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Basics.sml
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1689 lines (1372 loc) · 51.2 KB
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(*char property*)
(*Theorem char_pb:
∀A X a. is_mono a ∧ a∶ A → X ⇒
∃h1 h2.
(pb1 (char a) (i2 one one)) ∘ h1 = a ∧
a ∘ h2 = pb1 (char a) (i2 one one) ∧
h1∶ A → pbo (char a) (i2 one one) ∧
h2∶ pbo (char a) (i2 one one) → A ∧
h1 ∘ h2 = id (pbo (char a) (i2 one one)) ∧
h2 ∘ h1 = id A*)
val dom_1_mono = proved_th $
e0
(rpt strip_tac >> irule ismono_applied >> rpt strip_tac >>
qspecl_then ["X'","g","h"] accept_tac to1_unique)
(form_goal “!X f:1->X. ismono(f)”);
(*Theorem pb_fac_exists':
∀X Y Z f g.
f∶X → Z ∧ g∶Y → Z ⇒
∀A u v.
u∶A → X ∧ v∶A → Y ∧ f ∘ u = g ∘ v ⇒
∃a. a∶A → pbo f g ∧ pb1 f g ∘ a = u ∧ pb2 f g ∘ a = v
Proof
rw[] >>
‘pb1 f g∶pbo f g → X ∧ pb2 f g∶pbo f g → Y ∧
f ∘ pb1 f g = g ∘ pb2 f g ∧
∀A u v.
u∶A → X ∧ v∶A → Y ∧ f ∘ u = g ∘ v ⇒
∃!a. a∶A → pbo f g ∧ pb1 f g ∘ a = u ∧ pb2 f g ∘ a = v’
by (irule pb_thm >> metis_tac[]) >>
fs[EXISTS_UNIQUE_ALT] >> metis_tac[]
QED*)
val pb_fac_exists' = proved_th $
e0
(rpt strip_tac >> fs[ispb_def] >> first_x_assum drule >>
pop_assum strip_assume_tac >> qexistsl_tac ["a"] >>
first_x_assum (qspecl_then ["a"] assume_tac) >> arw[])
(form_goal “!X Z f:X->Z Y g:Y->Z Pb p:Pb->X q:Pb->Y.ispb(f:X->Z,g:Y->Z,p,q) ==>
!A u v. f o u = g o v ==> ?a:A->Pb. p o a = u & q o a = v”);
val char_pb = proved_th $
e0
(rpt strip_tac >> irule prop_2_corollary_as_subobj >> arw[]>>
drule pb_mono_mono >>
qspecl_then ["two","i2"] assume_tac dom_1_mono >>
first_x_assum drule >> arw[] >>
rpt strip_tac (* 2 *) >--
(by_tac
(rapf "?y:1->Pb. pb1:Pb->X o y = a:A->X o x:1->A & pb2:Pb->1 o y = id(1)") >--
(drule pb_fac_exists' >> (*TO-DO:irule bug,should use irule works*)
first_x_assum (qspecl_then ["1","a o x","id(1)"] assume_tac) >>
rev_drule char_def >> first_x_assum drule >>
(*TO-DO: once have pull exists test here test pass*)
first_x_assum (qspecl_then ["a o x"] assume_tac) >>
qsuff_tac ‘char(i1, i2, a) o a o x = i2’
>-- (strip_tac >> fs[idR] >> qexistsl_tac ["a'"] >> arw[]) >>
pop_assum (assume_tac o GSYM) >> arw[] >>
qexistsl_tac ["x"] >> rw[]) >>
pop_assum strip_assume_tac >> qexistsl_tac ["y"] >> arw[]) >>
rev_drule char_def >> first_x_assum drule >>
drule (ispb_def |> iffLR) >> pop_assum strip_assume_tac >> arw[] >>
qby_tac ‘char(i1, i2, a) o pb1 o y = i2 o pb2 o y’
>-- (rw[GSYM o_assoc] >> arw[]) >>
pop_assum mp_tac >> once_rw[one_to_one_id] >> rw[idR])
(form_goal
“!A X a.ismono(a:A->X) ==>
!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!Pb pb1 pb2. ispb(char(i1,i2,a),i2,pb1,pb2) ==>
?h1 h2.pb1 o h1 = a & a o h2 = pb1 & h1 o h2 = id(Pb) & h2 o h1 = id(A)”);
val char_square = proved_th $
e0
(rpt strip_tac >> irule fun_ext >> rpt strip_tac >>
drule char_def >> first_x_assum drule >>
first_x_assum (qspecl_then ["a o a'"] assume_tac) >> rw[o_assoc] >>
once_rw[one_to_one_id] >> rw[idR] >>
pop_assum (assume_tac o GSYM) >>
arw[] >> qexistsl_tac ["a'"] >> rw[])
(form_goal
“!two i1 i2.iscopr(i1:1->two,i2:1->two) ==>!A X a:A->X. ismono(a) ==> char(i1,i2,a) o a = i2 o to1(A,1)”);
val pb_ex = pb_exists
val char_is_pb = proved_th $
e0
(rpt strip_tac >> drule char_pb >> first_x_assum drule >>
(qspecl_then ["X","two","char(i1, i2, a)","1","i2"] assume_tac) pb_ex >>
pop_assum (x_choosel_then ["Pb","pb1","pb2"] assume_tac) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
drule (ispb_def |> iffLR) >> pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["A","a","to1(A,1)"] assume_tac) >>
rw[ispb_def] >>
qby_tac ‘char(i1, i2, a) o a = i2 o to1(A, 1)’
>-- (qby_tac ‘char(i1, i2, a) o pb1 o h1 = i2 o pb2 o h1’
>-- (arw[GSYM o_assoc]) >> rfs[] >>
qspecl_then ["A","pb2 o h1","to1(A,1)"] assume_tac to1_unique >>
arw[]) >>
arw[] >> rpt strip_tac >>
(*here the goal is ?(a' : A' -> A). !(a' : A' -> A). a o a'# = u & to1(A, 1) o a'# = v <=>
a'# = a'# ... same name*)
suffices_tac (rapf "?a':A'->A. a:A->X o a' = u")
>-- (strip_tac >> qexistsl_tac ["a'"] >> strip_tac >> dimp_tac >>
strip_tac (* 2 *)
>-- (irule ismono_property >> qexistsl_tac ["X","a"] >>
arw[]) >>
arw[] >>
qspecl_then ["A'","to1(A, 1) o a'","v"] assume_tac to1_unique >>
arw[]) >>
drule (ispb_def |> iffLR) >> pop_assum strip_assume_tac >>
last_x_assum drule >> pop_assum (K all_tac) (* drule this line is useless, just to kill the assumption*) >>
first_x_assum (qspecl_then ["A'","u","v"] assume_tac) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
qexistsl_tac ["h2 o a'"] >>
first_x_assum (qspecl_then ["a'"] assume_tac) >> fs[] >>
arw[GSYM o_assoc])
(form_goal
“ismono(a:A->X) ==> iscopr(i1:1->two,i2) ==> ispb(char(i1,i2,a),i2,a,to1(A,1))”);
val char_is_pb_unique = proved_th $
e0
(rpt strip_tac >> irule fun_ext >> rpt strip_tac >>
irule one_to_two_eq >> qexistsl_tac ["i1","i2"] >>
drule char_is_pb >> first_x_assum drule >>
drule (iffLR ispb_def) >> pop_assum strip_assume_tac >>
rev_drule (iffLR ispb_def) >> pop_assum strip_assume_tac >>
arw[] >> dimp_tac >> strip_tac (* 2 *)
>-- (qby_tac ‘c o a' = i2 o id(1)’ >-- arw[idR] >>
first_x_assum drule >>
pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["a''"] assume_tac) >> fs[] >>
qby_tac ‘char(i1, i2, a) o a o a'' = i2 o to1(A, 1) o a''’
>-- arw[GSYM o_assoc] >> pop_assum mp_tac >>
once_rw[one_to_one_id] >> rw[idR] >> once_arw[] >> rw[]) >>
qby_tac ‘char(i1, i2, a) o a' = i2 o id(1)’ >-- arw[idR] >>
last_x_assum drule >>
pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["a''"] assume_tac) >> fs[] >>
qby_tac ‘c o a o a'' = i2 o to1(A, 1) o a''’
>-- arw[GSYM o_assoc] >> pop_assum mp_tac >>
once_rw[one_to_one_id] >> rw[idR] >> once_arw[] >> rw[]
)
(form_goal
“!A X a:A->X. ismono(a) ==>
!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!c:X->two. ispb(c,i2,a,to1(A,1)) ==> c = char(i1,i2,a)”);
val iso_subobj_same_char = proved_th $
e0
(rpt strip_tac >> irule char_is_pb_unique >> arw[] >>
drule char_square >> first_x_assum drule >>
(*qspecl_then ["X","two","char(i1,i2,a)",""] assume_tac pb_ex*)
rw[ispb_def] >>
qby_tac ‘char(i1, i2, a) o b = i2 o to1(B, 1)’
>-- (qpick_x_assum ‘a o h2 = b’ (assume_tac o GSYM) >>
once_arw[] >> rev_drule char_def >>
first_x_assum drule >> irule fun_ext >> strip_tac >>
rw[o_assoc] >>
first_x_assum (qspecl_then ["a o h2 o a'"] assume_tac) >>
once_rw[one_to_one_id] >> rw[idR] >>
pop_assum (assume_tac o iffLR) >> first_x_assum irule >>
qexistsl_tac ["h2 o a'"] >> rw[]) >>
arw[] >> rpt strip_tac >>
rev_drule char_is_pb >> first_x_assum drule >>
drule (iffLR ispb_def) >> pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["A'"] assume_tac) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
qexistsl_tac ["h1 o a'"] >> strip_tac >> dimp_tac >> strip_tac (* 2 *)
>-- (irule ismono_property >> qexistsl_tac ["X","b"] >>
arw[] >> first_x_assum (qspecl_then ["a'"] assume_tac) >> fs[] >>
qpick_x_assum ‘a o a' = u’ mp_tac >>
qpick_x_assum ‘b o h1 = a’ (assume_tac o GSYM) >>
once_arw[] >> rw[o_assoc]) >>
qspecl_then ["A'"] assume_tac to1_unique >>
first_x_assum (qspecl_then ["v","to1(B,1) o a''"] assume_tac) >>
arw[] >>
first_x_assum (qspecl_then ["a'"] assume_tac) >> fs[] >>
qpick_x_assum ‘a o a' = u’ mp_tac >>
qpick_x_assum ‘b o h1 = a’ (assume_tac o GSYM) >>
once_arw[] >> rw[o_assoc]
)
(form_goal
“!a.ismono(a:A->X) ==>!b.ismono(b:B->X) ==>
!h1:A->B h2:B->A. h1 o h2 = id(B) & h2 o h1 = id(A) ==>
b o h1 = a & a o h2 = b ==>
!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
char(i1,i2,a) = char(i1,i2,b)”);
(* !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#, iNE#) ==>
isiso(copa(iEQ#, iNE#, pa(Nn#, nN#, id(N), id(N)), ne))]
ppbug , see ne has no #*)
val Posym_def = ex2fsym "*" ["A","B"] (iffRL $ eqT_intro $ spec_all pr_ex)
|> C mp (trueI []) |> gen_all
val pi1_def = ex2fsym "π1" ["A","B"] (iffRL $ eqT_intro $ spec_all Posym_def)
|> C mp (trueI []) |> gen_all
val pi2_def = ex2fsym "π2" ["A","B"] (iffRL $ eqT_intro $ spec_all pi1_def) |> C mp (trueI []) |> gen_all
val ispr_def = new_ax
“!A B AB p1:AB->A p2:AB->B. ispr(p1,p2) <=>
!X f:X->A g:X->B.
?fg0:X->AB. !fg. p1 o fg = f & p2 o fg = g <=> fg = fg0”
val pa_def0 =
ex2fsym "pa" ["p1","p2","f","g"]
(iffLR $ spec_all ispr_def |> strip_all_and_imp |>
disch_all)
val pa_def =
pa_def0 |> undisch
|> allI ("g",ar_sort (mk_ob "X") (mk_ob "B"))
|> allI ("f",ar_sort (mk_ob "X") (mk_ob "A"))
|> allI ("X",ob_sort) |> disch_all
|> allI ("p2",ar_sort (mk_ob "AB") (mk_ob "B"))
|> allI ("p1",ar_sort (mk_ob "AB") (mk_ob "A"))
|> allI ("AB",ob_sort)
|> allI ("B",ob_sort)
|> allI ("A",ob_sort)
val Pa_ex = proved_th $
e0
(rpt strip_tac >> assume_tac pi2_def >>
first_x_assum (qspecl_then ["A","B"] assume_tac) >>
fs[ispr_def])
(form_goal
“!X A f:X->A B g:X->B. ?fg:X->A * B.
!fg'. π1(A,B) o fg' = f & π2(A,B) o fg' = g <=> fg' = fg”)
val Pa_def = ex2fsym "Pa" ["f","g"] (iffRL $ eqT_intro $ spec_all Pa_ex) |> C mp (trueI []) |> gen_all
val isexp_def = new_ax
“!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.?h0:X-> A2B. !h. ev o pa(p1,p2,p1',h o p2') = f <=> h = h0”
val tp_def0 = ex2fsym "tp" ["p1","p2","ev","p1'","p2'","f"]
(iffLR $ spec_all isexp_def |> strip_all_and_imp |>
conjE2 |> strip_all_and_imp |> conj_all_assum |> disch_all)
val tp_def0' =
tp_def0 |> undisch |> gen_all |> split_assum
|> disch “ispr(p1':AX->A, p2':AX->X)”
|> allI ("p2'",ar_sort (mk_ob "AX") (mk_ob "X"))
|> allI ("p1'",ar_sort (mk_ob "AX") (mk_ob "A"))
|> allI ("AX",ob_sort)
|> allI ("X",ob_sort)
|> disch_all
|> allI ("ev",ar_sort (mk_ob "efs") (mk_ob "B"))
|> allI ("p2",ar_sort (mk_ob "efs") (mk_ob "A2B"))
|> allI ("p1",ar_sort (mk_ob "efs") (mk_ob "A"))
|> genl [("efs",ob_sort),("A2B",ob_sort)]
|> allI ("B",ob_sort) |> allI ("A",ob_sort)
val tp_def = proved_th $
e0
(rpt strip_tac >-- (drule $ iffLR isexp_def >> arw[]) >>
drule tp_def0' >> first_x_assum drule >> arw[])
(form_goal
“!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 pa_iso = proved_th $
e0
(rpt strip_tac >> rw[isiso_def] >>
qexists_tac "pa(p1,p2,p1',p2')" >>
strip_tac (* 2 *) >--
(rev_drule to_p_eq >> first_x_assum irule >>
rev_drule p12_of_pa >> arw[GSYM o_assoc] >>
drule p12_of_pa >> arw[idR]) >>
drule to_p_eq >> first_x_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc,idR] >>
rev_drule p12_of_pa >> arw[])
(form_goal
“!A B AB p1:AB->A p2:AB->B. ispr(p1,p2) ==>
!AB' p1':AB'->A p2':AB'->B. ispr(p1',p2') ==>
isiso(pa(p1',p2',p1,p2))”)
val exp_ex' = proved_th $
e0
(rpt strip_tac >> assume_tac exp_ex >>
first_x_assum (qspecl_then ["A","B"] strip_assume_tac) >>
qexists_tac "A2B" >> rpt strip_tac >>
drule exp_ispr >>
rev_drule pa_iso >> first_x_assum drule >>
qexists_tac "ev o pa(p1, p2, p1', p2')" >>
fs[isexp_def] >> rpt strip_tac >>
first_x_assum drule >>
first_x_assum (qspecl_then ["f"] assume_tac) >>
pop_assum strip_assume_tac >>
qexists_tac "h0" >>
rw[o_assoc] >>
qby_tac
‘!h:X -> A2B.
pa(p1, p2, p1'', h o p2'') =
pa(p1, p2, p1', p2') o pa(p1', p2', p1'', h o p2'')’ >--
(strip_tac >>
qpick_x_assum ‘ispr(p1,p2)’ assume_tac >>
drule to_p_eq >> first_x_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc] >>
rev_drule p12_of_pa >> arw[]) >>
pop_assum (assume_tac o GSYM) >> arw[]
)
(form_goal
“!A B.?A2B.
!efs p1:efs ->A p2:efs -> A2B.
ispr(p1,p2) ==> ?ev:efs ->B. isexp(p1,p2,ev)”)
val Exp_def = ex2fsym "Exp" ["A","B"] (iffRL $ eqT_intro $ spec_all exp_ex')
|> C mp (trueI []) |> gen_all
val Ev_ex = proved_th $
e0
(rpt strip_tac >> assume_tac pi2_def >>
first_x_assum (qspecl_then ["A","Exp(A,B)"] assume_tac) >>
drule Exp_def >> arw[])
(form_goal
“!A B.?ev: A * Exp(A,B) -> B.
isexp(π1(A,Exp(A,B)),π2(A,Exp(A,B)),ev)”)
(*
val isexp_th = proved_th $
e0
(form_goal
“!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.?h0:X-> A2B. !h. ev o pa(p1,p2,p1',h o p2') = f <=> h = h0”)
*)
val Ev_def = ex2fsym "Ev" ["A","B"] (iffRL $ eqT_intro $ spec_all Ev_ex)
|> C mp (trueI []) |> gen_all
val pa2Pa = proved_th $
e0
(rpt strip_tac >> irule $ iffLR Pa_def >>
assume_tac pi2_def >>
first_x_assum (qspecl_then ["A","B"] assume_tac) >>
drule p12_of_pa >> arw[]
)
(form_goal
“!A B X f:X->A g:X->B.pa(π1(A,B),π2(A,B),f,g) = Pa(f,g)”)
val Tp_ex = proved_th $
e0
(rpt strip_tac >> assume_tac Ev_def >>
fs[isexp_def] >>
first_x_assum (qspecl_then ["A","B"] assume_tac) >>
pop_assum strip_assume_tac >>
assume_tac pi2_def >>
first_x_assum (qspecl_then ["A","X"] assume_tac) >>
first_x_assum drule >>
first_x_assum (qspecl_then ["f"] assume_tac) >>
qby_tac
‘!h:X->Exp(A,B).
pa(π1(A, Exp(A, B)), π2(A, Exp(A, B)), π1(A, X), h o
π2(A, X)) =
Pa(π1(A, X), h o π2(A, X))’ >-- rw[pa2Pa] >>
fs[] >> qexists_tac "h0" >> rw[]
)
(form_goal
“!A B X f: A * X -> B. ?fBar0:X-> Exp(A,B).!fBar.
Ev(A,B) o Pa(π1(A,X),fBar o π2(A,X)) = f <=> fBar = fBar0”)
val Tp_def = ex2fsym "Tp" ["f"] (iffRL $ eqT_intro $ spec_all Tp_ex)
|> C mp (trueI []) |> gen_all
val const2_def = copr_ex |> allE one |> allE one |> eqT_intro
|> iffRL |> ex2fsym "2" [] |> C mp (trueI [])
val two = mk_fun "2" []
val FALSE_def = ex2fsym "FALSE" [] (iffRL $ eqT_intro $ spec_all const2_def)
|> C mp (trueI []) |> gen_all
val TRUE_def = ex2fsym "TRUE" [] (iffRL $ eqT_intro $ spec_all FALSE_def)
|> C mp (trueI []) |> gen_all
(*TODO: Char is_pb
or the alternative way of defining char
as below, or should add ismono(a) ==> ?Char? *)
val Char_ex = proved_th $
e0
(rpt strip_tac >> qexists_tac "char(FALSE,TRUE,a)" >> rw[])
(form_goal
“!A X a:A->X.
?ch. char(FALSE,TRUE,a) = ch”)
val Char_def = ex2fsym "Char" ["a"]
(iffRL $ eqT_intro $ spec_all Char_ex) |> C mp (trueI []) |> gen_all
val char_ispb =
char_is_pb |> strip_all_and_imp
|> gen_disch_all |> gen_all
val To1_ex = proved_th $
e0
(strip_tac >> qexists_tac "to1(X,1)" >> rw[])
(form_goal
“!X.?to1X:X->1. to1(X,1) = to1X”)
(*current version of ex2fsym cannot produce To1,except for as this...*)
val To1_def = To1_ex |> spec_all |> eqT_intro
|> iffRL |> ex2fsym "To1" ["X"] |> C mp (trueI []) |> gen_all
val True_ex = proved_th $
e0
(strip_tac >> qexists_tac "TRUE o To1(X)" >> rw[])
(form_goal
“!X. ?tX:X->2.TRUE o To1(X) = tX”)
val True_def = True_ex |> spec_all |> eqT_intro
|> iffRL |> ex2fsym "True" ["X"] |> C mp (trueI []) |> gen_all
val False_ex = proved_th $
e0
(strip_tac >> qexists_tac "FALSE o To1(X)" >> rw[])
(form_goal
“!X. ?fX:X->2.FALSE o To1(X) = fX”)
val False_def = False_ex |> spec_all |> eqT_intro
|> iffRL |> ex2fsym "False" ["X"] |> C mp (trueI []) |> gen_all
val Char_property = proved_th $
e0
(rpt strip_tac >>
assume_tac TRUE_def >>
drule char_ispb >>
first_x_assum drule >> fs[Char_def,To1_def]
)
(form_goal
“!A X a:A->X. ismono(a) ==>
ispb(Char(a),TRUE,a,To1(A))”)
val tp2Tp = proved_th $
e0
(rpt strip_tac >>
assume_tac Ev_def >>
first_x_assum (qspecl_then ["A","B"] assume_tac) >>
sym_tac >> drule is_tp >>
assume_tac pi2_def >>
first_x_assum (qspecl_then ["A","X"] assume_tac) >>
first_x_assum drule >>
first_x_assum irule >>
assume_tac Tp_def >>
first_x_assum
(qspecl_then ["A","B","X","f","Tp(f)"] assume_tac) >>
fs[] >> arw[pa2Pa])
(form_goal
“!A B X f:A * X -> B.
tp(π1(A,Exp(A,B)),π2(A,Exp(A,B)),Ev(A,B),
π1(A,X),π2(A,X),f) = Tp(f)”)
val Diag_ex = proved_th $
e0
(strip_tac >> qexists_tac "Pa(id(X),id(X))" >> rw[])
(form_goal
“!X.?dX:X->X * X. Pa(id(X),id(X)) = dX”)
val Diag_def = Diag_ex |> spec_all |> eqT_intro
|> iffRL |> ex2fsym "Diag" ["X"]
|> C mp (trueI []) |> gen_all
(*AQ: should abbrevation like this True_ex use the same schemata as well?*)
val Eq_ex = proved_th $
e0
(strip_tac >> qexists_tac "Char(Diag(X))" >> rw[])
(form_goal “!X.?eqX:X * X -> 2. Char(Diag(X)) = eqX”)
val Eq_def = Eq_ex |> spec_all |> eqT_intro
|> iffRL |> ex2fsym "Eq" ["X"]
|> C mp (trueI []) |> gen_all
val distr_to_pa =proved_th $
e0
(rpt strip_tac >> drule p12_of_pa >> drule to_p_eq >> first_x_assum irule >>
arw[GSYM o_assoc] )
(form_goal
“!A AA Aa:AA->A aA:AA->A. ispr(Aa,aA) ==>
!X0 X x:X0->X a1:X->A a2:X->A. pa(Aa,aA,a1,a2) o x =
pa(Aa,aA,a1 o x,a2 o x)”)
val distr_to_pa' =proved_th $
e0
(rpt strip_tac >> drule p12_of_pa >> drule to_p_eq >> first_x_assum irule >>
arw[GSYM o_assoc] )
(form_goal
“!A B AB Ab:AB->A aB:AB->B. ispr(Ab,aB) ==>
!X0 X x:X0->X a1:X->A a2:X->B. pa(Ab,aB,a1,a2) o x =
pa(Ab,aB,a1 o x,a2 o x)”)
val Pa_distr = proved_th $
e0
(rpt strip_tac >>
qspecl_then ["A","B"] assume_tac pi2_def >>
drule distr_to_pa' >> fs[pa2Pa])
(form_goal
“!A B X a1:X ->A a2:X->B X0 x:X0->X. Pa(a1,a2) o x =
Pa(a1 o x,a2 o x) ”)
val pi1_of_Pa = proved_th $
e0
(rpt strip_tac >>
qspecl_then ["A","B"] assume_tac pi2_def >>
drule p1_of_pa >> fs[pa2Pa])
(form_goal
“!A B X f:X->A g:X->B. π1(A,B) o Pa(f,g) = f”)
val pi2_of_Pa = proved_th $
e0
(rpt strip_tac >>
qspecl_then ["A","B"] assume_tac pi2_def >>
drule p2_of_pa >> fs[pa2Pa])
(form_goal
“!A B X f:X->A g:X->B. π2(A,B) o Pa(f,g) = g”)
val pi12_of_Pa =
conjI (spec_all pi1_of_Pa) (spec_all pi2_of_Pa) |> gen_all
val True2TRUE = proved_th $
e0
(rpt strip_tac >> rw[GSYM True_def,o_assoc] >>
once_rw[one_to_one_id] >> rw[idR])
(form_goal
“!X x:1->X. True(X) o x = TRUE”);
val pb_fac_iff = proved_th $
e0
(rpt strip_tac >> drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >>
dimp_tac >> strip_tac >--
(pop_assum mp_tac >> pop_assum (assume_tac o GSYM) >>
strip_tac >> pop_assum (assume_tac o GSYM) >>
arw[GSYM o_assoc]) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
first_x_assum (qspecl_then ["a"] assume_tac) >> fs[] >>
qexists_tac "a" >> arw[])
(form_goal
“!X Z f:X->Z Y g:Y->Z P p:P->X q.
ispb(f,g,p,q) ==>
!A u:A->X v:A->Y.
(?a:A->P. p o a = u & q o a = v) <=> f o u = g o v”);
val pb_fac_iff_1 = proved_th $
e0
(rpt strip_tac >> drule pb_fac_iff >>
first_x_assum
(qspecl_then ["1","u","id(1)"] assume_tac) >>
fs[idR] >> pop_assum (assume_tac o GSYM) >>
arw[] >> dimp_tac >> strip_tac (* 2 *)
>-- (qexists_tac "a" >> arw[] >> once_rw[one_to_one_id]) >>
qexists_tac "a" >> arw[])
(form_goal
“!X Z f:X->Z g:1->Z P p0:P->X q.
ispb(f,g,p0,q) ==>
!u:1->X.
(?a:1->P. p0 o a = u) <=> f o u = g”);
(*
fun readf f = (fvf f,[]:form list,f) *)
val pb_reorder = proved_th $
e0
(rw[ispb_def] >> rpt strip_tac
>-- (pop_assum (K all_tac) >> once_arw[] >> rw[]) >>
first_x_assum (qspecl_then ["A","v","u"] assume_tac) >>
pick_x_assum “g:Y->Z o u:A->Y = f:X->Z o v”
(assume_tac o GSYM) >>
first_x_assum drule >> pop_assum strip_assume_tac >>
qexists_tac "a" >> strip_tac >>
first_x_assum (qspecl_then ["a'"] assume_tac) >>
(*TODO: AQ: how to automatic on this?*)
qby_tac ‘q0 o a' = u & p0 o a' = v <=> p0 o a' = v & q0 o a' = u’ >--
(dimp_tac >> disch_tac >> arw[] >>
fs[]) >>
arw[]
)
(form_goal
“!X Z f:X->Z Y g:Y->Z P p0:P->X q0:P->Y.ispb(f,g,p0,q0) ==>
ispb(g,f,q0,p0)”)
val inc_ismono = proved_th $
e0
(rpt strip_tac (* 2 *) >--
(irule ismono_applied >> rpt strip_tac >>
irule fun_ext >> strip_tac >>
qby_tac ‘copa(iA,iB,id(A),g o a o to1(B,1)) o iA o g = copa(iA,iB,id(A),g o a o to1(B,1)) o iA o h’
>-- arw[] >> pop_assum mp_tac >>
drule i1_of_copa >> rw[GSYM o_assoc] >>
arw[idL] >> strip_tac >> arw[]) >>
irule ismono_applied >> rpt strip_tac >>
irule fun_ext >> strip_tac >>
qby_tac ‘copa(iA,iB,g o a o to1(A,1),id(B)) o iB o g =
copa(iA,iB,g o a o to1(A,1),id(B)) o iB o h’
>-- arw[] >> pop_assum mp_tac >>
drule i2_of_copa >> rw[GSYM o_assoc] >>
arw[idL] >> strip_tac >> arw[])
(form_goal
“!A B AB iA:A->AB iB:B->AB. iscopr(iA,iB) ==>
ismono(iA) & ismono(iB)”)
val ax7'=
ax7 |> strip_all_and_imp |> gen_all |>
split_assum |> disch_last |> gen_all |> disch_all |> gen_all
val ax7_const1 = proved_th $
e0
(rpt strip_tac >> drule ax7' >> assume_tac const1_def >>
first_x_assum drule >> fs[ismem0_def] >>
drule inc_ismono >> fs[] >> rfs[] >>
rw[ismem_def] >> arw[]
)
(form_goal
“!A AB iA:A->AB B iB:B->AB. iscopr(iA,iB) ==>
!f:1->AB. ismem(f,iA) | ismem(f,iB)”)
val copr_disjoint = proved_th $
e0
(rpt strip_tac >> drule prop_5_lemma >>
drule ax7_const1 >> drule inc_ismono >> fs[ismem_def] >>
first_x_assum (qspecl_then ["x"] assume_tac) >>
cases_on “?x0 : 1 -> A. x:1->AB = iA o x0” (* 2 *)
>-- (arw[] >> pop_assum strip_assume_tac >>
ccontra_tac >> pop_assum strip_assume_tac >>
qby_tac ‘iA o x0 = iB o x0'’
>-- (pop_assum mp_tac >>
pop_assum (assume_tac o GSYM) >>
strip_tac >> pop_assum (assume_tac o GSYM) >>
pick_xnth_assum 2 (K all_tac) >> arw[]) >>
rfs[]) >>
arw[] >> pop_assum_list (map_every strip_assume_tac) (* 2 *)
>-- (by_tac “?(x0 : 1 -> A). x = iA:A->AB o x0”
>-- (qexists_tac "x0" >> arw[]) >>
first_x_assum opposite_tac) >>
qexists_tac "x0" >> arw[])
(form_goal “!A B AB iA:A->AB iB:B->AB. iscopr(iA,iB) ==>
!x:1->AB. (~(?x0:1->A. x = iA o x0)) <=> (?x0:1->B. x = iB o x0)”)
val i1_xor_i2 = proved_th $
e0
(rpt strip_tac >> drule copr_disjoint >>
cases_on “x = i1:1->two” (* 2 *)
>-- (arw[] >> drule i1_ne_i2 >> first_x_assum accept_tac) >>
arw[] >> first_x_assum (qspecl_then ["x"] assume_tac) >>
by_tac “~(?x0.x = i1:1->two o x0:1->1)”
>-- (ccontra_tac >> pop_assum strip_assume_tac >>
pop_assum mp_tac >>
by_tac “x0 = id(1)”
>-- once_rw[one_to_one_id] >> arw[] >>
arw[idR]) >>
rfs[] >> once_rw[one_to_one_id] >> rw[idR])
(form_goal
“!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!x:1->two. x = i1 <=> ~(x = i2)”)
val two2two_cases = proved_th $
e0
(rpt strip_tac >> drule from_copr_components >>
first_x_assum (qspecl_then ["two","f"] assume_tac) >>
drule i1_xor_i2 >> once_arw[] >>
pop_assum mp_tac >> pop_assum (K all_tac) >>
strip_tac >>
first_assum (qspecl_then ["f o i1"] assume_tac) >>
first_x_assum (qspecl_then ["f o i2"] assume_tac) >>
cases_on “f:two ->two o i1:1->two = i2” (* 2 *)
>-- (once_arw[] >>
cases_on “f:two ->two o i2:1->two = i2” (*2 *) >--
(once_arw[] >> rw[]) >>
fs[]) >>
fs[] >> cases_on “f:two ->two o i2:1->two = i2” (*2 *) >--
(once_arw[] >> rw[]) >>
fs[])
(form_goal “!two i1:1->two i2:1->two.iscopr(i1,i2) ==>
!f:two->two. f = copa(i1,i2,i1,i1) | f = copa(i1,i2,i1,i2) | f = copa(i1,i2,i2,i2) | f = copa(i1,i2,i2,i1)”)
val iscopr_def = read_axiom $ q2str
‘!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'. (fg' o i1 = f & fg' o i2 = g) ==> fg' = fg)’
(*TODO: AQ: how does rename tac work?*)
val iso_copr_copr = proved_th $
e0
(rpt strip_tac >> rw[iscopr_def] >> rpt strip_tac >>
drule $ iffLR isiso_def>> drule copa_def >> fs[] >>
first_x_assum (qspecl_then ["X'","f","g"] assume_tac) >>
qexists_tac "copa(iA,iB,f,g) o f'" >> rw[o_assoc] >>
by_tac “f' o f0 = iA & f':X->AB o g0:B->X = iB:B->AB” >--
(qby_tac ‘f' o copa(iA, iB, f0, g0) o iA = id(AB) o iA &
f' o copa(iA, iB, f0, g0) o iB = id(AB) o iB’
>-- arw[GSYM o_assoc] >>
pop_assum mp_tac >> drule i12_of_copa >> arw[idL]) >>
arw[] >>
drule i12_of_copa >> arw[] >>
rpt strip_tac >> irule isepi_property >>
qexistsl_tac ["AB","copa(iA, iB, f0, g0)"] >>
drule iso_is_epi >> arw[o_assoc,idR] >>
drule from_cop_eq >> first_x_assum irule >>
drule i12_of_copa >> arw[o_assoc]
)
(form_goal “!A B AB iA:A->AB iB:B->AB. iscopr(iA,iB) ==>
!X f0:A->X g0:B->X. isiso(copa(iA,iB,f0,g0)) ==> iscopr(f0,g0)”);
val fac_diag_eq_iff = proved_th $
e0
(rpt strip_tac >> drule fac_diag_eq >>
dimp_tac >> strip_tac (* 2 *)
>-- (first_x_assum irule >> qexists_tac "a0" >>
drule to_p_eq >> first_x_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc]) >>
qexists_tac "aA o aa" >> drule to_p_eq >>
first_x_assum irule >> drule p12_of_pa >>
arw[GSYM o_assoc,idL])
(form_goal
“!A AA Aa:AA->A aA:AA->A. ispr(Aa,aA) ==>
!aa:1->AA.(?a0:1->A. aa = pa(Aa,aA,id(A),id(A)) o a0) <=>
Aa o aa = aA o aa”);
val SoE_lemma_2_5_5 = proved_th $
e0
(rw[iscopr_def] >> rpt strip_tac >>
qspecl_then ["N","X"] (x_choosel_then ["NX","Nx","nX"] assume_tac) pr_ex >>
qspecl_then ["NX","pa(Nx,nX,ZERO,f)","pa(Nx,nX,SUC,g) o Nx"] assume_tac constN_def >>
qexists_tac "nX o Nind(pa(Nx, nX, ZERO, f), pa(Nx, nX, SUC, g) o Nx)" >>
first_assum (qspecl_then ["Nind(pa(Nx, nX, ZERO, f), pa(Nx, nX, SUC, g) o Nx)"] assume_tac) >>
fs[] >>
by_tac
“Nx o Nind(pa(Nx:NX->N, nX:NX->X, ZERO, f), pa(Nx, nX, SUC, g) o Nx) = id(N)”
>-- (sym_tac >> irule comm_with_s_id >>
qby_tac
‘Nx o Nind(pa(Nx, nX, ZERO, f), (pa(Nx, nX, SUC, g) o Nx)) o SUC
= Nx o (pa(Nx, nX, SUC, g) o Nx) o
Nind(pa(Nx, nX, ZERO, f), pa(Nx, nX, SUC, g) o Nx)’ >-- arw[] >>
arw[o_assoc] >> drule p12_of_pa >>
arw[GSYM o_assoc]) >>
fs[o_assoc,idR] >>
by_tac
“nX o Nind(pa(Nx:NX->N, nX:NX->X, ZERO, f), (pa(Nx, nX, SUC, g:N->X) o Nx)) o ZERO = nX o pa(Nx, nX, ZERO, f)”
>-- arw[] >>
drule p2_of_pa >> arw[] >>
suffices_tac
“!fg:N->X. fg o ZERO = f:1->X & fg o SUC = g:N->X ==>
pa(Nx:NX->N,nX:NX->X,id(N),fg) = Nind(pa(Nx, nX, ZERO, f), pa(Nx, nX, SUC, g) o Nx)” >--
(strip_tac >> gen_tac >> disch_tac >>
first_assum drule >>
by_tac
“nX o pa(Nx:NX->N, nX:NX->X, id(N), fg':N->X) = nX o Nind(pa(Nx, nX, ZERO, f:1->X), pa(Nx, nX, SUC, g:N->X) o Nx)”
>-- (pop_assum mp_tac >> pop_assum_list (map_every (K all_tac)) >> strip_tac >> arw[]) >>
pop_assum mp_tac >> drule p2_of_pa >> arw[]) >>
rpt strip_tac >>
last_x_assum mp_tac >> last_x_assum (assume_tac o GSYM)>>
strip_tac >> arw[] >> strip_tac (* 2 *)
>-- (drule to_p_eq >> first_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc,idL]) >>
drule to_p_eq >> first_assum irule >>
drule p12_of_pa >> arw[GSYM o_assoc] >>
arw[o_assoc,idL,idR])
(form_goal “iscopr(ZERO,SUC)”);
val z_xor_s = proved_th $
e0
(assume_tac SoE_lemma_2_5_5 >>
drule copr_disjoint >>
strip_tac >>
first_x_assum (qspecl_then ["n"] assume_tac) >>
pop_assum (assume_tac o GSYM) >> arw[] >>
cases_on “n = ZERO” >> arw[] >>
ccontra_tac >> fs[] >> pop_assum mp_tac >>
rw[] >> once_rw[one_to_one_id] >>
arw[idR] >> qexists_tac "id(1)" >> rw[]
)
(form_goal
“!n:1->N. ~(n = ZERO) <=> ?n0:1->N. n = SUC o n0”);
val char_diag = proved_th $
e0
(rpt strip_tac >> drule fac_diag_eq_iff >>
first_x_assum (qspecl_then ["pa(Aa,aA,a1,a2)"] assume_tac) >>
drule p12_of_pa >> fs[] >> pop_assum (K all_tac) >>
pop_assum (assume_tac o GSYM) >> arw[] >>
drule diag_is_mono >> drule char_def >> first_x_assum drule >>
pop_assum (assume_tac o GSYM) >> arw[] >> dimp_tac >> rpt strip_tac
>-- (qexists_tac "x0" >> arw[]) >>
qexists_tac "a0" >> arw[])
(form_goal
“!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!A AA Aa:AA->A aA:AA ->A. ispr(Aa,aA) ==>
!a1:1->A a2:1->A. char(i1,i2,pa(Aa,aA,id(A),id(A))) o pa(Aa,aA,a1,a2) = i2 <=> a1 = a2”);
val ind_principle = proved_th $
e0
(rpt strip_tac >>
qspecl_then ["N","two","pred","1","i2"] (x_choosel_then ["A","a","At1"] assume_tac) pb_ex >>
drule pb_fac_iff_1 >>
by_tac “ismono(a:A->N)”
>-- (drule pb_mono_mono >> first_x_assum irule >>
once_rw[dom_1_mono]) >>
by_tac “pred = i2:1->two o to1(N,1) <=> isiso(a:A->N)” >-- (
dimp_tac >> strip_tac (* 2 *) >--
(irule Thm2_3' >> arw[] >> drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >> arw[ismem_def,o_assoc] >>
once_rw[one_to_one_id] >> rw[idR]) >>
fs[isiso_def] >> irule fun_ext >> strip_tac >>
rw[o_assoc] >> once_rw[one_to_one_id] >> rw[idR] >>
drule $ iffLR ispb_def >> pop_assum strip_assume_tac >>
by_tac
“pred o (a:A->N o f':N->A) o a' = i2:1->two o At1:A->1 o f':N->A o a':1->N”
>-- (rw[GSYM o_assoc] >> arw[]) >>
rfs[idL] >> once_rw[one_to_one_id] >> rw[idR]) >>
arw[] >> pop_assum (K all_tac) >> dimp_tac >> strip_tac (* 2 *) >--
(fs[isiso_def] >> drule $ iffLR ispb_def >>
pop_assum strip_assume_tac >>
by_tac
“!n0:1->N. pred o (a:A->N o f') o n0 = i2:1->two o At1:A->1 o f':N->A o n0”
>-- (strip_tac >> arw[GSYM o_assoc]) >>
rpt strip_tac (* 2 *)
>-- (first_x_assum (qspecl_then ["ZERO"] assume_tac) >>
rfs[idL] >> once_rw[one_to_one_id] >> rw[idR]) >>
first_x_assum (qspecl_then ["SUC o n"] assume_tac) >>
rfs[idL] >> once_rw[one_to_one_id] >> rw[idR]) >>
irule Thm2_3' >> arw[ismem_def])
(form_goal
“!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!pred:N->two. pred = i2 o to1(N,1) <=>
(pred o ZERO = i2 & (!n:1->N. pred o n = i2 ==> pred o SUC o n = i2))”);
val ind_principle_elements = proved_th $
e0
(rpt strip_tac >> drule ind_principle >> pop_assum (assume_tac o GSYM) >>
arw[] >> dimp_tac >> rpt strip_tac (* 2 *)
>-- (irule fun_ext >> rpt strip_tac >> rw[o_assoc] >>
once_rw[one_to_one_id] >> rw[idR] >> arw[]) >>
arw[] >> rw[o_assoc] >> once_rw[one_to_one_id] >> rw[idR]
)
(form_goal
“!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!pred:N->two. (!n.pred o n = i2) <=>
(pred o ZERO = i2 & (!n:1->N. pred o n = i2 ==> pred o SUC o n = i2))”);
val equality_ind = proved_th $
e0
(rpt strip_tac >>
qspecl_then ["1","1"] (x_choosel_then ["two","i1","i2"] assume_tac) copr_ex >>
qspecl_then ["A","A"] (x_choosel_then ["AA","Aa","aA"] assume_tac) pr_ex >>
drule char_diag >> first_x_assum drule >>
pop_assum (assume_tac o GSYM) >>
by_tac “(!n:1->N.f o pa(Xn:XN->X,xN:XN->N,x,n) = g o pa(Yn:YN->Y,yN:YN->N,y,n)) <=>
!n. char(i1,i2,pa(Aa,aA,id(A),id(A))) o
pa(Aa:AA->A,aA:AA->A,f:XN->A o pa(Xn:XN->X,xN:XN->N,x,n), g:YN->A o pa(Yn:YN->Y,yN:YN->N,y:1->Y,n)) = i2:1->two”
>-- (dimp_tac >> rpt strip_tac >--
(pop_assum mp_tac >> pop_assum (assume_tac o GSYM) >> arw[] >>
strip_tac >> arw[]) >>
arw[]) >>
arw[] >>
by_tac
“(!n. char(i1,i2,pa(Aa,aA,id(A),id(A))) o
pa(Aa:AA->A,aA:AA->A,f:XN->A o pa(Xn:XN->X,xN:XN->N,x,n), g:YN->A o pa(Yn:YN->Y,yN:YN->N,y:1->Y,n)) = i2:1->two) <=>
!n. char(i1,i2,pa(Aa,aA,id(A),id(A))) o
pa(Aa:AA->A,aA:AA->A,f:XN->A o pa(Xn:XN->X,xN:XN->N,x o to1(N,1),id(N)), g:YN->A o pa(Yn:YN->Y,yN:YN->N,y:1->Y o to1(N,1),id(N))) o n = i2:1->two” >--
(dimp_tac >> rpt strip_tac (* 2 *)
>-- (drule distr_to_pa >> rev_drule distr_to_pa' >>
pick_x_assum “ispr(Yn:YN->Y,yN:YN->N)” assume_tac >>
drule distr_to_pa' >> arw[o_assoc] >> once_rw[one_to_one_id] >>
rw[idL,idR] >> arw[]) >>
drule distr_to_pa >> rev_drule distr_to_pa' >>
pick_x_assum “ispr(Yn:YN->Y,yN:YN->N)” assume_tac >>
drule distr_to_pa' >> fs[o_assoc] >>
pick_xnth_assum 6 mp_tac(*to be edited*) >>
once_arw[one_to_one_id] >> rw[idL,idR] >>
strip_tac >> arw[]) >>
once_arw[] >> pop_assum (K all_tac) >>
drule ind_principle_elements >> rw[GSYM o_assoc] >>
first_x_assum (qspecl_then [" (char(i1, i2, pa(Aa, aA, id(A), id(A))) o pa(Aa, aA, f o pa(Xn, xN, x o to1(N, 1), id(N)), g o pa(Yn, yN, y o to1(N, 1), id(N))))"] assume_tac) >> once_arw[] >>
pop_assum (K all_tac) >>
drule distr_to_pa' >> rev_drule distr_to_pa' >>
pick_x_assum “ispr(Yn:YN->Y,yN:YN->N)” assume_tac >>
drule distr_to_pa' >> fs[o_assoc] >>
once_rw[one_to_one_id] >> rw[idL,idR])
(form_goal
“!X XN Xn:XN->X xN:XN->N. ispr(Xn,xN) ==>
!Y YN Yn:YN->Y yN:YN->N. ispr(Yn,yN) ==>
!A f:XN->A g:YN->A.
!x:1->X y:1->Y.(!n.f o pa(Xn,xN,x,n) = g o pa(Yn,yN,y,n)) <=>
f o pa(Xn,xN,x,ZERO) = g o pa(Yn,yN,y,ZERO) &
!n0:1->N. f o pa(Xn,xN,x,n0) = g o pa(Yn,yN,y,n0) ==>
f o pa(Xn,xN,x,SUC o n0) = g o pa(Yn,yN,y,SUC o n0)”);
val pxy_true = proved_th $
e0
(rpt strip_tac >> dimp_tac >> rpt strip_tac (* 2 *) >--
(arw[o_assoc] >> once_rw[one_to_one_id] >> rw[idR]) >>
irule fun_ext >> rpt strip_tac >> rw[o_assoc] >> once_rw[one_to_one_id] >>
rw[idR] >> drule to_p_components >>
first_x_assum (qspecl_then ["1","a"] assume_tac) >>
once_arw[] >> pop_assum (K all_tac) >> arw[])
(form_goal
“!two i1:1->two i2:1->two. iscopr(i1,i2) ==>
!XY X Y Xy:XY->X xY:XY->Y.ispr(Xy,xY) ==>
!X2t efs p1:efs->X p2:efs->X2t ev:efs ->two. isexp(p1,p2,ev) ==>
!pred.pred = i2 o to1(XY,1) <=> !x:1->X y:1->Y. pred o pa(Xy,xY,x,y) = i2”);
val pa_eq = proved_th $
e0
(rpt strip_tac >>
dimp_tac >> strip_tac (* 2 *) >--
(qby_tac
‘Ab o pa(Ab, aB, f1, g1) = Ab o pa(Ab, aB, f2, g2)’
>-- arw[] >>
qby_tac
‘aB o pa(Ab, aB, f1, g1) = aB o pa(Ab, aB, f2, g2)’
>-- arw[] >>
drule p12_of_pa >> fs[]) >>
arw[])
(form_goal
“!A B AB Ab:AB->A aB:AB->B. ispr(Ab,aB) ==>
!X f1:X->A g1 f2 g2. pa(Ab,aB,f1,g1) = pa(Ab,aB,f2,g2) <=>
f1 = f2 & g1 = g2”)
val Diag_ismono = proved_th $