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Narith.sml
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1466 lines (1291 loc) · 43.2 KB
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(*
fun pi1 A B = mk_fun "π1" [A,B]
fun pi2 A B = mk_fun "π2" [A,B]
*)
(*inv_suc_eq s_eq_iff_eq p_z_cases PRE_eq_0*)
val inv_suc_eq = proved_th $
e0
(assume_tac Thm2_2 >> drule ismono_property >>
rpt strip_tac >> dimp_tac >> strip_tac >--
(first_x_assum irule >> arw[]) >>
arw[])
(form_goal
“!m n:1->N. SUC o m = SUC o n <=> m = n”)
val s_eq_iff_eq = proved_th $
e0
(rpt strip_tac >> dimp_tac >> strip_tac >> arw[] >>
assume_tac Thm2_2 >> drule ismono_property >>
first_x_assum drule >> arw[])
(form_goal
“!n1:1->N n2. SUC o n1 = SUC o n2 <=> n1 = n2”);
val PRE_def = ex2fsym "PRE" [] $ iffRL $ eqT_intro pred_exists
|> C mp (trueI [])
val p_z_cases = proved_th $
e0
(assume_tac PRE_def >> strip_tac >>
cases_on “n = ZERO” >-- arw[] >>
arw[] >> assume_tac z_xor_s >>
first_x_assum (qspecl_then ["n"] assume_tac) >>
rfs[] >> arw[GSYM o_assoc,idL] >>
assume_tac s_eq_iff_eq >> arw[])
(form_goal
“!n:1->N. PRE o n = ZERO <=> (n = ZERO | n = SUC o ZERO)”)
val PRE_eq_0 = proved_th $
e0
(strip_tac >> assume_tac PRE_def >> cases_on “n = ZERO” >> arw[] >>
dimp_tac >> strip_tac (* 2 *) >--
(assume_tac z_xor_s >> first_x_assum (qspecl_then ["n"] assume_tac) >>
rfs[] >> fs[GSYM o_assoc,idL] >> rfs[idL] >> arw[] >> arw[GSYM o_assoc,idL]) >>
arw[GSYM o_assoc,idL])
(form_goal
“!n:1->N. PRE o n = ZERO <=> (n = ZERO | n = SUC o ZERO)”)
val ne_ex = proved_th $
e0
(rpt strip_tac >> drule diag_is_mono >> drule Thm5 >>
first_x_assum accept_tac)
(form_goal
“!NN Nn:NN->N nN:NN->N. ispr(Nn,nN) ==> ?NE ne:NE->NN. ismono(ne) &
!Sum iEQ:N -> Sum iNE:NE->Sum. iscopr(iEQ:N->Sum,iNE:NE->Sum) ==>
isiso(copa(iEQ,iNE,pa(Nn,nN,id(N),id(N)),ne))”)
val minus_uex = proved_th $
e0
(rpt strip_tac >> rev_drule Thm1 >> first_x_assum drule >>
first_x_assum drule >>
first_x_assum (qspecl_then ["id(N)","PRE o nnN"] assume_tac) >>
pop_assum strip_assume_tac >> qexistsl_tac ["f"] >> fs[idL,o_assoc])
(form_goal
“!NN Nn:NN->N nN:NN->N. ispr(Nn,nN) ==>
!NNN NNn:NNN->NN nnN:NNN->N.ispr(NNn,nnN) ==>
!N1 pN':N1->N pone:N1-> 1. ispr(pN',pone) ==>
?sub:NN->N.!sub':NN->N.sub' o pa(Nn,nN,pN',ZERO o pone) = pN' &
PRE o nnN o pa(NNn,nnN,id(NN),sub') = sub' o pa(Nn,nN,Nn,SUC o nN) <=> sub' = sub”)
val minus_ex = proved_th $
e0
(rpt strip_tac >> rev_drule minus_uex >> first_x_assum drule >>
first_x_assum drule >> pop_assum strip_assume_tac >>
qexistsl_tac ["sub"] >> first_x_assum (qspecl_then ["sub"] assume_tac) >>
fs[])
(form_goal
“!NN Nn:NN->N nN:NN->N. ispr(Nn,nN) ==>
!NNN NNn:NNN->NN nnN:NNN->N.ispr(NNn,nnN) ==>
!N1 pN':N1->N pone:N1-> 1. ispr(pN',pone) ==>
?sub:NN->N.
sub o pa(Nn,nN,pN',ZERO o pone) = pN' &
PRE o nnN o pa(NNn,nnN,id(NN),sub) = sub o pa(Nn,nN,Nn,SUC o nN)”)
(*
val N_Pr_N = prove_store("N_Pr_N",
e0
(rw[pi2_def])
(form_goal
“ispr(π1(N,N),π2(N,N))”));
*)
val N_Pr_N = proved_th $
e0
(rw[pi2_def])
(form_goal
“ispr(π1(N,N),π2(N,N))”);
val NN_Pr_N = proved_th $
e0
(rw[pi2_def])
(form_goal
“ispr(π1(N * N,N),π2(N * N,N))”);
(*ind_one_component*)
val INDUCT_one_component = proved_th $
e0
(rpt strip_tac >> assume_tac N_Pr_N >> drule equality_ind >>
first_x_assum drule >>
fs[pa2Pa])
(form_goal
“!f:N * N->N g:N * N->N.
!n0.(!n.f o Pa(n0,n) = g o Pa(n0,n)) <=>
f o Pa(n0,ZERO) = g o Pa(n0,ZERO) &
!n:1->N. f o Pa(n0,n) = g o Pa(n0,n) ==>
f o Pa(n0,SUC o n) = g o Pa(n0,SUC o n)”);
val SUB_def = minus_ex |> specl [Po N N,pi1 N N,pi2 N N] |> C mp N_Pr_N
|> specl [Po (Po N N) N,pi1 (Po N N) N,pi2 (Po N N) N] |> C mp NN_Pr_N
|> specl [N,rastt "id(N)",rastt "to1(N,1)"]
|> C mp (allE N pr_with_one)
|> eqT_intro |> iffRL |> ex2fsym "SUB" []
|> C mp (trueI []) |> rewr_rule[To1_def];
val SUB = mk_fun "SUB" [];
val LEQo_def = pb_ex |> specl [Po N N,N,SUB] |> specl [rastt "1",ZERO]
|> eqT_intro |> iffRL |> ex2fsym "LEQo" []
|> C mp (trueI []);
val LEQo = mk_fun "LEQo" [];
val LEQ_def = LEQo_def |> eqT_intro |> iffRL |> ex2fsym "LEQ" []
|> C mp (trueI []);
val LEQ = mk_fun "LEQ" [];
val LEQ_pb = proved_th $
e0
(strip_assume_tac LEQ_def >>
assume_tac (to1_unique |> specl [LEQo]) >>
first_x_assum (qspecl_then ["To1(LEQo)","q"] assume_tac) >>
fs[])
(form_goal “ispb(SUB,ZERO,LEQ,To1(LEQo))”);
val NEQo_def = ne_ex |> specl [Po N N,pi1 N N,pi2 N N] |> C mp N_Pr_N
|> eqT_intro |> iffRL
|> ex2fsym "NEQo" []
|> C mp (trueI []);
val NEQ_def = NEQo_def |> eqT_intro |> iffRL |> ex2fsym "NEQ" []
|> C mp (trueI [])
val NEQ = mk_fun "NEQ" []
val NEQo = mk_fun "NEQo" []
val LESSo_def = pb_ex |> specl [LEQo,Po N N,LEQ]
|> specl [NEQo,NEQ]
|> eqT_intro |> iffRL
|> ex2fsym "LESSo" []
|> C mp (trueI [])
val LESS2LEQ_def = LESSo_def |> eqT_intro |> iffRL |> ex2fsym "LESS2LEQ" []
|> C mp (trueI [])
val LESS2NEQ_def = LESS2LEQ_def |> eqT_intro |> iffRL |> ex2fsym "LESS2NEQ" []
|> C mp (trueI [])
val LESS_mono = proved_th $
e0
(irule o_mono_mono >>
by_tac “ismono(NEQ)” >--
accept_tac (conjE1 NEQ_def) >>
assume_tac LESS2NEQ_def >> arw[] >>
strip_assume_tac LEQ_def >>
by_tac “ismono(ZERO)”
>-- (qspecl_then ["N","ZERO"] accept_tac dom_1_mono) >>
drule pb_mono_mono >> first_x_assum drule >>
qby_tac ‘ispb(NEQ,LEQ, LESS2NEQ, LESS2LEQ)’
>-- (rev_drule pb_reorder >> first_x_assum accept_tac)>>
drule pb_mono_mono >> first_x_assum drule >>
first_x_assum accept_tac)
(form_goal “ismono(NEQ o LESS2NEQ)”)
val LESS_def0 = proved_th $
e0
(qexistsl_tac ["NEQ o LESS2NEQ"]>> rw[])
(form_goal “?LESS0. LESS0 = NEQ o LESS2NEQ”)
val LESS_def = LESS_def0 |> eqT_intro |> iffRL
|> ex2fsym "LESS" []
|> C mp (trueI []);
val NEQ_property = proved_th $
e0
(rpt strip_tac >> assume_tac NEQ_def >>
pop_assum strip_assume_tac >>
qspecl_then ["N","NEQo"] (x_choosel_then ["W","iN","iNEQo"] assume_tac) copr_ex >>
first_x_assum drule >>
drule iso_copr_copr >> first_x_assum drule >>
drule copr_disjoint >>
by_tac
“(?nnb : 1 -> NEQo. NEQ o nnb = nn:1->N * N) <=> (?nnb : 1 -> NEQo. nn:1->N * N= NEQ o nnb)”
(*TODO: the by above is stupid step*)>--
(dimp_tac >> strip_tac >> qexists_tac "nnb" >> arw[]) >>
(*TODO AQ: how to avoid this trivial steps?*)
arw[] >> pop_assum (K all_tac) >>
pop_assum (assume_tac o GSYM) >>
once_arw[] >>
assume_tac N_Pr_N >> drule fac_diag_eq_iff >>
first_x_assum (qspecl_then ["nn"] assume_tac) >>
arw[])
(form_goal
“!nn:1->N * N.(?nnb:1->NEQo. NEQ o nnb = nn) <=> ~
(π1(N,N) o nn = π2(N,N) o nn)”);
(*TODO: use char to define pred LESS,after removing all LE and LT etc*)
(*sub_z_iff_le*)
val SUB_EQ_00 = proved_th $
e0
(rw[GSYM pa2Pa] >> rpt strip_tac >>
assume_tac LEQ_def >> pop_assum strip_assume_tac >>
drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["1","pa(π1(N,N),π2(N,N),n1,n2)","id(1)"] assume_tac) >> fs[idR] >> dimp_tac >> strip_tac (* 2 *)
>-- (arw[GSYM o_assoc] >> rw[o_assoc] >>
once_rw[one_to_one_id] >> rw[idR]) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
qexists_tac "a" >>
first_x_assum (qspecl_then ["a"] assume_tac) >> fs[])
(form_goal
“!n1:1->N n2:1->N.
(?le0:1->LEQo. Pa(n1,n2) = LEQ o le0) <=>
SUB o Pa(n1,n2) = ZERO”);
(*sub_zero_id*)
val SUB_0_cj2 = proved_th $
e0
(strip_tac >> assume_tac SUB_def >>
pop_assum strip_assume_tac >>
by_tac “SUB o pa(π1(N,N), π2(N,N), id(N), ZERO o To1(N)) o n:1->N = id(N) o n” >-- (rw[GSYM o_assoc] >> arw[]) >>
fs[idL] >>
by_tac “pa(π1(N,N), π2(N,N), id(N), ZERO o To1(N)) o n:1->N = pa(π1(N,N), π2(N,N), n, ZERO)” >--
(assume_tac N_Pr_N >> drule to_p_eq >>
first_x_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc] >>
rw[o_assoc] >> once_rw[one_to_one_id] >>
rw[idL,idR]) >>
fs[pa2Pa])
(form_goal
“!n:1->N. SUB o Pa(n,ZERO) = n”);
(*le_z*)
val LESS_EQ_00 = proved_th $
e0
(rpt strip_tac >> assume_tac SUB_EQ_00 >>
first_x_assum (qspecl_then ["n0","ZERO"] assume_tac) >>
by_tac “?(le0 : 1 -> LEQo). Pa(n0, ZERO) = LEQ o le0”
>-- (qexists_tac "a" >> arw[]) >>
fs[] >>
assume_tac SUB_0_cj2 >> fs[]
)
(form_goal
“!n0:1->N a:1->LEQo. Pa(n0,ZERO) = LEQ o a ==>
n0 = ZERO”);
(*lt_le*)
val LESS_IMP_LESS_OR_EQ0 = proved_th $
e0
(rpt strip_tac >> assume_tac LESS_def >>
assume_tac LESS2NEQ_def >> drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >> fs[] >>
pick_x_assum “LEQ o LESS2LEQ = NEQ o LESS2NEQ” (assume_tac o GSYM) >>
fs[] >>
qexists_tac "LESS2LEQ o lt0" >> rw[o_assoc])
(form_goal
“
!n1:1->N n2:1->N.
(?lt0: 1->LESSo. Pa(n1,n2) = LESS o lt0) ==>
(?le0: 1->LEQo. Pa(n1,n2) = LEQ o le0) ”);
(*lt_ne0*)
val LESS_NOT_EQ00 = proved_th $
e0
(rpt strip_tac >> assume_tac LESS_def >>
assume_tac LESS2NEQ_def >> drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >> fs[] >>
qexists_tac "LESS2NEQ o lt0" >> rw[o_assoc])
(form_goal
“
!n1:1->N n2:1->N.
(?lt0: 1->LESSo. Pa(n1,n2) = LESS o lt0) ==>
(?ne0: 1->NEQo. Pa(n1,n2) = NEQ o ne0)”);
(*lt_ne*)
val LESS_NOT_EQ0 = proved_th $
e0
(strip_tac >> strip_tac >> disch_tac >>
assume_tac LESS_NOT_EQ00 >> first_x_assum drule >>
assume_tac NEQ_property >> pop_assum mp_tac >>
pop_assum (assume_tac o GSYM) >> strip_tac >>
pop_assum (assume_tac o iffLR) >> first_x_assum drule >>
pop_assum mp_tac >> assume_tac N_Pr_N >>
arw[pi12_of_Pa])
(form_goal
“
!n1:1->N n2:1->N.
(?lt0: 1->LESSo. Pa(n1,n2) = LESS o lt0) ==>
~(n1 = n2)”);
(*le_ne_lt*)
val LEQ_NEQ_LESS = proved_th $
e0
(
rpt strip_tac >>
assume_tac LESS_def >> assume_tac LESS2NEQ_def >>
drule $ iffLR ispb_def >> pop_assum strip_assume_tac >>
assume_tac NEQ_property >>
first_x_assum (qspecl_then ["Pa(n1,n2)"] assume_tac)>>
assume_tac N_Pr_N >> fs[pi12_of_Pa] >>
pop_assum (K all_tac) >>
pick_x_assum
“(?nnb : 1 -> NEQo. NEQ o nnb = Pa(n1:1->N, n2)) <=> ~(n1 = n2)” (assume_tac o GSYM) >> fs[] >>
first_x_assum (qspecl_then ["1","le0","nnb"] assume_tac) >>
rfs[] >> qexists_tac "a" >>
first_x_assum (qspecl_then ["a"] assume_tac) >> fs[] >>
arw[o_assoc])
(form_goal
“
!n1:1->N n2:1->N.
((?le0: 1->LEQo. Pa(n1,n2) = LEQ o le0) & ~(n1 = n2))
==> (?lt0: 1->LESSo. Pa(n1,n2) = LESS o lt0)”);
(*lt_iff_le_ne*)
val LESS_iff_LEQ_NEQ = proved_th $
e0
(rpt strip_tac >> dimp_tac >> disch_tac (* 2 *)
>-- (assume_tac LESS_NOT_EQ0 >> first_x_assum drule >>
assume_tac LESS_IMP_LESS_OR_EQ0 >> first_x_assum drule >> arw[]) >>
assume_tac LEQ_NEQ_LESS >> first_x_assum drule >> arw[])
(form_goal
“
!n1:1->N n2:1->N.
(?lt0: 1->LESSo. Pa(n1,n2) = LESS o lt0) <=>
((?le0: 1->LEQo. Pa(n1,n2) = LEQ o le0) & ~(n1 = n2))”);
(*clt_iff_le_ne*)
val CLESS_iff_LEQ_NEQ = proved_th $
e0
(rpt strip_tac >>
assume_tac LESS_iff_LEQ_NEQ >> pop_assum (assume_tac o GSYM) >>
arw[] >>
assume_tac LESS_mono >>
assume_tac $ GSYM LESS_def >> fs[] >>
drule char_def >> assume_tac TRUE_def >>
first_x_assum drule >>
pop_assum (assume_tac o GSYM) >> arw[] >>
fs[Char_def] >>
dimp_tac >> strip_tac (* 2 *)
>-- (qexists_tac "x0" >> arw[]) >>
qexists_tac "lt0" >> arw[])
(form_goal
“!n1:1->N n2:1->N.
(Char(LESS) o Pa(n1, n2) = TRUE) <=>
((?le0: 1->LEQo. Pa(n1,n2) = LEQ o le0) & ~(n1 = n2))”);
(*not_lt_z *)
val not_LESS_ZERO = proved_th $
e0
(rpt strip_tac >>
ccontra_tac >>
by_tac “Char(LESS) o Pa(n0, ZERO) = TRUE <=> (?a:1->LEQo.Pa(n0,ZERO) = LEQ o a) & ~(n0:1->N = ZERO)”
>-- rw[CLESS_iff_LEQ_NEQ] >> fs[] >>
drule LESS_EQ_00 >> fs[]
)
(form_goal
“!n0:1->N. ~(Char(LESS) o Pa(n0,ZERO) = TRUE)”);
(*sub_def'*)
val SUB_def' = proved_th $
e0
(assume_tac SUB_def >> fs[pi12_of_Pa,pa2Pa])
(form_goal
“SUB o Pa(id(N), ZERO o To1(N)) = id(N) &
PRE o SUB = SUB o Pa(π1(N,N), SUC o π2(N,N))”);
(*add_ex*)
val ADD_ex = proved_th $
e0
(assume_tac N_Pr_N >> assume_tac NN_Pr_N >>
assume_tac pr_with_one >>
first_x_assum (qspecl_then ["N"] assume_tac) >>
rev_drule Thm1 >> first_x_assum drule >> first_x_assum drule >>
first_x_assum (qspecl_then ["id(N)","SUC o π2(N * N,N)"] assume_tac) >>
pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["f"] assume_tac) >> fs[] >>
qexists_tac "f" >> fs[o_assoc,idL,pa2Pa,To1_def])
(form_goal
“?Add:N * N->N.Add o Pa(id(N),ZERO o To1(N)) = id(N) &
Add o Pa(π1(N,N),SUC o π2(N,N)) = SUC o π2(N * N,N) o Pa(id(N * N),Add)”);
(*add_def0*)
val ADD_def0 = ADD_ex |> eqT_intro |> iffRL |> ex2fsym "ADD" []
|> C mp (trueI [])
val ADD = mk_fun "ADD" []
(*add_def*)
val ADD_def = proved_th $
e0
(assume_tac ADD_def0 >> assume_tac NN_Pr_N >>
fs[pi12_of_Pa])
(form_goal
“ADD o Pa(id(N),ZERO o To1(N)) = id(N) &
ADD o Pa(π1(N,N),SUC o π2(N,N)) = SUC o ADD”);
(*add_elements*)
val ADD_el = proved_th $
e0
(rpt strip_tac >> assume_tac ADD_def (* 2 *)
>-- (by_tac “ADD o Pa(id(N), ZERO o To1(N)) o n:1->N = id(N) o n”
>-- arw[GSYM o_assoc] >>
assume_tac N_Pr_N >>
fs[Pa_distr] >>
pick_x_assum “ADD o Pa(id(N) o n:1->N, (ZERO o To1(N)) o n) =
id(N) o n” mp_tac >>
rw[o_assoc] >> once_rw[one_to_one_id] >> rw[idL,idR]) >>
by_tac “ADD o Pa(π1(N,N), SUC o π2(N,N)) o Pa(n, n0:1->N) = SUC o ADD o Pa(n, n0)” >-- arw[GSYM o_assoc] >>
fs[o_assoc,pi12_of_Pa,Pa_distr])
(form_goal
“!n:1->N. ADD o Pa(n,ZERO) = n &
!n0:1->N. ADD o Pa(n, SUC o n0) = SUC o ADD o Pa(n,n0)”);
(*sub_elements*)
val SUB_el = proved_th $
e0
(strip_assume_tac SUB_def' >> rpt strip_tac >--
(by_tac
“SUB o Pa(id(N), ZERO o To1(N)) o n:1->N = id(N) o n”
>-- arw[GSYM o_assoc] >> fs[Pa_distr,idL,o_assoc] >>
(* assume_tac nN_def >> drule distr_to_pa >> fs[idL] >> *)
pop_assum mp_tac >> once_rw[one_to_one_id] >> rw[idR]) >>
by_tac
“PRE o SUB o Pa(n:1->N, n0) =
SUB o Pa(π1(N,N), SUC o π2(N,N)) o Pa(n, n0)”
>-- arw[GSYM o_assoc] >> fs[o_assoc,Pa_distr,pi12_of_Pa])
(form_goal
“!n:1->N. SUB o Pa(n,ZERO) = n &
!n0.SUB o Pa(n,SUC o n0) = PRE o SUB o Pa(n,n0)”)
val ind_gen_principle = proved_th $
e0
(rpt strip_tac >> drule Uq_ex >>
qspecl_then ["X","two"] (x_choosel_then ["Xt2","efs","p1","p2","ev"] assume_tac) exp_ex >>
first_x_assum drule >> pop_assum strip_assume_tac >>
first_x_assum drule >>
drule pxy_true >> first_x_assum drule >>
first_x_assum drule >> once_arw[] >>
pop_assum (K all_tac) >>
first_x_assum (qspecl_then ["pred"] assume_tac) >>
dimp_tac
>-- (rpt strip_tac >-- arw[] >-- arw[]) >>
strip_tac >>
suffices_tac
“!y : 1 -> N x : 1 -> X. pred:XN->two o pa(Xn:XN->X, xN:XN->N, x, y) = i2”
>-- (strip_tac >> arw[]) >>
strip_tac >>
first_assum (qspecl_then ["y"] (assume_tac o GSYM)) >>
once_arw[] >>
suffices_tac
“ Uq :Xt2->two o tp(p1:efs->X, p2:efs->Xt2, ev:efs->two, Xn:XN->X, xN, pred) = i2:1->two o to1(N,1)”
>-- (strip_tac >>
qpick_x_assum
‘!y : 1 -> N. Uq o tp(p1:efs->X, p2:efs->Xt2, ev, Xn:XN->X, xN:XN->N, pred:XN->two) o y = i2 <=>
!x : 1 -> X. pred o pa(Xn, xN, x, y) = i2’
(K all_tac) >>
arw[GSYM o_assoc] >> rw[o_assoc] >>
once_rw[one_to_one_id] >> rw[idR]) >>
drule ind_principle >> arw[] >>
pop_assum mp_tac >> pop_assum (K all_tac) >> strip_tac >>
strip_tac (* 2 *)
>-- arw[o_assoc] >>
rpt strip_tac >> fs[o_assoc] >> rfs[] >>
strip_tac >> last_x_assum (qspecl_then ["x"] assume_tac) >>
pop_assum strip_assume_tac >> first_assum irule >> arw[]
)
(form_goal
“!X XN Xn:XN->X xN:XN->N. ispr(Xn,xN) ==>
!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!pred:XN->two. pred = i2 o to1(XN,1) <=>
(!x:1->X. pred o pa(Xn,xN,x,ZERO) = i2 &
(!n:1->N. pred o pa(Xn,xN,x,n) = i2 ==> pred o pa(Xn,xN,x, SUC o n) = i2))”)
val ind_gen_principle' = proved_th $
e0
(rpt strip_tac >> drule ind_gen_principle >>
first_x_assum drule >> once_arw[] >>
dimp_tac >> rpt strip_tac (* 4 *)
>-- arw[]
>-- (first_x_assum (qspecl_then ["x"] assume_tac) >>
fs[] >> first_x_assum rev_drule >>
first_x_assum accept_tac)
>-- arw[] >>
first_x_assum drule >> arw[])
(form_goal
“!X XN Xn:XN->X xN:XN->N. ispr(Xn,xN) ==>
!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!pred:XN->two. pred = i2 o to1(XN,1) <=>
(!x:1->X. pred o pa(Xn,xN,x,ZERO) = i2) &
(!x n:1->N. pred o pa(Xn,xN,x,n) = i2 ==> pred o pa(Xn,xN,x, SUC o n) = i2)”)
(*sub_mono_eq*)
val SUB_MONO_EQ = proved_th $
e0
(assume_tac N_Pr_N >>
drule ind_gen_principle >>
assume_tac TRUE_def >> first_assum drule >>
drule char_diag >> first_assum drule >>
pop_assum (assume_tac o GSYM) >> once_arw[] >>
last_x_assum drule >>
fs[Char_def,pa2Pa] >>
suffices_tac
“Char(Pa(id(N), id(N))) o
Pa(SUB o Pa(SUC o π1(N,N), SUC o π2(N,N)), SUB) = TRUE o To1(N * N)” >-- (rpt strip_tac >>
drule $ iffRL fun_ext_iff >>
first_assum (qspecl_then ["Pa(m,n)"] assume_tac) >>
fs[o_assoc,Pa_distr,pi12_of_Pa] >>
once_rw[one_to_one_id] >> rw[idR]) >>
fs[Pa_distr,o_assoc,To1_def,pi12_of_Pa] >>
assume_tac SUB_el >>
rpt strip_tac (* 2 *) >--
(arw[] >> rw[GSYM o_assoc,PRE_def,idL] >>
pop_assum (K all_tac) >> pop_assum (K all_tac) >>
pop_assum (assume_tac o GSYM) >> arw[]) >>
fs[Pa_distr,pi12_of_Pa] >>
pick_xnth_assum 4 (assume_tac o GSYM) >> fs[] >>
qsuff_tac
‘PRE o SUB o Pa(SUC o x, n) = SUB o Pa(x,n)’
>-- (strip_tac >> arw[]) >>
qby_tac
‘PRE o SUB o Pa(SUC o x, n) = SUB o Pa(SUC o x, SUC o n)’
>-- (pop_assum mp_tac >> pop_assum mp_tac >> arw[]) >>
arw[]
)
(form_goal
“!m:1->N n:1->N.
SUB o Pa(SUC o m, SUC o n) = SUB o Pa(m,n)”)
(*add_sub0*)
val ADD_SUB0 = proved_th $
e0
(rw[INDUCT_one_component] >>
rw[o_assoc,Pa_distr,pi12_of_Pa] >>
rw[ADD_el,SUB_MONO_EQ] >> rw[SUB_el])
(form_goal
“!a:1->N. (!c:1->N. (SUB o Pa(ADD,π2(N,N))) o Pa(a,c) = π1(N,N) o Pa(a,c))”)
(*add_sub*)
val ADD_SUB = proved_th $
e0
(assume_tac ADD_SUB0 >> fs[o_assoc,Pa_distr,pi12_of_Pa])
(form_goal
“!a:1->N c:1->N. SUB o Pa(ADD o Pa(a,c),c) = a”)
val ind_N_element = proved_th $
e0
(rpt strip_tac >> assume_tac INDUCT_one_component >>
first_x_assum (qspecl_then ["f o π2(N,N)","g o π2(N,N)","ZERO"] assume_tac) >>
fs[o_assoc,pa2Pa,pi12_of_Pa])
(form_goal
“!f:N->N g:N->N. (!n:1->N.f o n = g o n) <=>
f o ZERO = g o ZERO &
!n:1->N. f o n = g o n ==> f o SUC o n = g o SUC o n”)
(*add_z_n0*)
val ADD_ZERO_n0 = proved_th $
e0
(rw[ind_N_element,o_assoc,Pa_distr,idL] >>
once_rw[one_to_one_id] >> rw[idR] >>
rw[ADD_el] >> rpt strip_tac >> arw[])
(form_goal
“!n:1->N. (ADD o Pa(ZERO o To1(N),id(N))) o n = id(N) o n”)
(*add_z_n*)
val ADD_ZERO_n = proved_th $
e0
(assume_tac ADD_ZERO_n0 >> fs[Pa_distr,idL,o_assoc] >>
pop_assum mp_tac >> once_rw[one_to_one_id] >>
rw[idR])
(form_goal
“!n:1->N. ADD o Pa(ZERO,n) = n”)
(*add_clauses1*)
val ADD_CLAUSES1 = ADD_ZERO_n
(*sub_equal_0*)
val SUB_equal_0 = proved_th $
e0
(strip_tac >> assume_tac ADD_SUB >>
first_x_assum (qspecl_then ["ZERO","n"] assume_tac) >>
fs[ADD_ZERO_n])
(form_goal
“!n. SUB o Pa(n,n) = ZERO”)
(*n_sub_n_z*)
val n_SUB_n_z = SUB_equal_0
val z_mono = proved_th $
e0
(assume_tac dom_1_mono >> once_arw[])
(form_goal “ismono(ZERO)”)
(*le_mono*)
val LEQ_mono = proved_th $
e0
(assume_tac LEQ_def >> pop_assum strip_assume_tac >>
drule pb_mono_mono >> first_x_assum irule >>
accept_tac z_mono)
(form_goal “ismono(LEQ)”)
(*le_refl*)
val LEQ_refl = proved_th $
e0
(rpt strip_tac >> assume_tac LEQ_mono >>
drule char_def >>
assume_tac TRUE_def >>
first_x_assum drule >> pop_assum (assume_tac o GSYM) >>
fs[Char_def] >>
arw[] >> strip_assume_tac LEQ_def >>
drule pb_fac_iff_1 >> arw[] >>
assume_tac n_SUB_n_z >>
arw[]
)
(form_goal
“!x:1->N. Char(LEQ) o Pa(x, x) = TRUE”)
(*le_z_z*)
val LEQ_ZERO_ZERO = proved_th $
e0
(rpt strip_tac >> assume_tac LEQ_mono >>
drule char_def >> assume_tac TRUE_def >>
first_x_assum drule >>
pop_assum (assume_tac o GSYM) >>
last_x_assum mp_tac >> fs[Char_def] >>
disch_tac >> strip_assume_tac LEQ_def >>
drule pb_fac_iff_1 >> fs[] >>
assume_tac SUB_0_cj2 >> fs[])
(form_goal
“!n0:1->N. Char(LEQ) o Pa(n0, ZERO) = TRUE ==> n0 = ZERO”)
(*le_cases*)
val LEQ_cases = proved_th $
e0
(rpt strip_tac >> cases_on “n0 = n:1->N” (* 2 *)
>-- arw[] >>
assume_tac CLESS_iff_LEQ_NEQ >>
arw[] >> assume_tac LEQ_mono >> drule char_def >>
assume_tac TRUE_def >>
first_x_assum drule >>
fs[Char_def] >> pop_assum (assume_tac o GSYM) >>
fs[] >> qexists_tac "x0" >> arw[])
(form_goal
“!n0:1->N n:1->N. Char(LEQ) o Pa(n0, n) = TRUE ==>
Char(LESS) o Pa(n0, n) = TRUE | n0 = n”)
(*sub_s*)
val SUB_SUC = proved_th $
e0
(rpt strip_tac >> assume_tac SUB_def' >>
by_tac
“PRE o SUB o Pa(a:1->N, b:1->N) =
SUB o Pa(π1(N,N), SUC o π2(N,N)) o Pa(a, b)”
>-- arw[GSYM o_assoc] >>
fs[o_assoc,Pa_distr,pi12_of_Pa])
(form_goal
“!a:1->N b:1->N. SUB o Pa(a,SUC o b) =
PRE o SUB o Pa(a,b)”)
(*double_ind*)
val double_IND = proved_th $
e0
(rpt strip_tac >>
assume_tac TRUE_def >> drule ind_principle_elements >>
rw[GSYM All_def,GSYM o_assoc] >> arw[] >>
qsuff_tac
‘(!n : 1 -> N.
(All(N) o Tp(pred)) o n = TRUE ==>
(All(N) o Tp(pred)) o SUC o n = TRUE) <=>
(!n : 1 -> N.
(All(N) o Tp(pred)) o n = TRUE ==>
pred o Pa(ZERO, SUC o n) = TRUE &
!m : 1 -> N.
pred o Pa(m, SUC o n) = TRUE ==>
pred o Pa(SUC o m, SUC o n) = TRUE)’
>-- (strip_tac >> arw[]) >>
qsuff_tac
‘!n:1->N.
(All(N) o Tp(pred)) o n = TRUE ==>
((All(N) o Tp(pred)) o SUC o n = TRUE <=>
pred o Pa(ZERO, SUC o n) = TRUE &
!m : 1 -> N.
pred o Pa(m, SUC o n) = TRUE ==>
pred o Pa(SUC o m, SUC o n) = TRUE)’
>-- (strip_tac >> dimp_tac >> strip_tac >> strip_tac >>
strip_tac >-- (last_x_assum drule >>
first_x_assum drule >> fs[]) >>
first_x_assum drule >> first_x_assum drule >> fs[]) >>
rpt strip_tac >>
first_x_assum (qspecl_then ["pred o Pa(id(N),SUC o n o To1(N))"] assume_tac) >>
fs[o_assoc,Pa_distr] >> pop_assum mp_tac >>
once_rw[one_to_one_id] >> rw[idL,idR] >>
rw[All_def])
(form_goal
“!pred:N * N-> 2.(!n m:1->N. pred o Pa(m,n) = TRUE) <=>
(!m.pred o Pa(m,ZERO) = TRUE) &
(!n.(!m.pred o Pa(m,n) = TRUE)
==>
pred o Pa(ZERO,SUC o n) = TRUE &
(!m.pred o Pa(m,SUC o n) = TRUE ==> pred o Pa(SUC o m, SUC o n) = TRUE))”)
val All_Pr = proved_th $
e0
(cheat)
(form_goal
“!X Y Z pxyz: (X * Y) * Z -> 2 z0:1->Z. All(X * Y) o Tp(pxyz) o z0 = TRUE <=> !x:1->X y:1->Y.pxyz o Pa(Pa(x,y),z0) = TRUE”)
val All_Pr' = proved_th $
e0
(cheat)
(form_goal
“!X Y Z pxyz: (X * Y) * Z -> 2 z0:1->Z. All(X * Y) o Tp(pxyz) o z0 = TRUE <=> !y:1->Y x:1->X.pxyz o Pa(Pa(x,y),z0) = TRUE”)
val triple_IND = proved_th $
e0
(rpt strip_tac >> assume_tac TRUE_def >>
drule ind_principle_elements >>
qby_tac
‘(!a:1->N m:1-> N n:1->N.
pred o Pa(Pa(n, m), a) = TRUE) <=>
(!a:1->N. All(N * N) o Tp(pred) o a = TRUE)’ >--
(rw[All_def] >> dimp_tac >> rpt strip_tac (* 2 *) >--
(first_x_assum (qspecl_then ["a","π2(N,N) o x","π1(N,N) o x"] assume_tac) >> fs[to_Pr_components]) >> arw[]) >>
arw[GSYM o_assoc] >>
rw[o_assoc,All_Pr'] >>
qsuff_tac
‘!a:1->N.(!m:1->N n:1->N.pred o Pa(Pa(n,m),a) = TRUE) ==>
((All(N * N) o Tp(pred)) o SUC o a = TRUE <=>
(!n:1->N.pred o Pa(Pa(n,ZERO),SUC o a) = TRUE) &
!m:1->N.
(!n:1->N. pred o Pa(Pa(n,m),SUC o a) = TRUE) ==>
pred o Pa(Pa(ZERO,SUC o m),SUC o a) = TRUE &
!n:1->N. pred o Pa(Pa(n,SUC o m),SUC o a) = TRUE ==>
pred o Pa(Pa(SUC o n,SUC o m),SUC o a) = TRUE)’
>-- (rw[GSYM All_Pr'] >> strip_tac >>
dimp_tac >> strip_tac (* 2 *) >--
(arw[] >> strip_tac >> strip_tac >>
first_x_assum drule >>
pop_assum mp_tac >> first_x_assum drule >>
fs[GSYM o_assoc]) >>
arw[] >> strip_tac >> strip_tac >>
first_x_assum drule >> first_x_assum drule >>
fs[o_assoc]) >>
rpt strip_tac >> arw[o_assoc] >>
assume_tac double_IND >>
first_x_assum (qspecl_then ["pred o Pa(id(N * N),(SUC o a) o To1(N * N))"] assume_tac) >>
rw[All_Pr'] >> pop_assum mp_tac >>
rw[o_assoc] >> once_rw[Pa_distr] >>
rw[o_assoc] >> once_rw[one_to_one_id] >>
once_rw[idL] >> once_rw[idR] >> rw[])
(form_goal
“!pred:(N * N) * N-> 2.
(!a:1->N m n. pred o Pa(Pa(n,m),a) = TRUE) <=>
(!m:1->N n. pred o Pa(Pa(n,m),ZERO) = TRUE) &
(!a:1->N.
(!m:1->N n. pred o Pa(Pa(n,m),a) = TRUE)==>
(!n.pred o Pa(Pa(n,ZERO),SUC o a) = TRUE) &
(!m.(!n.pred o Pa(Pa(n,m),SUC o a) = TRUE) ==>
pred o Pa(Pa(ZERO,SUC o m),SUC o a) = TRUE &
(!n. pred o Pa(Pa(n,SUC o m),SUC o a) = TRUE ==>
pred o Pa(Pa(SUC o n,SUC o m),SUC o a) = TRUE)))”)
(*le_sub*)
val LEQ_SUB = proved_th $
e0
(rpt strip_tac >> assume_tac LEQ_def >>
pop_assum strip_assume_tac >> assume_tac LEQ_mono >>
drule char_def >> assume_tac TRUE_def >>
first_x_assum drule >>
pop_assum (assume_tac o GSYM) >>
once_arw[] >>
drule pb_fac_iff_1 >> arw[] >> fs[Char_def])
(form_goal
“(!m:1->N n. Char(LEQ) o Pa(m,n) = TRUE <=>
SUB o Pa(m,n) = ZERO)”)
(*TODO automatic rw A ==> B ==>C <=> A & B ==>C*)
val CANCEL_SUB_pred = proved_th $
e0
(rpt strip_tac >>
qexists_tac
$ q2str
‘IMP o
Pa(CONJ o
Pa(Char(LEQ) o Pa(π2(N * N,N),π1(N,N) o π1(N * N,N)),
Char(LEQ) o Pa(π2(N * N,N),π2(N,N) o π1(N * N,N))),
IFF o
Pa(Eq(N) o
Pa(SUB o Pa(π1(N,N) o π1(N * N,N),π2(N * N,N)),
SUB o Pa(π2(N,N) o π1(N * N,N),π2(N * N,N))),
Eq(N) o
Pa(π1(N,N) o π1(N * N,N),π2(N,N) o π1(N * N,N))))’ >>
rpt strip_tac >>
rw[o_assoc,Pa_distr,IMP_def] >>
rw[pi12_of_Pa] >>
rw[CONJ_def] >> rw[IFF_def] >>
rw[GSYM True1TRUE] >> rw[GSYM Eq_property] >>
dimp_tac >> rpt strip_tac >> fs[])
(form_goal
“?pred:(N * N) * N-> 2.
!a:1->N m n.(Char(LEQ) o Pa(a,n) = TRUE ==>
Char(LEQ) o Pa(a,m) = TRUE ==>
(SUB o Pa(n,a) = SUB o Pa(m,a) <=> n = m)) <=>
pred o Pa(Pa(n,m),a) = TRUE”)
(*cancel_sub00*)
val CANCEL_SUB00 = proved_th $
e0
(strip_assume_tac CANCEL_SUB_pred >> arw[] >>
rw[triple_IND] >> pop_assum (assume_tac o GSYM) >> arw[] >>
strip_tac (* 2 *) >-- rw[SUB_0_cj2] >>
strip_tac >> strip_tac >> strip_tac (* 2 *) >--
rw[LEQ_SUB,SUB_0_cj2,Thm2_1] >>
strip_tac >> strip_tac >> strip_tac (* 2 *) >--
rw[LEQ_SUB,SUB_0_cj2,Thm2_1] >>
rpt strip_tac >> rw[SUB_MONO_EQ,inv_suc_eq] >>
first_x_assum irule >> fs[SUB_MONO_EQ,LEQ_SUB]
)
(form_goal
“!a m n. Char(LEQ) o Pa(a,n) = TRUE ==>
Char(LEQ) o Pa(a,m) = TRUE ==>
(SUB o Pa(n,a) = SUB o Pa(m,a) <=> n = m)”)
(*cancel_sub00'*)
val CANCEL_SUB00' = proved_th $
e0
(rpt strip_tac >> rev_drule CANCEL_SUB00 >>
first_x_assum drule >> arw[]
)
(form_goal
“!a n. Char(LEQ) o Pa(a,n) = TRUE ==>
!m. Char(LEQ) o Pa(a,m) = TRUE ==>
(SUB o Pa(n,a) = SUB o Pa(m,a) <=> n = m)”)
(*sub_0*)
val SUB_0 = proved_th $
e0
(suffices_tac
“!n:1->N. (SUB o Pa(ZERO o To1(N), id(N))) o n = ZERO o To1(N) o n” >--
(rpt strip_tac >>
pop_assum mp_tac >> rw[o_assoc,Pa_distr] >>
once_rw[one_to_one_id] >> rw[idL,idR] >> rpt strip_tac >>
arw[]) >>
rw[GSYM o_assoc] >> rw[ind_N_element] >>
assume_tac SUB_el >> rw[o_assoc,Pa_distr] >>
once_rw[one_to_one_id] >> rw[idL,idR] >> arw[] >>
rpt strip_tac >> arw[PRE_def])
(form_goal
“!n:1->N. SUB o Pa(ZERO,n) = ZERO”)
(*zero_less_eq*)
val ZERO_LESS_EQ = proved_th $
e0
(rw[LEQ_SUB,SUB_0])
(form_goal
“!x. Char(LEQ) o Pa(ZERO, x) = TRUE”)
(*less_eq_mono*)
val LESS_EQ_MONO = proved_th $
e0
(rw[SUB_MONO_EQ,LEQ_SUB])
(form_goal
“(!m n.Char(LEQ) o Pa(SUC o m, SUC o n) = TRUE <=>
Char(LEQ) o Pa(m, n) = TRUE)”)
(*lt_char_LT*)
val LESS_Char_LESSo = proved_th $
e0
(rpt strip_tac >> assume_tac $ GSYM LESS_def >>
assume_tac LESS_mono >> rfs[] >> drule char_def >>
assume_tac TRUE_def >>
first_x_assum drule >> pop_assum (assume_tac o GSYM) >>
fs[Char_def] >> dimp_tac >> rpt strip_tac (* 2 *)
>-- (qexists_tac "x0" >> arw[]) >>
qexists_tac "x0" >> arw[])
(form_goal
“!x:1->N * N. (?(x0 : 1 -> LESSo). x = LESS o x0) <=>
Char(LESS) o x = TRUE”)
(*le_char_LE*)
val LEQ_Char_LEQo = proved_th $
e0
(rpt strip_tac >> strip_assume_tac LEQ_def >>
assume_tac LEQ_mono >> drule char_def >>
assume_tac TRUE_def >>
first_x_assum drule >> pop_assum (assume_tac o GSYM) >>
fs[Char_def] >> dimp_tac >> rpt strip_tac (* 2 *)
>-- (qexists_tac "x0" >> arw[]) >>
qexists_tac "x0" >> arw[])
(form_goal
“!x:1->N * N. (?(x0 : 1 -> LEQo). x = LEQ o x0) <=>
Char(LEQ) o x = TRUE”)
(*less_0*)
val LESS_0 = proved_th $
e0
(rpt strip_tac >>
rw[GSYM LESS_Char_LESSo] >>
rw[LESS_iff_LEQ_NEQ] >>
rw[GSYM Thm2_1] >>
rw[LEQ_Char_LEQo] >> rw[ZERO_LESS_EQ]
)
(form_goal
“!n. Char(LESS) o Pa(ZERO, SUC o n) = TRUE”)
(*less_mono_eq*)
val LESS_MONO_EQ = proved_th $
e0
(assume_tac LESS_mono >>
(*assume_tac LESS_def >> *)drule char_def >>
assume_tac TRUE_def >> first_x_assum drule >>
fs[Char_def] >> fs[GSYM LESS_def] >>
rw[GSYM LESS_Char_LESSo] >>
rw[LESS_iff_LEQ_NEQ] >>
rw[inv_suc_eq] >> assume_tac LEQ_Char_LEQo >>
arw[LESS_EQ_MONO])
(form_goal
“(!m n.Char(LESS) o Pa(SUC o m, SUC o n) = TRUE <=>
Char(LESS) o Pa(m, n) = TRUE)”)
(*less_cases_pred*)
val LESS_cases_pred = proved_th $
e0
(rpt strip_tac >>
qexists_tac $ q2str
‘DISJ o
Pa(Char(LESS),Char(LEQ) o Pa(π2(N,N),π1(N,N)))’ >>
rw[o_assoc,Pa_distr,DISJ_def,pi12_of_Pa])
(form_goal