(* Title : HOL/Real/PNat.ML
ID : $Id$
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
The positive naturals -- proofs mainly as in theory Nat.
*)
Goal "mono(%X. {Suc 0} Un Suc`X)";
by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
qed "pnat_fun_mono";
bind_thm ("pnat_unfold", pnat_fun_mono RS (pnat_def RS def_lfp_unfold));
Goal "Suc 0 : pnat";
by (stac pnat_unfold 1);
by (rtac (singletonI RS UnI1) 1);
qed "one_RepI";
Addsimps [one_RepI];
Goal "i: pnat ==> Suc(i) : pnat";
by (stac pnat_unfold 1);
by (etac (imageI RS UnI2) 1);
qed "pnat_Suc_RepI";
Goal "Suc (Suc 0) : pnat";
by (rtac (one_RepI RS pnat_Suc_RepI) 1);
qed "two_RepI";
(*** Induction ***)
val major::prems = Goal
"[| i: pnat; P(Suc 0); \
\ !!j. [| j: pnat; P(j) |] ==> P(Suc(j)) |] ==> P(i)";
by (rtac ([pnat_def, pnat_fun_mono, major] MRS def_lfp_induct) 1);
by (blast_tac (claset() addIs prems) 1);
qed "PNat_induct";
val prems = Goalw [pnat_one_def,pnat_Suc_def]
"[| P(1); \
\ !!n. P(n) ==> P(pSuc n) |] ==> P(n)";
by (rtac (Rep_pnat_inverse RS subst) 1);
by (rtac (Rep_pnat RS PNat_induct) 1);
by (REPEAT (ares_tac prems 1
ORELSE eresolve_tac [Abs_pnat_inverse RS subst] 1));
qed "pnat_induct";
(*Perform induction on n. *)
fun pnat_ind_tac a i =
induct_thm_tac pnat_induct a i THEN rename_last_tac a [""] (i+1);
val prems = Goal
"[| !!x. P x 1; \
\ !!y. P 1 (pSuc y); \
\ !!x y. [| P x y |] ==> P (pSuc x) (pSuc y) \
\ |] ==> P m n";
by (res_inst_tac [("x","m")] spec 1);
by (pnat_ind_tac "n" 1);
by (rtac allI 2);
by (pnat_ind_tac "x" 2);
by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
qed "pnat_diff_induct";
(*Case analysis on the natural numbers*)
val prems = Goal
"[| n=1 ==> P; !!x. n = pSuc(x) ==> P |] ==> P";
by (subgoal_tac "n=1 | (EX x. n = pSuc(x))" 1);
by (fast_tac (claset() addSEs prems) 1);
by (pnat_ind_tac "n" 1);
by (rtac (refl RS disjI1) 1);
by (Blast_tac 1);
qed "pnatE";
(*** Isomorphisms: Abs_Nat and Rep_Nat ***)
Goal "inj_on Abs_pnat pnat";
by (rtac inj_on_inverseI 1);
by (etac Abs_pnat_inverse 1);
qed "inj_on_Abs_pnat";
Addsimps [inj_on_Abs_pnat RS inj_on_iff];
Goal "inj(Rep_pnat)";
by (rtac inj_inverseI 1);
by (rtac Rep_pnat_inverse 1);
qed "inj_Rep_pnat";
Goal "0 ~: pnat";
by (stac pnat_unfold 1);
by Auto_tac;
qed "zero_not_mem_pnat";
(* 0 : pnat ==> P *)
bind_thm ("zero_not_mem_pnatE", zero_not_mem_pnat RS notE);
Addsimps [zero_not_mem_pnat];
Goal "x : pnat ==> 0 < x";
by (dtac (pnat_unfold RS subst) 1);
by Auto_tac;
qed "mem_pnat_gt_zero";
Goal "0 < x ==> x: pnat";
by (stac pnat_unfold 1);
by (dtac (gr_implies_not0 RS not0_implies_Suc) 1);
by (etac exE 1 THEN Asm_simp_tac 1);
by (induct_tac "m" 1);
by (auto_tac (claset(),simpset()
addsimps [one_RepI]) THEN dtac pnat_Suc_RepI 1);
by (Blast_tac 1);
qed "gt_0_mem_pnat";
Goal "(x: pnat) = (0 < x)";
by (blast_tac (claset() addDs [mem_pnat_gt_zero,gt_0_mem_pnat]) 1);
qed "mem_pnat_gt_0_iff";
Goal "0 < Rep_pnat x";
by (rtac (Rep_pnat RS mem_pnat_gt_zero) 1);
qed "Rep_pnat_gt_zero";
Goalw [pnat_add_def] "(x::pnat) + y = y + x";
by (simp_tac (simpset() addsimps [add_commute]) 1);
qed "pnat_add_commute";
(** alternative definition for pnat **)
(** order isomorphism **)
Goal "pnat = {x::nat. 0 < x}";
by (auto_tac (claset(), simpset() addsimps [mem_pnat_gt_0_iff]));
qed "Collect_pnat_gt_0";
(*** Distinctness of constructors ***)
Goalw [pnat_one_def,pnat_Suc_def] "pSuc(m) ~= 1";
by (rtac (inj_on_Abs_pnat RS inj_on_contraD) 1);
by (rtac (Rep_pnat_gt_zero RS Suc_mono RS less_not_refl2) 1);
by (REPEAT (resolve_tac [Rep_pnat RS pnat_Suc_RepI, one_RepI] 1));
qed "pSuc_not_one";
bind_thm ("one_not_pSuc", pSuc_not_one RS not_sym);
AddIffs [pSuc_not_one,one_not_pSuc];
bind_thm ("pSuc_neq_one", (pSuc_not_one RS notE));
bind_thm ("one_neq_pSuc", pSuc_neq_one RS pSuc_neq_one);
(** Injectiveness of pSuc **)
Goalw [pnat_Suc_def] "inj(pSuc)";
by (rtac injI 1);
by (dtac (inj_on_Abs_pnat RS inj_onD) 1);
by (REPEAT (resolve_tac [Rep_pnat, pnat_Suc_RepI] 1));
by (dtac (inj_Suc RS injD) 1);
by (etac (inj_Rep_pnat RS injD) 1);
qed "inj_pSuc";
bind_thm ("pSuc_inject", inj_pSuc RS injD);
Goal "(pSuc(m)=pSuc(n)) = (m=n)";
by (EVERY1 [rtac iffI, etac pSuc_inject, etac arg_cong]);
qed "pSuc_pSuc_eq";
AddIffs [pSuc_pSuc_eq];
Goal "n ~= pSuc(n)";
by (pnat_ind_tac "n" 1);
by (ALLGOALS Asm_simp_tac);
qed "n_not_pSuc_n";
bind_thm ("pSuc_n_not_n", n_not_pSuc_n RS not_sym);
Goal "n ~= 1 ==> EX m. n = pSuc m";
by (rtac pnatE 1);
by (REPEAT (Blast_tac 1));
qed "not1_implies_pSuc";
Goal "pSuc m = m + 1";
by (auto_tac (claset(),simpset() addsimps [pnat_Suc_def,
pnat_one_def,Abs_pnat_inverse,pnat_add_def]));
qed "pSuc_is_plus_one";
Goal
"(Rep_pnat x + Rep_pnat y): pnat";
by (cut_facts_tac [[Rep_pnat_gt_zero,
Rep_pnat_gt_zero] MRS add_less_mono,Collect_pnat_gt_0] 1);
by (etac ssubst 1);
by Auto_tac;
qed "sum_Rep_pnat";
Goalw [pnat_add_def]
"Rep_pnat x + Rep_pnat y = Rep_pnat (x + y)";
by (simp_tac (simpset() addsimps [sum_Rep_pnat RS
Abs_pnat_inverse]) 1);
qed "sum_Rep_pnat_sum";
Goalw [pnat_add_def]
"(x + y) + z = x + (y + (z::pnat))";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [sum_Rep_pnat RS
Abs_pnat_inverse,add_assoc]) 1);
qed "pnat_add_assoc";
Goalw [pnat_add_def] "x + (y + z) = y + (x + (z::pnat))";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [sum_Rep_pnat RS
Abs_pnat_inverse,add_left_commute]) 1);
qed "pnat_add_left_commute";
(*Addition is an AC-operator*)
bind_thms ("pnat_add_ac", [pnat_add_assoc, pnat_add_commute, pnat_add_left_commute]);
Goalw [pnat_add_def] "((x::pnat) + y = x + z) = (y = z)";
by (auto_tac (claset() addDs [inj_on_Abs_pnat RS inj_onD,
inj_Rep_pnat RS injD],simpset() addsimps [sum_Rep_pnat]));
qed "pnat_add_left_cancel";
Goalw [pnat_add_def] "(y + (x::pnat) = z + x) = (y = z)";
by (auto_tac (claset() addDs [inj_on_Abs_pnat RS inj_onD,
inj_Rep_pnat RS injD],simpset() addsimps [sum_Rep_pnat]));
qed "pnat_add_right_cancel";
Goalw [pnat_add_def] "!(y::pnat). x + y ~= x";
by (rtac (Rep_pnat_inverse RS subst) 1);
by (auto_tac (claset() addDs [inj_on_Abs_pnat RS inj_onD]
addSDs [add_eq_self_zero],
simpset() addsimps [sum_Rep_pnat, Rep_pnat,Abs_pnat_inverse,
Rep_pnat_gt_zero RS less_not_refl2]));
qed "pnat_no_add_ident";
(***) (***) (***) (***) (***) (***) (***) (***) (***)
(*** pnat_less ***)
Goalw [pnat_less_def] "~ y < (y::pnat)";
by Auto_tac;
qed "pnat_less_not_refl";
bind_thm ("pnat_less_irrefl",pnat_less_not_refl RS notE);
Goalw [pnat_less_def]
"x < (y::pnat) ==> x ~= y";
by Auto_tac;
qed "pnat_less_not_refl2";
Goal "~ Rep_pnat y < 0";
by Auto_tac;
qed "Rep_pnat_not_less0";
(*** Rep_pnat < 0 ==> P ***)
bind_thm ("Rep_pnat_less_zeroE",Rep_pnat_not_less0 RS notE);
Goal "~ Rep_pnat y < Suc 0";
by (auto_tac (claset(),simpset() addsimps [less_Suc_eq,
Rep_pnat_gt_zero,less_not_refl2]));
qed "Rep_pnat_not_less_one";
(*** Rep_pnat < 1 ==> P ***)
bind_thm ("Rep_pnat_less_oneE",Rep_pnat_not_less_one RS notE);
Goalw [pnat_less_def]
"x < (y::pnat) ==> Rep_pnat y ~= Suc 0";
by (auto_tac (claset(),simpset()
addsimps [Rep_pnat_not_less_one] delsimps [less_Suc0]));
qed "Rep_pnat_gt_implies_not0";
Goalw [pnat_less_def]
"(x::pnat) < y | x = y | y < x";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (claset() addIs [inj_Rep_pnat RS injD]) 1);
qed "pnat_less_linear";
Goalw [le_def] "Suc 0 <= Rep_pnat x";
by (rtac Rep_pnat_not_less_one 1);
qed "Rep_pnat_le_one";
Goalw [pnat_less_def]
"!! (z1::nat). z1 < z2 ==> EX z3. z1 + Rep_pnat z3 = z2";
by (dtac less_imp_add_positive 1);
by (force_tac (claset() addSIs [Abs_pnat_inverse],
simpset() addsimps [Collect_pnat_gt_0]) 1);
qed "lemma_less_ex_sum_Rep_pnat";
(*** pnat_le ***)
(*** alternative definition for pnat_le ***)
Goalw [pnat_le_def,pnat_less_def]
"((m::pnat) <= n) = (Rep_pnat m <= Rep_pnat n)";
by (auto_tac (claset() addSIs [leI] addSEs [leD],simpset()));
qed "pnat_le_iff_Rep_pnat_le";
Goal "!!k::pnat. (k + m <= k + n) = (m<=n)";
by (simp_tac (simpset() addsimps [pnat_le_iff_Rep_pnat_le,
sum_Rep_pnat_sum RS sym]) 1);
qed "pnat_add_left_cancel_le";
Goalw [pnat_less_def] "!!k::pnat. (k + m < k + n) = (m<n)";
by (simp_tac (simpset() addsimps [sum_Rep_pnat_sum RS sym]) 1);
qed "pnat_add_left_cancel_less";
Addsimps [pnat_add_left_cancel, pnat_add_right_cancel,
pnat_add_left_cancel_le, pnat_add_left_cancel_less];
Goalw [pnat_less_def] "i+j < (k::pnat) ==> i<k";
by (auto_tac (claset() addEs [add_lessD1],
simpset() addsimps [sum_Rep_pnat_sum RS sym]));
qed "pnat_add_lessD1";
Goal "!!i::pnat. ~ (i+j < i)";
by (rtac notI 1);
by (etac (pnat_add_lessD1 RS pnat_less_irrefl) 1);
qed "pnat_not_add_less1";
Goal "!!i::pnat. ~ (j+i < i)";
by (simp_tac (simpset() addsimps [pnat_add_commute, pnat_not_add_less1]) 1);
qed "pnat_not_add_less2";
AddIffs [pnat_not_add_less1, pnat_not_add_less2];
Goal "m + k <= n --> m <= (n::pnat)";
by (simp_tac (simpset() addsimps [pnat_le_iff_Rep_pnat_le,
sum_Rep_pnat_sum RS sym]) 1);
qed_spec_mp "pnat_add_leD1";
Goal "!!n::pnat. m + k <= n ==> k <= n";
by (full_simp_tac (simpset() addsimps [pnat_add_commute]) 1);
by (etac pnat_add_leD1 1);
qed_spec_mp "pnat_add_leD2";
Goal "!!n::pnat. m + k <= n ==> m <= n & k <= n";
by (blast_tac (claset() addDs [pnat_add_leD1, pnat_add_leD2]) 1);
bind_thm ("pnat_add_leE", result() RS conjE);
Goalw [pnat_less_def]
"!!k l::pnat. [| k < l; m + l = k + n |] ==> m < n";
by (rtac less_add_eq_less 1 THEN assume_tac 1);
by (auto_tac (claset(),simpset() addsimps [sum_Rep_pnat_sum]));
qed "pnat_less_add_eq_less";
(* ordering on positive naturals in terms of existence of sum *)
(* could provide alternative definition -- Gleason *)
Goalw [pnat_less_def,pnat_add_def]
"((z1::pnat) < z2) = (EX z3. z1 + z3 = z2)";
by (rtac iffI 1);
by (res_inst_tac [("t","z2")] (Rep_pnat_inverse RS subst) 1);
by (dtac lemma_less_ex_sum_Rep_pnat 1);
by (etac exE 1 THEN res_inst_tac [("x","z3")] exI 1);
by (auto_tac (claset(),simpset() addsimps [sum_Rep_pnat_sum,Rep_pnat_inverse]));
by (res_inst_tac [("t","Rep_pnat z1")] (add_0_right RS subst) 1);
by (auto_tac (claset(),simpset() addsimps [sum_Rep_pnat_sum RS sym,
Rep_pnat_gt_zero] delsimps [add_0_right]));
qed "pnat_less_iff";
Goal "(EX (x::pnat). z1 + x = z2) | z1 = z2 \
\ |(EX x. z2 + x = z1)";
by (cut_facts_tac [pnat_less_linear] 1);
by (asm_full_simp_tac (simpset() addsimps [pnat_less_iff]) 1);
qed "pnat_linear_Ex_eq";
Goal "!!(x::pnat). x + y = z ==> x < z";
by (rtac (pnat_less_iff RS iffD2) 1);
by (Blast_tac 1);
qed "pnat_eq_lessI";
(*** Monotonicity of Addition ***)
Goal "1 * Rep_pnat n = Rep_pnat n";
by (Asm_simp_tac 1);
qed "Rep_pnat_mult_1";
Goal "Rep_pnat n * 1 = Rep_pnat n";
by (Asm_simp_tac 1);
qed "Rep_pnat_mult_1_right";
Goal "(Rep_pnat x * Rep_pnat y): pnat";
by (cut_facts_tac [[Rep_pnat_gt_zero,
Rep_pnat_gt_zero] MRS mult_less_mono1,Collect_pnat_gt_0] 1);
by (etac ssubst 1);
by Auto_tac;
qed "mult_Rep_pnat";
Goalw [pnat_mult_def]
"Rep_pnat x * Rep_pnat y = Rep_pnat (x * y)";
by (simp_tac (simpset() addsimps [mult_Rep_pnat RS Abs_pnat_inverse]) 1);
qed "mult_Rep_pnat_mult";
Goalw [pnat_mult_def] "m * n = n * (m::pnat)";
by (full_simp_tac (simpset() addsimps [mult_commute]) 1);
qed "pnat_mult_commute";
Goalw [pnat_mult_def,pnat_add_def] "(m + n)*k = (m*k) + ((n*k)::pnat)";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [mult_Rep_pnat RS
Abs_pnat_inverse,sum_Rep_pnat RS
Abs_pnat_inverse, add_mult_distrib]) 1);
qed "pnat_add_mult_distrib";
Goalw [pnat_mult_def,pnat_add_def] "k*(m + n) = (k*m) + ((k*n)::pnat)";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [mult_Rep_pnat RS
Abs_pnat_inverse,sum_Rep_pnat RS
Abs_pnat_inverse, add_mult_distrib2]) 1);
qed "pnat_add_mult_distrib2";
Goalw [pnat_mult_def]
"(x * y) * z = x * (y * (z::pnat))";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [mult_Rep_pnat RS
Abs_pnat_inverse,mult_assoc]) 1);
qed "pnat_mult_assoc";
Goalw [pnat_mult_def] "x * (y * z) = y * (x * (z::pnat))";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (simp_tac (simpset() addsimps [mult_Rep_pnat RS
Abs_pnat_inverse,mult_left_commute]) 1);
qed "pnat_mult_left_commute";
Goalw [pnat_mult_def] "x * (Abs_pnat (Suc 0)) = x";
by (full_simp_tac (simpset() addsimps [one_RepI RS Abs_pnat_inverse,
Rep_pnat_inverse]) 1);
qed "pnat_mult_1";
Goal "Abs_pnat (Suc 0) * x = x";
by (full_simp_tac (simpset() addsimps [pnat_mult_1,
pnat_mult_commute]) 1);
qed "pnat_mult_1_left";
(*Multiplication is an AC-operator*)
bind_thms ("pnat_mult_ac",
[pnat_mult_assoc, pnat_mult_commute, pnat_mult_left_commute]);
Goal "!!i::pnat. i<j ==> k*i < k*j";
by (asm_full_simp_tac (simpset() addsimps [pnat_less_def,
mult_Rep_pnat_mult RS sym,Rep_pnat_gt_zero,mult_less_mono2]) 1);
qed "pnat_mult_less_mono2";
Goal "!!i::pnat. i<j ==> i*k < j*k";
by (dtac pnat_mult_less_mono2 1);
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [pnat_mult_commute])));
qed "pnat_mult_less_mono1";
Goalw [pnat_less_def] "(m*(k::pnat) < n*k) = (m<n)";
by (asm_full_simp_tac (simpset() addsimps [mult_Rep_pnat_mult
RS sym,Rep_pnat_gt_zero]) 1);
qed "pnat_mult_less_cancel2";
Goalw [pnat_less_def] "((k::pnat)*m < k*n) = (m<n)";
by (asm_full_simp_tac (simpset() addsimps [mult_Rep_pnat_mult
RS sym,Rep_pnat_gt_zero]) 1);
qed "pnat_mult_less_cancel1";
Addsimps [pnat_mult_less_cancel1, pnat_mult_less_cancel2];
Goalw [pnat_mult_def] "(m*(k::pnat) = n*k) = (m=n)";
by (cut_inst_tac [("x","k")] Rep_pnat_gt_zero 1);
by (auto_tac (claset() addSDs [inj_on_Abs_pnat RS inj_onD,
inj_Rep_pnat RS injD]
addIs [mult_Rep_pnat],
simpset() addsimps [mult_cancel2]));
qed "pnat_mult_cancel2";
Goal "((k::pnat)*m = k*n) = (m=n)";
by (rtac (pnat_mult_cancel2 RS subst) 1);
by (auto_tac (claset () addIs [pnat_mult_commute RS subst],simpset()));
qed "pnat_mult_cancel1";
Addsimps [pnat_mult_cancel1, pnat_mult_cancel2];
Goal "!!(z1::pnat). z2*z3 = z4*z5 ==> z2*(z1*z3) = z4*(z1*z5)";
by (auto_tac (claset() addIs [pnat_mult_cancel1 RS iffD2],
simpset() addsimps [pnat_mult_left_commute]));
qed "pnat_same_multI2";
val [prem] = Goal
"(!!u. z = Abs_pnat(u) ==> P) ==> P";
by (cut_inst_tac [("x1","z")]
(rewrite_rule [pnat_def] (Rep_pnat RS Abs_pnat_inverse)) 1);
by (res_inst_tac [("u","Rep_pnat z")] prem 1);
by (dtac (inj_Rep_pnat RS injD) 1);
by (Asm_simp_tac 1);
qed "eq_Abs_pnat";
(** embedding of naturals in positive naturals **)
(* pnat_one_eq! *)
Goalw [pnat_of_nat_def,pnat_one_def]"1 = pnat_of_nat 0";
by (Full_simp_tac 1);
qed "pnat_one_iff";
Goalw [pnat_of_nat_def,pnat_one_def,
pnat_add_def] "1 + 1 = pnat_of_nat 1";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (auto_tac (claset() addIs [(gt_0_mem_pnat RS Abs_pnat_inverse RS ssubst)],
simpset()));
qed "pnat_two_eq";
Goal "inj(pnat_of_nat)";
by (rtac injI 1);
by (rewtac pnat_of_nat_def);
by (dtac (inj_on_Abs_pnat RS inj_onD) 1);
by (auto_tac (claset() addSIs [gt_0_mem_pnat],simpset()));
qed "inj_pnat_of_nat";
Goal "0 < n + (1::nat)";
by Auto_tac;
qed "nat_add_one_less";
Goal "0 < n1 + n2 + (1::nat)";
by Auto_tac;
qed "nat_add_one_less1";
(* this worked with one call to auto_tac before! *)
Goalw [pnat_add_def,pnat_of_nat_def,pnat_one_def]
"pnat_of_nat n1 + pnat_of_nat n2 = \
\ pnat_of_nat (n1 + n2) + 1";
by (res_inst_tac [("f","Abs_pnat")] arg_cong 1);
by (rtac (gt_0_mem_pnat RS Abs_pnat_inverse RS ssubst) 1);
by (rtac (gt_0_mem_pnat RS Abs_pnat_inverse RS ssubst) 2);
by (rtac (gt_0_mem_pnat RS Abs_pnat_inverse RS ssubst) 3);
by (rtac (gt_0_mem_pnat RS Abs_pnat_inverse RS ssubst) 4);
by (auto_tac (claset(),
simpset() addsimps [sum_Rep_pnat_sum,
nat_add_one_less,nat_add_one_less1]));
qed "pnat_of_nat_add";
Goalw [pnat_of_nat_def,pnat_less_def]
"(n < m) = (pnat_of_nat n < pnat_of_nat m)";
by (auto_tac (claset(),simpset()
addsimps [Abs_pnat_inverse,Collect_pnat_gt_0]));
qed "pnat_of_nat_less_iff";
Addsimps [pnat_of_nat_less_iff RS sym];
Goalw [pnat_mult_def,pnat_of_nat_def]
"pnat_of_nat n1 * pnat_of_nat n2 = \
\ pnat_of_nat (n1 * n2 + n1 + n2)";
by (auto_tac (claset(),simpset() addsimps [mult_Rep_pnat_mult,
pnat_add_def,Abs_pnat_inverse,gt_0_mem_pnat]));
qed "pnat_of_nat_mult";