hcju/isabelleps huggingface dataset:
pr_iso_test_rec_corr: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "[] \<notin> set Fs\<^sub>2"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "pr_iso_test_rec [] Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2)"
coprime_mult_right_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one c (times a b) = (algebraic_semidom.coprime times one c a \<and> algebraic_semidom.coprime times one c b)"
ldropWhile_eq_ldrop: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "ldropWhile P xs = ldrop (llength (ltakeWhile P xs)) xs"
mult_left_le: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq c one"   and "less_eq zero a" shows "less_eq (times a c) a"
numeral_less_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (numeral.numeral one plus m) (numeral.numeral one plus n) = (m < n)"
inject_5: fixes x61 :: "'c"   and x62 :: "('a, 'b, 'c) Stmt"   and y61 :: "'c"   and y62 :: "('a, 'b, 'c) Stmt" shows "(While x61 x62 = While y61 y62) = (x61 = y61 \<and> x62 = y62)"
b: shows "x:M \<in> NEGn B (\<parallel><B>\<parallel>)"
inf_aci_3: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
balanced: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int"   and ps :: "'b set" assumes "t_wise_balance point_set block_collection grouping index"   and "ps \<subseteq> point_set"   and "int (card ps) = grouping" shows "int (block_collection index ps) = index"
bigo_def: fixes f :: "'c \<Rightarrow> 'd" shows "O(f) \<equiv> {h. \<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"
soundness_and_entailment: fixes Rule :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and S :: "'at Literal set set" assumes "Sound Rule"   and "Rule P1 P2 C"   and "P1 \<in> S"   and "P2 \<in> S" shows "entails S C"
fps_inv_right: fixes a :: "'a fps" assumes "a $ 0 = (0::'a)"   and "a $ 1 \<noteq> (0::'a)" shows "a oo fps_inv a = fps_X"
SB: shows "nths (xs @ [x]) B = nths xs B @ nths [x] {j. j + length xs \<in> B}"
mI_vals_nonempty: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and j :: "nat" assumes "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "j \<le> n" shows "\<nu>\<^bsub>K P j\<^esub> ` I \<noteq> {}"
second_def: fixes x :: "('c, 'b) OrdSum" shows "second x = snd (Rep_OrdSum x)"
hd_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []" shows "lead_coeff (map f xs) = f (lead_coeff xs)"
not_eq_order_implies_strict: fixes a :: "('b, 'c) Term.term"   and b :: "('b, 'c) Term.term" assumes "a \<noteq> b"   and "a \<unrhd> b" shows "a \<rhd> b"
iso_dghm_ArrMap_eqE: fixes f :: "V"   and g :: "V" assumes "\<FF>\<lparr>ArrMap\<rparr>\<lparr>f\<rparr> = \<FF>\<lparr>ArrMap\<rparr>\<lparr>g\<rparr>"   and "f \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"   and "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" obtains "f = g"
assms_8: shows "component_of_term (lt r) = component_of_term (lt p)"
the_\<theta>_props_1: shows "\<guillemotleft>the_\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
second_inner_invariant_when_e_not_bot: fixes j :: "'a"   and f :: "'a"   and h :: "'a"   and g :: "'a"   and d :: "'a" assumes "boruvka_inner_invariant j f h g d"   and "j \<noteq> bot"   and "selected_edge h j g \<le> - forest_components f"   and "selected_edge h j g \<noteq> bot" shows "boruvka_inner_invariant (j \<sqinter> - choose_component (forest_components h) j) (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion> (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion> selected_edge h j g) h g (d \<squnion> selected_edge h j g)"
exp_times_self_gt: fixes x :: "real" assumes "x \<noteq> - 1" shows "- exp (- 1) < x * exp x"
mp_remdups': fixes c :: "'c"   and f :: "'c \<Rightarrow> 'd"   and xs :: "'c list" assumes "c \<in> set (remdups' f xs)" shows "c \<in> set xs"
continuous_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f \<equiv> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
m_zmax_pdsI_hom_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set"   and I :: "'c set" shows "m_zmax_pdsI_hom K P I = (\<lambda>j. tna (AMin (\<nu>\<^bsub>K P j\<^esub> ` I)))"
symrun_interp_absorb1: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and \<Gamma>\<^sub>2 :: "'b constr list" assumes "set \<Gamma>\<^sub>1 \<subseteq> set \<Gamma>\<^sub>2" shows "\<lbrakk>\<lbrakk> \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m = \<lbrakk>\<lbrakk> \<Gamma>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m"
induct: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'a" assumes "\<And>a b. (b \<noteq> (0::'a) \<Longrightarrow> P b (a mod b)) \<Longrightarrow> P a b" shows "P a0 a1"
simps_2: fixes w :: "brick list"   and ws :: "wall" shows "kauff_mat (w * ws) = rat_poly.matrix_mult (blockmat w) (kauff_mat ws)"
lincomb_Nil: fixes rs :: "'f list"   and ms :: "'v list" assumes "rs = [] \<or> ms = []" shows "lincomb' rs ms = (0::'v)"
pw_nres_rel_iff: fixes a :: "'a nres"   and b :: "'b nres"   and A :: "('a \<times> 'b) set" shows "((a, b) \<in> \<langle>A\<rangle>nres_rel) = (nofail (\<Down> A b) \<longrightarrow> nofail a \<and> (\<forall>x. inres a x \<longrightarrow> inres (\<Down> A b) x))"
ideal_d: shows "ideal_generated {a, b} = ideal_generated {d}"
nfoldli_alt_eq: fixes l :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli_alt l c f s = (case l of [] \<Rightarrow> RETURN s | x # ls \<Rightarrow> let b = c s in if b then f x s \<bind> nfoldli_alt ls c f else RETURN s)"
assms: shows "\<k> \<le> des.\<v> - 1"
card_of_image: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "|f ` A| \<le>o |A|"
intro: fixes grouping :: "int"   and u_block_size :: "int" assumes "grouping \<le> u_block_size" shows "t_design_axioms u_block_size grouping"
has_integral_I0: shows "((\<lambda>x. x powr - (1 / 2) * sqrt (4 - x)) has_integral 2 * pi) {0..4}"
nullhomotopic_into_contractible: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "contractible T" obtains c :: "'c" where "homotopic_with_canon (\<lambda>h. True) S T f (\<lambda>x. c)"
simps_1: fixes x1 :: "lname"   and y1 :: "lname" shows "(LVar x1 = LVar y1) = (x1 = y1)"
unfold: fixes g :: "('d \<Rightarrow> bool) \<times> ('d \<Rightarrow> 'e \<times> 'd)"   and s :: "'d" assumes "fst g s"   and "snd (snd g s) \<in> terminates_on g" shows "s \<in> terminates_on g"
app_invimageI: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "b \<in>\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"
q64_upto_def: shows "q64 = (\<lambda>k. k\<^sup>2 mod 64) ` {..<64}"
telescope_sums: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "f \<longlonglongrightarrow> c" shows "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
SPEC_rule: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Phi>' :: "'c \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Phi>' x" shows "SPEC \<Phi> \<le> SPEC \<Phi>'"
HARootStates_notmem_Chi: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and T :: "'a" assumes "S \<in> HAStates A"   and "T \<in> States (HARoot A)" shows "T \<notin> Chi A S"
ta_prod_correct_aux1: fixes TA1 :: "('e, 'd, 'g) tree_automaton_rec_scheme"   and TA2 :: "('f, 'd, 'h) tree_automaton_rec_scheme" shows "ta_lang (ta_prod TA1 TA2) = ta_lang TA1 \<inter> ta_lang TA2"
chine_assoc_props_1: shows "\<guillemotleft>chine_assoc : \<mu>\<nu>_\<pi>.chine \<rightarrow>\<^sub>C \<mu>\<nu>\<pi>.chine\<guillemotright>"
q_def: shows "q = (\<lambda>b. h (q0 b))"
mk_disjoint_insert: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "\<exists>B. A = insert a B \<and> a \<notin> B"
mult_less_cancel_right1: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times b c) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
WRP: fixes i :: "nat" shows "{} \<turnstile> WRP (ORD_OF i) \<guillemotleft>ORD_OF i\<guillemotright>"
image2_eqI: fixes b :: "'a"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and c :: "'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "b = f x"   and "c = g x"   and "x \<in> A" shows "(b, c) \<in> BNF_Greatest_Fixpoint.image2 A f g"
satisfies_properties_sys1: shows "satisfies_properties_R p qs1 subsets1 signs1 matrix1"
well_foundedI_pf: assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<subseteq> ((\<sqsubset>) ``` X) \<inter> A\<rbrakk> \<Longrightarrow> X = {}" shows "well_founded A (\<sqsubset>)"
head_\<omega>_\<omega>: shows "head_\<omega> \<omega> = \<omega>"
rel_sum_simps_3: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a2 :: "'d"   and b1 :: "'c" shows "rel_sum R1 R2 (Inr a2) (Inl b1) = False"
SpecI: fixes H :: "fm set"   and A :: "fm"   and i :: "name" shows "H \<turnstile> A IMP SyntaxN.Ex i A"
ub_add_nat: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a"   and p :: "nat" assumes "m \<le> n + 1" shows "sum g {m..n + p} = sum g {m..n} + sum g {n + 1..n + p}"
inj_map_strong: fixes x :: "'g list"   and xa :: "'g list"   and f :: "'g \<Rightarrow> 'h"   and fa :: "'g \<Rightarrow> 'h" assumes "\<And>z za. \<lbrakk>z \<in> set x; za \<in> set xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map f x = map fa xa" shows "x = xa"
inj_vimage_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "f -` f ` A = A"
mult_ant_def: fixes z :: "ant"   and w :: "ant" shows "z * w \<equiv> Abs_Ainteg (zag_t (Rep_Ainteg z) (Rep_Ainteg w))"
trivial_ker_imp_inj: assumes "a_kernel R S h = {\<zero>}" shows "inj_on h (carrier R)"
rel_mono_strong: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'g \<rightharpoonup>\<^sub>f 'e"   and y :: "'g \<rightharpoonup>\<^sub>f 'f"   and Ra :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "fmrel R x y"   and "\<And>z yb. \<lbrakk>z \<in> fmran' x; yb \<in> fmran' y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "fmrel Ra x y"
preserves_hom: fixes f :: "'b \<times> 'b"   and a :: "'b \<times> 'b"   and b :: "'b \<times> 'b" assumes "B.VV.in_hom f a b" shows "\<guillemotleft>fst f \<star>\<^sub>B snd f : fst a \<star>\<^sub>B snd a \<rightarrow>\<^sub>B fst b \<star>\<^sub>B snd b\<guillemotright>"
induct: fixes P :: "(('b \<rightarrow> 'c) \<rightarrow> ['b] \<rightarrow> ['c]) \<Rightarrow> bool" assumes "adm P"   and "P \<bottom>"   and "\<And>x. P x \<Longrightarrow> P (\<Lambda> f v. Fixrec.run\<cdot>(match_Nil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>[]) +++ match_Cons\<cdot>v\<cdot>(\<Lambda> xa xs. Fixrec.succeed\<cdot>(f\<cdot>xa : x\<cdot>f\<cdot>xs))))" shows "P map"
has_integral_substitution_general_: fixes s :: "real set"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real"   and c :: "real"   and d :: "real"   and f :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> real" assumes "finite s"   and "a \<le> b"   and "g ` {a..b} \<subseteq> {c..d}"   and "f integrable_on {c..d}"   and "continuous_on ({c..d} - g ` s) f"   and "continuous_on {a..b} g"   and "inj_on g ({a..b} \<union> s)"   and "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_real_derivative g' x) (at x within {a..b})" shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral integral {g a..g b} f - integral {g b..g a} f) {a..b}"
combine_sub_sys_alt: shows "sub_incidence_system \<V>' \<B>' \<V>\<^sup>+ \<B>\<^sup>+"
cases: fixes a1 :: "nat"   and a2 :: "nat \<Rightarrow>f value option"   and a3 :: "((nat \<times> nat) \<times> transition) fset"   and a4 :: "nat"   and a5 :: "nat \<Rightarrow>f value option"   and a6 :: "(String.literal \<times> value list) list" assumes "obtains a1 a2 a3 a4 a5 a6" obtains s :: "nat" and r :: "nat \<Rightarrow>f value option" and e :: "((nat \<times> nat) \<times> transition) fset" where "a1 = s__" and "a2 = r__" and "a3 = e__" and "a4 = s__" and "a5 = r__" and "a6 = []"   | e :: "((nat \<times> nat) \<times> transition) fset" and s' :: "nat" and r' :: "nat \<Rightarrow>f value option" and l :: "String.literal" and i :: "value list" and s :: "nat" and r :: "nat \<Rightarrow>f value option" and t :: "(String.literal \<times> value list) list" where "a1 = s" and "a2 = r" and "a3 = e" and "a4 = s'" and "a5 = r'" and "a6 = (l, i) # t" and "\<exists>(s'', T)|\<in>|possible_steps e s' r' l i. obtains s r e s'' (evaluate_updates T i r') t"
ps2_P_d1: shows "ps2_P.d1 = N"
SPEC_refine: fixes S :: "'a nres"   and R :: "('a \<times> 'b) set"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "S \<le> SPEC (\<lambda>x. \<exists>x'. (x, x') \<in> R \<and> \<Phi> x')" shows "S \<le> \<Down> R (SPEC \<Phi>)"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "CCC.prod (g \<cdot> f) b = CCC.prod g b \<cdot> CCC.prod f b"
fold_once_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list" shows "fold_once f xs = (if rsp_fold f then fold f (remdups xs) else id)"
assms_1: shows "\<forall>(a, b)\<in>r. \<exists>c. (b, c) \<in> r"
strategyproofness_1: shows "(pmf (sds R2) a < pmf (sds R28) a \<or> pmf (sds R2) a + pmf (sds R2) c < pmf (sds R28) a + pmf (sds R28) b) \<or> pmf (sds R2) a = pmf (sds R28) a \<and> pmf (sds R2) a + pmf (sds R2) c = pmf (sds R28) a + pmf (sds R28) b"
distinct_17: fixes x21 :: "('a, 'b, 'c) expr"   and x22 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Apply x21 x22 \<noteq> Ref x4"
mult_less_cancel_left: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times c a) (times c b) = ((less_eq zero c \<longrightarrow> less a b) \<and> (less_eq c zero \<longrightarrow> less b a))"
absorb2: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" assumes "b \<preceq> a" shows "linordered_monoid.min a b = b"
addfunset_def: fixes A :: "'a set"   and M :: "'b set" shows "addfunset A M \<equiv> {f. supp f \<subseteq> A \<and> range f \<subseteq> M \<and> (\<forall>x\<in>A. \<forall>y\<in>A. f (x + y) = f x + f y)}"
2: shows "inj_on And\<^sub>n (fset ` min_dnf \<phi>)"
induct: fixes P :: "'b rexp \<Rightarrow> 'b trm \<Rightarrow> bool"   and a0 :: "'b rexp"   and a1 :: "'b trm" assumes "\<And>r rexp. P r (Lam rexp)"   and "\<And>r X rexp. P r (Trn X rexp)" shows "P a0 a1"
inv_func: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "f\<inverse>\<^bsub>B,A\<^esub> \<in> B \<rightarrow> A"
coinduct: fixes R :: "('b, 'c) dds \<Rightarrow> ('b, 'c) dds \<Rightarrow> bool"   and dds :: "('b, 'c) dds"   and dds' :: "('b, 'c) dds" assumes "R dds dds'"   and "\<And>dds dds'. R dds dds' \<Longrightarrow> ((=) ===> rel_prod (=) R) (run dds) (run dds')" shows "dds = dds'"
card_H: shows "real n / 2 \<le> real (card (uverts H))"
subcls1_irrefl_lemma2: assumes "ws_prog" shows "\<forall>x. (x, x) \<notin> subcls1\<^sup>+"
simps_2: fixes m :: "nat"   and a :: "'a"   and ts :: "'a FOL_Fitting.term list" shows "closedt m (App a ts) = closedts m ts"
pow_eq_div2: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier \<G>"   and "x [^] m = x [^] n" shows "x [^] (m - n) = \<one>"
is_full: fixes a :: "'b"   and a' :: "'b"   and g :: "'c" assumes "F.B.ide a"   and "F.B.ide a'"   and "\<guillemotleft>g : G a' \<rightarrow> G a\<guillemotright>" shows "\<exists>f. F.B.in_hom f a' a \<and> G f = g"
wfP_sim21_size_aux: fixes n :: "nat" shows "wfP (sim21_size_aux n)"
le_nrows: fixes n :: "nat"   and A :: "'b matrix" shows "(n \<le> nrows A) = (\<forall>m. (\<forall>j i. m \<le> j \<longrightarrow> Rep_matrix A j i = (0::'b)) \<longrightarrow> n \<le> m)"
has_integral_cos_nx: fixes n :: "int" shows "((\<lambda>x. cos (real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {- pi..pi}"
is_ring_morphism: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e"   and U :: "'c set" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U" shows "ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)"
skip: fixes s :: "'a"   and s' :: "'a" assumes "g s = Skip s'" shows "(s', s) \<in> R"
one_closed: fixes H :: "'e set"   and G :: "('e, 'f) monoid_scheme" assumes "submonoid H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
assms_3: shows "es\<^sub>2 = map (\<lambda>v. (v, False)) (delete_effects_of op)"
2: fixes x :: "'b"   and \<sigma> :: "'a" shows "f x \<sigma> \<noteq> dSUCCEED"
assms_2: shows "ref \<in> set (interp_ins (xs @ [(oid, Some ref)] @ ys))"
adopt_node_def: fixes document :: "(_) document_ptr"   and node :: "(_) node_ptr" shows "adopt_node document node = Heap_Error_Monad.bind (get_ancestors_di (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document)) (\<lambda>ancestors. if cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node \<in> set ancestors then error HierarchyRequestError else adopt_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M document node)"
trg_dom: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>C (C.dom \<mu>) = trg\<^sub>C \<mu>"
Gauss_Jordan_in_ij_det_P_iarrays_def: fixes A :: "'b iarray iarray"   and i :: "nat"   and j :: "nat" shows "Gauss_Jordan_in_ij_det_P_iarrays A i j = (let n = least_non_zero_position_of_vector_from_index (column_iarray j A) i in (if i = n then (1::'b) / A !! i !! j else - (1::'b) / A !! n !! j, Gauss_Jordan_in_ij_iarrays A i j))"
l9_5: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and C :: "'a"   and R :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.Col Bet R P Q"   and "Tarski_neutral_dimensionless.Out Bet R A B" shows "Tarski_neutral_dimensionless.TS Bet P Q B C"
a2cb: shows "a\<^sup>2 = c\<^sup>2 - b\<^sup>2"
gtr: shows "(\<Sum>w\<in>a ` (Is p \<inter> {Suc r..<m}). ls p w * (cs q \<bullet> w)) = (0::'a)"
nn_integral_def: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "integral\<^sup>N M f = Sup (integral\<^sup>S M ` {g. simple_function M g \<and> g \<le> f})"
fin: fixes g :: "graph" assumes "final g" shows "succs g = []"
null_eqI: fixes n :: "'a \<times> 'a" assumes "\<And>f. B.VV.comp n f = n \<and> B.VV.comp f n = n" shows "n = B.VV.null"
countable_Sigma: shows "countable {x \<in> Sigma A B. (0::'c) \<noteq> f x}"
ureal: shows "\<forall>\<^sub>F n in F. u n = ereal (real_of_ereal (u n))"
simps_3: fixes Ks :: "'c list" shows "S.combine Ks [] = \<zero>\<^bsub>S\<^esub>"
Int_Diff: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A \<inter> B - C = A \<inter> (B - C)"
CollectI: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a" shows "a \<in> {x. P x}"
hom_0_iff: fixes x :: "'a poly" shows "(poly_lift x = 0) = (x = 0)"
quasisym_not_greater: fixes a :: "'a"   and b :: "'a" assumes "cmp a b \<noteq> Greater"   and "cmp b a \<noteq> Greater" shows "cmp a b = Equiv"
partially_vector_differentiable_def: fixes F :: "'a \<Rightarrow> 'b"   and base_vec :: "'a"   and p :: "'a" shows "partially_vector_differentiable F base_vec p \<equiv> \<exists>F'. has_partial_vector_derivative F base_vec F' p"
halloc_xcpt: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and oi :: "obj_tag"   and a :: "loc"   and s' :: "state" assumes "G\<turnstile>(Some xc, s) \<midarrow>halloc oi\<succ>a\<rightarrow> s'" shows "s' = (Some xc, s)"