(* 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_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];
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";
val surj_pair = prove_goal Prod.thy "? x y. z = (x, y)" (K [
rtac exI 1, rtac exI 1, rtac surjective_pairing 1]);
Addsimps [surj_pair];
(* lemmas for splitting paired `!!' *)
local
val lemma1 = prove_goal Prod.thy "(!!x. PROP P x) ==> (!!a b. PROP P(a,b))"
(fn prems => [resolve_tac prems 1]);
val psig = sign_of Prod.thy;
val pT = Sign.read_typ (psig, K None) "?'a*?'b=>prop";
val PeqP = reflexive(read_cterm psig ("P", pT));
val psplit = zero_var_indexes(read_instantiate [("p","x")]
surjective_pairing RS eq_reflection);
val adhoc = combination PeqP psplit;
val lemma = prove_goal Prod.thy "(!!a b. PROP P(a,b)) ==> PROP P x"
(fn prems => [rewtac adhoc, resolve_tac prems 1]);
val lemma2 = prove_goal Prod.thy "(!!a b. PROP P(a,b)) ==> (!!x. PROP P x)"
(fn prems => [rtac lemma 1, resolve_tac prems 1]);
in
val split_paired_all = equal_intr lemma1 lemma2
end;
bind_thm("split_paired_all", split_paired_all);
(*
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 *)
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];
Goal "split Pair p = p";
by (pair_tac "p" 1);
by (Simp_tac 1);
qed "split_Pair";
(*unused: val surjective_pairing2 = split_Pair RS sym;*)
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*)
qed_goal "split_weak_cong" Prod.thy
"p=q ==> split c p = split c q"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
qed_goal "split_eta" Prod.thy "(%(x,y). f(x,y)) = f"
(K [rtac ext 1, split_all_tac 1, rtac split 1]);
qed_goal "cond_split_eta" Prod.thy
"!!f. (!!x y. f x y = g(x,y)) ==> (%(x,y). f x y) = g"
(K [asm_simp_tac (simpset() addsimps [split_eta]) 1]);
(*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*)
local
val split_eta_pat = Thm.read_cterm (sign_of thy)
("split (%x. split (%y. f x y))", HOLogic.termTVar);
val split_eta_meta_eq = standard (meta_eq cond_split_eta);
fun Pair_pat 0 (Bound 0) = true
| Pair_pat k (Const ("Pair", _) $ Bound m $ t) =
m = k andalso Pair_pat (k-1) t
| Pair_pat _ _ = false;
fun split_pat k (Abs (_, _, f $
(Const ("Pair",_) $ Bound m $
(Const ("Pair",_) $ Bound n $ t)))) =
if m = k andalso n = k-1 andalso Pair_pat (k-2) t
then Some f else None
| split_pat k (Const ("split", _) $ Abs (_, _, t)) = split_pat (k+1) t
| split_pat k _ = None;
fun proc _ _ (s as
(Const ("split", _) $ Abs (_, _,
(Const ("split", _) $ Abs (_, _, t))))) = (case split_pat 2 t of
Some f => (let val fvar = Free ("f", fastype_of f);
fun atom_fun t = if t = f then fvar else atom t
and atom (Abs (x, T, t)) = Abs (x, T,atom_fun t)
| atom (t $ u) = atom_fun t $ atom_fun u
| atom x = x;
val ct = cterm_of (sign_of thy) (HOLogic.mk_Trueprop
(HOLogic.mk_eq (atom_fun s,fvar)));
val ss = HOL_basic_ss addsimps [cond_split_eta];
in Some (meta_eq (prove_goalw_cterm [] ct (K [simp_tac ss 1]))) end)
| None => None)
| proc _ _ _ = None;
in
val split_eta_proc = Simplifier.mk_simproc "split_eta" [split_eta_pat] proc;
end;
Addsimprocs [split_eta_proc];
qed_goal "split_beta" Prod.thy "(%(x,y). P x y) z = P (fst z) (snd z)"
(K [stac surjective_pairing 1, stac split 1, rtac refl 1]);
(*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 "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 splitE2 = prove_goal Prod.thy
"[|Q (split P z); !!x y. [|z = (x, y); Q (P x y)|] ==> R|] ==> R" (fn prems => [
REPEAT (resolve_tac (prems@[surjective_pairing]) 1),
rtac (split_beta RS subst) 1,
rtac (hd prems) 1]);
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, mem_splitI, mem_splitI2];
AddSEs [splitE, mem_splitE];
(* 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";
(*** 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)) ]);
AddSIs [SigmaI];
(*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) ]);
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";
qed_goal "Sigma_empty1" Prod.thy "Sigma {} B = {}"
(fn _ => [ (Blast_tac 1) ]);
qed_goal "Sigma_empty2" Prod.thy "A Times {} = {}"
(fn _ => [ (Blast_tac 1) ]);
Addsimps [Sigma_empty1,Sigma_empty2];
Goal "((a,b): Sigma A B) = (a:A & b:B(a))";
by (Blast_tac 1);
qed "mem_Sigma_iff";
AddIffs [mem_Sigma_iff];
val Collect_split = prove_goal Prod.thy
"{(a,b). P a & Q b} = Collect P Times Collect Q" (K [Blast_tac 1]);
Addsimps [Collect_split];
(*Suggested by Pierre Chartier*)
Goal "(UN (a,b):(A Times B). E a Times F b) = (UNION A E) Times (UNION B F)";
by (Blast_tac 1);
qed "UNION_Times_distrib";
(*** Domain of a relation ***)
Goalw [image_def] "(a,b) : r ==> a : fst``r";
by (rtac CollectI 1);
by (rtac bexI 1);
by (rtac (fst_conv RS sym) 1);
by (assume_tac 1);
qed "fst_imageI";
val major::prems = Goal
"[| 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 ***)
Goalw [image_def] "(a,b) : r ==> b : snd``r";
by (rtac CollectI 1);
by (rtac bexI 1);
by (rtac (snd_conv RS sym) 1);
by (assume_tac 1);
qed "snd_imageI";
val major::prems = Goal "[| 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 [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";
AddIs [fst_imageI, snd_imageI, prod_fun_imageI];
AddSEs [fst_imageE, snd_imageE, prod_fun_imageE];
(*simplification procedure for unit_eq.
Cannot use this rule directly -- it loops!*)
local
val unit_pat = Thm.cterm_of (sign_of thy) (Free ("x", HOLogic.unitT));
val unit_meta_eq = standard (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];
(*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;