hcju/isabelleps huggingface dataset:
rupture_dimension: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (A [X])"   and "p \<in> carrier (K [X]\<^bsub>A [X]\<^esub>)"   and "0 < degree p" shows "ring.dimension (Rupt\<^bsub>A [X]\<^esub> K p) (degree p) (UP_A.rupture_surj K p ` UP_A.poly_of_const ` K) (carrier (Rupt\<^bsub>A [X]\<^esub> K p))"
assms: shows "as1 @ as2 @ as3 = filter f1 (map f2 p)"
b_e_type_call_2: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and i :: "nat" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Call i" shows "\<exists>ts'' tf1 tf2. ts = ts'' @ tf1 \<and> ts' = ts'' @ tf2 \<and> func_t \<C> ! i = tf1 _> tf2"
open_countable_Union_open_box: fixes S :: "'b set" assumes "open S" obtains \<D> :: "'b set set" where "countable \<D>" and "\<D> \<subseteq> Pow S" and "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = box a b" and "\<Union> \<D> = S"
ag_l_zero: fixes x :: "'a" assumes "x \<in> carrier A" shows "\<zero> \<plusminus> x = x"
adjacent_atomics: fixes i :: "nat"   and j :: "nat" shows "adjacent (i, j) (local.atomics i j)"
prod_2_id_def: fixes p :: "'b \<Rightarrow> 'c decision option"   and q :: "'b \<Rightarrow> 'd decision option" shows "p \<Otimes>\<^sub>2\<^sub>I q = p \<Otimes>\<^sub>2 q \<circ> (\<lambda>x. (x, x))"
bad_def: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "bad p x = (\<exists>f. inf_chain p f \<and> f 0 = x)"
T_in_def: fixes x :: "'b"   and h :: "'b tree" shows "T_in x h = T_part x h"
elims: fixes x :: "('b, 'c) Combinators list"   and y :: "('b, 'c) Combinators" assumes "list2policyR x = y" obtains xa :: "('b, 'c) Combinators" where "x = [xa__]" and "y = xa__"   | xa :: "('b, 'c) Combinators" and v :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators list" where "x = xa # v # va" and "y = list2policyR (v # va) \<oplus> xa"   | "x = []" and "y = undefined"
int_shiftl0: fixes x :: "int" shows "x << 0 = x"
powr_smallo_iff: fixes g :: "'b \<Rightarrow> real"   and F :: "'b filter"   and p :: "real"   and q :: "real" assumes "filterlim g at_top F"   and "F \<noteq> bot" shows "((\<lambda>x. g x powr p) \<in> o[F](\<lambda>x. g x powr q)) = (p < q)"
mult_onel: fixes x :: "'b \<times> 'c" shows "times_prod' one_prod x = x"
refine: fixes xsi :: "'b list"   and xs :: "real list set"   and E :: "nat" assumes "(xsi, xs) \<in> appr_rell"   and "DIM_precond TYPE('c) E" shows "(op_eucl_of_list_image_pad E xsi, op_eucl_of_list_image $ xs) \<in> appr_rel"
im_uminus: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and g :: "'a" assumes "GroupHom G T" shows "T (- g) = - T g"
Base_def: fixes x1 :: "trm" shows "Base x1 \<equiv> Abs_btrm (Base_Rep (Rep_trm x1))"
elem_in: fixes x :: "'a" assumes "x \<in># p1" shows "x \<in># A"
map_cong: fixes x :: "'b multiset"   and ya :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in># ya \<Longrightarrow> f z = g z" shows "image_mset f x = image_mset g ya"
isomorphic_trans: fixes smult'' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set" assumes "\<exists>T. FGModuleIso G (\<star>) W smult'' T X" shows "isomorphic smult'' X"
measure_lborel_box_eq: fixes l :: "'a"   and u :: "'a" shows "content (box l u) = (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
fixp_def: fixes f :: "'b \<Rightarrow> 'b" shows "b.fixp f = lubb (b.iterates f)"
field_differentiable_imp_differentiable: fixes f :: "'b \<Rightarrow> 'b"   and F :: "'b filter" assumes "f field_differentiable F" shows "f differentiable F"
hyps_3: assumes "\<And>i. i \<in> I \<Longrightarrow> poly_fun (f i)" shows "poly_fun (\<lambda>x. \<Prod>i\<in>I. f i x)"
induct: fixes P :: "'b prog"   and x1 :: "char list"   and x2 :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option"   and Pa :: "char list \<Rightarrow> (char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'b) \<times> char list) option) \<Rightarrow> bool" assumes "P \<turnstile> x1 sees_methods x2"   and "\<And>D fs ms Mm. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms\<rbrakk> \<Longrightarrow> Pa Object Mm"   and "\<And>C D fs ms Mm Mm'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D sees_methods Mm; Pa D Mm; Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)\<rbrakk> \<Longrightarrow> Pa C Mm'" shows "Pa x1 x2"
effective_matrix_tensor_elements: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and i :: "nat"   and M1 :: "'b list list"   and M2 :: "'b list list"   and j :: "nat" assumes "Matrix_Tensor.mult id f"   and "(i < mult.row_length M1 * mult.row_length M2 \<and> j < length M1 * length M2) \<and> mat (mult.row_length M1) (length M1) M1 \<and> mat (mult.row_length M2) (length M2) M2" shows "mult.Tensor f M1 M2 ! j ! i = f (M1 ! (j div length M2) ! (i div mult.row_length M2)) (M2 ! (j mod length M2) ! (i mod mult.row_length M2))"
maximal_drop: fixes P :: "'a llist"   and n :: "nat" assumes "maximal_path P" shows "maximal_path (ldropn n P)"
lbk_effect_compute: fixes pr :: "real"   and matur :: "nat" shows "(\<Sum>w\<in>range (pseudo_proj_True matur). prod (prob_component pr w) {0..<matur} * discounted_value r (\<lambda>m. lbk_option matur) matur w) = (\<Sum>y\<in>gener_bool_list matur. lprob_comp pr y * det_discount r matur * ((MAX i\<in>{0..matur}. lgeom_proc u d init (take i y)) - lgeom_proc u d init y))"
is_basis_cart_basis': shows "is_basis (set_of_vector cart_basis')"
distinct_adj_singleton: fixes x :: "'a" shows "distinct_adj [x]"
min_less_iff_disj: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(ord.min (\<sqsubseteq>) x y \<sqsubset> z) = (x \<sqsubset> z \<or> y \<sqsubset> z)"
if_polynomial_0_evaluate_notex: fixes wd :: "nat \<Rightarrow> real"   and ws :: "nat \<Rightarrow> real" assumes "polynomial_f wd \<noteq> 0" shows "\<nexists>weights_shallow Z. Z < r ^ N_half \<and> (\<forall>inputs. input_sizes (deep_model_l rs) = map dim_vec inputs \<longrightarrow> evaluate_net (insert_weights shared_weights (deep_model_l rs) wd) inputs = evaluate_net (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws) inputs)"
u': fixes x :: "'a" assumes "x \<in> space M" shows "(\<lambda>i. u i x) \<longlonglongrightarrow> u' x"
CAS_\<tau>red1r_xt1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and D :: "String.literal"   and F :: "String.literal"   and e2 :: "'addr expr1"   and e3 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (e\<bullet>compareAndSwap(D\<bullet>F, e2, e3), xs) (e'\<bullet>compareAndSwap(D\<bullet>F, e2, e3), xs')"
noop_returns_heap: fixes h :: "'a"   and h' :: "'a" shows "h \<turnstile> noop \<rightarrow>\<^sub>h h' = (h = h')"
prj_joint_monic: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and h' :: "'a" assumes "cospan f g"   and "seq \<p>\<^sub>1[f, g] h"   and "seq \<p>\<^sub>1[f, g] h'"   and "\<p>\<^sub>0[f, g] \<cdot> h = \<p>\<^sub>0[f, g] \<cdot> h'"   and "\<p>\<^sub>1[f, g] \<cdot> h = \<p>\<^sub>1[f, g] \<cdot> h'" shows "h = h'"
var_not_snd: fixes x :: "'b"   and P :: "'b trm" shows "Var x \<noteq> Snd P"
arb_pf_def: shows "arb_pf = (\<lambda>x n w. 0)(stk := \<lambda>n w. - 1, risk_free_asset := \<lambda>n. geom_proc 0)"
evalE: shows "sinvar \<lparr>nodes = N, edges = E\<rparr> nP"
sf_2: fixes a :: "'a poly"   and i :: "nat" assumes "(a, i) \<in> set bs" shows "square_free a \<and> degree a \<noteq> 0"
asm: shows "LIM x F. f x * g x :> at_top"
set_iterator_intro: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "set_iterator_genord it S0 R" shows "set_iterator it S0"
CallRedsNull: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>null,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>2\<rangle>"
SCallNonStatic\<^sub>1: fixes P :: "nat exp prog"   and ps :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>"   and "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = body in D" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,s\<^sub>2\<rangle>"
cases: fixes a :: "('g \<times> 'g) set" assumes "asym a" obtains R :: "('g \<times> 'g) set" where "a = R" and "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R"
linorder_less_wlog: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "\<And>a b. a < b \<Longrightarrow> P a b"   and "\<And>a. P a a"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
assms: shows "r = memory_write asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
not_less_zero: fixes n :: "'e multiset" shows "\<not> n \<subset># {#}"
vsv_dg_prod_Cod: shows "vsv (dg_prod I \<AA>\<lparr>Cod\<rparr>)"
idealprod_whole_r: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" assumes "Ring R"   and "ideal R I" shows "I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> carrier R = I"
ord_isom_segment_mem: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c"   and x :: "'c" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "a \<in> carrier D"   and "x \<in> segment D a" shows "f x \<in> segment E (f a)"
iso_is_retraction: fixes f :: "'a arrow_of_spans_data" assumes "V.iso f" shows "V.retraction f"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "int \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> pol \<Rightarrow> 'a"   and f3 :: "pol \<Rightarrow> nat \<Rightarrow> pol \<Rightarrow> 'a"   and pol :: "pol" shows "P (case pol of Pc x \<Rightarrow> f1 x | Pinj x xa \<Rightarrow> f2 x xa | PX x xa xb \<Rightarrow> f3 x xa xb) = ((\<forall>x1. pol = Pc x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. pol = Pinj x21 x22 \<longrightarrow> P (f2 x21 x22)) \<and> (\<forall>x31 x32 x33. pol = PX x31 x32 x33 \<longrightarrow> P (f3 x31 x32 x33)))"
before_dist_r1: fixes vs :: "'a list"   and r1 :: "'a"   and r2 :: "'a" assumes "distinct vs"   and "before vs r1 r2" shows "r1 \<in> set (fst (splitAt r2 vs))"
DirProds_one': fixes i :: "'d"   and I :: "'d set"   and G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" assumes "i \<in> I" shows "\<one>\<^bsub>DirProds G I\<^esub> i = \<one>\<^bsub>G i\<^esub>"
simps: fixes \<phi> :: "'c stream \<Rightarrow> 'b"   and xs :: "'c stream" shows "nxt \<phi> xs = \<phi> (stl xs)"
comp_def: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'a \<times> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "product_category.comp C1 C2 g f = (if (partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f)) \<and> (partial_magma.arr C1 (fst g) \<and> partial_magma.arr C2 (snd g)) \<and> (if partial_magma.arr C1 (fst f) \<and> partial_magma.arr C2 (snd f) then (partial_magma.cod C1 (fst f), partial_magma.cod C2 (snd f)) else (partial_magma.null C1, partial_magma.null C2)) = (if partial_magma.arr C1 (fst g) \<and> partial_magma.arr C2 (snd g) then (partial_magma.dom C1 (fst g), partial_magma.dom C2 (snd g)) else (partial_magma.null C1, partial_magma.null C2)) then (C1 (fst g) (fst f), C2 (snd g) (snd f)) else (partial_magma.null C1, partial_magma.null C2))"
primitive_dchar_iff: shows "primitive_dchar n \<chi> = (\<not> (\<exists>d<n. induced_modulus d))"
Y: fixes H :: "('a, 'b) pre_digraph" assumes "induced_subgraph H G"   and "strongly_connected H"   and "\<forall>H'. strongly_connected H' \<longrightarrow> induced_subgraph H' G \<longrightarrow> \<not> verts H \<subset> verts H'" shows "max_subgraph strongly_connected H"
entropy_commute: fixes X :: "'key \<times> 'message list \<Rightarrow> 'a"   and Y :: "'key \<times> 'message list \<Rightarrow> 'b" shows "\<H>( \<lambda>x. (X x, Y x) ) = \<H>( \<lambda>x. (Y x, X x) )"
wellformed_\<G>: fixes \<Sigma> :: "'b set set"   and \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set" assumes "ltl_FG_to_rabin \<Sigma> \<G> w" shows "Only_G \<G>"
intro: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and G :: "('b, 'c, 'e, 'f) functor"   and u :: "'b \<Rightarrow> 'f" assumes "two_cats AA BB"   and "natural_transformation_axioms AA BB F G u" shows "u : F \<Rightarrow> G in Func (AA , BB )"
mult_inv_character: fixes k :: "'a \<Rightarrow> complex" shows "principal_char (Characters G) k * inv_character (principal_char (Characters G)) k = principal_char (Characters G) k"
c_def: shows "\<C>' = \<C>\<lparr>label := [tm] @ label \<C>\<rparr>"
Dg2_is_gamma4_pathimg: shows "path_image gamma4 = {p. \<exists>x. x \<in> cbox a b \<and> p = (g2 x, x)}"
iNextStrong_iff: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iNextStrong t0 I P = (iNext t0 I P \<and> I \<down>> t0 \<noteq> {} \<and> t0 \<in> I)"
simps_2: fixes P :: "'f \<Rightarrow> bool"   and s :: "'f" shows "transfer (P)\<^sub>\<surd> s = s"
A: fixes b :: "'a" assumes "b \<in> B" shows "\<exists>a\<in>A. b \<le> a"
HSem_merge: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "atom ` domA \<Gamma> \<sharp>* \<Delta>" shows "\<lbrace>\<Gamma>\<rbrace>\<lbrace>\<Delta>\<rbrace>\<rho> = \<lbrace>\<Gamma> @ \<Delta>\<rbrace>\<rho>"
dgrad_p_set_le_except: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "'t set" shows "dgrad_p_set_le d {except p S} {p}"
evaln_max2: fixes c1 :: "com"   and s1 :: "state"   and n1 :: "nat"   and t1 :: "state"   and c2 :: "com"   and s2 :: "state"   and n2 :: "nat"   and t2 :: "state" assumes "<c1,s1> -n1-> t1"   and "<c2,s2> -n2-> t2" shows "\<exists>n. <c1,s1> -n-> t1 \<and> <c2,s2> -n-> t2"
igAbsIPresIGWls_def: fixes MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igAbsIPresIGWls MOD \<equiv> \<forall>xs s x X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igWlsAbs MOD (xs, s) (igAbs MOD xs x X)"
sum_mono2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'b set"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> less_eq zero (f b)" shows "less_eq (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero f B)"
strict_sorted_interact_I: fixes ys :: "'a list list"   and xs :: "'a list list" assumes "length ys \<le> length xs"   and "length xs \<le> Suc (length ys)"   and "\<And>x. x \<in> list.set xs \<Longrightarrow> strict_sorted x"   and "\<And>y. y \<in> list.set ys \<Longrightarrow> strict_sorted y"   and "\<And>n. n < length ys \<Longrightarrow> xs ! n < ys ! n"   and "\<And>n. Suc n < length xs \<Longrightarrow> ys ! n < xs ! Suc n"   and "xs \<in> lists (- {[]})"   and "ys \<in> lists (- {[]})" shows "strict_sorted (interact xs ys)"
U_l: shows "\<forall>(t, n)\<in>set (\<Upsilon> (rlfm (And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). numbound0 t \<and> 0 < n"
sep_add_commute': fixes x :: "'b"   and y :: "'b" assumes "x ## y" shows "sep_add' x y = sep_add' y x"
wf_imp_SN: fixes A :: "('d \<times> 'd) set" assumes "wf (A\<inverse>)" shows "SN A"
igFreshIGOpSTR_imp_igFreshIGOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'e, 'f, 'g, 'h) model_scheme" assumes "igFreshIGOpSTR MOD" shows "igFreshIGOp MOD"
pNode_1: fixes p :: "ref"   and bdt1 :: "bdt" assumes "Dag p low high lt"   and "bdt lt var = Some bdt1"   and "Dag no low high not"   and "no \<in> set_of lt" shows "\<exists>bdt2. bdt not var = Some bdt2"
adds_ppI: fixes v :: "'t"   and t :: "'a"   and u :: "'t" assumes "v = t \<oplus> u" shows "t adds\<^sub>p v"
cCons_def: fixes x :: "'a"   and xs :: "'a list" shows "cCons x xs = (if xs = [] \<and> x = (0::'a) then [] else x # xs)"
prognosis_Var_lam: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and u :: "Arity"   and as :: "Arity list"   and S :: "stack_elem list"   and a :: "Arity" assumes "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u"   and "isVal e" shows "prognosis ae as u (\<Gamma>, e, S) \<sqsubseteq> record_call x\<cdot>(prognosis ae as a (\<Gamma>, Var x, S))"
r_neg: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_group G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"
le: shows "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
canceled_def: fixes l :: "(bool \<times> 'd) list" shows "canceled l = (\<not> Domainp cancels_to_1 l)"
that_2: assumes "\<paragraph>''Var pres''" shows "(v s', v s) \<in> R"
induces_monoidal_category: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and tensor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unity :: "'b"   and lunit :: "'b \<Rightarrow> 'b"   and runit :: "'b \<Rightarrow> 'b"   and assoc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "elementary_monoidal_category C tensor unity lunit runit assoc" shows "monoidal_category C (elementary_monoidal_category.T C tensor) (elementary_monoidal_category.\<alpha> C tensor assoc) (elementary_monoidal_category.\<iota> unity lunit)"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_subst_fv_subset: fixes x :: "'c"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "x \<in> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F" shows "fv (\<theta> x) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
double_orthogonal_complement_id: fixes M :: "'a set" assumes "closed_csubspace M" shows "orthogonal_complement (orthogonal_complement M) = M"
simps_4: fixes m :: "nat"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "closed m (And p q) = (closed m p \<and> closed m q)"
prj0_ext: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "\<not> (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g)" shows "prj0 f g = partial_magma.null C"
dim_span: fixes S :: "('b, 'c) vec set" shows "vec.dim (vec.span S) = vec.dim S"
3: shows "set_integrable lborel (einterval 0 1) (\<lambda>x. (1 - (1 - q powr log q (1 - x)) ^ n) * (- 1 / (ln q * (1 - x))))"
rel_FGf_def: fixes L1 :: "'m \<Rightarrow> 'w \<Rightarrow> bool"   and L2 :: "'n \<Rightarrow> 'x \<Rightarrow> bool"   and Co1 :: "'p \<Rightarrow> 'y \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1 :: "'s \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'t \<Rightarrow> 'ab \<Rightarrow> bool" shows "rel_FGf L1 L2 Co1 Co2 Contra1 Contra2 = rel_F L1 L2 (=) Co1 Co2 (=) Contra1 Contra2 (=)"
imonad2: fixes M :: "'a nres" shows "M \<bind> RETURN = M"
Fourier_sum_limit_pair: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>n. \<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l = (\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l"
IH: shows "\<forall>x\<in>fv t. \<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var y) = \<Gamma> (Var x)"
simps_2: shows "OUT sA2 = {data4, data5, data11, data12}"
par: shows "par (\<l>[cod f] \<cdot> (\<I> \<otimes> f)) (f \<cdot> \<l>[local.dom f])"
spanning_tree_def: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" shows "spanning_tree H G \<equiv> tree H \<and> spanning H G"
cblinfun_compose_id_right: fixes U :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "U o\<^sub>C\<^sub>L id_cblinfun = U"
assms_2: shows "v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"
simps_11: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and b2 :: "bool" shows "local.less_gexp_aux (Gt e1 e2) (Bc b2) = False"
axioms_1: fixes K :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "subfield K R" shows "subdomain K R"
disjointFrame_L1: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and rcv :: "nat \<Rightarrow> 'a Frame list"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and i :: "nat"   and t :: "nat"   and j :: "nat" assumes "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "\<forall>i<n. FlexRayController (nReturn i) rcv (nC i) (nStore i) (nSend i) (nGet i)"   and "nSend i t \<noteq> []"   and "i < n"   and "j < n"   and "i \<noteq> j" shows "nSend j t = []"
insert_commute: fixes x :: "'g"   and y :: "'g"   and A :: "'g set" shows "insert x (insert y A) = insert y (insert x A)"
distinct_28: fixes x4 :: "nat"   and x3 :: "nat" shows "Pan x4 \<noteq> Nonce x3"
strict_order_letter_less: fixes r :: "('a \<times> 'a) set" assumes "strict_order r" shows "strict_order (letter_less r)"
sublist_def: fixes A :: "'b list"   and B :: "'b list" shows "sublist A B \<equiv> \<forall>x. List.member A x \<longrightarrow> List.member B x"
get_shadow_root_ptr_simp1: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme"   and h :: "(_) heap" shows "get shadow_root_ptr (put shadow_root_ptr shadow_root h) = Some shadow_root"
algebraic_add_cancel_left: fixes x :: "'a"   and y :: "'a" assumes "algebraic x" shows "algebraic (x + y) = algebraic y"
ide_char: fixes a :: "'a \<times> 'a" shows "VoV.ide a = (VoV.arr a \<and> VxV.ide a)"
AE_alw_accD: fixes P :: "'c cfg stream \<Rightarrow> bool"   and s :: "'c cfg"   and s' :: "'c cfg" assumes "Measurable.pred (stream_space (count_space UNIV)) P"   and "almost_everywhere (MC.T s) (alw P)"   and "(s, s') \<in> MC.acc" shows "almost_everywhere (MC.T s') (alw P)"