(* Title: HOL/NatDef.ML
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
*)
goal thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
qed "Nat_fun_mono";
val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
(* Zero is a natural number -- this also justifies the type definition*)
goal thy "Zero_Rep: Nat";
by (stac Nat_unfold 1);
by (rtac (singletonI RS UnI1) 1);
qed "Zero_RepI";
val prems = goal thy "i: Nat ==> Suc_Rep(i) : Nat";
by (stac Nat_unfold 1);
by (rtac (imageI RS UnI2) 1);
by (resolve_tac prems 1);
qed "Suc_RepI";
(*** Induction ***)
val major::prems = goal thy
"[| i: Nat; P(Zero_Rep); \
\ !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |] ==> P(i)";
by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
by (fast_tac (!claset addIs prems) 1);
qed "Nat_induct";
val prems = goalw thy [Zero_def,Suc_def]
"[| P(0); \
\ !!k. P(k) ==> P(Suc(k)) |] ==> P(n)";
by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*)
by (rtac (Rep_Nat RS Nat_induct) 1);
by (REPEAT (ares_tac prems 1
ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
qed "nat_induct";
(*Perform induction on n. *)
fun nat_ind_tac a i =
EVERY [res_inst_tac [("n",a)] nat_induct i,
rename_last_tac a ["1"] (i+1)];
(*A special form of induction for reasoning about m<n and m-n*)
val prems = goal thy
"[| !!x. P x 0; \
\ !!y. P 0 (Suc y); \
\ !!x y. [| P x y |] ==> P (Suc x) (Suc y) \
\ |] ==> P m n";
by (res_inst_tac [("x","m")] spec 1);
by (nat_ind_tac "n" 1);
by (rtac allI 2);
by (nat_ind_tac "x" 2);
by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
qed "diff_induct";
(*Case analysis on the natural numbers*)
val prems = goal thy
"[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P";
by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
by (fast_tac (!claset addSEs prems) 1);
by (nat_ind_tac "n" 1);
by (rtac (refl RS disjI1) 1);
by (Fast_tac 1);
qed "natE";
(*** Isomorphisms: Abs_Nat and Rep_Nat ***)
(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
since we assume the isomorphism equations will one day be given by Isabelle*)
goal thy "inj(Rep_Nat)";
by (rtac inj_inverseI 1);
by (rtac Rep_Nat_inverse 1);
qed "inj_Rep_Nat";
goal thy "inj_onto Abs_Nat Nat";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Nat_inverse 1);
qed "inj_onto_Abs_Nat";
(*** Distinctness of constructors ***)
goalw thy [Zero_def,Suc_def] "Suc(m) ~= 0";
by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
by (rtac Suc_Rep_not_Zero_Rep 1);
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
qed "Suc_not_Zero";
bind_thm ("Zero_not_Suc", Suc_not_Zero RS not_sym);
AddIffs [Suc_not_Zero,Zero_not_Suc];
bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
val Zero_neq_Suc = sym RS Suc_neq_Zero;
(** Injectiveness of Suc **)
goalw thy [Suc_def] "inj(Suc)";
by (rtac injI 1);
by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
by (dtac (inj_Suc_Rep RS injD) 1);
by (etac (inj_Rep_Nat RS injD) 1);
qed "inj_Suc";
val Suc_inject = inj_Suc RS injD;
goal thy "(Suc(m)=Suc(n)) = (m=n)";
by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]);
qed "Suc_Suc_eq";
AddIffs [Suc_Suc_eq];
goal thy "n ~= Suc(n)";
by (nat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
qed "n_not_Suc_n";
bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
(*** nat_case -- the selection operator for nat ***)
goalw thy [nat_case_def] "nat_case a f 0 = a";
by (fast_tac (!claset addIs [select_equality]) 1);
qed "nat_case_0";
goalw thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
by (fast_tac (!claset addIs [select_equality]) 1);
qed "nat_case_Suc";
(** Introduction rules for 'pred_nat' **)
goalw thy [pred_nat_def] "(n, Suc(n)) : pred_nat";
by (Fast_tac 1);
qed "pred_natI";
val major::prems = goalw thy [pred_nat_def]
"[| p : pred_nat; !!x n. [| p = (n, Suc(n)) |] ==> R \
\ |] ==> R";
by (rtac (major RS CollectE) 1);
by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
qed "pred_natE";
goalw thy [wf_def] "wf(pred_nat)";
by (strip_tac 1);
by (nat_ind_tac "x" 1);
by (fast_tac (!claset addSEs [mp, pred_natE]) 2);
by (fast_tac (!claset addSEs [mp, pred_natE]) 1);
qed "wf_pred_nat";
(*** nat_rec -- by wf recursion on pred_nat ***)
(* The unrolling rule for nat_rec *)
goal thy
"(%n. nat_rec c d n) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))";
by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
bind_thm("nat_rec_unfold", wf_pred_nat RS
((result() RS eq_reflection) RS def_wfrec));
(*---------------------------------------------------------------------------
* Old:
* bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec)));
*---------------------------------------------------------------------------*)
(** conversion rules **)
goal thy "nat_rec c h 0 = c";
by (rtac (nat_rec_unfold RS trans) 1);
by (simp_tac (!simpset addsimps [nat_case_0]) 1);
qed "nat_rec_0";
goal thy "nat_rec c h (Suc n) = h n (nat_rec c h n)";
by (rtac (nat_rec_unfold RS trans) 1);
by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
qed "nat_rec_Suc";
(*These 2 rules ease the use of primitive recursion. NOTE USE OF == *)
val [rew] = goal thy
"[| !!n. f(n) == nat_rec c h n |] ==> f(0) = c";
by (rewtac rew);
by (rtac nat_rec_0 1);
qed "def_nat_rec_0";
val [rew] = goal thy
"[| !!n. f(n) == nat_rec c h n |] ==> f(Suc(n)) = h n (f n)";
by (rewtac rew);
by (rtac nat_rec_Suc 1);
qed "def_nat_rec_Suc";
fun nat_recs def =
[standard (def RS def_nat_rec_0),
standard (def RS def_nat_rec_Suc)];
(*** Basic properties of "less than" ***)
(** Introduction properties **)
val prems = goalw thy [less_def] "[| i<j; j<k |] ==> i<(k::nat)";
by (rtac (trans_trancl RS transD) 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
qed "less_trans";
goalw thy [less_def] "n < Suc(n)";
by (rtac (pred_natI RS r_into_trancl) 1);
qed "lessI";
AddIffs [lessI];
(* i<j ==> i<Suc(j) *)
bind_thm("less_SucI", lessI RSN (2, less_trans));
Addsimps [less_SucI];
goal thy "0 < Suc(n)";
by (nat_ind_tac "n" 1);
by (rtac lessI 1);
by (etac less_trans 1);
by (rtac lessI 1);
qed "zero_less_Suc";
AddIffs [zero_less_Suc];
(** Elimination properties **)
val prems = goalw thy [less_def] "n<m ==> ~ m<(n::nat)";
by (fast_tac (!claset addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
qed "less_not_sym";
(* [| n<m; m<n |] ==> R *)
bind_thm ("less_asym", (less_not_sym RS notE));
goalw thy [less_def] "~ n<(n::nat)";
by (rtac notI 1);
by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
qed "less_not_refl";
(* n<n ==> R *)
bind_thm ("less_irrefl", (less_not_refl RS notE));
goal thy "!!m. n<m ==> m ~= (n::nat)";
by (fast_tac (!claset addEs [less_irrefl]) 1);
qed "less_not_refl2";
val major::prems = goalw thy [less_def]
"[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \
\ |] ==> P";
by (rtac (major RS tranclE) 1);
by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
eresolve_tac (prems@[pred_natE, Pair_inject])));
by (rtac refl 1);
qed "lessE";
goal thy "~ n<0";
by (rtac notI 1);
by (etac lessE 1);
by (etac Zero_neq_Suc 1);
by (etac Zero_neq_Suc 1);
qed "not_less0";
AddIffs [not_less0];
(* n<0 ==> R *)
bind_thm ("less_zeroE", not_less0 RS notE);
val [major,less,eq] = goal thy
"[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P";
by (rtac (major RS lessE) 1);
by (rtac eq 1);
by (Fast_tac 1);
by (rtac less 1);
by (Fast_tac 1);
qed "less_SucE";
goal thy "(m < Suc(n)) = (m < n | m = n)";
by (fast_tac (!claset addSIs [lessI]
addEs [less_trans, less_SucE]) 1);
qed "less_Suc_eq";
val prems = goal thy "m<n ==> n ~= 0";
by (res_inst_tac [("n","n")] natE 1);
by (cut_facts_tac prems 1);
by (ALLGOALS Asm_full_simp_tac);
qed "gr_implies_not0";
Addsimps [gr_implies_not0];
qed_goal "zero_less_eq" thy "0 < n = (n ~= 0)" (fn _ => [
rtac iffI 1,
etac gr_implies_not0 1,
rtac natE 1,
contr_tac 1,
etac ssubst 1,
rtac zero_less_Suc 1]);
(** Inductive (?) properties **)
val [prem] = goal thy "Suc(m) < n ==> m<n";
by (rtac (prem RS rev_mp) 1);
by (nat_ind_tac "n" 1);
by (ALLGOALS (fast_tac (!claset addSIs [lessI RS less_SucI]
addEs [less_trans, lessE])));
qed "Suc_lessD";
val [major,minor] = goal thy
"[| Suc(i)<k; !!j. [| i<j; k=Suc(j) |] ==> P \
\ |] ==> P";
by (rtac (major RS lessE) 1);
by (etac (lessI RS minor) 1);
by (etac (Suc_lessD RS minor) 1);
by (assume_tac 1);
qed "Suc_lessE";
goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
by (fast_tac (!claset addEs [lessE, Suc_lessD] addIs [lessI]) 1);
qed "Suc_less_SucD";
goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
by (etac rev_mp 1);
by (nat_ind_tac "n" 1);
by (ALLGOALS (fast_tac (!claset addSIs [lessI]
addEs [less_trans, lessE])));
qed "Suc_mono";
goal thy "(Suc(m) < Suc(n)) = (m<n)";
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
qed "Suc_less_eq";
Addsimps [Suc_less_eq];
goal thy "~(Suc(n) < n)";
by (fast_tac (!claset addEs [Suc_lessD RS less_irrefl]) 1);
qed "not_Suc_n_less_n";
Addsimps [not_Suc_n_less_n];
goal thy "!!i. i<j ==> j<k --> Suc i < k";
by (nat_ind_tac "k" 1);
by (ALLGOALS (asm_simp_tac (!simpset)));
by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
by (fast_tac (!claset addDs [Suc_lessD]) 1);
qed_spec_mp "less_trans_Suc";
(*"Less than" is a linear ordering*)
goal thy "m<n | m=n | n<(m::nat)";
by (nat_ind_tac "m" 1);
by (nat_ind_tac "n" 1);
by (rtac (refl RS disjI1 RS disjI2) 1);
by (rtac (zero_less_Suc RS disjI1) 1);
by (fast_tac (!claset addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
qed "less_linear";
qed_goal "nat_less_cases" thy
"[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
( fn prems =>
[
(res_inst_tac [("m1","n"),("n1","m")] (less_linear RS disjE) 1),
(etac disjE 2),
(etac (hd (tl (tl prems))) 1),
(etac (sym RS hd (tl prems)) 1),
(etac (hd prems) 1)
]);
(*Can be used with less_Suc_eq to get n=m | n<m *)
goal thy "(~ m < n) = (n < Suc(m))";
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (ALLGOALS Asm_simp_tac);
qed "not_less_eq";
(*Complete induction, aka course-of-values induction*)
val prems = goalw thy [less_def]
"[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)";
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
by (eresolve_tac prems 1);
qed "less_induct";
qed_goal "nat_induct2" thy
"[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
cut_facts_tac prems 1,
rtac less_induct 1,
res_inst_tac [("n","n")] natE 1,
hyp_subst_tac 1,
atac 1,
hyp_subst_tac 1,
res_inst_tac [("n","x")] natE 1,
hyp_subst_tac 1,
atac 1,
hyp_subst_tac 1,
resolve_tac prems 1,
dtac spec 1,
etac mp 1,
rtac (lessI RS less_trans) 1,
rtac (lessI RS Suc_mono) 1]);
(*** Properties of <= ***)
goalw thy [le_def] "(m <= n) = (m < Suc n)";
by (rtac not_less_eq 1);
qed "le_eq_less_Suc";
goalw thy [le_def] "0 <= n";
by (rtac not_less0 1);
qed "le0";
goalw thy [le_def] "~ Suc n <= n";
by (Simp_tac 1);
qed "Suc_n_not_le_n";
goalw thy [le_def] "(i <= 0) = (i = 0)";
by (nat_ind_tac "i" 1);
by (ALLGOALS Asm_simp_tac);
qed "le_0_eq";
Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
Suc_n_not_le_n,
n_not_Suc_n, Suc_n_not_n,
nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
(*
goal thy "(Suc m < n | Suc m = n) = (m < n)";
by (stac (less_Suc_eq RS sym) 1);
by (rtac Suc_less_eq 1);
qed "Suc_le_eq";
this could make the simpset (with less_Suc_eq added again) more confluent,
but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
*)
(*Prevents simplification of f and g: much faster*)
qed_goal "nat_case_weak_cong" thy
"m=n ==> nat_case a f m = nat_case a f n"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
qed_goal "nat_rec_weak_cong" thy
"m=n ==> nat_rec a f m = nat_rec a f n"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
qed_goal "expand_nat_case" thy
"P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
(fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
by (resolve_tac prems 1);
qed "leI";
val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
by (resolve_tac prems 1);
qed "leD";
val leE = make_elim leD;
goal thy "(~n<m) = (m<=(n::nat))";
by (fast_tac (!claset addIs [leI] addEs [leE]) 1);
qed "not_less_iff_le";
goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
by (Fast_tac 1);
qed "not_leE";
goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);
by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);
qed "lessD";
goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
qed "Suc_leD";
(* stronger version of Suc_leD *)
goalw thy [le_def]
"!!m. Suc m <= n ==> m < n";
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
by (cut_facts_tac [less_linear] 1);
by (Fast_tac 1);
qed "Suc_le_lessD";
goal thy "(Suc m <= n) = (m < n)";
by (fast_tac (!claset addIs [lessD, Suc_le_lessD]) 1);
qed "Suc_le_eq";
goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
by (fast_tac (!claset addDs [Suc_lessD]) 1);
qed "le_SucI";
Addsimps[le_SucI];
bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
by (fast_tac (!claset addEs [less_asym]) 1);
qed "less_imp_le";
goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);
qed "le_imp_less_or_eq";
goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (!claset addEs [less_irrefl,less_asym]) 1);
by (flexflex_tac);
qed "less_or_eq_imp_le";
goal thy "(x <= (y::nat)) = (x < y | x=y)";
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
qed "le_eq_less_or_eq";
goal thy "n <= (n::nat)";
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
qed "le_refl";
val prems = goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
by (dtac le_imp_less_or_eq 1);
by (fast_tac (!claset addIs [less_trans]) 1);
qed "le_less_trans";
goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
by (dtac le_imp_less_or_eq 1);
by (fast_tac (!claset addIs [less_trans]) 1);
qed "less_le_trans";
goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
rtac less_or_eq_imp_le, fast_tac (!claset addIs [less_trans])]);
qed "le_trans";
val prems = goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
fast_tac (!claset addEs [less_irrefl,less_asym])]);
qed "le_anti_sym";
goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
qed "Suc_le_mono";
AddIffs [Suc_le_mono];
(* Axiom 'order_le_less' of class 'order': *)
goal thy "(m::nat) < n = (m <= n & m ~= n)";
br iffI 1;
br conjI 1;
be less_imp_le 1;
be (less_not_refl2 RS not_sym) 1;
by(fast_tac (!claset addSDs [le_imp_less_or_eq]) 1);
qed "nat_less_le";
(** LEAST -- the least number operator **)
val [prem1,prem2] = goalw thy [Least_def]
"[| P(k::nat); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
by (rtac select_equality 1);
by (fast_tac (!claset addSIs [prem1,prem2]) 1);
by (cut_facts_tac [less_linear] 1);
by (fast_tac (!claset addSIs [prem1] addSDs [prem2]) 1);
qed "Least_equality";
val [prem] = goal thy "P(k::nat) ==> P(LEAST x.P(x))";
by (rtac (prem RS rev_mp) 1);
by (res_inst_tac [("n","k")] less_induct 1);
by (rtac impI 1);
by (rtac classical 1);
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
by (assume_tac 1);
by (assume_tac 2);
by (Fast_tac 1);
qed "LeastI";
(*Proof is almost identical to the one above!*)
val [prem] = goal thy "P(k::nat) ==> (LEAST x.P(x)) <= k";
by (rtac (prem RS rev_mp) 1);
by (res_inst_tac [("n","k")] less_induct 1);
by (rtac impI 1);
by (rtac classical 1);
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
by (assume_tac 1);
by (rtac le_refl 2);
by (fast_tac (!claset addIs [less_imp_le,le_trans]) 1);
qed "Least_le";
val [prem] = goal thy "k < (LEAST x.P(x)) ==> ~P(k::nat)";
by (rtac notI 1);
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
by (rtac prem 1);
qed "not_less_Least";
qed_goalw "Least_Suc" thy [Least_def]
"!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
(fn _ => [
rtac select_equality 1,
fold_goals_tac [Least_def],
safe_tac (!claset addSEs [LeastI]),
rename_tac "j" 1,
res_inst_tac [("n","j")] natE 1,
Fast_tac 1,
fast_tac (!claset addDs [Suc_less_SucD, not_less_Least]) 1,
rename_tac "k n" 1,
res_inst_tac [("n","k")] natE 1,
Fast_tac 1,
hyp_subst_tac 1,
rewtac Least_def,
rtac (select_equality RS arg_cong RS sym) 1,
safe_tac (!claset),
dtac Suc_mono 1,
Fast_tac 1,
cut_facts_tac [less_linear] 1,
safe_tac (!claset),
atac 2,
Fast_tac 2,
dtac Suc_mono 1,
Fast_tac 1]);
(*** Instantiation of transitivity prover ***)
structure Less_Arith =
struct
val nat_leI = leI;
val nat_leD = leD;
val lessI = lessI;
val zero_less_Suc = zero_less_Suc;
val less_reflE = less_irrefl;
val less_zeroE = less_zeroE;
val less_incr = Suc_mono;
val less_decr = Suc_less_SucD;
val less_incr_rhs = Suc_mono RS Suc_lessD;
val less_decr_lhs = Suc_lessD;
val less_trans_Suc = less_trans_Suc;
val leI = lessD RS (Suc_le_mono RS iffD1);
val not_lessI = leI RS leD
val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
(fn _ => [etac swap2 1, etac leD 1]);
val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
(fn _ => [etac less_SucE 1,
fast_tac (HOL_cs addSDs [Suc_less_SucD] addSEs [less_irrefl]
addDs [less_trans_Suc]) 1,
atac 1]);
val leD = le_eq_less_Suc RS iffD1;
val not_lessD = nat_leI RS leD;
val not_leD = not_leE
val eqD1 = prove_goal thy "!!n. m = n ==> m < Suc n"
(fn _ => [etac subst 1, rtac lessI 1]);
val eqD2 = sym RS eqD1;
fun is_zero(t) = t = Const("0",Type("nat",[]));
fun nnb T = T = Type("fun",[Type("nat",[]),
Type("fun",[Type("nat",[]),
Type("bool",[])])])
fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
| decomp_Suc t = (t,0);
fun decomp2(rel,T,lhs,rhs) =
if not(nnb T) then None else
let val (x,i) = decomp_Suc lhs
val (y,j) = decomp_Suc rhs
in case rel of
"op <" => Some(x,i,"<",y,j)
| "op <=" => Some(x,i,"<=",y,j)
| "op =" => Some(x,i,"=",y,j)
| _ => None
end;
fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
| negate None = None;
fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
| decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
negate(decomp2(rel,T,lhs,rhs))
| decomp _ = None
end;
structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
open Trans_Tac;
(*** eliminates ~= in premises, which trans_tac cannot deal with ***)
qed_goal "nat_neqE" thy
"[| (m::nat) ~= n; m < n ==> P; n < m ==> P |] ==> P"
(fn major::prems => [cut_facts_tac [less_linear] 1,
REPEAT(eresolve_tac ([disjE,major RS notE]@prems) 1)]);
(* add function nat_add_primrec *)
val (_, nat_add_primrec) = Datatype.add_datatype
([], "nat", [("0", [], Mixfix ("0", [], max_pri)), ("Suc", [dtTyp ([], "nat")], NoSyn)]) HOL.thy;