Thm* (AH ~ 8)  ((AH {1...8}) ~ 64) | [chessboard_example] |
Thm* p:( a {k} ). {i: a| p(i) } ~ k | [card_st_sized_bool] |
Thm* p:( a  ). {x: a| p(x) } ~ ( size( a)(p)) | [card_st_vs_boolsize] |
Thm* f:( a  2).
Thm* ( i: a. P(i)  f(i) = 1 2)  ({x: a| P(x) } ~ (Msize(f))) | [card_st_vs_msize] |
Thm* ( a {k} ) ~ ( a {k} 2) | [card_boolset_vs_mset] |
Thm* ( a bij a) ~ ( a  a) | [nsub_bij_ooc_invpair] |
Thm* ( k inj k) ~ ( k onto k) | [nsub_inj_ooc_nsub_surj] |
Thm* ( k inj k) ~ ( k bij k) | [nsub_inj_ooc_nsub_bij] |
Thm* ( k onto k) ~ ( k bij k) | [nsub_surj_ooc_nsub_bij] |
Thm* (k:  k  ) ~  | [card_nat_vs_nat_tuples_all] |
Thm* (k:   k  ) ~  | [card_nat_vs_nat_tuples] |
Thm* k: . ( k  ) ~  | [card_nat_vs_nat_tuple] |
Thm* (  ) ~  | [card_nat_vs_nat_nat] |
Thm* ( (A B)) ~ (A ~ B) | [inv_pair_inv2] |
Thm* (A B) ~ (A ~ B) | [inv_pair_inv] |
Thm* a: , b:( a  ). (i: a b(i)) ~ ( i: a. b(i)) | [card_sigma_vs_nsub_sigma] |
Thm* a,b: . ( a+ b) ~ (a+b) | [nsub_add_rw] |
Thm* (A B C) ~ (B A C) | [card_curry_simple2] |
Thm* a,b: . ( a  b) ~ (b a) | [card_fun_vs_nsub_exp] |
Thm* a: , b:( a  ). (i: a  b(i)) ~ ( i: a. b(i)) | [card_pi_vs_nsub_pi] |
Thm* ( k inj k) ~ (k!) | [nsub_inj_factorial] |
Thm* ( a inj b) ~ (b!a) | [nsub_inj_factorial_tail] |
Thm* ( b inj b) ~ (b!) | [nsub_inj_factorial2] |
Thm* a,b: . (A ~ a)  (B ~ b)  ((A B) ~ (a b)) | [finite_indep_sum_card] |
Thm* a,b: . ( a b) ~ (a b) | [nsub_mul_rw] |
Thm* a,b,c: . a b = c  (( a b) ~ c) | [nsub_mul] |
Thm* c,a,b: . a+b = c  (( a+ b) ~ c) | [nsub_add] |
Thm* a,b: , c:{a...b}. ({a..c }+{c..b }) ~ {a..b } | [intseg_split] |
Thm* a,b: . {a...b} ~ (1+b-a) | [nsub_intiseg_rw] |
Thm* a: , b: . {a...b} ~ {a..(b+1) } | [intiseg_intseg_plus] |
Thm* a,b',b: . b = b'+1  ({a...b'} ~ {a..b }) | [intiseg_intseg] |
Thm* a,b: . {a..b } ~ (b-a) | [nsub_intseg_rw] |
Thm* c,a: , b: . c = b-a  ({a..b } ~ c) | [nsub_intseg] |
Thm* ~ 2 | [nsub_bool] |
Thm* Unit ~ 1 | [nsub_unit] |
Thm* a: , b,c: . {a..b } ~ {(a+c)..(b+c) } | [intseg_shift_by] |
Thm* a,b,a',b': . b-a = b'-a'  ({a..b } ~ {a'..b' }) | [intseg_shift] |
Thm* Void ~ 0 | [nsub_void] |
Thm* a,b: . b a  (Void ~ {a..b }) | [intseg_void] |
Thm* (xy:(x:A B(x)) C(xy)) ~ (x:A y:B(x) C(<x,y>)) | [card_sigma_distr] |
Thm* (z:(x:A B(x)) C(z)) ~ (x:A y:B(x) C(<x,y>)) | [card_curry] |
Thm* (A B C) ~ (A B C) | [card_curry_simple] |
Thm* (A ~ A')  (B ~ B')  ((A B) ~ (A' B')) | [function_functionality_wrt_one_one_corr] |
Thm* B,B':(A Type). ( x:A. B(x) ~ B'(x))  ((x:A B(x)) ~ (x:A B'(x))) | [function_functionality_wrt_one_one_corr_B] |
Thm* (x:A. B(x)) ~ (x:A'. B'(x))  ((x:A B(x)) ~ (x:A' B'(x))) | [card_pi] |
Thm* a,b: , B:({a..b } Type).
Thm* a<b  ((i:{a..b } B(i)) ~ ((i:{a..(b-1) } B(i))+B(b-1))) | [card_split_end_sum_intseg_family] |
Thm* a,b: , c:{a...b}, B:({a..b } Type).
Thm* (i:{a..b } B(i)) ~ ((i:{a..c } B(i))+(i:{c..b } B(i))) | [card_split_sum_intseg_family] |
Thm* ( x:A. Dec( P(x)))  (A ~ ({x:A| P(x) }+{x:A| P(x) })) | [card_split_decbl_squash] |
Thm* (A+ 0) ~ A | [card_nsub0_union_2] |
Thm* ( 0+A) ~ A | [card_nsub0_union] |
Thm* B:(A Type), P:(A Prop).
Thm* ( i:A. Dec(P(i)))
Thm* 
Thm* ((i:A B(i)) ~ ((i:{i:A| P(i) } B(i))+(i:{i:A| P(i) } B(i)))) | [card_split_sigma_dom_decbl] |
Thm* B:(A Type), P:(A Prop).
Thm* (i:{i:A| P(i) } B(i)) ~ {v:(i:A B(i))| P(v/x,y. x) } | [card_sigma_st_dom] |
Thm* ( x:A. Dec(P(x)))  (A ~ ({x:A| P(x) }+{x:A| P(x) })) | [card_split_decbl] |
Thm* (A+B) ~ (i: 2 if i= 0 A else B fi) | [card_union_vs_sigma] |
Thm* (A+B) ~ (B+A) | [card_union_swap] |
Thm* a,b: , c:{a...b}, B:({a..b } Type).
Thm* (i:{a..b } B(i)) ~ ((i:{a..c } B(i)) (i:{c..b } B(i))) | [card_split_prod_intseg_family] |
Thm* (A ~ A')  (B ~ B')  ((A B) ~ (A' B')) | [product_functionality_wrt_one_one_corr] |
Thm* (q:A/E {x:A| q = x A/E }) ~ A | [card_quotient] |
Thm* B,B':(A Type). ( x:A. B(x) ~ B'(x))  ((x:A B(x)) ~ (x:A B'(x))) | [product_functionality_wrt_one_one_corr_B] |
Thm* (x:A. B(x)) ~ (x:A'. B'(x))  ((x:A B(x)) ~ (x:A' B'(x))) | [card_sigma] |
Thm* ( x:A. B(x)  B'(x))  ({x:A| B(x) } ~ {x:A| B'(x) }) | [set_functionality_wrt_one_one_corr_n_pred] |
Thm* ( x:A. B(x)  B'(x))  ({x:A| B(x) } ~ {x:A| B'(x) }) | [set_functionality_wrt_one_one_corr_n_pred2] |
Thm* Bij(A; A'; f)
Thm* 
Thm* ( x:A. B(x)  B'(f(x)))  ({x:A| B(x) } ~ {x:A'| B'(x) }) | [card_settype_sq] |
Thm* Bij(A; A'; f)
Thm* 
Thm* ( x:A. B(x)  B'(f(x)))  ({x:A| B(x) } ~ {x:A'| B'(x) }) | [card_settype] |
Thm* (A B) ~ (B A) | [card_product_swap] |
Thm* (A ~ A')  (B ~ B')  ((A+B) ~ (A'+B')) | [union_functionality_wrt_one_one_corr] |
Thm* (A ~ A')  (B ~ B')  (:A. B) ~ (:A'. B') | [one_one_corr_fams_indep_if_one_one_corr] |
Thm* ( x:A. B(x) ~ B'(x))  (x:A. B(x)) ~ (x:A. B'(x)) | [one_one_corr_fams_if_one_one_corr_B] |
Thm* P:(A B Prop). ( x:A. !y:B. P(x;y))  (A ~ (y:B {x:A| P(x;y) })) | [partition_type] |
Thm* a,b: . ( a ~ b)  a = b | [fin_card_vs_nat_eq] |
Thm* (A ~ m)  (A ~ k)  m = k | [counting_is_unique] |
Thm* ((   ) ~ ) | [not_nat_occ_natfuns] |
Thm* ( f:(A B). Bij(A; B; f))  (B ~ A) | [bij_iff_1_1_corr_version2a] |
Thm* (A ~ B)  Infinite(A)  Infinite(B) | [ooc_preserves_infiniteness] |
Thm* (A ~ B)  Unbounded(A)  Unbounded(B) | [ooc_preserves_unb_inf] |
Thm* ~  | [int_ooc_nat] |
Thm* ( f:(A B). Bij(A; B; f))  (A ~ B) | [bij_iff_1_1_corr_version2] |
Thm* (A ~ B)  (B ~ C)  (A ~ C) | [one_one_corr_2_transitivity] |
Thm* (A ~ B)  (B ~ A) | [one_one_corr_2_inversion] |
Thm* A ~ A | [one_one_corr_2_reflex] |
Thm* (A ~ A')  (B ~ B')  ((A ~ B)  (A' ~ B')) | [one_one_corr_2_functionality_wrt_one_one_corr] |
Thm* (A ~ B)  (( A)  ( B)) | [inhabited_functionality_wrt_one_one_corr] |
Thm* (A ~ B)  (A  B) | [iff_weakening_wrt_one_one_corr_2] |
Thm* EquivRel X,Y:Type. X ~ Y | [one_one_corr_is_equiv_rel] |
Thm* (A ~ A')  (B ~ B')  ((A inj B) ~ (A' inj B')) | [injection_type_functionality_wrt_ooc] |
Thm* (A ~ B)  Dedekind-Infinite(A)  Dedekind-Infinite(B) | [ooc_preserves_dededkind_inf] |
Thm* (A ~ A')  (B ~ B')  ((A onto B) ~ (A' onto B')) | [surjection_type_functionality_wrt_ooc] |
Thm* ( x:A. Dec(P(x)))
Thm* 
Thm* ((x:A B(x)) ~ ((x:A st P(x) B(x)) (x:A st P(x) B(x)))) | [card_split_prod_intseg_family_decbl] |
Thm* A:Type, B:Type. (A ~~ B)  (A ~ B) | [one_one_corr_iff_one_one_corr_2] |
Thm* (A ~ A')  (B ~ B')  ((A B) ~ (A' B')) | [inv_pair_functionality_wrt_one_one_corr] |
Thm* ( (A B))  (A ~ B) | [inv_pair_iff_ooc] |
Thm* a,a': . a'-a = 1  ( B:({a..a' } Type). (i:{a..a' } B(i)) ~ B(a)) | [card_sum_family_intseg_singleton_elim] |
Thm* B:({a:A} Type). (i:{a:A} B(i)) ~ B(a) | [card_sum_family_singleton_elim] |
Thm* (A ~ 0)  ( A) | [card0_iff_uninhabited] |
Thm* A = B  (A ~ B) | [one_one_corr_2_weakening_wrt] |
Thm* {x:A| B(x) } ~ {x:A| B(x) } | [subset_sq_remove_card] |
Thm* a,a': . a'-a = 1  ( B:({a..a' } Type). (i:{a..a' } B(i)) ~ B(a)) | [card_prod_family_intseg_singleton_elim] |
Thm* (i:{a:A} B(i)) ~ B(a) | [card_prod_family_singleton_elim] |
Thm* a,b: . ( a inj b) ~ ( b ( (a-1) inj (b-1))) | [card_nsub_inj_vs_lopped] |
Thm* k: , j: k. {i: k| i = j } ~ (k-1) | [nsub_delete_rw] |
Thm* k: , j: k, m: . m = k-1  ({i: k| i = j } ~ m) | [nsub_delete] |
Thm* ( 0 A) ~ 1 | [card_void_dom_fun_unit] |
Thm* (A ~ 1)  ( ! A) | [card1_iff_inhabited_uniquely] |
Thm* a,a': .
Thm* a'-a = 1  ( B:({a..a' } Type). (i:{a..a' } B(i)) ~ ({a: } B(a))) | [card_sum_family_singleton_vs_intseg] |
Thm* {x:{x:A| P(x) }| Q(x) } ~ {x:A| P(x) & Q(x) } | [card_subset_vs_and] |
Thm* {x:A| if b P(x) else Q(x) fi } ~ if b {x:A| P(x) } else {x:A| Q(x) } fi | [ifthenelse_distr_subtype_ooc] |
Thm* (A bij B) ~ ((A inj B) (A onto B)) | [bijtype_ooc_inj_isect_surjtype] |
Thm* (A B C) ~ ((A B) C) | [simple_card_cross_assoc] |
Thm* (A ~ B)  Finite(A)  Finite(B) | [ooc_preserves_finiteness] |
Thm*  ~  | [nat_plus_ooc_nat] |
Thm* k: . ( k+ ) ~  | [fin_plus_nat_ooc_nat] |
Thm* A = A'  B = B'  (A ~ B) = (A' ~ B') | [one_one_corr_2_functionality_wrt_eq] |
Def Finite(A) == n: . A ~ n | [is_finite_type] |