (split_all_tac 1);
by (Asm_simp_tac 1);
qed "mem_splitI2";
val prems = Goalw [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";
AddSIs [splitI, splitI2, splitI2', mem_splitI, mem_splitI2];
AddSEs [splitE, splitE', mem_splitE];
Goal "(%u. ? x y. u = (x, y) & P (x, y)) = P";
by (rtac ext 1);
by (Fast_tac 1);
qed "split_eta_SetCompr";
Addsimps [split_eta_SetCompr];
Goal "(%u. ? x y. u = (x, y) & P x y) = split P";
br ext 1;
by (Fast_tac 1);
qed "split_eta_SetCompr2";
Addsimps [split_eta_SetCompr2];
(* allows simplifications of nested splits in case of independent predicates *)
Goal "(%(a,b). P & Q a b) = (%ab. P & split Q ab)";
by (rtac ext 1);
by (Blast_tac 1);
qed "split_part";
Addsimps [split_part];
Goal "(@(x',y'). x = x' & y = y') = (x,y)";
by (Blast_tac 1);
qed "Eps_split_eq";
Addsimps [Eps_split_eq];
(*
the following would be slightly more general,
but cannot be used as rewrite rule:
### Cannot add premise as rewrite rule because it contains (type) unknowns:
### ?y = .x
Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)";
by (rtac select_equality 1);
by ( Simp_tac 1);
by (split_all_tac 1);
by (Asm_full_simp_tac 1);
qed "Eps_split_eq";
*)
(*** prod_fun -- action of the product functor upon functions ***)
Goalw [prod_fun_def] "prod_fun f g (a,b) = (f(a),g(b))";
by (rtac split 1);
qed "prod_fun";
Addsimps [prod_fun];
Goal "prod_fun (f1 o f2) (g1 o g2) = ((prod_fun f1 g1) o (prod_fun f2 g2))";
by (rtac ext 1);
by (pair_tac "x" 1);
by (Asm_simp_tac 1);
qed "prod_fun_compose";
Goal "prod_fun (%x. x) (%y. y) = (%z. z)";
by (rtac ext 1);
by (pair_tac "z" 1);
by (Asm_simp_tac 1);
qed "prod_fun_ident";
Addsimps [prod_fun_ident];
Goal "(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 (assume_tac 1);
qed "prod_fun_imageI";
val major::prems = Goal
"[| 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 (Blast_tac 2);
by (blast_tac (claset() addIs [prod_fun]) 1);
qed "prod_fun_imageE";
AddIs [prod_fun_imageI];
AddSEs [prod_fun_imageE];
(*** Disjoint union of a family of sets - Sigma ***)
Goalw [Sigma_def] "[| a:A; b:B(a) |] ==> (a,b) : Sigma A B";
by (REPEAT (ares_tac [singletonI,UN_I] 1));
qed "SigmaI";
AddSIs [SigmaI];
(*The general elimination rule*)
val major::prems = Goalw [Sigma_def]
"[| c: Sigma A B; \
\ !!x y.[| x:A; y:B(x); c=(x,y) |] ==> P \
\ |] ==> P";
by (cut_facts_tac [major] 1);
by (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ;
qed "SigmaE";
(** Elimination of (a,b):A*B -- introduces no eigenvariables **)
Goal "(a,b) : Sigma A B ==> a : A";
by (etac SigmaE 1);
by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ;
qed "SigmaD1";
Goal "(a,b) : Sigma A B ==> b : B(a)";
by (etac SigmaE 1);
by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ;
qed "SigmaD2";
val [major,minor]= goal Prod.thy
"[| (a,b) : Sigma A B; \
\ [| a:A; b:B(a) |] ==> P \
\ |] ==> P";
by (rtac minor 1);
by (rtac (major RS SigmaD1) 1);
by (rtac (major RS SigmaD2) 1) ;
qed "SigmaE2";
AddSEs [SigmaE2, SigmaE];
val prems = Goal
"[| A<=C; !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D";
by (cut_facts_tac prems 1);
by (blast_tac (claset() addIs (prems RL [subsetD])) 1);
qed "Sigma_mono";
Goal "Sigma {} B = {}";
by (Blast_tac 1) ;
qed "Sigma_empty1";
Goal "A <*> {} = {}";
by (Blast_tac 1) ;
qed "Sigma_empty2";
Addsimps [Sigma_empty1,Sigma_empty2];
Goal "UNIV <*> UNIV = UNIV";
by Auto_tac;
qed "UNIV_Times_UNIV";
Addsimps [UNIV_Times_UNIV];
Goal "- (UNIV <*> A) = UNIV <*> (-A)";
by Auto_tac;
qed "Compl_Times_UNIV1";
Goal "- (A <*> UNIV) = (-A) <*> UNIV";
by Auto_tac;
qed "Compl_Times_UNIV2";
Addsimps [Compl_Times_UNIV1, Compl_Times_UNIV2];
Goal "((a,b): Sigma A B) = (a:A & b:B(a))";
by (Blast_tac 1);
qed "mem_Sigma_iff";
AddIffs [mem_Sigma_iff];
Goal "x:C ==> (A <*> C <= B <*> C) = (A <= B)";
by (Blast_tac 1);
qed "Times_subset_cancel2";
Goal "x:C ==> (A <*> C = B <*> C) = (A = B)";
by (blast_tac (claset() addEs [equalityE]) 1);
qed "Times_eq_cancel2";
Goal "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))";
by (Fast_tac 1);
qed "SetCompr_Sigma_eq";
(*** Complex rules for Sigma ***)
Goal "{(a,b). P a & Q b} = Collect P <*> Collect Q";
by (Blast_tac 1);
qed "Collect_split";
Addsimps [Collect_split];
(*Suggested by Pierre Chartier*)
Goal "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)";
by (Blast_tac 1);
qed "UN_Times_distrib";
Goal "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))";
by (Fast_tac 1);
qed "split_paired_Ball_Sigma";
Addsimps [split_paired_Ball_Sigma];
Goal "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))";
by (Fast_tac 1);
qed "split_paired_Bex_Sigma";
Addsimps [split_paired_Bex_Sigma];
Goal "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))";
by (Blast_tac 1);
qed "Sigma_Un_distrib1";
Goal "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))";
by (Blast_tac 1);
qed "Sigma_Un_distrib2";
Goal "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))";
by (Blast_tac 1);
qed "Sigma_Int_distrib1";
Goal "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))";
by (Blast_tac 1);
qed "Sigma_Int_distrib2";
Goal "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))";
by (Blast_tac 1);
qed "Sigma_Diff_distrib1";
Goal "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))";
by (Blast_tac 1);
qed "Sigma_Diff_distrib2";
Goal "Sigma (Union X) B = (UN A:X. Sigma A B)";
by (Blast_tac 1);
qed "Sigma_Union";
(*Non-dependent versions are needed to avoid the need for higher-order
matching, especially when the rules are re-oriented*)
Goal "(A Un B) <*> C = (A <*> C) Un (B <*> C)";
by (Blast_tac 1);
qed "Times_Un_distrib1";
Goal "(A Int B) <*> C = (A <*> C) Int (B <*> C)";
by (Blast_tac 1);
qed "Times_Int_distrib1";
Goal "(A - B) <*> C = (A <*> C) - (B <*> C)";
by (Blast_tac 1);
qed "Times_Diff_distrib1";
(** Exhaustion rule for unit -- a degenerate form of induction **)
Goalw [Unity_def]
"u = ()";
by (stac (rewrite_rule [unit_def] Rep_unit RS singletonD RS sym) 1);
by (rtac (Rep_unit_inverse RS sym) 1);
qed "unit_eq";
(*simplification procedure for unit_eq.
Cannot use this rule directly -- it loops!*)
local
val unit_pat = Thm.cterm_of (Theory.sign_of thy) (Free ("x", HOLogic.unitT));
val unit_meta_eq = standard (mk_meta_eq unit_eq);
fun proc _ _ t =
if HOLogic.is_unit t then None
else Some unit_meta_eq;
in
val unit_eq_proc = Simplifier.mk_simproc "unit_eq" [unit_pat] proc;
end;
Addsimprocs [unit_eq_proc];
Goal "P () ==> P x";
by (Simp_tac 1);
qed "unit_induct";
(*This rewrite counters the effect of unit_eq_proc on (%u::unit. f u),
replacing it by f rather than by %u.f(). *)
Goal "(%u::unit. f()) = f";
by (rtac ext 1);
by (Simp_tac 1);
qed "unit_abs_eta_conv";
Addsimps [unit_abs_eta_conv];
(*Attempts to remove occurrences of split, and pair-valued parameters*)
val remove_split = rewrite_rule [split RS eq_reflection] o
rule_by_tactic (TRYALL split_all_tac);
local
(*In ap_split S T u, term u expects separate arguments for the factors of S,
with result type T. The call creates a new term expecting one argument
of type S.*)
fun ap_split (Type ("*", [T1, T2])) T3 u =
HOLogic.split_const (T1, T2, T3) $
Abs("v", T1,
ap_split T2 T3
((ap_split T1 (HOLogic.prodT_factors T2 ---> T3) (incr_boundvars 1 u)) $
Bound 0))
| ap_split T T3 u = u;
(*Curries any Var of function type in the rule*)
fun split_rule_var' (t as Var (v, Type ("fun", [T1, T2])), rl) =
let val T' = HOLogic.prodT_factors T1 ---> T2
val newt = ap_split T1 T2 (Var (v, T'))
val cterm = Thm.cterm_of (#sign (rep_thm rl))
in
instantiate ([], [(cterm t, cterm newt)]) rl
end
| split_rule_var' (t, rl) = rl;
in
val split_rule_var = standard o remove_split o split_rule_var';
(*Curries ALL function variables occurring in a rule's conclusion*)
fun split_rule rl = remove_split (foldr split_rule_var' (term_vars (concl_of rl), rl))
|> standard;
end;