hcju/isabelleps huggingface dataset:
transpose_mat_def: fixes A :: "'a mat" shows "A\<^sup>t \<equiv> mat (dim_col A) (dim_row A) (\<lambda>(i, j). A $$ (j, i))"
Red_F_subs_Red_F_diff_Red_F: fixes N :: "'f set" shows "Red_F N \<subseteq> Red_F (N - Red_F N)"
Cons_1: assumes "\<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0" shows "\<exists>y'>q. \<forall>a\<in>set leq. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f \<le> 0"
arr_cod: fixes f :: "'t" assumes "S'.arr f" shows "S'.arr (S'.cod f)"
vp2: shows "cls_val_process Mkt pf2 (Suc n) w = (\<Sum>x\<in>support_set pf1 \<union> support_set pf2. prices Mkt x (Suc n) w * pf2 x (Suc n) w)"
prv_cnj_imp_monoR2: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> (imp \<chi> \<psi>))" shows "prv (imp (cnj \<phi> \<chi>) \<psi>)"
completeness_game_def: fixes h :: "'g \<times> 'a"   and w :: "'h + 'b"   and e :: "'e" shows "\<Sigma>_OR.completeness_game h w e = init_OR h w \<bind> (\<lambda>(r, a). response_OR r w e \<bind> (\<lambda>z. return_spmf (check_OR h a e z)))"
cross_ratio_01inf_id: fixes z :: "complex_homo" shows "cross_ratio z 0\<^sub>h 1\<^sub>h \<infinity>\<^sub>h = z"
trss_env_increasing_s: fixes s :: "'f list"   and c :: "'f list multiset"   and e :: "('g, 'h) label"   and s' :: "'f list"   and c' :: "'f list multiset"   and fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme" assumes "((s, c), e, s', c') \<in> trss fg" shows "c \<subseteq># c'"
OUTfromVCorrect1_data14: shows "OUTfromVCorrect1 data14"
comp_cod_arr: fixes f :: "'c"   and b :: "'c" assumes "C.arr f"   and "C.cod f = b" shows "b \<cdot> f = f"
IH: fixes gpv :: "('c, 'a, 'b) gpv" shows "c * expectation_gpv' gpv = expectation_gpv'' gpv"
weak_le_iff_join: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> y) = (x .=\<^bsub>L\<^esub> x \<sqinter>\<^bsub>L\<^esub> y)"
fls_inverse_base_factor: fixes f :: "'a fls" shows "inverse (fls_base_factor f) = fls_base_factor (inverse f)"
steps_or: fixes L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and q :: "bool list"   and w :: "'b list" shows "((start (or L R), q) \<in> steps (or L R) w) = (w = [] \<and> q = start (or L R) \<or> w \<noteq> [] \<and> (\<exists>p. q = True # p \<and> (start L, p) \<in> steps L w \<or> q = False # p \<and> (start R, p) \<in> steps R w))"
Gram_Schmidt_upt_k_efficient_induction: fixes A :: "((real, 'a) vec, 'b) vec"   and k :: "nat" assumes "Gram_Schmidt_upt_k A k = foldl Gram_Schmidt_column_k_efficient A [0..<Suc k]"   and "Suc k < ncols A" shows "Gram_Schmidt_column_k (Gram_Schmidt_upt_k A k) (Suc k) = Gram_Schmidt_column_k_efficient (Gram_Schmidt_upt_k A k) (Suc k)"
uI_int2: fixes n :: "nat" shows "I.expectation (uI n) = expectation (u n)"
initS_def: shows "initS = {u. fst (u 0) = 0}"
assms_1: shows "\<I>'',map_\<I> f' g' \<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"
sublens_pres_indep: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and Z :: "'h \<Longrightarrow> 'f" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<bowtie> Z" shows "X \<bowtie> Z"
stepResultsinR_def: fixes p :: "'a option"   and p' :: "'a option"   and R :: "('a list \<times> 'a list) set" shows "WHATWHERE.stepResultsinR p p' R \<equiv> p = None \<and> p' = None \<or> (\<exists>c c'. p = Some c \<and> p' = Some c' \<and> ([c], [c']) \<in> R)"
z: shows "z \<in> interior (convex hull {a - complex_of_real e, c, a + complex_of_real e})"
q_1: fixes x :: "'a set" assumes "x \<in> p" shows "q x division_of cbox a b"
projective_family_axioms: shows "projective_family I P (\<lambda>_. borel)"
v: shows "(s \<cdot> r) \<^sup>@ Suc i = v"
bind_return_val': fixes x :: "val"   and t :: "pdf_type"   and f :: "val \<Rightarrow> val"   and t' :: "pdf_type" assumes "val_type x = t"   and "f \<in> stock_measure t \<rightarrow>\<^sub>M stock_measure t'" shows "return_val x \<bind> (\<lambda>x. return_val (f x)) = return_val (f x)"
step_3: fixes T :: "ty" assumes "P,E,hp s',shp s' \<turnstile> e' : T"   and "iconf (shp s') e'"   and "P,E \<turnstile> s' \<surd>" shows "P,E \<turnstile> s' \<surd>"
1: shows "local.set (local.dom f) \<subseteq> Univ \<and> |local.set (local.dom f)| <o \<AA> \<and> local.set (cod f) \<subseteq> Univ \<and> |local.set (cod f)| <o \<AA> \<and> ide (local.dom f) \<and> ide (cod f) \<and> Fun f \<in> extensional (local.set (local.dom f)) \<inter> (local.set (local.dom f) \<rightarrow> local.set (cod f))"
mem_not_refl: fixes i :: "V" shows "i \<notin> elts i"
compositionality: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and multiset :: "'c multiset" shows "image_mset f (image_mset g multiset) = image_mset (f \<circ> g) multiset"
vec_add: fixes x :: "'a"   and y :: "'a" shows "vec (x + y) = vec x + vec y"
image_insert: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and B :: "'c set" shows "f ` insert a B = insert (f a) (f ` B)"
target_alt_def_2: fixes io :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "length io = length tr"   and "0 < length io" shows "target (io || tr) q = last tr"
mult_less_cancel1_hmset: fixes k :: "hmultiset"   and m :: "hmultiset"   and n :: "hmultiset" shows "(k * m < k * n) = (0 < k \<and> m < n)"
in_fold_Deriv: fixes v :: "'b list"   and w :: "'b list"   and L :: "'b list set" shows "(v \<in> fold Deriv w L) = (w @ v \<in> L)"
span_eq: fixes S :: "'a set"   and T :: "'a set" shows "(cspan S = cspan T) = (S \<subseteq> cspan T \<and> T \<subseteq> cspan S)"
f1: shows "\<forall>f fa fb. (((prv (imp f (dsj fa fb)) \<or> \<not> prv (imp f (dsj fb fa))) \<or> f \<notin> fmla) \<or> fa \<notin> fmla) \<or> fb \<notin> fmla"
simps_2: fixes x21 :: "nat"   and x22 :: "nat"   and y21 :: "nat"   and y22 :: "nat" shows "(Eq x21 x22 = Eq y21 y22) = (x21 = y21 \<and> x22 = y22)"
elims_2: fixes x :: "'v"   and xa :: "('v \<times> 'w \<times> 'v) list"   and xb :: "'v" assumes "is_path x xa xb" obtains v :: "'v" and v' :: "'v" where "x = vb__" and "xa = []" and "xb = v'a__" and "vb__ = v'a__ \<and> v'a__ \<in> V"   | v :: "'v" and v1 :: "'v" and w :: "'w" and v2 :: "'v" and p :: "('v \<times> 'w \<times> 'v) list" and v' :: "'v" where "x = v" and "xa = (v1, w, v2) # p" and "xb = v'" and "v = v1 \<and> (v1, w, v2) \<in> E \<and> is_path v2 p v'"
bij_betw_arr_sets: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "bij_betw F (Collect (partial_magma.arr B)) (Collect (partial_magma.arr A))"
wts_def: fixes OS :: "Store" shows "wts OS = (\<forall>l. \<tau> OS@@l \<le> ltype l)"
Im_Arcsin_of_real: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "Im (Arcsin (complex_of_real x)) = 0"
assert_bound_nolhs_sat: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "\<not> \<U> (assert_bound a s)" shows "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s (assert_bound a s) \<and> \<diamond> (assert_bound a s)"
a_set_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "local.a_set_tag_name_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name_update)"
Sup_ne_def: fixes X :: "'a set" shows "Sup_ne X = FOLD_bin_ne sup X"
imageI: fixes x :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'd" assumes "x \<in> A" shows "f x \<in> f ` A"
subsLR: shows "vertices (fst R) \<subseteq> vertices (snd R)"
scale_left_imp_eq: fixes a :: "'c"   and x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" assumes "a \<noteq> (0::'c)"   and "a *s x = a *s y" shows "x = y"
list_emb_ConsD: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "list_emb P (x # xs) ys" shows "\<exists>us v vs. ys = us @ v # vs \<and> P x v \<and> list_emb P xs vs"
det30_plus_scaled2: fixes a :: "real \<times> real"   and x :: "real"   and b :: "real \<times> real" shows "coll 0 (a + x *\<^sub>R a) b = (if x = - 1 then True else coll 0 a b)"
sets_return: fixes M :: "'b measure"   and x :: "'b" shows "sets (return M x) = sets M"
heyting_stone: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and implies :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.heyting_stone_algebra uminus inf less_eq less sup bot top implies" shows "sup (uminus x) (uminus (uminus x)) = top"
balance_bdinv: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "bd_inv l"   and "bd_inv r"   and "black_depth l = black_depth r" shows "bd_inv (balance l k v r)"
ttree_restr_mono: fixes x :: "'b ttree"   and y :: "'b ttree"   and S :: "'b set" assumes "x \<sqsubseteq> y" shows "ttree_restr S x \<sqsubseteq> ttree_restr S y"
cases: fixes a :: "nat set" assumes "a \<in> i_set0_ind" obtains "a = {}"   | I :: "nat set" where "a = I" and "I \<in> i_set_ind"
that: fixes inner :: "complex set"   and outer :: "complex set" assumes "inner \<noteq> {}"   and "open inner"   and "connected inner"   and "outer \<noteq> {}"   and "open outer"   and "connected outer"   and "bounded inner"   and "\<not> bounded outer"   and "inner \<inter> outer = {}"   and "inner \<union> outer = - path_image c"   and "frontier inner = path_image c"   and "frontier outer = path_image c" shows "thesis"
Rel_def: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Rel r \<equiv> r"
aPQ: shows "\<forall>a. \<forall>r\<in>P a. Q r"
effect_updE: fixes i :: "nat"   and v :: "'b"   and a :: "'b array"   and h :: "heap"   and h' :: "heap"   and r :: "'b array" assumes "effect (a.(i) \<leftarrow> v) h h' r" obtains "r = a" and "h' = Array.update a i v h" and "i < Array.length h a"
rep_eq: fixes x :: "'d"   and xa :: "'d" shows "set_bset (bdoubleton x xa) = {x, xa}"
FrameStatEqRefl: fixes F :: "'b frame" shows "F \<simeq>\<^sub>F F"
bind_eq_Some_conv: fixes f :: "'e option"   and g :: "'e \<Rightarrow> 'd option"   and x :: "'d" shows "(Option.bind f g = Some x) = (\<exists>y. f = Some y \<and> g y = Some x)"
rep_eq: fixes x :: "'b fset" shows "fcard x = card \<langle>x\<rangle>\<^sub>f"
assms: fixes f :: "'a"   and g :: "'a"   and h :: "'a" shows "b (b f g) h = b f (b g h)"
callee_def: shows "callee \<equiv> \<lambda>s c. lift_spmf (p s c)"
weakPsiCongruence_def: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<doteq> Q \<equiv> \<Psi> \<rhd> P \<approx> Q \<and> \<Psi> \<rhd> P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q \<and> \<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
that: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> real" assumes "\<And>x y. 0 \<le> h x y"   and "\<And>x y. 0 < h x y \<Longrightarrow> (x, y) \<in> R"   and "\<And>x. h x sums f x"   and "\<And>y. (\<lambda>x. h x y) sums g y" shows "thesis"
ReadMem: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ad :: "'b"   and al :: "addr_loc"   and v :: "'b val" assumes "a \<in> actions E"   and "action_obs E a = NormalAction (ReadMem ad al v)" shows "read_actionsp E a"
validSummand_3: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "valid P"   and "hnf P"   and "a<x>.P' \<in> summands P" shows "valid P'"
bind_spmf_assoc: fixes x :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf"   and g :: "'d \<Rightarrow> 'c spmf" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
comm_group_def: fixes G :: "('e, 'f) monoid_scheme" shows "comm_group G \<equiv> Group.comm_monoid G \<and> Group.group G"
ide_cod: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.ide (C.VV.cod f)"
x0_hb_bound4': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "bs ! i / 2 \<le> bs ! i + (hs ! i) x / x"
distinct_36: fixes x71 :: "nat"   and x72 :: "msg"   and x4 :: "nat" shows "Crypt x71 x72 \<noteq> Key x4"
RModuleI: fixes R :: "'a set"   and M :: "'b set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Ring1 R"   and "Group M"   and "RModule_axioms R smult M" shows "RModule R smult M"
inv_solve_right: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a"   and c :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P"   and "c \<in> carrier P" shows "(a = b \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> c) = (b = a \<oplus>\<^bsub>P\<^esub> c)"
two_comparisons_into_case_order_2: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and Q :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if le_of_comp comp x y then if y = x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
bv_uminus_returntype: fixes w :: "bit list" shows "norm_signed (bv_uminus w) = bv_uminus w"
pmf_integral_pmf_set_integral: fixes p :: "'b pmf"   and f :: "'b \<Rightarrow> real" shows "pmf_integral p f = pmf_set_integral p f (set_pmf p)"
finite_span_complete: fixes A :: "'d set" assumes "finite A" shows "complete (span A)"
distinct_map_of_rev: fixes xs :: "('a \<times> 'b) list" assumes "distinct (map fst xs)" shows "map_of (rev xs) = map_of xs"
monad_fail_axioms_def: fixes bind :: "'d \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> 'd"   and fail :: "'d" shows "monad_fail_axioms bind fail \<equiv> \<forall>f. bind fail f = fail"
termdiffs_sums_strong: fixes K :: "real"   and c :: "nat \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and z :: "'a" assumes "\<And>z. norm z < K \<Longrightarrow> (\<lambda>n. c n * z ^ n) sums f z"   and "(f has_field_derivative f') (at z)"   and "norm z < K" shows "(\<lambda>n. diffs c n * z ^ n) sums f'"
main_step_aux1: fixes qs :: "rat poly list" assumes "fst (factorize_polys qs) = []" shows "set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"
prems_1: shows " \<Gamma> \<subseteq> \<Gamma>' "
Poly_list_of_vec_0: shows "Poly (list_of_vec (0\<^sub>v 0)) = [:0::'b:]"
ctx_dcl_mem_path: fixes P :: "cld list"   and ctx :: "ctx"   and dcl :: "char list"   and path :: "(ctx \<times> cld) list" assumes "find_path_f P ctx (cl_fqn (fqn_def dcl)) = Some path" shows "(ctx, dcl) \<in> (\<lambda>(ctx, cld). (ctx, class_name_f cld)) ` set path"
neg2: shows "Negligible.negligible (\<lambda>n. local.ddh.advantage n (local.ddh.DDH_\<A>' n (P1_DDH_mal_adv_\<sigma>_true n M z \<A> D)))"
test_mult_lb1: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p \<cdot> q \<le> p"
subseqs_length_simple_False: fixes b :: "'a list"   and xs :: "'a list" assumes "b \<in> set (subseqs xs)"   and "length xs < length b" shows "False"
sinvar_all_edges_normal_form_not_refl_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "sinvar_all_edges_normal_form_not_refl P \<equiv> \<forall>G nP. sinvar G nP = (\<forall>(e1, e2)\<in>edges G. e1 \<noteq> e2 \<longrightarrow> P (nP e1) (nP e2))"
prems: shows "Branching T1' T2' = Branching T1 T2"
pre_ranks_element: fixes q' :: "'b"   and n :: "nat"   and r :: "nat"   and q :: "'b" assumes "state_rank q' n = Some r"   and "q = \<delta> q' (w n)" shows "r \<in> pre_ranks (\<lambda>q. state_rank q n) (w n) q"
simps_44: shows "params_fm \<^bold>\<bottom> = {}"
restriction_mon1: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> restrict u f c \<le> restrict v f c"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and P :: "'a prog" assumes "P \<turnstile> a1 \<prec>\<^sup>1 a2" obtains C :: "char list" and D :: "char list" and rest :: "fdecl list \<times> 'a mdecl list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
SA_all: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "POPLmarkRecord.type"   and S\<^sub>1 :: "POPLmarkRecord.type"   and S\<^sub>2 :: "POPLmarkRecord.type"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1"   and "TVarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2" shows "\<Gamma> \<turnstile> (\<forall><:S\<^sub>1. S\<^sub>2) <: (\<forall><:T\<^sub>1. T\<^sub>2)"
ll_on_open: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and S :: "'b set"   and t\<^sub>0 :: "real" assumes "picard_lindeloef f T S t\<^sub>0" shows "ll_on_open T f S"
urel32_def: fixes pp :: "uint32"   and x :: "uint32"   and y :: "int" assumes "p = int_of_uint32 pp"   and "p \<le> 65535" shows "urel32 x y = (y = int_of_uint32 x \<and> y < p)"
R2P_P2R: fixes \<phi> :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "R2P (P2R \<phi>) = \<phi>"
eq: fixes f :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "f \<in> F" shows "dehomogenize x (homogenize x f) = f"
assms_3: fixes loc :: "'loc" shows "c_work c loc = {#}\<^sub>z"
card_Diff2_less: fixes A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "finite A"   and "x \<in> A"   and "y \<in> A" shows "card (A - {x} - {y}) < card A"
elims_3: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'd"   and xa :: "('a, 'b) lit" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP"   and "\<not> M.Struct.satL intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>__" and "xa = Pos a__" and "\<not> M.Struct.satA intF intP \<xi>__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and a :: "('a, 'b) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> M.Struct.satA intF intP \<xi> a"
simps_2: fixes h :: "'b hp option" shows "exec Del_min [h] = del_min h"
fully_faithful_functor_axioms: shows "fully_faithful_functor (local.Left a') (local.Left a') (H\<^sub>L a')"
extendID: fixes S :: "'a sequent"   and i :: "nat"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form multiset" assumes "extend S ( \<LM> At i  \<RM> \<Rightarrow>* \<LM> At i  \<RM>) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "At i \<in># \<Gamma> \<and> At i \<in># \<Delta>"
Abs_status_inverse: fixes y :: "('d, 'e, 'f) hierauto \<times> 'd set \<times> 'e set \<times> 'f data" assumes "y \<in> status" shows "Rep_status (Abs_status y) = y"
per_not_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "Per A B C" shows "\<not> Col A B C"
powr_minus_divide: fixes x :: "'a"   and a :: "'a" shows "x powr - a = (1::'a) / x powr a"
v_TS_defs_2: shows "v_init = {\<lparr>next_round = 0, votes = \<lambda>r. Map.empty, decisions = Map.empty\<rparr>}"
finite_Lim_measure_decseq: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> sets N"   and "decseq A" shows "(\<lambda>n. Sigma_Algebra.measure N (A n)) \<longlonglongrightarrow> Sigma_Algebra.measure N (\<Inter> (range A))"
wt_app_eff_imp_wt_err: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and app :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and ts :: "'a list" assumes "wt_app_eff r app step ts"   and "bounded (err_step (length ts) app step) (length ts)" shows "wt_err_step r (err_step (length ts) app step) (map OK ts)"
hurwitz_zeta_neg_of_nat: fixes a :: "real"   and n :: "nat" assumes "0 < a" shows "hurwitz_zeta a (- of_nat n) = complex_of_real (- bernpoly (Suc n) a / real (Suc n))"
NOT_mask_AND_mask: fixes w :: "'a"   and n :: "nat" shows "(w AND mask n) AND NOT (mask n) = (0::'a)"
Rcd: fixes fs :: "(char list \<times> trm) list" assumes "\<forall>(l, t)\<in>set fs. t \<in> value" shows "Rcd fs \<in> value"
HS_Sub: fixes pp :: "L"   and GG :: "Var \<Rightarrow> L"   and c :: "IMP"   and HH :: "Var \<Rightarrow> L"   and p :: "L"   and G :: "Var \<Rightarrow> L"   and H :: "Var \<Rightarrow> L" assumes "(pp, GG, c, HH) \<in> HS"   and "LEQ p pp"   and "\<forall>x. LEQ (G x) (GG x)"   and "\<forall>x. LEQ (HH x) (H x)" shows "(p, G, c, H) \<in> HS"
sup_bot_right: fixes x :: "'b" shows "x \<sqinter> top = x"
induct: fixes P :: "(Vars.var \<times> exp) list \<Rightarrow> bool"   and a0 :: "(Vars.var \<times> exp) list" assumes "P []"   and "\<And>v e \<Gamma>. P \<Gamma> \<Longrightarrow> P ((v, e) # \<Gamma>)" shows "P a0"
coprime_dvd_mult_left_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 a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a c" shows "dvd.dvd times a (times b c) = dvd.dvd times a b"
fundamental_theorem_algebra_factorized: fixes p :: "complex poly" shows "\<exists>as. Polynomial.smult (lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = degree p"
psi: fixes x :: "nat"   and y :: "nat" shows "eval psi [x, y] = eval r_phi [if x \<in> I then j\<^sub>1 else j\<^sub>2, y]"
subgraph_refl: fixes G :: "('c, 'd) pre_digraph" assumes "wf_digraph G" shows "subgraph G G"
monom_eq_0_iff: fixes a :: "'a"   and n :: "nat" shows "(monom a n = 0) = (a = (0::'a))"
leg1_commutes: shows "src.cod.leg1 \<cdot> src.chine = src.leg1"
comp_cod_arr: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "cod f = b" shows "local.comp b f = f"
Seq_def: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "a ;; b \<equiv> \<lambda>ab. a ab \<circ> b ab"
PRECOND_tag_def: fixes P :: "'a" shows "PRECOND_tag P \<equiv> P"
col_permutation_1: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col B C A"
Pi_I: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and B :: "'e \<Rightarrow> 'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
select_convs_2: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "sp_impl_subj_obj \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = sp_impl_subj_obj"
swapidseq_ext_finite_expansion: fixes B :: "'c set"   and A :: "'c set"   and n :: "nat"   and p :: "'c \<Rightarrow> 'c" assumes "finite B"   and "swapidseq_ext A n p" shows "swapidseq_ext (A \<union> B) n p"