(* Title: HOL/NatDef.ML
ID: $Id$
Author: Tobias Nipkow, Cambridge University Computer Laboratory
Copyright 1991 University of Cambridge
*)
Addsimps [One_nat_def];
val rew = rewrite_rule [symmetric Nat_def];
(*** Induction ***)
val prems = Goalw [Zero_nat_def,Suc_def]
"[| P(0); \
\ !!n. P(n) ==> P(Suc(n)) |] ==> P(n)";
by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*)
by (rtac (Rep_Nat RS rew 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 =
res_inst_tac [("n",a)] nat_induct i THEN rename_last_tac a [""] (i+1);
(*A special form of induction for reasoning about m<n and m-n*)
val prems = Goal
"[| !!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";
(*** 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 "inj(Rep_Nat)";
by (rtac inj_inverseI 1);
by (rtac Rep_Nat_inverse 1);
qed "inj_Rep_Nat";
Goal "inj_on Abs_Nat Nat";
by (rtac inj_on_inverseI 1);
by (etac Abs_Nat_inverse 1);
qed "inj_on_Abs_Nat";
(*** Distinctness of constructors ***)
Goalw [Zero_nat_def,Suc_def] "Suc(m) ~= 0";
by (rtac (inj_on_Abs_Nat RS inj_on_contraD) 1);
by (rtac Suc_Rep_not_Zero_Rep 1);
by (REPEAT (resolve_tac [Rep_Nat, rew Nat'.Suc_RepI, rew Nat'.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));
bind_thm ("Zero_neq_Suc", sym RS Suc_neq_Zero);
(** Injectiveness of Suc **)
Goalw [Suc_def] "inj(Suc)";
by (rtac injI 1);
by (dtac (inj_on_Abs_Nat RS inj_onD) 1);
by (REPEAT (resolve_tac [Rep_Nat, rew Nat'.Suc_RepI] 1));
by (dtac (inj_Suc_Rep RS injD) 1);
by (etac (inj_Rep_Nat RS injD) 1);
qed "inj_Suc";
bind_thm ("Suc_inject", inj_Suc RS injD);
Goal "(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 "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);
Goal "(ALL x. x = (0::nat)) = False";
by Auto_tac;
qed "nat_not_singleton";
(*** Basic properties of "less than" ***)
Goalw [wf_def, pred_nat_def] "wf pred_nat";
by (Clarify_tac 1);
by (nat_ind_tac "x" 1);
by (ALLGOALS Blast_tac);
qed "wf_pred_nat";
Goalw [less_def] "wf {(x,y::nat). x<y}";
by (rtac (wf_pred_nat RS wf_trancl RS wf_subset) 1);
by (Blast_tac 1);
qed "wf_less";
Goalw [less_def] "((m,n) : pred_nat^+) = (m<n)";
by (rtac refl 1);
qed "less_eq";
(** Introduction properties **)
Goalw [less_def] "[| i<j; j<k |] ==> i<(k::nat)";
by (rtac (trans_trancl RS transD) 1);
by (assume_tac 1);
by (assume_tac 1);
qed "less_trans";
Goalw [less_def, pred_nat_def] "n < Suc(n)";
by (simp_tac (simpset() addsimps [r_into_trancl]) 1);
qed "lessI";
AddIffs [lessI];
(* i<j ==> i<Suc(j) *)
bind_thm("less_SucI", lessI RSN (2, less_trans));
Goal "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 **)
Goalw [less_def] "n<m ==> ~ m<(n::nat)";
by (blast_tac (claset() addIs [wf_pred_nat, wf_trancl RS wf_asym])1);
qed "less_not_sym";
(* [| n<m; ~P ==> m<n |] ==> P *)
bind_thm ("less_asym", less_not_sym RS contrapos_np);
Goalw [less_def] "~ n<(n::nat)";
by (rtac (wf_pred_nat RS wf_trancl RS wf_not_refl) 1);
qed "less_not_refl";
(* n<n ==> R *)
bind_thm ("less_irrefl", less_not_refl RS notE);
AddSEs [less_irrefl];
Goal "n<m ==> m ~= (n::nat)";
by (Blast_tac 1);
qed "less_not_refl2";
(* s < t ==> s ~= t *)
bind_thm ("less_not_refl3", less_not_refl2 RS not_sym);
val major::prems = Goalw [less_def, pred_nat_def]
"[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \
\ |] ==> P";
by (rtac (major RS tranclE) 1);
by (ALLGOALS Full_simp_tac);
by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
eresolve_tac (prems@[asm_rl, Pair_inject])));
qed "lessE";
Goal "~ n < (0::nat)";
by (blast_tac (claset() addEs [lessE]) 1);
qed "not_less0";
AddIffs [not_less0];
(* n<0 ==> R *)
bind_thm ("less_zeroE", not_less0 RS notE);
val [major,less,eq] = Goal
"[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P";
by (rtac (major RS lessE) 1);
by (rtac eq 1);
by (Blast_tac 1);
by (rtac less 1);
by (Blast_tac 1);
qed "less_SucE";
Goal "(m < Suc(n)) = (m < n | m = n)";
by (blast_tac (claset() addSEs [less_SucE] addIs [less_trans]) 1);
qed "less_Suc_eq";
Goal "(n < (1::nat)) = (n = 0)";
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
qed "less_one";
AddIffs [less_one];
Goal "(n < Suc 0) = (n = 0)";
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
qed "less_Suc0";
AddIffs [less_Suc0];
Goal "m<n ==> Suc(m) < Suc(n)";
by (etac rev_mp 1);
by (nat_ind_tac "n" 1);
by (ALLGOALS (fast_tac (claset() addEs [less_trans, lessE])));
qed "Suc_mono";
(*"Less than" is a linear ordering*)
Goal "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 (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
qed "less_linear";
Goal "!!m::nat. (m ~= n) = (m<n | n<m)";
by (cut_facts_tac [less_linear] 1);
by (Blast_tac 1);
qed "nat_neq_iff";
val [major,eqCase,lessCase] = Goal
"[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m";
by (rtac (less_linear RS disjE) 1);
by (etac disjE 2);
by (etac lessCase 1);
by (etac (sym RS eqCase) 1);
by (etac major 1);
qed "nat_less_cases";
(** Inductive (?) properties **)
Goal "[| m<n; Suc m ~= n |] ==> Suc(m) < n";
by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
qed "Suc_lessI";
Goal "Suc(m) < n ==> m<n";
by (etac 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
"[| 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 "Suc(m) < Suc(n) ==> m<n";
by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
qed "Suc_less_SucD";
Goal "(Suc(m) < Suc(n)) = (m<n)";
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
qed "Suc_less_eq";
AddIffs [Suc_less_eq];
(*Goal "~(Suc(n) < n)";
by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
qed "not_Suc_n_less_n";
Addsimps [not_Suc_n_less_n];*)
Goal "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 (blast_tac (claset() addDs [Suc_lessD]) 1);
qed_spec_mp "less_trans_Suc";
(*Can be used with less_Suc_eq to get n=m | n<m *)
Goal "(~ 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 [less_def]
"[| !!n. [| ALL 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 "nat_less_induct";
(*** Properties of <= ***)
(*Was le_eq_less_Suc, but this orientation is more useful*)
Goalw [le_def] "(m < Suc n) = (m <= n)";
by (rtac (not_less_eq RS sym) 1);
qed "less_Suc_eq_le";
(* m<=n ==> m < Suc n *)
bind_thm ("le_imp_less_Suc", less_Suc_eq_le RS iffD2);
Goalw [le_def] "(0::nat) <= n";
by (rtac not_less0 1);
qed "le0";
AddIffs [le0];
Goalw [le_def] "~ Suc n <= n";
by (Simp_tac 1);
qed "Suc_n_not_le_n";
Goalw [le_def] "!!i::nat. (i <= 0) = (i = 0)";
by (nat_ind_tac "i" 1);
by (ALLGOALS Asm_simp_tac);
qed "le_0_eq";
AddIffs [le_0_eq];
Goal "(m <= Suc(n)) = (m<=n | m = Suc n)";
by (simp_tac (simpset() delsimps [less_Suc_eq_le]
addsimps [less_Suc_eq_le RS sym, less_Suc_eq]) 1);
qed "le_Suc_eq";
(* [| m <= Suc n; m <= n ==> R; m = Suc n ==> R |] ==> R *)
bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
Goalw [le_def] "~n<m ==> m<=(n::nat)";
by (assume_tac 1);
qed "leI";
Goalw [le_def] "m<=n ==> ~ n < (m::nat)";
by (assume_tac 1);
qed "leD";
bind_thm ("leE", make_elim leD);
Goal "(~n<m) = (m<=(n::nat))";
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
qed "not_less_iff_le";
Goalw [le_def] "~ m <= n ==> n<(m::nat)";
by (Blast_tac 1);
qed "not_leE";
Goalw [le_def] "(~n<=m) = (m<(n::nat))";
by (Simp_tac 1);
qed "not_le_iff_less";
Goalw [le_def] "m < n ==> Suc(m) <= n";
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
qed "Suc_leI"; (*formerly called lessD*)
Goalw [le_def] "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 [le_def] "Suc m <= n ==> m < n";
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
by (cut_facts_tac [less_linear] 1);
by (Blast_tac 1);
qed "Suc_le_lessD";
Goal "(Suc m <= n) = (m < n)";
by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
qed "Suc_le_eq";
Goalw [le_def] "m <= n ==> m <= Suc n";
by (blast_tac (claset() addDs [Suc_lessD]) 1);
qed "le_SucI";
(*bind_thm ("le_Suc", not_Suc_n_less_n RS leI);*)
Goalw [le_def] "m < n ==> m <= (n::nat)";
by (blast_tac (claset() addEs [less_asym]) 1);
qed "less_imp_le";
(*For instance, (Suc m < Suc n) = (Suc m <= n) = (m<n) *)
bind_thms ("le_simps", [less_imp_le, less_Suc_eq_le, Suc_le_eq]);
(** Equivalence of m<=n and m<n | m=n **)
Goalw [le_def] "m <= n ==> m < n | m=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
qed "le_imp_less_or_eq";
Goalw [le_def] "m<n | m=n ==> m <=(n::nat)";
by (cut_facts_tac [less_linear] 1);
by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
qed "less_or_eq_imp_le";
Goal "(m <= (n::nat)) = (m < n | m=n)";
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
qed "le_eq_less_or_eq";
(*Useful with Blast_tac. m=n ==> m<=n *)
bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
Goal "n <= (n::nat)";
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
qed "le_refl";
Goal "[| i <= j; j < k |] ==> i < (k::nat)";
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
addIs [less_trans]) 1);
qed "le_less_trans";
Goal "[| i < j; j <= k |] ==> i < (k::nat)";
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
addIs [less_trans]) 1);
qed "less_le_trans";
Goal "[| i <= j; j <= k |] ==> i <= (k::nat)";
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
addIs [less_or_eq_imp_le, less_trans]) 1);
qed "le_trans";
Goal "[| m <= n; n <= m |] ==> m = (n::nat)";
(*order_less_irrefl could make this proof fail*)
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
addSEs [less_irrefl] addEs [less_asym]) 1);
qed "le_anti_sym";
Goal "(Suc(n) <= Suc(m)) = (n <= m)";
by (simp_tac (simpset() addsimps le_simps) 1);
qed "Suc_le_mono";
AddIffs [Suc_le_mono];
(* Axiom 'order_less_le' of class 'order': *)
Goal "((m::nat) < n) = (m <= n & m ~= n)";
by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
by (blast_tac (claset() addSEs [less_asym]) 1);
qed "nat_less_le";
(* [| m <= n; m ~= n |] ==> m < n *)
bind_thm ("le_neq_implies_less", [nat_less_le, conjI] MRS iffD2);
(* Axiom 'linorder_linear' of class 'linorder': *)
Goal "(m::nat) <= n | n <= m";
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
by (cut_facts_tac [less_linear] 1);
by (Blast_tac 1);
qed "nat_le_linear";
Goal "~ n < m ==> (n < Suc m) = (n = m)";
by (blast_tac (claset() addSEs [less_SucE]) 1);
qed "not_less_less_Suc_eq";
(*Rewrite (n < Suc m) to (n=m) if ~ n<m or m<=n hold.
Not suitable as default simprules because they often lead to looping*)
bind_thms ("not_less_simps", [not_less_less_Suc_eq, leD RS not_less_less_Suc_eq]);
(** Re-orientation of the equations 0=x and 1=x.
No longer added as simprules (they loop)
but via reorient_simproc in Bin **)
(*Polymorphic, not just for "nat"*)
Goal "(0 = x) = (x = 0)";
by Auto_tac;
qed "zero_reorient";
Goal "(1 = x) = (x = 1)";
by Auto_tac;
qed "one_reorient";