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4013 lines (3413 loc) · 171 KB
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(*ETCS axioms*)
fun is_ar t =
case view_sort (sort_of t) of
vo => false | _ => true
fun is_ar_ss (n,s) = is_ar (var (n,s))
fun aspec tl th =
let val (b,vs) = strip_forall $ concl th
val ars = List.filter is_ar_ss vs
val env = match_tl essps (List.map var ars) tl emptyvd
val tl' = List.map (inst_term env) (List.map var vs)
in specl tl' th
end
fun check_wffv fvs =
case fvs of
[] => true
| h :: t => if ill_formed_fv h then
raise ERR ("ill-formed free variable",[snd h],[var h],[])
else check_wffv t
fun wffv_ok f = check_wffv (HOLset.listItems (fvf f))
fun new_ax f =
let val _ = wffv_ok f orelse
raise ERR ("formula contains ill-formed free variable(s)",[],[],[])
val _ = HOLset.equal(fvf f,essps) orelse
raise simple_fail"formula has free variables"
in
mk_thm essps [] f
end
fun read_axiom thstr =
let
val f = rapf thstr
val _ = HOLset.equal(fvf f,essps) orelse
raise simple_fail"formula has free variables"
in
mk_thm essps [] f
end
val idL = read_axiom "!B A f:B -> A. id (A) o f = f";
val idR = read_axiom "!A B f: A -> B. f o id(A) = f";
val o_assoc = read_axiom "!A.!B.!f: A -> B.!C.!g:B -> C.!D.!h: C -> D.(h o g) o f = h o g o f";
val to1_def = read_axiom "!one.is1(one) ==> !X.!t1x:X->one.t1x =to1(X,one)";
val const1_def = read_axiom "is1(1)";
val ax1_1 = const1_def |> existsI ("one",mk_ob_sort) one (rapf "is1(one)");
val from0_def = read_axiom "!zero.is0(zero) ==> !X.!f0x:zero->X.f0x=from0(zero,X)";
val const0_def = read_axiom "is0(0)";
val ax1_2 = const0_def |> existsI ("zero",mk_ob_sort) zero (rapf "is0(zero)");
val pr_def = read_axiom "!A.!B.!AB.!p1:AB->A.!p2:AB->B.ispr(p1,p2)<=>!X.!f:X->A.!g:X->B.?fg:X->AB.!fg':X->AB.p1 o fg' = f & p2 o fg' = g <=> fg' = fg";
val pa_def = read_axiom "!A.!B.!AB.!p1:AB->A.!p2:AB->B.ispr(p1,p2)==>!X.!f:X->A.!g:X->B.!fg:X->AB.p1 o fg = f & p2 o fg = g <=> fg = pa(p1,p2,f,g)";
val pr_ex = read_axiom "!A.!B.?AB.?p1:AB->A.?p2:AB->B.ispr(p1,p2)";
val ax1_3 = pr_ex;
val copa_def = read_axiom "!A.!B.!AB.!i1:A->AB.!i2:B->AB.iscopr(i1,i2)==>!X.!f:A->X.!g:B->X.!fg:AB->X.fg o i1 = f & fg o i2 = g <=> fg = copa(i1,i2,f,g)";
val copr_ex = read_axiom "!A.!B.?AB.?i1:A->AB.?i2:B->AB.iscopr(i1,i2)";
val ax1_4 = copr_ex;
val eqind_def = read_axiom "!A.!B.!f:A->B.!g:A->B.!E.!e:E->A.iseq(e,f,g)==> f o e = g o e & !X.!x:X->A.f o x = g o x ==> (!x0:X->E.e o x0 = x <=> x0 = eqind(e,f,g,x))";
val eq_ex = read_axiom "!A.!B.!f:A->B.!g:A->B.?E.?e:E->A.iseq(e,f,g)";
val ax1_5 = eq_ex
val coeqind_def = read_axiom "!A.!B.!f:A->B.!g:A->B.!cE.!ce:B->cE.iscoeq(ce,f,g)==> ce o f = ce o g & !X.!x:B->X. x o f = x o g ==> (!x0:cE->X.x0 o ce = x <=> x0 = coeqind(ce,f,g,x))"
val coeq_ex = read_axiom "!A.!B.!f:A->B.!g:A->B.?cE.?ce:B->cE.iscoeq(ce,f,g)"
val ax1_6 = coeq_ex
val tp_def = read_axiom "!A.!B.!A2B.!efs.!p1:efs ->A.!p2:efs ->A2B.!ev:efs->B.isexp(p1,p2,ev)==> ispr(p1,p2) & !X.!AX.!p1':AX->A.!p2':AX->X. ispr(p1',p2') ==> !f:AX->B.!h:X->A2B. ev o pa(p1,p2,p1',h o p2') = f <=> h = tp(p1,p2,ev,p1',p2',f)"
val exp_ex = read_axiom "!A.!B.?A2B efs p1:efs ->A p2:efs ->A2B ev:efs->B.isexp(p1,p2,ev)"
val ax2 = exp_ex
val Nind0_def = read_axiom "!N0 one z0:one -> N0 s0:N0->N0. isN0(z0,s0) ==> is1(one) & !X x0:one -> X t:X -> X x:N0 -> X. x o z0 = x0 & x o s0 = t o x <=> x = Nind0(z0,s0,x0,t)";
val _ = new_fun "SUC" (ar_sort N N,[])
val _ = new_fun "ZERO" (ar_sort one N,[])
val constN_def = read_axiom "!X x0:1->X t:X ->X x:N->X.(x o ZERO = x0 & x o SUC = t o x) <=> x = Nind(x0,t)"
(*to be edited, switch the ordrr of s0 and z0*)
val ax3 = constN_def
val ax_wp = read_axiom "!A B f: A -> B g:A ->B.(~(f = g)) ==> ?a: 1 -> A. ~(f o a = g o a)";
val ax4 = ax_wp
val ismono_def = read_axiom "!A B f: A -> B. ismono(f) <=> !X g:X -> A h. f o g = f o h ==> h = g"
val ax_el = read_axiom "!X.(~is0(X)) ==> ?x:1->X.T";
val ax6 = ax_el
val ac = read_axiom "!A one a: one -> A B f: A -> B. is1(one) ==> ?g : B -> A. f o g o f = f"
val ax5 = ac
val ismem_def = read_axiom "!A x:1-> A A0 a:A0 -> A. ismem(x,a) <=> ismono(a) & ?x0:1 -> A0. a o x0 = x";
new_pred "ismem0" ([("x",mk_ar_sort (mk_ob "one") (mk_ob "A")),("a",mk_ar_sort (mk_ob "A0") (mk_ob "A"))]);
val ismem0_def = read_axiom "!A one x:one-> A A0 a:A0 -> A. ismem0(x,a) <=> is1(one) & ismono(a) & ?x0:one -> A0. a o x0 = x";
val ax_incfac0 = read_axiom "!A B AB i1:A->AB i2:B->AB one f:one -> AB. iscopr(i1,i2) & is1(one) ==> ismem0(f,i1)| ismem0(f,i2)";
val ax7 = ax_incfac0
val ax_2el = read_axiom "?X x1: 1 -> X x2: 1 -> X. ~ (x1 = x2)"
val ax8 = ax_2el
val eq_eqn = eqind_def |> strip_all_and_imp
|> conjE2 |> iffRL |> strip_all_and_imp
|> rewr_rule [assume (rapf "x0 = eqind(e:E->A, f:A->B, g, x:X->A)")] |> disch (rapf "x0 = eqind(e:E->A, f:A->B, g, x:X->A)")
|> gen_all
|> allE (rastt "eqind(e:E->A, f:A->B, g, x:X->A)")
|> C mp $ refl (rastt "eqind(e:E->A, f:A->B, g, x:X->A)")
|> gen_disch_all
val coeq_eqn = coeqind_def |> strip_all_and_imp
|> conjE2 |> iffRL |> strip_all_and_imp
|> rewr_rule [assume (rapf "x0 = coeqind(ce:B->cE, f:A->B, g, x:B->X)")] |> disch (rapf "x0 = coeqind(ce:B->cE, f:A->B, g, x:B->X)")
|> gen_all
|> allE (rastt "coeqind(ce:B->cE, f:A->B, g, x:B->X)")
|> C mp $ refl (rastt "coeqind(ce:B->cE, f:A->B, g, x:B->X)")
|> gen_disch_all
(*∀A B X f. f∶A × X → B ⇒ ev A B ∘ ⟨p1 A X, tp f ∘ p2 A X⟩ = f*)
val ev_of_tp =
tp_def |> strip_all_and_imp |> conjE2 |> strip_all_and_imp |> iffRL |>
undisch |> arw_rule [] |> disch (rapf "h = tp(p1:efs->A, p2:efs->A2B, ev:efs->B, p1':AX->A, p2':AX->X, f)") |> allI ("h",mk_ar_sort (mk_ob "X") (mk_ob "A2B"))|>
allE $ rastt "tp(p1:efs->A, p2:efs->A2B, ev:efs->B, p1':AX->A, p2':AX->X, f)"|>
C mp $ refl $ rastt "tp(p1:efs->A, p2:efs->A2B, ev:efs->B, p1':AX->A, p2':AX->X, f)" |>
gen_disch_all |> gen_all
(*∀A B C f g. f∶ A×B → C ∧ g∶ A×B → C ⇒ (tp f = tp g ⇔ f = g)*)
fun mp_canon th =
let val th0 = strip_all_and_imp th
val th1 = conj_all_assum th0
in th1 |> gen_disch_all |> gen_all
end
val tp_eq = proved_th $
e0
(strip_tac >> drule ev_of_tp >> first_x_assum drule >>
first_assum (specl_then [rastt "f:AX->B"] assume_tac) >>
first_assum (specl_then [rastt "g:AX->B"] assume_tac) >>
suffices_tac (rapf "ev o pa(p1:efs->A, p2:efs->A2B, p1':AX->A, tp(p1, p2, ev, p1', p2':AX->X, f:AX->B) o p2') = ev o pa(p1, p2, p1', tp(p1, p2, ev, p1', p2', g) o p2')")
>-- (once_arw_tac[] >> rw_tac[]) >>
pop_assum (K all_tac) >> pop_assum (K all_tac) >> pop_assum (K all_tac) >>
arw_tac[])
(rapg "isexp(p1:efs->A,p2:efs->A2B,ev:efs->B) & ispr(p1':AX->A,p2':AX->X) & tp(p1, p2, ev, p1', p2', f) = tp(p1, p2, ev, p1', p2', g) ==> f = g")
(*
∀A B X f h. f∶ A × X → B ∧ h∶ X → exp A B ∧
(ev A B) o ⟨p1 A X, h o (p2 A X)⟩ = f ⇒
h = tp f
*)
val is_tp = tp_def |> strip_all_and_imp |> conjE2 |> iffLR
|> gen_disch_all |> gen_all
(*
∀A B X f. f∶ (po A X) → B ⇒
(∀h. (h∶ X → (exp A B) ∧
(ev A B) o (pa (p1 A X) (h o (p2 A X))) = f) ⇔
h = tp f)
*)
val ax2_conj2 = tp_def |> strip_all_and_imp |> conjE2 |> gen_disch_all |> gen_all
(*
∀X x0 t. x0∶ one → X ∧ t∶ X → X ⇒
!x. (x∶ N → X ∧ x o z = x0 ∧ x o s = t o x) ⇔
x = N_ind x0 t
ax3_conj2
*)
(*∀A B X f g. f∶ X →(exp A B) ∧ g∶X → (exp A B) ∧
(ev A B) o ⟨p1 A X,f o (p2 A X)⟩ =
(ev A B) o ⟨p1 A X,g o (p2 A X)⟩ ⇒
f = g
*)
val exp_ispr = tp_def |> strip_all_and_imp |> conjE1 |> disch_all |> gen_all
val ev_eq_eq = proved_th $
e0
(repeat strip_tac >>
suffices_tac (rapf "f = tp (p1:efs->A,p2:efs->A2B,ev:efs->B,p1':AX->A,p2':AX->X,ev o pa(p1, p2, p1', f o p2')) & g = tp (p1,p2,ev,p1',p2',ev o pa(p1, p2, p1', f o p2'))")
>-- (strip_tac >> once_arw_tac[] >> rw_tac[]) >> strip_tac
>> (irule is_tp >> arw_tac[]))
(rapg "!A A2B B efs ev:efs->B p1:efs -> A p2:efs->A2B. isexp(p1,p2,ev) ==> !AX p1':AX->A X p2':AX->X. ispr(p1',p2') ==> !f: X -> A2B g. ev o pa(p1, p2, p1', f o p2')=ev o pa(p1, p2, p1', g o p2')==> f = g");
(*
∀f g X A B. f∶ X → (A×B) ∧ g∶ X → (A × B) ⇒
((p1 A B) o f = (p1 A B) o g ∧ (p2 A B) o f = (p2 A B) o g ⇔ f = g)
*)
val to_p_component = proved_th $
e0
(repeat strip_tac >> match_mp_tac (pa_def |> iffLR |> ir_canon) >>
arw_tac[])
(rapg "!A B AB p1:AB->A p2:AB->B. ispr(p1, p2) ==>!X f:X->AB. f = pa(p1, p2, p1 o f, p2 o f)");
(*
val it =
A , AB , B , X ,
(f : X -> AB), (g : X -> AB), (p1 : AB -> A), (p2 : AB -> B)
1.ispr(p1, p2)
2.p1 o f = p1 o g
3.p2 o f = p2 o g
4.f = pa(p1, p2, p1 o f, p2 o f)
5.g = pa(p1, p2, p1 o g, p2 o g)
----------------------------------------------------------------------
pa(p1, p2, p1 o f, p2 o f) = pa(p1, p2, p1 o g, p2 o g)
*)
val to_p_eq = proved_th $
e0
(repeat strip_tac >>
suffices_tac (rapf "f = pa(p1:AB->A,p2:AB->B,p1 o (f:X->AB),p2 o f) & g = pa(p1:AB->A,p2:AB->B,p1 o (g:X->AB),p2 o g)")
>-- (strip_tac >> once_arw_tac[] >>
pop_assum (K all_tac) >> pop_assum (K all_tac) >> arw[]) >> strip_tac
>> match_mp_tac to_p_component >> arw_tac[])
(form_goal
“!A B AB p1:AB->A p2:AB->B.ispr(p1,p2) ==> !X f:X ->AB g. p1 o f = p1 o g & p2 o f = p2 o g ==> f = g”);
val from_copr_components = proved_th $
e0
(repeat strip_tac >> match_mp_tac (copa_def |> iffLR |> ir_canon) >>
arw_tac[])
(rapg "!A B AB i1:A->AB i2:B->AB. iscopr(i1, i2) ==>!X f:AB->X. f = copa(i1, i2, f o i1, f o i2)");
(*∀A B X f g. f∶ A → X ∧ g∶ B → X ⇒ copa f g o i1 A B = f*)
val i12_of_copa = copa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:A->X","g:B->X","copa(i1:A->AB,i2:B->AB,f:A->X,g)"]) |> C mp $ refl (rastt "copa(i1:A->AB,i2:B->AB,f:A->X,g)")
|> gen_all |> disch_all |> gen_all;
val i1_of_copa = copa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:A->X","g:B->X","copa(i1:A->AB,i2:B->AB,f:A->X,g)"]) |> C mp $ refl (rastt "copa(i1:A->AB,i2:B->AB,f:A->X,g)")
|> conjE1 |> gen_all |> disch_all |> gen_all;
val i2_of_copa = copa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:A->X","g:B->X","copa(i1:A->AB,i2:B->AB,f:A->X,g)"]) |> C mp $ refl (rastt "copa(i1:A->AB,i2:B->AB,f:A->X,g)")
|> conjE2 |> gen_all |> disch_all |> gen_all;
(*may add once full simp *)
val i1_ne_i2 = proved_th $
e0
(repeat strip_tac >> ccontra_tac >>
x_choosel_then ["X","x1","x2"] assume_tac ax8 >>
qby_tac ‘copa(i1,i2,x1,x2) o i1 = x1 &copa(i1,i2,x1,x2) o i2 = x2’
>-- (drule i12_of_copa >> arw_tac[]) >>
pop_assum (assume_tac o GSYM) >>
rev_full_simp_tac[] >>
qsuff_tac ‘x1 = x2’
>-- (arw_tac[]) >>
qpick_x_assum ‘~(x1 = x2)’ (K all_tac) >> once_arw_tac[] >> rw_tac[] >>
rw_tac[])
(rapg "!oneone i1:1 -> oneone i2:1 -> oneone. iscopr(i1,i2) ==> ~(i1 = i2)");
(*∀A B x. ¬(x∶one → (A + B) ∧ (∃x0 x0'. x0∶one → A ∧ x0'∶one → B ∧
i1 A B ∘ x0 = x ∧
i2 A B ∘ x0' = x))
*)
val eq_to1 = mp (allE one to1_def) const1_def;
val to1_unique = specl [rastt "X",rastt "f:X->1"] eq_to1 |> GSYM
|> trans (specl [rastt "X",rastt "g:X->1"] eq_to1) |> gen_all;
val prop_5_lemma = proved_th $
e0
(repeat strip_tac >> x_choosel_then ["oneone","one1","one2"] assume_tac (specl (List.map rastt ["1","1"]) copr_ex) >> ccontra_tac >>
match_mp_tac (i1_ne_i2|> spec_all |> undisch|> eqF_intro |> iffLR |> undisch|> conj_all_assum |> disch_all|> gen_all) >>
exists_tac (rastt "oneone") >> exists_tac (rastt "one1:1->oneone") >>
exists_tac (rastt "one2:1->oneone") >>
arw_tac[] >> pop_assum mp_tac >> pop_assum mp_tac >> drule i12_of_copa >>
first_x_assum (specl_then (List.map rastt ["oneone","(one1:1->oneone) o (to1(A,1))",
"(one2:1->oneone) o (to1(B,1))"]) assume_tac) >>
repeat strip_tac >>
suffices_tac (rapf "(one1:1->oneone) o (to1(A,1)) o (x0:1->A) = (one2:1->oneone) o (to1(B,1)) o (x0':1->B)")
>-- (assume_tac (specl (List.map rastt ["1","id(1)","to1(A, 1) o (x0:1->A)"]) to1_unique) >> assume_tac (specl (List.map rastt ["1","id(1)","to1(B, 1) o (x0':1->B)"]) to1_unique)>>
arw_tac[] >> rw_tac[idR]) >>
suffices_tac (rapf "copa(i1:A->AB, i2:B->AB, ((one1:1->oneone) o to1(A, 1)), ((one2:1->oneone) o to1(B, 1))) o i1 o (x0:1->A) = copa(i1, i2, (one1 o to1(A, 1)), (one2 o to1(B, 1))) o i2 o x0'")
>-- (rw_tac[GSYM o_assoc] >> arw_tac[]) >>
arw_tac[]
)
(rapg "!A B AB i1:A->AB i2:B->AB. iscopr(i1,i2) ==> !x0:1->A x0':1->B.~(i1 o x0 = i2 o x0')");
val from_cop_eq = proved_th $
e0
(repeat strip_tac >>
suffices_tac (rapf "f = copa(i1:A->AB,i2:B->AB,(f:AB->X) o i1,f o i2) & g = copa(i1,i2,(g:AB->X) o i1,g o i2)")
>-- (strip_tac >> once_arw_tac[] >>
pop_assum (K all_tac) >> pop_assum (K all_tac) >> arw_tac[]) >> strip_tac
>> match_mp_tac from_copr_components >> arw_tac[])
(form_goal “!A B AB i1:A->AB i2:B->AB.iscopr(i1,i2) ==> !X f:AB ->X g. f o i1 = g o i1 & f o i2 = g o i2 ==> f = g”);
(*!X t. t∶ one → X ==> i1 X X o t <> i2 X X o t*)
val i1_i2_disjoint = proved_th $
e0
(repeat strip_tac >> match_mp_tac prop_5_lemma >> arw_tac[])
(rapg "!X XX i1:X->XX i2:X->XX. iscopr(i1,i2) ==> !t:1->X. ~(i1 o t = i2 o t)");
val pr_with_one = proved_th $
e0
(strip_tac >> rw_tac[pr_def] >> repeat strip_tac >> exists_tac (rastt "f:X->A") >>
repeat strip_tac >> dimp_tac (* 2 *)
>-- (strip_tac >> full_simp_tac[idL]) >>
strip_tac >> arw_tac[idL] >> rw_tac[specl (List.map rastt ["X","g:X->1"]) to1_unique])
(rapg "!A. ispr(id(A),to1(A,1))");
(*
∀A B C D P Q f g i j. f∶ A → C ∧ g∶ B → D ∧ i∶ C → P ∧ j∶ D → Q ⇒
⟨i o p1 C D,j o p2 C D⟩ o ⟨f o p1 A B, g o p2 A B⟩ =
⟨i o f o p1 A B, j o g o p2 A B⟩*)
fun dest_form_view fv = case fv of vQ("?",n,s,b) => (n,s,b)
val p12_of_pa = pa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:X->A","g:X->B","pa(p1:AB->A,p2:AB->B,f:X->A,g)"]) |> C mp $ refl (rastt "pa(p1:AB->A,p2:AB->B,f:X->A,g)")
|> gen_all |> disch_all |> gen_all
val p1_of_pa = pa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:X->A","g:X->B","pa(p1:AB->A,p2:AB->B,f:X->A,g)"]) |> C mp $ refl (rastt "pa(p1:AB->A,p2:AB->B,f:X->A,g)")
|> conjE1 |> gen_all |> disch_all |> gen_all
val p2_of_pa = pa_def |> iffRL |> spec_all |> undisch
|> specl (List.map rastt ["X","f:X->A","g:X->B","pa(p1:AB->A,p2:AB->B,f:X->A,g)"]) |> C mp $ refl (rastt "pa(p1:AB->A,p2:AB->B,f:X->A,g)")
|> conjE2 |> gen_all |> disch_all |> gen_all
(*AQ: massive conditional rw here *)
(*AQ: why is pb proof so slow*)
val parallel_p_compose = proved_th $
e0
(strip_tac >> irule to_p_eq >>
exists_tac (rastt "P") >> exists_tac (rastt "pP:PQ->P") >> exists_tac (rastt "Q") >>
exists_tac (rastt "pQ:PQ->Q") >> strip_tac (* 2 *)
>-- (drule p2_of_pa >> arw_tac[] >> arw_tac[GSYM o_assoc] >>
pick_x_assum (rapf "ispr(pC:CD->C, pD:CD->D)") assume_tac >>
drule p2_of_pa >> arw_tac[o_assoc]) >>
drule p1_of_pa >> arw_tac[GSYM o_assoc] >>
pick_x_assum (rapf "ispr(pC:CD->C, pD:CD->D)") assume_tac >>
drule p1_of_pa >> arw_tac[o_assoc]
)
(form_goal “ispr(pM:MN->M,pN:MN->N1)& ispr(pC:CD->C,pD:CD->D) & ispr(pP:PQ->P,pQ:PQ->Q) ==> pa(pP,pQ, i o pC,j o pD) o pa(pC,pD,f o pM,g o pN) = pa(pP,pQ,i o f o pM,j o g o pN)”);
val parallel_p_one_side =
proved_th $
e0
(strip_tac >> by_tac (rapf "ispr(pA:AB ->A,pB:AB->B) & ispr(pA':AC->A,pC:AC->C) & ispr(pA'':AD->A,pD:AD->D)")
>-- arw_tac[] >>
drule (parallel_p_compose |> undisch |> gen_all |> disch_all |> gen_all) >>
first_x_assum (specl_then (List.map rastt ["id(A)","f:B->C","id(A)","g:C->D"]) assume_tac)>>
full_simp_tac[idL])
(rapg "ispr(pA:AB ->A,pB:AB->B) & ispr(pA':AC->A,pC:AC->C) & ispr(pA'':AD->A,pD:AD->D) ==>pa(pA'',pD,pA,(g:C->D) o (f:B->C) o pB) = pa(pA'',pD,pA',g o pC) o pa(pA',pC,pA,f o pB)");
val parallel_p_one_side' = proved_th $
e0
(strip_tac >> assume_tac (undisch parallel_p_one_side) >> arw_tac[o_assoc])
(rapg "ispr(pA:AB ->A,pB:AB->B) & ispr(pA':AC->A,pC:AC->C) & ispr(pA'':AD->A,pD:AD->D) ==>pa(pA'',pD,pA,((g:C->D) o (f:B->C)) o pB) = pa(pA'',pD,pA',g o pC) o pa(pA',pC,pA,f o pB)");
(*∀A B C D E f g h.
f∶B → C ∧ g∶C → D /\ h ∶ D → E ⇒
⟨p1 A B,(h o g ∘ f) ∘ p2 A B⟩ =
⟨p1 A D, h ∘ p2 A D⟩ o ⟨p1 A C,g ∘ p2 A C⟩ ∘ ⟨p1 A B,f ∘ p2 A B⟩
*)
val parallel_p_one_side = parallel_p_one_side |> undisch|> gen_all |> disch_all |> gen_all
val parallel_p_one_side_three = proved_th $
e0
(strip_tac >> by_tac (rapf "ispr(pA:AB ->A,pB:AB->B) & ispr(pA'':AD->A,pD:AD->D) & ispr(pA''':AE->A,pE:AE->E)") >-- arw_tac[] >>
drule parallel_p_one_side >> first_x_assum (specl_then (List.map rastt ["(g:C->D) o (f:B->C)","h:D->E"]) assume_tac) >> full_simp_tac[o_assoc] >>
suffices_tac (rapf "pa(pA'':AD->A, pD:AD->D, pA:AB->A, (g:C->D) o (f:B->C) o (pB:AB->B)) = pa(pA'', pD, pA':AC->A, (g o pC)) o pa(pA', pC:AC->C, pA, f o pB)")
>-- (strip_tac >> arw_tac[]) >>
by_tac (rapf "ispr(pA:AB ->A,pB:AB->B) & ispr(pA':AC->A,pC:AC->C) & ispr(pA'':AD->A,pD:AD->D)") >-- arw_tac[] >>
drule parallel_p_one_side >> arw_tac[])
(rapg "ispr(pA:AB ->A,pB:AB->B) & ispr(pA':AC->A,pC:AC->C) & ispr(pA'':AD->A,pD:AD->D) & ispr(pA''':AE->A,pE:AE->E)==>pa(pA''',pE,pA,(h:D->E) o (g:C->D) o (f:B->C) o pB) = pa(pA''',pE,pA'',h o pD) o pa(pA'',pD,pA',g o pC) o pa(pA',pC,pA,f o pB)");
(*
∀X A B C f g. (f∶X → ((A×B)×C) ∧ g∶X → ((A×B)×C) ∧
(p1 A B) o (p1 (A×B) C) o f = (p1 A B) o (p1 (A×B) C) o g ∧
(p2 A B) o (p1 (A×B) C) o f = (p2 A B) o (p1 (A×B) C) o g ∧
(p2 (A×B) C) o f = (p2 (A×B) C) o g) ⇒ f = g
*)
val iterated_p_eq_applied = proved_th $
e0
(repeat strip_tac >> irule to_p_eq >>
exists_tac (rastt "PQ") >> exists_tac (rastt "pPQ:PQC->PQ") >> exists_tac (rastt "C") >>
exists_tac (rastt "pC:PQC->C") >> arw_tac[] >> irule to_p_eq >>
exists_tac (rastt "P") >>
exists_tac (rastt "pP:PQ->P")>>
exists_tac (rastt "Q") >> exists_tac (rastt "pQ:PQ->Q") >> arw_tac[])
(form_goal “ispr(pPQ:PQC->PQ,pC:PQC->C) & ispr(pP:PQ->P,pQ:PQ->Q) ==> !X f:X->PQC g. pP o pPQ o f = pP o pPQ o g & pQ o pPQ o f = pQ o pPQ o g & pC o f = pC o g ==> f = g”);
(*id N = N_ind z s*)
val N_ind_Z_s_id = constN_def |> specl (List.map rastt ["N","ZERO","SUC","id(N)"])
|> rewr_rule [idL,idR]
(*∀f. f∶ N → N ∧ f o z = z ∧ f o s = s o f ⇒ f = id N*)
val comm_with_s_id =
constN_def |> specl (List.map rastt ["N","ZERO","SUC","f:N->N"])
|> iffLR |> undisch |> GSYM |> trans N_ind_Z_s_id |> disch_all |> gen_all
(*∀A B f g. f∶ A → B ∧ g∶ A → B ∧ ⟨id A,f⟩ = ⟨id A,g⟩ ⇒ f = g*)
val to_p_components = to_p_component;
(*AQ: once_arw_tac[] can cause : # # # # # Exception- ERR ("extra variable involved", [A -> A], [f], []) raised, how to prevent this?*)
val to_p_eq_one_side = proved_th $
e0
(repeat strip_tac >> drule p2_of_pa >> pop_assum (assume_tac o GSYM) >>
first_assum (specl_then (List.map rastt ["A","id(A)","f:A->B"]) assume_tac) >>
first_x_assum (specl_then (List.map rastt ["A","id(A)","g:A->B"]) assume_tac) >>
once_arw_tac[] >> pop_assum (K all_tac) >> pop_assum (K all_tac) >> arw_tac[])
(rapg "ispr(p1:AB ->A,p2:AB->B) ==> !f:A->B g. pa(p1,p2,id(A),f) = pa(p1,p2,id(A),g) ==> f = g")
val _ = new_pred "isinc" [("a",mk_ar_sort (mk_ob "A") (mk_ob "X")),
("b",mk_ar_sort (mk_ob "B") (mk_ob "X"))]
(*is_inc a b A ⇔ is_subset a A ∧ is_subset b A ∧ ∃h. h∶(dom a) → (dom b) ∧ b o h = a*)
val isinc_def = read_axiom "!X A B a b.isinc(a:A->X,b:B->X) <=> ismono(a) & ismono(b) & ?h:A->B. b o h = a"
val ismono_applied = ismono_def |> iffRL
val ismono_property = ismono_def |> iffLR
(*∀A B m i. m∶ A → B ∧ i∶ B → A ∧ (i o m) = id A ⇒ is_mono m*)
val post_inv_mono = proved_th $
e0
(repeat strip_tac >> match_mp_tac ismono_applied >> repeat strip_tac >>
by_tac (rapf "(i:B->A) o (m:A->B) o (g:X->A) = i o m o h") >-- arw_tac[] >>
pop_assum mp_tac >> rw_tac[GSYM o_assoc] >> arw_tac[idL] >> strip_tac >>
arw_tac[])
(rapg "!A B m:A->B i:B->A. i o m = id(A) ==> ismono(m)")
(*is_epi f ⇔ ∀g1 g2. dom g1 = dom g2 ∧ cod g1 = cod g2 ∧ dom g1 = cod f ∧ g1 o f = g2 o f ⇒ g1 = g2*)
val _ = new_pred "isepi" [("e",mk_ar_sort (mk_ob "A") (mk_ob "B"))]
val isepi_def = read_axiom "!A B e:A->B. isepi(e) <=> !X f:B->X g. f o e = g o e ==> f = g"
val isepi_applied = isepi_def |> iffRL
val isepi_property = isepi_def |> iffLR
(*∀A B e i. e∶ A → B ∧ i∶ B → A ∧ e o i = id B ⇒ is_epi e*)
val pre_inv_epi = proved_th $
e0
(repeat strip_tac >> match_mp_tac isepi_applied >> repeat strip_tac >>
by_tac (rapf "(f:B->X) o (e:A->B) o (i:B->A) = g o e o i") >-- arw_tac[GSYM o_assoc] >>
pop_assum mp_tac >> arw_tac[idR])
(rapg "!A B e:A->B i:B->A. e o i = id(B) ==> isepi(e)")
(*is_pb P p q f g <=> cod f = cod g /\ p∶ P → dom f ∧ q∶ P → dom g /\
f o p = g o q ∧
(∀A u v. u∶ A → dom f ∧ v∶ A → dom g ∧ f o u = g o v ⇒
∃!a. a∶ A → P ∧ p o a = u ∧ q o a = v)*)
val _ = new_pred "ispb" [("f",mk_ar_sort (mk_ob "X") (mk_ob "Z")),
("g",mk_ar_sort (mk_ob "Y") (mk_ob "Z")),
("p",mk_ar_sort (mk_ob "P") (mk_ob "X")),
("q",mk_ar_sort (mk_ob "P") (mk_ob "Y"))]
val ispb_def = read_axiom "!X Z f:X->Z Y g:Y->Z P p:P->X q:P ->Y. ispb(f,g,p,q) <=> f o p = g o q & (!A u:A->X v:A->Y. f o u = g o v ==> ?a: A ->P. !a':A->P. p o a' = u & q o a' = v <=> a' = a)"
(*∀A B f g.
f∶A → B ∧ g∶A → B ⇒ f ∘ eqa f g = g ∘ eqa f g*)
val eq_equality = eqind_def |> strip_all_and_imp |> conjE1 |> disch_all |> gen_all
val coeq_equality = coeqind_def |> strip_all_and_imp |> conjE1 |> disch_all |> gen_all
(*f A B. f∶ A → B ==> ?ki. ki∶ coeqo f f → B /\ ki o (coeqa f f) = id B*)
val iscoeq_def = read_axiom "!A B f:A->B g:A->B cE ce:B->cE. iscoeq(ce,f,g) <=> ce o f = ce o g & !X x:B->X. x o f = x o g ==> (?x0:cE ->X. !x0'. x0' o ce = x <=> x0' = x0)"
(*TO-DO:
A , B , X ,
(f : A -> B), (x : B -> X), (x0' : B -> X)
----------------------------------------------------------------------
x0' = x <=> x0' = x
cannot be solved by rw_tac
rw_tac[frefl (mk_fvar "f0")]
loops
understand why it is finished by strip_tac solved
*)
val coeq_of_equal = proved_th $
e0
(rw_tac[iscoeq_def] >> repeat strip_tac >> exists_tac (rastt "x:B->X") >>
strip_tac >> rw_tac[idR])
(rapg "iscoeq(id(B),f:A->B,f:A->B)")
(*∀A B f g. f∶ A → B ∧ g∶ A → B ⇒ is_mono (eqa f g)*)
val is_eqind = eqind_def |> strip_all_and_imp |> conjE2 |> iffLR |> strip_all_and_imp |> gen_disch_all
val is_coeqind = coeqind_def |> strip_all_and_imp |> conjE2 |> iffLR |> strip_all_and_imp |> gen_disch_all
val eqa_is_mono = proved_th $
e0
(rw_tac[ismono_def] >> repeat strip_tac >>
suffices_tac (rapf "h:X->E0 = eqind(e0:E0->A0,f0:A0->B0,g0, e0 o h) & g:X->E0 = eqind(e0:E0->A0,f0:A0->B0,g0, e0 o h) ")
>-- (strip_tac >> once_arw_tac[] >> rw_tac[]) >>
drule eq_equality >>
strip_tac (* 2 *)
>> (irule is_eqind >> arw_tac[] >> arw_tac[GSYM o_assoc])
)
(rapg "iseq(e0:E0->A0,f0:A0->B0,g0) ==> ismono(e0)")
(*∀A B f g. f∶ A → B ∧ g∶ A → B ⇒ is_epi (coeqa f g)*)
val coeqa_is_epi = proved_th $
e0
(rw_tac[isepi_def] >> repeat strip_tac >>
suffices_tac (rapf "f:cE0->X = coeqind(ce0:B0->cE0,f0:A0->B0,g0, g o ce0) & g:cE0->X = coeqind(ce0:B0->cE0,f0:A0->B0,g0, g o ce0)")
>-- (strip_tac >> once_arw_tac[] >> rw_tac[]) >>
drule coeq_equality >>
strip_tac (* 2 *)
>> (irule is_coeqind >> arw_tac[o_assoc])
)
(rapg "iscoeq(ce0:B0->cE0,f0:A0->B0,g0) ==> isepi(ce0)")
(*∀X Y Z f g. f∶ X → Z ∧ g∶ Y → Z ⇒ ∃P p q. p∶ P → X ∧ q∶ P → Y ∧ f o p = g o q ∧
(∀A u v. u∶ A → X ∧ v∶ A → Y ∧ f o u = g o v ⇒
∃!a. a∶ A → P ∧ p o a = u ∧ q o a = v)*)
val eqind_def' = eqind_def |> strip_all_and_imp |> conjE2 |> disch_all |> gen_all
(* TO-DO: drule bug:
first_x_assum (specl_then (List.map rastt ["A","pa(pX:XY->X, pY:XY->Y, u:A->X, v)"]) assume_tac) >> first_x_assum drule--cannot find it...
*)
val ispb_def_alt = proved_th $
e0
(repeat strip_tac >> rw_tac[ispb_def] >> dimp_tac >> strip_tac >> arw_tac[] >>
repeat strip_tac >> first_x_assum drule >>
first_x_assum (x_choose_then "a" assume_tac) >> exists_tac (rastt "a:A->P") >>
repeat strip_tac (* 4 *)
>-- (pop_assum (assume_tac o (fn th => th |> allE (rastt "a:A->P")
|> (C dimp_mp_r2l) (refl (rastt "a:A->P")))) >>
arw_tac[])
>-- (pop_assum (assume_tac o (fn th => th |> allE (rastt "a:A->P")
|> (C dimp_mp_r2l) (refl (rastt "a:A->P")))) >>
arw_tac[])
>-- (suffices_tac (rapf "a1 = a & a2 = a:A->P")
>-- (strip_tac >> arw_tac[]) >>
strip_tac >> first_x_assum (match_mp_tac o iffLR) >> arw_tac[]) >>
dimp_tac >> strip_tac >> arw_tac[] >> pop_assum_list (map_every strip_assume_tac) >>
first_x_assum match_mp_tac >> arw_tac[])
(rapg "!X Z f:X -> Z Y g : Y -> Z P p : P -> X q : P -> Y.\
\ ispb(f, g, p, q) <=> \
\ f o p = g o q & \
\ !A u : A -> X v : A -> Y. \
\ f o u = g o v ==> ?a : A -> P. p o a = u & q o a = v & !a1 : A -> P a2:A->P. p o a1 = u & q o a1 = v& p o a2 = u & q o a2 = v ==> a1 = a2")
val long_induced_map = rastt "eqind(e:E->XY, (f:X->Z) o pX, (g:Y->Z) o pY, pa(pX:XY->X, pY:XY->Y, u:A->X, v:A->Y))"
(*TO-DO: match_mp_bug e o a1 = e o a2 ==> a1 = a2 ismono_property h is double bind to a1 and a2 because ismono_property is not in correct order!!!!!*)
val pb_exists = proved_th $ (*val (ct,asl,w) = cg $*)
e0
(rw_tac[ispb_def_alt] >> repeat strip_tac >>
(specl_then (List.map rastt ["X","Y"])
(x_choosel_then ["XY","pX","pY"] assume_tac)) pr_ex >>
(specl_then (List.map rastt ["XY","Z","(f:X->Z)o (pX:XY->X)","(g:Y->Z)o (pY:XY->Y)"])
(x_choosel_then ["E","e"] assume_tac)) eq_ex >>
exists_tac (rastt "E") >> exists_tac (rastt "(pX:XY->X) o (e:E->XY)") >>
exists_tac (rastt "(pY:XY->Y) o (e:E->XY)") >>
by_tac (rapf "(f:X->Z) o pX o e = (g:Y->Z) o pY o (e:E->XY)")
>-- (drule eq_equality >> arw_tac[GSYM o_assoc]) >>
arw_tac[] >> repeat strip_tac >> rw_tac[o_assoc] >>
by_tac (rapf "!c:A->XY. (pX:XY->X) o c = u:A->X & (pY:XY->Y) o c = v:A->Y <=> c = pa(pX,pY,u,v)")
>-- (drule pa_def >> strip_tac >> arw_tac[] (*>> dimp_tac >> rw_tac[]*)) >>
drule eqind_def' >>
by_tac (rapf "((f:X->Z) o pX) o pa(pX:XY->X, pY:XY->Y, u:A->X, v:A->Y) = (g o pY) o pa(pX, pY, u, v)")
>-- (rw_tac[o_assoc] >> drule p12_of_pa >> arw_tac[]) >>
first_x_assum (specl_then (List.map rastt ["A","pa(pX:XY->X, pY:XY->Y, u:A->X, v)"]) assume_tac) >> first_x_assum drule >>
exists_tac long_induced_map >>
by_tac (rapf "pX o e o eqind(e:E->XY, (f:X->Z) o pX, (g:Y->Z) o pY, pa(pX:XY->X, pY:XY->Y, u:A->X, v:A->Y)) = u & pY o e o eqind(e:E->XY, (f:X->Z) o pX, (g:Y->Z) o pY, pa(pX:XY->X, pY:XY->Y, u:A->X, v:A->Y)) = v")
>-- (pop_assum (assume_tac o (fn th => th |> allE long_induced_map |> (C dimp_mp_r2l) (refl long_induced_map))) >> arw_tac[]) >> repeat strip_tac (* 3 *)
>-- arw_tac[]
>-- arw_tac[] >>
suffices_tac (rapf "e o a1 = (e:E->XY) o (a2:A->E)")
>-- (suffices_tac (rapf "ismono(e:E->XY)")
>-- (strip_tac >> drule ismono_property >>
strip_tac >> first_x_assum drule >> arw_tac[]) >>
drule eqa_is_mono >> arw_tac[]) >>
suffices_tac (rapf "(e:E->XY) o a1 = pa(pX:XY->X, pY:XY->Y, u:A->X, v:A->Y) & e o a2 = pa(pX, pY, u, v)")
>-- (strip_tac >> arw_tac[]) >>
strip_tac (* 2 *)
>> (first_x_assum (match_mp_tac o iffLR) >> arw_tac[])
)
(rapg "!X Z f:X->Z Y g:Y->Z.?P p:P->X q. ispb(f,g,p,q)");
(*∀X Y Z f g. g∶ Y → Z ∧ f∶ X → Z ⇒ ∃P p q. p∶ P → X ∧ q∶ P → Y ∧ f o p = g o q ∧
(∀A u v. u∶ A → X ∧ v∶ A → Y ∧ f o u = g o v ⇒
∃a. a∶ A → P ∧ p o a = u ∧ q o a = v)*)
val pb_fac_exists = proved_th $
e0
(repeat strip_tac >>
x_choosel_then ["P","p","q"] assume_tac (pb_exists |> rewr_rule [ispb_def_alt] |> strip_all_and_imp) >>
exists_tac (rastt "P") >> exists_tac (rastt "p:P->X") >> exists_tac (rastt "q:P->Y") >>
pop_assum strip_assume_tac >> arw_tac[] >> repeat strip_tac >> first_x_assum drule >>
arw_tac[] >> pop_assum (x_choosel_then ["a"] assume_tac) >> exists_tac (rastt "a:A->P")>>
arw_tac[])
(rapg "!X Z f:X->Z Y g:Y->Z.?P p:P->X q:P->Y. f o p = g o q & !A u:A->X v:A->Y. f o u = g o v ==> ?a:A->P. p o a = u & q o a = v");
val ispb_def_alt' = proved_th $
e0
(repeat strip_tac >> rw_tac[ispb_def_alt] >> dimp_tac >> strip_tac >> arw_tac[] >>
repeat strip_tac >> first_x_assum drule (* 3 *)
>-- (pop_assum (x_choosel_then ["a"] assume_tac) >> exists_tac (rastt "a:A->P") >>
arw_tac[])
>-- (pop_assum (x_choosel_then ["a"] strip_assume_tac) >>
first_x_assum match_mp_tac >> arw_tac[]) >>
pop_assum strip_assume_tac >> exists_tac (rastt "a:A->P") >>
arw_tac[])
(rapg "!X Z f:X -> Z Y g : Y -> Z P p : P -> X q : P -> Y. ispb(f, g, p, q) <=> f o p = g o q & !A u : A -> X v : A -> Y. f o u = g o v ==> (?a : A -> P. p o a = u & q o a = v) & !a1 : A -> P a2:A->P. p o a1 = u & q o a1 = v& p o a2 = u & q o a2 = v ==> a1 = a2")
(*!P p q f g. is_pb P p q f g /\ is_mono g ==> is_mono p*)
val pb_equality = ispb_def_alt' |> iffLR |> strip_all_and_imp
|> conjE1 |> disch_all|> gen_all
val pb_fac_unique =
ispb_def_alt' |> iffLR |> strip_all_and_imp |> conjE2
|> strip_all_and_imp |> conjE2
|> disch (rapf "(f:X->Z) o (u:A->X) = (g:Y->Z) o v")
|> gen_all |> disch_all |> gen_all
val pb_mono_mono = proved_th $
e0
(repeat strip_tac >> match_mp_tac ismono_applied >> repeat strip_tac >>
by_tac (rapf "(q:P->Y) o (g':X'->P) = q o h")
>-- (suffices_tac (rapf "(g:Y->Z) o (q:P->Y) o (g':X'->P) = g o q o h")
>-- (drule ismono_property >> strip_tac >> first_x_assum drule >>
arw_tac[]) >>
drule (GSYM pb_equality) >> arw_tac[GSYM o_assoc] >> arw_tac[o_assoc]) >>
drule pb_fac_unique >>
suffices_tac (rapf "(f:X->Z) o (p:P->X) o (h:X'->P) = (g:Y->Z) o q o h")
>-- (strip_tac >> first_x_assum drule >> first_x_assum match_mp_tac >> arw_tac[]) >>
drule pb_equality >> arw_tac[GSYM o_assoc]
(*drule ismono_property >> *))
(rapg "ispb(f:X->Z,g:Y->Z,p:P->X,q:P->Y) ==> ismono(g) ==> ismono(p)")
(*∀A B f. f∶ A → B ∧ ¬(A ≅ zero) ⇒ ∃g. g∶B → A ∧ f ∘ g ∘ f = f*)
val non_zero_pinv = proved_th $
e0
(strip_tac >> drule ax6 >> pop_assum strip_assume_tac >> strip_tac >>
match_mp_tac ax5 >> exists_tac one >> exists_tac (rastt "x:1->A") >>
rw_tac[const1_def])
(rapg "~is0(A)==>!B f:A->B.?g:B->A. f o g o f = f")
(*∀A B f g. f∶ A → B ∧ g∶B → A ∧ is_epi f ∧ f ∘ g ∘ f = f ⇒ f o g = id B*)
val epi_pinv_pre_inv = proved_th $
e0
(repeat strip_tac >> drule isepi_property >> first_x_assum match_mp_tac >>
arw_tac[o_assoc,idL])
(rapg "!A B f:A->B. isepi(f) ==> !g:B->A. f o g o f = f ==> f o g = id(B)")
(*∀A B f g. f∶ A → B ∧ g∶B → A ∧ is_mono f ∧ f ∘ g ∘ f = f ⇒
g o f = id A*)
val mono_pinv_post_inv = proved_th $
e0
(repeat strip_tac >> drule ismono_property >> first_x_assum match_mp_tac >>
arw_tac[o_assoc,idR])
(rapg "!A B f:A->B. ismono(f) ==> !g:B->A. f o g o f = f ==> g o f = id(A)")
(*∀A B f. f∶ A → B ∧ is_epi f ∧ ¬(A ≅ zero) ⇒ ∃g. g∶ B → A ∧ f o g = id B*)
val epi_non_zero_pre_inv = proved_th $
e0
(repeat strip_tac >> drule non_zero_pinv >>
first_x_assum (specl_then (List.map rastt ["B","f:A->B"]) strip_assume_tac) >>
drule epi_pinv_pre_inv >> first_x_assum drule >> exists_tac (rastt "g:B->A") >>
arw_tac[])
(rapg "!A B f:A->B. isepi(f) ==> (~is0(A)) ==> ?g:B->A. f o g = id(B)")
(*∀A B f. f∶ A → B ∧ is_mono f ∧ ¬(A ≅ zero) ⇒ ∃g. g∶ B → A ∧ g o f = id A*)
val mono_non_zero_post_inv = proved_th $
e0
(repeat strip_tac >> drule non_zero_pinv >>
first_x_assum (specl_then (List.map rastt ["B","f:A->B"]) strip_assume_tac) >>
drule mono_pinv_post_inv >> first_x_assum drule >> exists_tac (rastt "g:B->A") >>
arw_tac[])
(rapg "!A B f:A->B. ismono(f) ==> ~is0(A) ==> ?g:B->A. g o f = id(A)")
(*∀A B C f g. is_mono f ∧ is_mono g ∧ f∶ A → B ∧ g∶ B → C ⇒ is_mono (g o f)*)
val o_mono_mono = proved_th $
e0
(repeat strip_tac >> match_mp_tac ismono_applied >> repeat strip_tac >>
drule ismono_property >> full_simp_tac[o_assoc] >> first_x_assum drule >>
pick_x_assum (rapf "ismono(g:B->C)") (K all_tac) >>
drule ismono_property >> first_x_assum drule >> arw_tac[])
(rapg "!A B f:A->B. ismono(f) ==> !C g:B->C. ismono(g) ==> ismono(g o f)")
(*∀f A B. f∶ A → B ⇒
(is_iso f ⇔
∃f'. f'∶ B → A ∧ f' o f = id A ∧ f o f' = id B)*)
val _ = new_pred "isiso" [("f",mk_ar_sort (mk_ob "A") (mk_ob "B"))]
val isiso_def = read_axiom "!A B f:A->B. isiso(f) <=> ?f':B->A. f' o f = id(A) & f o f' = id(B)"
(*!A B X f i. is_iso i /\ is_mono f /\ f∶ A → B /\ i∶ X → A ==> (is_mono (f o i))*)
val iso_is_mono = proved_th $
e0
(rw_tac[isiso_def,ismono_def] >> repeat strip_tac >>
suffices_tac (rapf "(f':B->A) o (f:A->B) o (g:X->A) = f' o f o h")
>-- (arw_tac[GSYM o_assoc,idL] >> strip_tac >> arw_tac[]) >>
arw_tac[])
(rapg "isiso(f) ==> ismono(f)")
val mono_o_iso_mono = proved_th $
e0
(repeat strip_tac >> irule o_mono_mono >> drule iso_is_mono >>
arw_tac[])
(rapg "!X A i:X->A.isiso(i) ==> !B f:A->B. ismono(f) ==> ismono(f o i)")
(*∀A B C f m. f∶ A → B ∧ m∶ B → C ∧ is_mono (m o f) ⇒ is_mono f*)
val o_mono_imp_mono = proved_th $
e0
(repeat strip_tac >> match_mp_tac ismono_applied >> repeat strip_tac >>
drule ismono_property >> first_x_assum match_mp_tac >> arw_tac[o_assoc])
(rapg "!A B f:A->B C m:B->C. ismono(m o f) ==> ismono(f)")
(*∀A B C f e. f∶ A → B ∧ e∶ C → A ∧ is_epi (f o e) ⇒ is_epi f*)
val o_epi_imp_epi =
proved_th $
e0
(repeat strip_tac >> match_mp_tac isepi_applied >> repeat strip_tac >>
drule isepi_property >> first_x_assum match_mp_tac >> arw_tac[GSYM o_assoc])
(rapg "!A B f:A->B C e:C->A. isepi(f o e) ==> isepi(f)")
(*∀A B f g. f∶ A → B ∧ g∶ A → B ∧ (∀a. a∶ one → A ⇒ f o a = g o a) ⇒ f = g*)
val fun_ext = proved_th $
e0
(repeat strip_tac >> ccontra_tac >> drule ax4 >> pop_assum strip_assume_tac >>
first_x_assum (specl_then [rastt "a:1->A"] assume_tac) >> full_simp_tac[])
(rapg "!A B f:A->B g. (!a:1->A. f o a = g o a) ==> f = g")
(*∀A B f. f∶ A → B ∧ (∀b. b∶ one → B ⇒ ∃b0. b0∶ one → A ∧ f o b0 = b) ⇒ is_epi f*)
val surj_is_epi = proved_th $
e0
(repeat strip_tac >> match_mp_tac isepi_applied >> repeat strip_tac >>
match_mp_tac fun_ext >> strip_tac >>
first_x_assum (specl_then [rastt "a:1->B"] assume_tac) >>
pop_assum (strip_assume_tac o GSYM) >> arw_tac[GSYM o_assoc])
(rapg "!A B f:A->B. (!b:1->B. ?b0:1->A. f o b0 = b) ==> isepi(f)")
(*edited SUC ...*)
(*∀A B. A ≅ B ⇔ ∃f. f∶ A → B ∧ is_iso f*)
val _ = new_pred "areiso" [("A",mk_ob_sort),("B",mk_ob_sort)]
val areiso_def = read_axiom "!A B. areiso(A,B) <=> ?f:A->B g:B->A. f o g = id(B) & g o f = id(A)"
val areiso_isiso = proved_th $
e0
(rw_tac[areiso_def,isiso_def] >> dimp_tac >> strip_tac >>
pop_assum strip_assume_tac (*2 *)
>-- (exists_tac (rastt "f:A->B") >> exists_tac (rastt "g:B->A") >> arw_tac[]) >>
exists_tac (rastt "f:A->B") >> exists_tac (rastt "f':B->A") >> arw_tac[])
(rapg "areiso(A,B) <=> ?f:A->B. isiso(f)")
(*∀A B. A ≅ B ⇔ ∃f. f∶ B → A ∧ is_iso f*)
val areiso_isiso' = proved_th $
e0
(rw_tac[areiso_def,isiso_def] >> dimp_tac >> strip_tac >>
pop_assum strip_assume_tac (*2 *)
>-- (exists_tac (rastt "g:B->A") >> exists_tac (rastt "f:A->B") >> arw_tac[]) >>
exists_tac (rastt "f':A->B") >> exists_tac (rastt "f:B->A") >> arw_tac[])
(rapg "areiso(A,B) <=> ?f:B->A. isiso(f)")
val iso_is_epi = proved_th $
e0
(rw_tac[isiso_def,isepi_def] >> repeat strip_tac >>
suffices_tac (rapf "(f'':B->X) o (f:A->B) o (f':B->A) = g o f o f'")
>-- arw_tac[idR] >>
arw_tac[GSYM o_assoc])
(rapg "isiso(f) ==> isepi(f)")
(*∀A B X f g i. is_iso i ∧ f∶ A→ B ∧ g∶ A → B ∧ i∶ X → A ∧ f o i = g o i ⇒ f = g*)
val o_iso_eq_eq = proved_th $
e0
(rw_tac[isiso_def] >> repeat strip_tac >>
suffices_tac (rapf "(f:A->B) o i o (f':A->X) = g o i o f'")
>-- arw_tac[idR] >> arw_tac[GSYM o_assoc])
(rapg "!X A i:X->A. isiso(i) ==> !B f:A->B g. f o i = g o i ==> f = g")
(*TO-DO: !B f:A->B g. f o i = g o i ==> f = g if g is not quantified, there will be a free variable g whose sort is bounded variables, is that bad?- yes, should edit parser*)
(*∀A B f g. A≅ zero ∧ f∶ A → B ∧ g∶ A → B ⇒ f = g*)
val is0_def = read_axiom "!zero. is0(zero) <=> !X. ?f0x:zero ->X.!f0x':zero->X. f0x'= f0x"
(*
AQ
(f : A -> 0), (f0x : 0 -> X), (f0x' : A -> X), (g : 0 -> A)
1.f o g = id(0)
2.g o f = id(A)
3.!(f0x' : 0 -> X). f0x'# = f0x
----------------------------------------------------------------------
f0x' o g o f = f0x o f o g o f ==> f0x' = f0x o f
arw[idR] on this loops, in iso_0_is0. below suff
*)
val iso_0_is0 = proved_th $
e0
(rw_tac[areiso_def,is0_def] >> strip_tac >> strip_tac >>
strip_assume_tac (const0_def |> rewr_rule [is0_def] |> allE (rastt "X")) >>
exists_tac (rastt "(f0x:0->X) o (f:A->0)") >> strip_tac >>
suffices_tac (rapf "f0x' o g o f = (f0x:0->X) o (f:A->0) o (g:0->A) o (f:A->0)")
>-- once_arw_tac[] >> rw_tac[idR] >>
rw_tac[GSYM o_assoc] >> once_arw[] >> rw[])
(form_goal “areiso(A,0) ==> is0(A)”)
(*
A , B ,
(f : A -> B), (g : A -> B)
1.areiso(A, 0)
2.is0(A)
3.!X (f0x : A -> X#). f0x# = from0(A, X#)
----------------------------------------------------------------------
f = g
arw_tac[]
loops
*)
val from_iso_zero_eq = proved_th $
e0
(repeat strip_tac >> drule iso_0_is0 >> drule (is0_def|> iffLR) >>
first_x_assum (specl_then [rastt "B"] strip_assume_tac) >>
once_arw_tac[] >> rw[])
(form_goal “!A. areiso(A,0) ==> !B f:A->B g. f = g”)
val iso_zero_is_zero = proved_th $
e0
(rw_tac[areiso_def,is0_def] >> strip_tac >> dimp_tac >> repeat strip_tac
>-- (assume_tac const0_def >> drule (iffLR is0_def) >>
first_x_assum (specl_then [rastt "X"] strip_assume_tac) >>
exists_tac (rastt "(f0x:0->X) o (f:A->0)") >> strip_tac >>
suffices_tac (rapf "f0x' o g o f = (f0x:0->X) o (f:A->0) o (g:0->A) o f")
>-- (once_arw[] >> rw[idR]) >>
rw[GSYM o_assoc] >> once_arw[] >> rw[]) >>
first_assum (specl_then [zero] strip_assume_tac) >>
exists_tac (rastt "f0x:A->0") >> assume_tac const0_def >> drule (iffLR is0_def) >>
first_assum (specl_then [rastt "A"] strip_assume_tac) >>
exists_tac (rastt "f0x':0->A") >>
first_assum (specl_then [zero] strip_assume_tac) >> once_arw_tac[] >>
last_assum (specl_then [rastt "A"] strip_assume_tac) >> once_arw_tac[] >>
rw_tac[])
(form_goal “!A.areiso(A,0) <=> is0(A)”)
(*∀f. f∶ one → one ⇒ f = id one*)
val is1_def = read_axiom "!one. is1(one) <=> !X. ?t1x:X ->one.!t1x':X->one. t1x' = t1x"
val one_to_one_id = proved_th $
e0
(strip_tac >> assume_tac const1_def >> drule (iffLR is1_def) >>
first_x_assum (specl_then [one] strip_assume_tac) >>
once_arw[] >> rw[])
(form_goal “!f:1->1.f = id(1)”)
(*∀f A B. is_epi f ∧ f∶ A → B ∧ ¬(B≅ zero) ⇒ ¬(A ≅ zero)*)
val no_epi_from_zero = proved_th $
e0
(repeat strip_tac >> ccontra_tac >>
specl_then [one,one] (x_choose_then "oneone" strip_assume_tac) copr_ex >>
suffices_tac (rapf "i1:1->oneone = i2")
>-- (drule i1_ne_i2 >> strip_tac) >>
assume_tac const1_def >> drule (iffLR is1_def) >> pop_assum strip_assume_tac >>
first_x_assum (specl_then [rastt "B"] strip_assume_tac) >>
suffices_tac (rapf "i1 o t1x = (i2:1->oneone) o (t1x:B->1)")
>-- (drule ax6 >> pop_assum strip_assume_tac >>
strip_tac >>
by_tac (rapf "(t1x:B->1) o x = id(1)")
>-- accept_tac (one_to_one_id|> allE (rastt "(t1x:B->1) o (x:1->B)")) >>
suffices_tac (rapf "i1 o t1x o (x:1->B) = (i2:1->oneone) o (t1x:B->1) o x")
>-- (once_arw[] >> rw[idR]) >>
rw[GSYM o_assoc] >> once_arw[] >> rw[]) >>
drule isepi_property >> first_x_assum match_mp_tac >>
drule (iffLR is0_def) >> first_x_assum (specl_then [rastt "oneone"] strip_assume_tac) >> once_arw[] >> rw[]
)
(form_goal “!A B f:A->B. isepi(f) ==> ~is0(B) ==> ~is0(A)”)