(* Title: HOL/nat
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
For nat.thy. Type nat is defined as a set (Nat) over the type ind.
*)
open Nat;
goal Nat.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 Nat.thy "Zero_Rep: Nat";
by (rtac (Nat_unfold RS ssubst) 1);
by (rtac (singletonI RS UnI1) 1);
qed "Zero_RepI";
val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
by (rtac (Nat_unfold RS ssubst) 1);
by (rtac (imageI RS UnI2) 1);
by (resolve_tac prems 1);
qed "Suc_RepI";
(*** Induction ***)
val major::prems = goal Nat.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 (set_cs addIs prems) 1);
qed "Nat_induct";
val prems = goalw Nat.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 Nat.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 Nat.thy
"[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P";
by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
by (fast_tac (HOL_cs addSEs prems) 1);
by (nat_ind_tac "n" 1);
by (rtac (refl RS disjI1) 1);
by (fast_tac HOL_cs 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 Nat.thy "inj(Rep_Nat)";
by (rtac inj_inverseI 1);
by (rtac Rep_Nat_inverse 1);
qed "inj_Rep_Nat";
goal Nat.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 Nat.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));
Addsimps [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 Nat.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 Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]);
qed "Suc_Suc_eq";
goal Nat.thy "n ~= Suc(n)";
by (nat_ind_tac "n" 1);
by (ALLGOALS(asm_simp_tac (!simpset addsimps [Suc_Suc_eq])));
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 Nat.thy [nat_case_def] "nat_case a f 0 = a";
by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
qed "nat_case_0";
goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
by (fast_tac (set_cs addIs [select_equality]
addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
qed "nat_case_Suc";
(** Introduction rules for 'pred_nat' **)
goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat";
by (fast_tac set_cs 1);
qed "pred_natI";
val major::prems = goalw Nat.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 Nat.thy [wf_def] "wf(pred_nat)";
by (strip_tac 1);
by (nat_ind_tac "x" 1);
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject,
make_elim Suc_inject]) 2);
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
qed "wf_pred_nat";
(*** nat_rec -- by wf recursion on pred_nat ***)
(* The unrolling rule for nat_rec *)
goal Nat.thy
"(%n. nat_rec n c d) = 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 Nat.thy "nat_rec 0 c h = c";
by (rtac (nat_rec_unfold RS trans) 1);
by (simp_tac (!simpset addsimps [nat_case_0]) 1);
qed "nat_rec_0";
goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)";
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 Nat.thy
"[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c";
by (rewtac rew);
by (rtac nat_rec_0 1);
qed "def_nat_rec_0";
val [rew] = goal Nat.thy
"[| !!n. f(n) == nat_rec n c h |] ==> 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 Nat.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 Nat.thy [less_def] "n < Suc(n)";
by (rtac (pred_natI RS r_into_trancl) 1);
qed "lessI";
Addsimps [lessI];
(* i<j ==> i<Suc(j) *)
val less_SucI = lessI RSN (2, less_trans);
goal Nat.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";
Addsimps [zero_less_Suc];
(** Elimination properties **)
val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";
by (fast_tac (HOL_cs 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 Nat.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 Nat.thy "!!m. n<m ==> m ~= (n::nat)";
by (fast_tac (HOL_cs addEs [less_irrefl]) 1);
qed "less_not_refl2";
val major::prems = goalw Nat.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 Nat.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";
Addsimps [not_less0];
(* n<0 ==> R *)
bind_thm ("less_zeroE", (not_less0 RS notE));
val [major,less,eq] = goal Nat.thy
"[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P";
by (rtac (major RS lessE) 1);
by (rtac eq 1);
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
by (rtac less 1);
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
qed "less_SucE";
goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
by (fast_tac (HOL_cs addSIs [lessI]
addEs [less_trans, less_SucE]) 1);
qed "less_Suc_eq";
val prems = goal Nat.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];
(** Inductive (?) properties **)
val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
by (rtac (prem RS rev_mp) 1);
by (nat_ind_tac "n" 1);
by (rtac impI 1);
by (etac less_zeroE 1);
by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
addSDs [Suc_inject]
addEs [less_trans, lessE]) 1);
qed "Suc_lessD";
val [major,minor] = goal Nat.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";
val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
by (rtac (major RS lessE) 1);
by (REPEAT (rtac lessI 1
ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
qed "Suc_less_SucD";
val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
by (fast_tac (HOL_cs addIs prems) 1);
by (nat_ind_tac "n" 1);
by (rtac impI 1);
by (etac less_zeroE 1);
by (fast_tac (HOL_cs addSIs [lessI]
addSDs [Suc_inject]
addEs [less_trans, lessE]) 1);
qed "Suc_mono";
goal Nat.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 Nat.thy "~(Suc(n) < n)";
by (fast_tac (HOL_cs addEs [Suc_lessD RS less_irrefl]) 1);
qed "not_Suc_n_less_n";
Addsimps [not_Suc_n_less_n];
goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";
by (nat_ind_tac "k" 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
qed_spec_mp "less_trans_Suc";
(*"Less than" is a linear ordering*)
goal Nat.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 (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
qed "less_linear";
(*Can be used with less_Suc_eq to get n=m | n<m *)
goal Nat.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 Nat.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";
(*** Properties of <= ***)
goalw Nat.thy [le_def] "0 <= n";
by (rtac not_less0 1);
qed "le0";
goalw Nat.thy [le_def] "~ Suc n <= n";
by (Simp_tac 1);
qed "Suc_n_not_le_n";
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";
by (nat_ind_tac "i" 1);
by (ALLGOALS Asm_simp_tac);
qed "le_0";
Addsimps [less_not_refl,
less_Suc_eq, le0, le_0,
Suc_Suc_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];
(*Prevents simplification of f and g: much faster*)
qed_goal "nat_case_weak_cong" Nat.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" Nat.thy
"m=n ==> nat_rec m a f = nat_rec n a f"
(fn [prem] => [rtac (prem RS arg_cong) 1]);
val prems = goalw Nat.thy [le_def] "~n<m ==> m<=(n::nat)";
by (resolve_tac prems 1);
qed "leI";
val prems = goalw Nat.thy [le_def] "m<=n ==> ~ n < (m::nat)";
by (resolve_tac prems 1);
qed "leD";
val leE = make_elim leD;
goal Nat.thy "(~n<m) = (m<=(n::nat))";
by (fast_tac (HOL_cs addIs [leI] addEs [leE]) 1);
qed "not_less_iff_le";
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
by (fast_tac HOL_cs 1);
qed "not_leE";
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
by (Simp_tac 1);
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
qed "lessD";
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
by (Asm_full_simp_tac 1);
by (fast_tac HOL_cs 1);
qed "Suc_leD";
goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
qed "le_SucI";
Addsimps[le_SucI];
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
by (fast_tac (HOL_cs addEs [less_asym]) 1);
qed "less_imp_le";
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
qed "le_imp_less_or_eq";
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
by (flexflex_tac);
qed "less_or_eq_imp_le";
goal Nat.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 Nat.thy "n <= (n::nat)";
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
qed "le_refl";
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
by (dtac le_imp_less_or_eq 1);
by (fast_tac (HOL_cs addIs [less_trans]) 1);
qed "le_less_trans";
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
by (dtac le_imp_less_or_eq 1);
by (fast_tac (HOL_cs addIs [less_trans]) 1);
qed "less_le_trans";
goal Nat.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 (HOL_cs addIs [less_trans])]);
qed "le_trans";
val prems = goal Nat.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 (HOL_cs addEs [less_irrefl,less_asym])]);
qed "le_anti_sym";
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
qed "Suc_le_mono";
Addsimps [le_refl,Suc_le_mono];
(** LEAST -- the least number operator **)
val [prem1,prem2] = goalw Nat.thy [Least_def]
"[| P(k); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
by (rtac select_equality 1);
by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1);
by (cut_facts_tac [less_linear] 1);
by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1);
qed "Least_equality";
val [prem] = goal Nat.thy "P(k) ==> 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 HOL_cs 1);
qed "LeastI";
(*Proof is almost identical to the one above!*)
val [prem] = goal Nat.thy "P(k) ==> (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 (HOL_cs addIs [less_imp_le,le_trans]) 1);
qed "Least_le";
val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)";
by (rtac notI 1);
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
by (rtac prem 1);
qed "not_less_Least";