hcju/isabelleps huggingface dataset:
cyclic_on_FOO: fixes f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and g :: "'b \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "cyclic_on f S"   and "\<And>x. x \<in> S \<Longrightarrow> g (h x) = h (f x)" shows "cyclic_on g (h ` S)"
sup_loc_trans: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a :: "ty err list"   and b :: "ty err list"   and c :: "ty err list" assumes "P \<turnstile> a [\<le>\<^sub>\<top>] b"   and "P \<turnstile> b [\<le>\<^sub>\<top>] c" shows "P \<turnstile> a [\<le>\<^sub>\<top>] c"
restrictP_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b" shows "(r |` A) a b = (r a b \<and> a \<in> A \<and> b \<in> A)"
fin_a_star_reduce: fixes M :: "trm"   and x :: "name"   and M' :: "trm" assumes "fin M x"   and "M \<longrightarrow>\<^sub>a* M'" shows "fin M' x"
nat_term_order_eq_def: fixes to1 :: "'b nat_term_order"   and to2 :: "'b nat_term_order"   and dg :: "bool"   and ps :: "bool" shows "nat_term_order_eq to1 to2 dg ps = (\<forall>u v. (dg \<longrightarrow> deg_pp (fst (rep_nat_term u)) = deg_pp (fst (rep_nat_term v))) \<longrightarrow> (ps \<longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)) \<longrightarrow> nat_term_compare to1 u v = nat_term_compare to2 u v)"
bind_tm_def: fixes s :: "'d tm"   and f :: "'d \<Rightarrow> 'c tm" shows "s \<bind> f = (case s of ??.Time_Monad.tm.TM u m \<Rightarrow> case f u of ??.Time_Monad.tm.TM v n \<Rightarrow> ??.Time_Monad.tm.TM v (m + n))"
card_of_ordIso: fixes A :: "'b set"   and B :: "'c set" shows "(\<exists>f. bij_betw f A B) = (|A| =o |B|)"
simps_1: fixes statename :: "nat option"   and datastate :: "nat \<Rightarrow>f value option"   and action :: "String.literal \<times> value list"   and "output" :: "value option list"   and more :: "'a" shows "statename \<lparr>statename = statename, datastate = datastate, action = action, output = output, \<dots> = more\<rparr> = statename"
isometry_iterates: fixes f :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "isometry f" shows "isometry (f ^^ n)"
flowsto_to_empty_iff: fixes a :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and t :: "real set"   and b :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" shows "flowsto a t b {} = (a = {})"
relcomppI: fixes r :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and a :: "'n"   and b :: "'o"   and s :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and c :: "'p" assumes "r a b"   and "s b c" shows "(r OO s) a c"
right_cancel: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(y \<otimes>\<^bsub>G\<^esub> x = z \<otimes>\<^bsub>G\<^esub> x) = (y = z)"
nhds_core: fixes U :: "'b set"   and T :: "'b set set"   and x :: "'b" assumes "U \<in> nhds\<^bsub>T\<^esub> x" shows "x \<in> core\<^bsub>T\<^esub> U"
that: fixes n :: "real" assumes "n \<in> \<int>"   and "x < n * a \<and> n * a < y \<or> y < n * a \<and> n * a < x" shows "thesis"
CLAMP'_def: fixes f :: "nat \<Rightarrow> real"   and x :: "nat" shows "CLAMP' f x = (if x < 3 then 0 else f x)"
I'_def: shows "I' = \<lparr>pverts = verts I, parcs = arcs I\<rparr>"
pp_star_elim: fixes x :: "'a ppath"   and X :: "'a ppath set" shows "(x \<in> pp_star X) = (\<exists>n. x \<in> ppath_dioid.power X n)"
assms_6: fixes A :: "'a trm" assumes "P S A" shows "P S (Fst A)"
Confidentiality_S: fixes A :: "agent"   and K' :: "nat"   and Tk :: "nat"   and B :: "agent"   and K :: "nat"   and Ticket :: "msg"   and evs :: "event list" assumes "Says Server A (Crypt K' \<lbrace>Number Tk, Agent B, Key K, Ticket\<rbrace>) \<in> set evs"   and "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerb_gets" shows "Key K \<notin> analz (knows Spy evs)"
distrib_imp2: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "\<And>x y z. sup_prod x (inf_prod y z) = inf_prod (sup_prod x y) (sup_prod x z)" shows "inf_prod x (sup_prod y z) = sup_prod (inf_prod x y) (inf_prod x z)"
exhaust: fixes y :: "'g +\<^sub>\<bottom> 'h" obtains "y = Bottom"   | x2 :: "'g" where "y = sum_bot.Left x2"   | x3 :: "'h" where "y = sum_bot.Right x3"
coboundedI2: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "b \<le> c" shows "a \<sqinter> b \<le> c"
find_first_correct_1: fixes x :: "'b"   and xs :: "'b list"   and i :: "nat" assumes "find_first x xs = Some i" shows "i < length xs"
second_vertex_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" assumes "DisjointPaths G v0 v1 paths" shows "DisjointPaths.second_vertex \<equiv> \<lambda>xs. hd (tl xs)"
merges_incr_lemma: fixes ps :: "(nat \<times> nat list) list" shows "\<forall>xs. xs \<in> list n A \<longrightarrow> distinct (map fst ps) \<longrightarrow> (\<forall>(p, x)\<in>set ps. p < length xs \<and> x \<in> A) \<longrightarrow> xs [\<sqsubseteq>\<^bsub>r\<^esub>] merges f ps xs"
adjoint_def: fixes I :: "('sf, 'sc, 'sz) interp"   and \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "local.adjoint I \<sigma> \<nu> = \<lparr>Functions = \<lambda>f. case SFunctions \<sigma> f of None \<Rightarrow> Functions I f | Some f' \<Rightarrow> \<lambda>R. dterm_sem (extendf I R) f' \<nu>, Predicates = \<lambda>p. case SPredicates \<sigma> p of None \<Rightarrow> Predicates I p | Some p' \<Rightarrow> \<lambda>R. \<nu> \<in> fml_sem (extendf I R) p', Contexts = \<lambda>c. case SContexts \<sigma> c of None \<Rightarrow> Contexts I c | Some c' \<Rightarrow> \<lambda>R. fml_sem (extendc I R) c', Programs = \<lambda>a. case SPrograms \<sigma> a of None \<Rightarrow> Programs I a | Some a' \<Rightarrow> prog_sem I a', ODEs = \<lambda>ode. case SODEs \<sigma> ode of None \<Rightarrow> ODEs I ode | Some ode' \<Rightarrow> ODE_sem I ode', ODEBV = \<lambda>ode. case SODEs \<sigma> ode of None \<Rightarrow> ODEBV I ode | Some ode' \<Rightarrow> ODE_vars I ode'\<rparr>"
iso_set_sym: fixes G :: "('f, 'g) monoid_scheme"   and h :: "'f \<Rightarrow> 'h"   and H :: "('h, 'i) monoid_scheme" assumes "Group.group G"   and "h \<in> Group.iso G H" shows "inv_into (carrier G) h \<in> Group.iso H G"
igFreshIGAbs1_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igFreshIGAbs1 MOD \<equiv> \<forall>ys y s X. isInBar (ys, s) \<and> igWls MOD s X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD ys y X)"
approx_slp_outer_sing: fixes p :: "nat"   and fa :: "floatarith"   and XS :: "(real \<times> real pdevs) list"   and R :: "(real \<times> real pdevs) list" shows "(approx_slp_outer p (Suc 0) [fa] XS = Some R) = (\<exists>Y. approx_floatarith p fa (map (\<lambda>x. (x, 0)) XS) = Some Y \<and> [aform_err_to_aform Y (max (degree_aforms XS) (degree_aform_err Y))] = R)"
st_hypreal_of_real: fixes x :: "real" shows "st (hypreal_of_real x) = hypreal_of_real x"
parallel_HSem_ind: fixes P :: "(Vars.var \<Rightarrow> 'Value) \<Rightarrow> (Vars.var \<Rightarrow> 'Value) \<Rightarrow> bool"   and \<rho>\<^sub>1 :: "Vars.var \<Rightarrow> 'Value"   and \<Gamma>\<^sub>1 :: "(Vars.var \<times> exp) list"   and \<rho>\<^sub>2 :: "Vars.var \<Rightarrow> 'Value"   and \<Gamma>\<^sub>2 :: "(Vars.var \<times> exp) list" assumes "adm (\<lambda>\<rho>'. P (fst \<rho>') (snd \<rho>'))"   and "P \<bottom> \<bottom>"   and "\<And>y z. P y z \<Longrightarrow> P (\<rho>\<^sub>1 ++\<^bsub>domA \<Gamma>\<^sub>1\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>1 \<^bold>\<rbrakk>\<^bsub>y\<^esub>) (\<rho>\<^sub>2 ++\<^bsub>domA \<Gamma>\<^sub>2\<^esub> \<^bold>\<lbrakk> \<Gamma>\<^sub>2 \<^bold>\<rbrakk>\<^bsub>z\<^esub>)" shows "P (\<lbrace>\<Gamma>\<^sub>1\<rbrace>\<rho>\<^sub>1) (\<lbrace>\<Gamma>\<^sub>2\<rbrace>\<rho>\<^sub>2)"
assms: shows "\<lless>\<pi>, e\<ggreater> I\<rightarrow>i \<lless>\<pi>', e'\<ggreater>"
VInter_lower: fixes x :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "\<Inter>\<^sub>\<circ> A \<subseteq>\<^sub>\<circ> x"
Compl_eq: fixes A :: "'e set" shows "- A = {x. x \<notin> A}"
global_separability_condition: shows "(\<forall>n>0. separable n) = (\<forall>k>0. \<not> coprime k n \<longrightarrow> gauss_sum k = 0)"
h2: shows "step0 (steps0 (1, tp) A stp') A = (0, tp')"
Exit_Use_empty: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and V :: "'d" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "V \<in> Use Exit" shows "False"
term_is_le_rel_minus: fixes rel :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and s :: "'a"   and t :: "'a"   and u :: "'t"   and v :: "'t" assumes "ord_term_lin.is_le_rel rel"   and "s adds t" shows "rel ((t - s) \<oplus> u) v = rel (t \<oplus> u) (s \<oplus> v)"
arr_assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "C.arr \<a>\<^sub>C[a, b, c]"
ok_extend_imp_ok_project: fixes F :: "'a program"   and G :: "'c program" assumes "extend h F ok G" shows "F ok project h UNIV G"
assms: shows "\<exists>x'. (x, x') \<in> R \<and> \<Phi> x'"
decr_grading_p_idI: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and m :: "nat" assumes "hom_grading d"   and "p \<in> dgrad_p_set d m" shows "decr_grading_p d m p = p"
OF_lm_noa_none_iff: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and ft :: "('c, 'b) flow_entry_match list"   and p :: "'d" shows "(OF_match_linear \<gamma> ft p = NoAction) = (\<forall>e\<in>set ft. \<not> \<gamma> (ofe_fields e) p)"
permutes_inv_eq: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and y :: "'c"   and x :: "'c" assumes "p permutes S" shows "(inv p y = x) = (p x = y)"
simps_2: fixes f :: "'b \<Rightarrow> 'a option set"   and x :: "'b" shows "map_option_set f (Some x) = f x"
S9: shows "2 ^ (x - y) = 2 ^ (x - y - 1) * 2"
cases: fixes a1 :: "call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a"   and a2 :: "call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a" assumes "a1 \<lessapprox> a2" obtains ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" and ve' :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set" and c :: "call" and \<beta> :: "nat \<Rightarrow> 'a option" and b :: "'a" where "a1 = (c, \<beta>, ve, b)" and "a2 = (c, \<beta>, ve', b)" and "ve \<lessapprox> ve'"
basechambers_half_chamber_systems_2: shows "D0 \<in> g \<turnstile> folding_g.\<C>"
AKcryptSK_not_AKcryptSK: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and K :: "nat" assumes "AKcryptSK authK servK evs"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK servK K evs"
invKey_K: fixes K :: "key" assumes "K \<in> symKeys" shows "invKey K = K"
conc_partitions_elim: fixes x :: "'a list"   and A :: "'a list set"   and B :: "'a list set" assumes "x \<in> A \<cdot> B" shows "\<exists>(u, v)\<in>Partitions x. u \<in> A \<and> v \<in> B"
birkhoff_sum_def: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and x :: "'a" shows "A.birkhoff_sum f n x = (\<Sum>i<n. f ((induced_map A ^^ i) x))"
disjoint_def: fixes ts :: "'b set set" shows "disjoint ts \<equiv> \<forall>A\<in>ts. \<forall>B\<in>ts. A \<noteq> B \<longrightarrow> A \<inter> B = {}"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst_cases': fixes x :: "'c"   and s :: "('d, 'c) stateful_strand_step"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "x \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (s \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta>)" shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p s \<or> x \<in> fv\<^sub>s\<^sub>e\<^sub>t (\<theta> ` vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p s)"
safe_assignment_future_bounded: fixes X :: "nat set"   and \<phi> :: "Formula.formula" assumes "safe_assignment X \<phi>" shows "Formula.future_bounded \<phi>"
all_union_of: fixes P :: "'c set set \<Rightarrow> bool"   and Q :: "'c set \<Rightarrow> bool"   and R :: "'c set \<Rightarrow> bool" shows "(\<forall>S. (P union_of Q) S \<longrightarrow> R S) = (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R (\<Union> T))"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>L 'b" shows "blinfun_apply (x - xa) = (\<lambda>xb. blinfun_apply x xb - blinfun_apply xa xb)"
assms_6: fixes x :: "'a option list" assumes "wf_tuple n A x"   and "P x"   and "Q x" shows "Q' (x[y := Some (f x)])"
asubg_test: fixes A :: "('e, 'f) aGroup_scheme"   and H :: "'e set" assumes "aGroup A"   and "H \<subseteq> carrier A"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b \<in> H" shows "A +> H"
K3_def: fixes upds :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set) list"   and C :: "acom"   and Q :: "'b" shows "K3 upds C Q = (\<forall>(a, b, c)\<in>set upds. K2 C b c Q)"
assms_1: fixes i :: "nat" shows "\<exists>x. seq (Suc i) = x # seq i"
connected_component_intermediate_subset: fixes U :: "'a set"   and a :: "'a"   and T :: "'a set" assumes "connected_component_set U a \<subseteq> T"   and "T \<subseteq> U" shows "connected_component_set T a = connected_component_set U a"
hom_dim_gen_eigenspace: fixes A :: "real mat"   and n :: "nat"   and x :: "real" assumes "A \<in> carrier_mat n n" shows "dim_gen_eigenspace (of_real_hom.mat_hom A) (of_real x) = dim_gen_eigenspace A x"
ln_minus_ln_floor_bigo: shows "(\<lambda>x. ln x - ln (real_of_int \<lfloor>x\<rfloor>)) \<in> O((/) 1)"
max_def: fixes a :: "'e word"   and b :: "'e word" shows "signed.max a b = (if a \<le>s b then b else a)"
assms_1: shows "\<forall>\<^sub>F x in net. norm (f x) \<le> g x"
trapped_forward_def: fixes x :: "'a"   and K :: "'a set" shows "trapped_forward x K = (flow0 x ` (existence_ivl0 x \<inter> {0..}) \<subseteq> K)"
unsigned_word_eqI: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and v :: "'b word"   and w :: "'b word" assumes "class.semiring_char_0 one times plus zero"   and "semiring_1.unsigned one plus zero v = semiring_1.unsigned one plus zero w" shows "v = w"
goedel_first_theHardHalf_pos: assumes "prv (neg \<phi>G)" shows "prv fls"
rroots3: fixes p :: "real poly" assumes "degree p = 3" shows "set (rroots3 p) = {x. poly p x = 0}"
E_linear_diff2: fixes A :: "'c pmf"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "finite (set_pmf A)" shows "E (map_pmf f A) - E (map_pmf g A) = E (map_pmf (\<lambda>x. f x - g x) A)"
U'_sf: fixes i :: "nat" shows "simple_function M (U' i)"
semiring_axioms_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.semiring_axioms plus times \<equiv> (\<forall>a b c. times (plus a b) c = plus (times a c) (times b c)) \<and> (\<forall>a b c. times a (plus b c) = plus (times a b) (times a c))"
f2: shows "cpu_reg_val WIM (snd (fst (get_curr_win () s1))) = cpu_reg_val WIM (snd (fst (get_curr_win () s2)))"
pairwiseI: fixes M :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> P m n" shows "Compiler_Utils.pairwise P M"
lt_sum_distinct_eq_Max: fixes I :: "'d set"   and p :: "'d \<Rightarrow> 't \<Rightarrow>\<^sub>0 'e" assumes "finite I"   and "sum p I \<noteq> 0"   and "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; p i1 \<noteq> 0; p i2 \<noteq> 0; lt (p i1) = lt (p i2)\<rbrakk> \<Longrightarrow> i1 = i2" shows "lt (sum p I) = ord_term_lin.Max (lt_set (p ` I))"
assms_1: fixes T :: "'a set" assumes "T \<in> \<F>" shows "compact T"
strand_subst_comp: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "range_vars \<delta> \<inter> bvars\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<delta> \<circ>\<^sub>s \<theta> = (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<cdot>\<^sub>s\<^sub>t \<theta>"
all_opaque: shows "set (opaque x) \<subseteq> {0..<n}"
EmptyMap_hierauto: fixes D :: "'d data"   and SA :: "('e, 'f, 'd) seqauto" shows "(D, {SA}, SAEvents SA, EmptyMap (States SA)) \<in> hierauto"
LSC_weaken_set: fixes \<Gamma> :: "'b formula multiset"   and \<Theta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow>\<^sub>n" shows "\<Gamma> + \<Theta> \<Rightarrow>\<^sub>n"
left_assoc: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ac_operator f" shows "f a (f b c) = f (f a b) c"
glueing: fixes S :: "'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 U :: "'c set"   and I :: "real set"   and V :: "real \<Rightarrow> 'c set"   and s :: "real \<Rightarrow> 'd" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "open_cover_of_open_subset S is_open U I V"   and "\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> \<FF> (V i)"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> \<rho> (V i) (V i \<inter> V j) (s i) = \<rho> (V j) (V i \<inter> V j) (s j)" shows "\<exists>t. t \<in> \<FF> U \<and> (\<forall>i. i \<in> I \<longrightarrow> \<rho> U (V i) t = s i)"
shiftl_shiftr1: fixes c :: "nat"   and b :: "nat"   and a :: "'b word" assumes "c \<le> b" shows "a << b >> c = a AND mask (size a - b) << b - c"
mod_one_cancel: fixes y :: "nat"   and z :: "int"   and x :: "int"   and y' :: "int" assumes "[int y * z * x = y' * x] (mod int (order \<G>))"   and "[z * x = 1] (mod int (order \<G>))" shows "[int y = y' * x] (mod int (order \<G>))"
assms_5: shows "E1 = (\<lambda>s. 1 + (4 + 3 * nat (s ''b'')) * (nat (s ''a'') - nat (s ''z'')))"
Arg2pi_eq_iff: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0" shows "(Arg2pi w = Arg2pi z) = (\<exists>x>0. w = complex_of_real x * z)"
assms_1: shows "r\<lparr>a\<rparr> = b"
abs_eq: shows "0\<^sub>h\<^sub>c = Abs_complex_homo_coords 0\<^sub>v"
env_le_def: fixes \<rho> :: "(nat \<times> val) list"   and \<rho>' :: "(nat \<times> val) list" shows "\<rho> \<sqsubseteq> \<rho>' \<equiv> \<forall>x v. lookup \<rho> x = Some v \<longrightarrow> (\<exists>v'. lookup \<rho>' x = Some v' \<and> v \<sqsubseteq> v')"
conj_comm: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "(P \<and> Q) = (Q \<and> P)"
cmf: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set" assumes "class.arelations meets \<I> e m b ov d s f" shows "m O f \<subseteq> s \<union> ov \<union> d"
C00: shows "mat n n (\<lambda>(i, j). if i = 0 \<and> j = 0 then e else (0::'a)) $$ (0, 0) = e"
assms_2: shows "x \<notin> set (map fst ys)"
strict_implies_order: fixes b :: "'a"   and a :: "'a" assumes "b < a" shows "b \<le> a"
mono_fMax_commute: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a fset" assumes "mono f"   and "A \<noteq> {||}" shows "f (fMax A) = fMax (f |`| A)"
wcount_def: fixes f :: "'k \<Rightarrow> nat"   and M :: "'k multiset" shows "wcount f M = (\<lambda>x. count M x * Suc (f x))"
rev: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "'a list" assumes "comm_monoid_list f z" shows "monoid_list.F f z (rev xs) = monoid_list.F f z xs"
edge_space_def: fixes p :: "real" shows "edge_space p \<equiv> valid_prob p"
bdd_lookup_subsetbdd': fixes bdds :: "bool list bdd list"   and q :: "bool list"   and ws :: "bool list"   and obdd :: "bool list bdd"   and c :: "nat"   and a :: "nat" assumes "length bdds = length q"   and "\<forall>x\<in>set bdds. bddh (length ws) x"   and "bddh (length ws) obdd"   and "\<And>bs w. \<lbrakk>bs \<in> set bdds; length w = length ws\<rbrakk> \<Longrightarrow> length (bdd_lookup bs w) = c"   and "\<And>w. length w = length ws \<Longrightarrow> length (bdd_lookup obdd w) = c"   and "a < c" shows "bdd_lookup (subsetbdd bdds q obdd) ws ! a = ((\<exists>i<length q. q ! i \<and> bdd_lookup (bdds ! i) ws ! a) \<or> bdd_lookup obdd ws ! a)"
linear_power_nonzero: fixes a :: "'c"   and n :: "nat" shows "[:a, 1::'c:] ^ n \<noteq> 0"
iso_assoc': fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.iso (Span.\<a>' f g h)"
ord0_less_trans: fixes x :: "ord0"   and y :: "ord0"   and z :: "ord0" assumes "x < y"   and "y < z" shows "x < z"
update_convs_15: fixes sip' :: "nat \<Rightarrow> nat"   and ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "sip_update sip' \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip' sip, handled = handled, \<dots> = more\<rparr>"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> Matrix.mat n n (\<lambda>(i, j). if i = j then 1::'a else (0::'a))"
Span_is_add_subgroup: fixes K :: "'a set"   and Us :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R" shows "subgroup (Span K Us) (add_monoid R)"
front_tickFree_mono: fixes t :: "'b event list"   and r :: "'b event list" shows "front_tickFree (t @ r) \<and> r \<noteq> [] \<longrightarrow> tickFree t \<and> front_tickFree r"
steps: shows "x \<Rightarrow>\<^sub>d\<^sup>* y"
cases: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and a1 :: "'d spmf"   and a2 :: "'g spmf" assumes "trace_callee_closure callee1 callee2 A p q a1 a2" obtains xs :: "('e \<times> 'f) list" where "a1 = trace_callee_aux callee1 p xs" and "a2 = trace_callee_aux callee2 q xs" and "set xs \<subseteq> A \<times> UNIV"
pick_psubst_qPsubst: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "good X"   and "goodEnv rho" shows "pick X #[rho] #= pick X #[[pickE rho]]"
cod_closed: fixes f :: "'c" assumes "Arr f" shows "Arr (C.cod f)"
list_all_append: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"