--- a/src/HOL/NatDef.ML Mon Aug 05 12:00:51 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,469 +0,0 @@
-(* 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";
--- a/src/HOL/NatDef.thy Mon Aug 05 12:00:51 2002 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,70 +0,0 @@
-(* Title: HOL/NatDef.thy
- ID: $Id$
- Author: Tobias Nipkow, Cambridge University Computer Laboratory
- Copyright 1991 University of Cambridge
-
-Definition of types ind and nat.
-
-Type nat is defined as a set Nat over type ind.
-*)
-
-NatDef = Wellfounded_Recursion +
-
-(** type ind **)
-
-types ind
-arities ind :: type
-
-consts
- Zero_Rep :: ind
- Suc_Rep :: ind => ind
-
-rules
- (*the axiom of infinity in 2 parts*)
- inj_Suc_Rep "inj(Suc_Rep)"
- Suc_Rep_not_Zero_Rep "Suc_Rep(x) ~= Zero_Rep"
-
-
-
-(** type nat **)
-
-(* type definition *)
-
-consts
- Nat' :: "ind set"
-
-inductive Nat'
-intrs
- Zero_RepI "Zero_Rep : Nat'"
- Suc_RepI "i : Nat' ==> Suc_Rep i : Nat'"
-
-global
-
-typedef (Nat)
- nat = "Nat'" (Nat'.Zero_RepI)
-
-instance
- nat :: {ord, zero, one}
-
-
-(* abstract constants and syntax *)
-
-consts
- Suc :: nat => nat
- pred_nat :: "(nat * nat) set"
-
-local
-
-defs
- Zero_nat_def "0 == Abs_Nat(Zero_Rep)"
- Suc_def "Suc == (%n. Abs_Nat(Suc_Rep(Rep_Nat(n))))"
- One_nat_def "1 == Suc 0"
-
- (*nat operations*)
- pred_nat_def "pred_nat == {(m,n). n = Suc m}"
-
- less_def "m<n == (m,n):trancl(pred_nat)"
-
- le_def "m<=(n::nat) == ~(n<m)"
-
-end