hcju/isabelleps huggingface dataset:
induct: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f" obtains "\<And>x. d_OUT f x \<le> weight \<Gamma> x" and "\<And>x. d_IN f x \<le> weight \<Gamma> x" and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT f x \<le> d_IN f x" and "\<And>a. a \<in> A \<Gamma> \<Longrightarrow> d_IN f a = 0" and "\<And>b. b \<in> B \<Gamma> \<Longrightarrow> d_OUT f b = 0" and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<Longrightarrow> f e = 0"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "infinite A" shows "F.F g A = \<^bold>1"
UN_empty: fixes B :: "'f \<Rightarrow> 'e set" shows "\<Union> (B ` {}) = {}"
simps_4: fixes e :: "expr"   and r :: "ref_ty" shows "constVal (e InstOf r) = None"
wlog_perpendicular_axes: fixes u :: "complex_homo"   and v :: "complex_homo"   and z :: "complex_homo"   and H1 :: "circline"   and H2 :: "circline"   and P :: "complex_homo \<Rightarrow> complex_homo \<Rightarrow> complex_homo \<Rightarrow> bool" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "z \<in> unit_disc"   and "is_poincare_line H1"   and "is_poincare_line H2"   and "perpendicular H1 H2"   and "z \<in> circline_set H1 \<inter> circline_set H2"   and "u \<in> circline_set H1"   and "v \<in> circline_set H2"   and "\<And>x y. \<lbrakk>is_real x; 0 \<le> Re x; Re x < 1; is_imag y; 0 \<le> Im y; Im y < 1\<rbrakk> \<Longrightarrow> P 0\<^sub>h (of_complex x) (of_complex y)"   and "\<And>M u v w. \<lbrakk>unit_disc_fix M; u \<in> unit_disc; v \<in> unit_disc; w \<in> unit_disc; P (moebius_pt M u) (moebius_pt M v) (moebius_pt M w)\<rbrakk> \<Longrightarrow> P u v w"   and "\<And>u v w. \<lbrakk>u \<in> unit_disc; v \<in> unit_disc; w \<in> unit_disc; P (conjugate u) (conjugate v) (conjugate w)\<rbrakk> \<Longrightarrow> P u v w" shows "P z u v"
row_transpose: fixes j :: "nat"   and A :: "'b mat" assumes "j < dim_col A" shows "Matrix.row A\<^sup>T j = col A j"
LAssThrow: fixes V :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>V:=throw e,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
\<mu>_small_def: fixes n :: "nat"   and fs :: "int vec list"   and i :: "nat" shows "LLL.\<mu>_small n fs i = (\<forall>j<i. \<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j\<bar> \<le> 1 / 2)"
fset_of_list_simps_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "fset_of_list (x21 # x22) = finsert x21 (fset_of_list x22)"
strong_induct: fixes x1 :: "ty"   and x2 :: "tyS"   and P :: "'a \<Rightarrow> ty \<Rightarrow> tyS \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<prec> x2"   and "\<And>T y. P y T (Ty T)"   and "\<And>X T' T S y. \<lbrakk>X \<sharp> y; X \<sharp> T'; T \<prec> S; \<And>z. P z T S\<rbrakk> \<Longrightarrow> P y (T[X::=T']) (\<forall>[X].S)" shows "P z x1 x2"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : r\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : r\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : r\<^sub>1 \<star> w \<Rightarrow> r\<^sub>1 \<star> w'\<guillemotright>"   and "(r \<star> \<theta>) \<cdot> \<a>[r, r\<^sub>0, w] \<cdot> (\<rho> \<star> w) = ((r \<star> \<theta>') \<cdot> \<a>[r, r\<^sub>0, w'] \<cdot> (\<rho> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = r\<^sub>1 \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (r\<^sub>0 \<star> \<gamma>)"
Eps_psimp: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "P x"   and "\<not> P y"   and "Eps P = y" shows "Eps P = x"
zero_order_1: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less_eq zero x"
quadratic1: shows "\<forall>f\<in>set (map (quadratic_sub var (- isolate_variable_sparse p var (Suc 0)) 1 ((isolate_variable_sparse p var (Suc 0))\<^sup>2 - 4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) (2 * isolate_variable_sparse p var 2)) L @ map (quadratic_sub_fm var (- isolate_variable_sparse p var (Suc 0)) 1 ((isolate_variable_sparse p var (Suc 0))\<^sup>2 - 4 * isolate_variable_sparse p var 2 * isolate_variable_sparse p var 0) (2 * isolate_variable_sparse p var 2)) F). freeIn var f"
intros_13: fixes \<tau> :: "'a OCL_Types.type" shows "Sequence \<tau> \<sqsubset> Collection \<tau>"
STL2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
SubstTermP_cong: fixes H :: "fm set"   and v :: "tm"   and v' :: "tm"   and i :: "tm"   and i' :: "tm"   and t :: "tm"   and t' :: "tm"   and u :: "tm"   and u' :: "tm" assumes "H \<turnstile> v EQ v'"   and "H \<turnstile> i EQ i'"   and "H \<turnstile> t EQ t'"   and "H \<turnstile> u EQ u'" shows "H \<turnstile> SubstTermP v i t u IFF SubstTermP v' i' t' u'"
simps_23: fixes e :: "('b, 'c, 'd) exp"   and C :: "String.literal"   and V :: "'b"   and e' :: "('b, 'c, 'd) exp" shows "block_types (try e catch(C V) e') = block_types e @ Class C # block_types e'"
\<nu>LTL_simp: fixes \<phi> :: "'a ltln" shows "(\<phi> \<in> \<nu>LTL) = is_\<nu>LTL \<phi>"
rep_eq: fixes x :: "moebius_mat"   and xa :: "moebius_mat" shows "moebius_mat_eq x xa = moebius_cmat_eq (Rep_moebius_mat x) (Rep_moebius_mat xa)"
mprod_expR_memTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and e :: "nat \<Rightarrow> nat" shows "(\<forall>j\<le>n. f j \<in> carrier K) \<longrightarrow> mprod_expR K e f n \<in> carrier K"
I\<^sub>1_def: shows "I\<^sub>1 \<equiv> {(Some a, None)}\<^sup>="
wset_Suspend_ok_def: fixes I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set" shows "wset_Suspend_ok I = {s \<in> I. \<forall>t\<in>dom (wset s). \<exists>s0\<in>I. \<exists>s1\<in>I. \<exists>ttas x x0 ta w' ln' ln''. s0 -t\<triangleright>ta\<rightarrow> s1 \<and> s1 -\<triangleright>ttas\<rightarrow>* s \<and> thr s0 t = \<lfloor>(x0, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x0, shr s0\<rangle> -ta\<rightarrow> \<langle>x, shr s1\<rangle> \<and> Suspend w' \<in> set \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> actions_ok s0 t ta \<and> thr s1 t = \<lfloor>(x, ln')\<rfloor> \<and> thr s t = \<lfloor>(x, ln'')\<rfloor>}"
inv_solve_right': fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.group G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "(b \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> c = a) = (b = a \<otimes>\<^bsub>G\<^esub> c)"
comb_seq_Suc: fixes n :: "nat"   and \<omega> :: "nat \<Rightarrow> 'b"   and \<omega>' :: "nat \<Rightarrow> 'b" shows "comb_seq (Suc n) \<omega> \<omega>' = comb_seq n \<omega> (case_nat (\<omega> n) \<omega>')"
sfg_agroup: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e" assumes "sfg A a" shows "aGroup A"
undual_dual: fixes x :: "'b" shows "undual (dual x) = x"
times_assn_def: fixes P :: "assn"   and Q :: "assn" shows "P * Q = Abs_assn (Assn (\<lambda>h. \<exists>as1 as2. addrOf h = as1 \<union> as2 \<and> as1 \<inter> as2 = {} \<and> aseval (Rep_assn P) (pHeap (heapOf h) as1) \<and> aseval (Rep_assn Q) (pHeap (heapOf h) as2)))"
same_together: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and H :: "'e set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e" assumes "R module M"   and "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> H" shows "\<exists>t\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> A. \<exists>g\<in>{j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} \<rightarrow> f ` {j. j \<le> n}. surj_to g {j. j \<le> card (f ` {j. j \<le> n}) - Suc 0} (f ` {j. j \<le> n}) \<and> \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s\<^bsub>M\<^esub> f j) n = \<Sigma>\<^sub>e M (\<lambda>k. t k \<cdot>\<^sub>s\<^bsub>M\<^esub> g k) (card (f ` {j. j \<le> n}) - Suc 0)"
fresh_permute_iff: fixes p :: "perm"   and a :: "atom"   and x :: "'b" shows "p \<bullet> a \<sharp> p \<bullet> x = a \<sharp> x"
1: shows "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g"
Un_insert_right: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A \<union> Set.insert a B = Set.insert a (A \<union> B)"
intro: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme" assumes "h \<in> ring_hom R S" shows "ring_hom_cring_axioms R S h"
SubobjsR_isSubobj: fixes 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 C :: "char list"   and Cs :: "char list list" assumes "Subobjs\<^sub>R P C Cs" shows "is_subobj P (C, Cs)"
valid_Entry_path_cases: fixes as :: "'edge list"   and n :: "'node" assumes "(_Entry_) -as\<rightarrow>\<^sub>\<surd>* n"   and "as \<noteq> []" shows "(\<exists>a' as'. as = as' @ [a'] \<and> intra_kind (kind a')) \<or> (\<exists>a' as' Q r p fs. as = as' @ [a'] \<and> kind a' = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs) \<or> (\<exists>as' as'' n'. as = as' @ as'' \<and> as'' \<noteq> [] \<and> n' -as''\<rightarrow>\<^bsub>sl\<^esub>* n)"
simps_2: fixes x :: "'b"   and hs :: "'b heap list" shows "mset_heap (Hp x hs) = {#x#} + \<Sum>\<^sub># (mset (map mset_heap hs))"
code: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and w :: "'c llist"   and a :: "'b" shows "lscan f w a = (case w of <> \<Rightarrow> a | _ \<Rightarrow> a) % (case w of <> \<Rightarrow> <> | x % xs \<Rightarrow> lscan f xs (f x a))"
f_def: shows "is_increasing f \<and> s' = take_subseq s f"
gen_binomial_real': fixes x :: "real"   and y :: "real"   and a :: "real" assumes "\<bar>x\<bar> < y" shows "(\<lambda>n. (a gchoose n) * x ^ n * y powr (a - real n)) sums (x + y) powr a"
filtration_measurable: fixes f :: "'a \<Rightarrow> 'd"   and t :: "'b"   and N :: "'d measure" assumes "f \<in> F t \<rightarrow>\<^sub>M N" shows "random_variable N f"
isShortestPath_min_dist_def: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isShortestPath u p v = (isPath u p v \<and> length p = min_dist u v)"
total_ordered_set_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "total_ordered_set A less_eq \<equiv> total_quasi_ordered_set A less_eq \<and> antisymmetric A less_eq"
iso_iff: shows "((\<lambda>x j. if j \<in> J then h (g j) (x (g j)) else undefined) \<in> Group.iso (DirProds Gs I) (DirProds Hs J)) = (carrier (DirProds Hs J) \<subseteq> (\<lambda>x j. if j \<in> J then h (g j) (x (g j)) else undefined) ` carrier (DirProds Gs I) \<and> (\<forall>x\<in>carrier (DirProds Gs I). (\<lambda>j. if j \<in> J then h (g j) (x (g j)) else undefined) = \<one>\<^bsub>DirProds Hs J\<^esub> \<longrightarrow> x = \<one>\<^bsub>DirProds Gs I\<^esub>))"
simps_1: fixes \<zeta> :: "('a, 'b) hd" shows "head0 (Hd \<zeta>) = \<zeta>"
strict_implies_order: fixes b :: "'b list"   and a :: "'b list" assumes "b <p a" shows "b \<le>p a"
gen_iapath_def: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "gen_iapath V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> apath u p v \<and> set (inner_verts p) \<inter> V = {} \<and> p \<noteq> []"
HPhase0Read_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" assumes "HPhase0Read s s' p d"   and "HInv6 s"   and "Inv2c s" shows "HInv6 s'"
nan_not_le: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "\<not> x \<le> y"
strict_linear_order_def: fixes r :: "('a \<times> 'a) set" shows "strict_linear_order r = (strict_partial_order r \<and> total r)"
surjective: fixes r :: "('e, 'f, 'g, 'h) KripkeStructure_scheme" shows "r = \<lparr>worlds = worlds r, relations = relations r, valuation = valuation r, \<dots> = more r\<rparr>"
ldistinct_lSup: fixes Y :: "'c llist set" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) Y"   and "\<forall>xs\<in>Y. ldistinct xs" shows "ldistinct (lSup Y)"
set_mult_hom: fixes h :: "'c \<Rightarrow> 'd"   and G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme"   and I :: "'c set"   and J :: "'c set" assumes "h \<in> hom G H"   and "I \<subseteq> carrier G"   and "J \<subseteq> carrier G" shows "h ` (I <#>\<^bsub>G\<^esub> J) = h ` I <#>\<^bsub>H\<^esub> h ` J"
closed_open_segment_iff: fixes a :: "'b"   and b :: "'b" shows "closed (open_segment a b) = (a = b)"
induct: fixes P :: "('d \<times> 'e) list \<Rightarrow> bool"   and a0 :: "('d \<times> 'e) list" assumes "P []"   and "\<And>p ps. P (delete (fst p) ps) \<Longrightarrow> P (p # ps)" shows "P a0"
simps_2: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "insort_wrt r x (y # ys) = (if r x y then x # y # ys else y # insort_wrt r x ys)"
absorb_iff1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<sqinter> b = a)"
ZObis_mC_ZOC: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state" assumes "c \<approx>01 d"   and "s \<approx> t" shows "\<exists>I0 P F. mC_ZOC ZObis c d s t I0 P F"
HVZK0: fixes x0 :: "'g"   and w0 :: "'h"   and x1 :: "'a" assumes "(x0, w0) \<in> Rel0" shows "\<forall>c\<in>challenge_space. \<Sigma>_OR.R (x0, x1) (Inl w0) c = \<Sigma>_OR.S (x0, x1) c"
weak_unit_iff_self_source: fixes a :: "'a" shows "weak_unit a = (a \<in> sources a)"
emeasure_count_space_eq_0: fixes A :: "'b set"   and X :: "'b set" shows "(emeasure (count_space A) X = 0) = (X \<subseteq> A \<longrightarrow> X = {})"
finfun_comp_comp_collapse: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and h :: "('a, 'd) finfun" shows "finfun_comp f (finfun_comp g h) = finfun_comp (f \<circ> g) h"
SEND'_rec: shows "SEND' = (left`?`x \<rightarrow> mid`!`x \<rightarrow> ack \<rightarrow> SEND')"
T2: shows "T o\<^sub>L (f' (x, y) o\<^sub>L embed2_blinfun) = 1\<^sub>L"
openin_continuous_map_preimage: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and U :: "'b set" assumes "continuous_map X Y f"   and "openin Y U" shows "openin X {x \<in> topspace X. f x \<in> U}"
dom_comp_finite: fixes n :: "'c \<Rightarrow> 'd option"   and m :: "'d \<Rightarrow> 'e option" assumes "finite (dom n)" shows "finite (dom (m \<circ>\<^sub>m n))"
Sym: fixes H :: "fmla set"   and t :: "trm"   and u :: "trm" assumes "H \<turnstile> t EQ u" shows "H \<turnstile> u EQ t"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'e list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
bounded_cball: fixes x :: "'a"   and e :: "real" shows "bounded (cball x e)"
us_facts_2: shows "set us \<inter> Fvars \<phi> = {}"
measurable_split_conv: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "('c \<times> 'd) measure"   and B :: "'e measure" shows "((\<lambda>(x, y). f x y) \<in> A \<rightarrow>\<^sub>M B) = ((\<lambda>x. f (fst x) (snd x)) \<in> A \<rightarrow>\<^sub>M B)"
integral_iff: fixes a :: "nat \<Rightarrow> 'a"   and b :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier P"   and "b \<in> carrier P" shows "(a \<otimes>\<^bsub>P\<^esub> b = \<zero>\<^bsub>P\<^esub>) = (a = \<zero>\<^bsub>P\<^esub> \<or> b = \<zero>\<^bsub>P\<^esub>)"
pref_prod_short: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v \<cdot> w"   and "\<^bold>|u\<^bold>| \<le> \<^bold>|v\<^bold>|" shows "u \<le>p v"
h1: shows "(\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)) \<longrightarrow> (\<exists>(a', b', c')\<in>set eq. (a' = 0 \<and> b' \<noteq> 0) \<and> (\<forall>(d, e, f)\<in>set eq. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * (- c' / b')\<^sup>2 + e * (- c' / b') + f < 0) \<or> a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> ((\<forall>(d, e, f)\<in>set eq. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0) \<or> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0))) \<or> (\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and R :: "('procS, 'procT) Proc" assumes "P \<sim>\<lbrakk>\<cdot>\<rbrakk> Q"   and "Q \<sim>\<lbrakk>\<cdot>\<rbrakk> R" shows "P \<sim>\<lbrakk>\<cdot>\<rbrakk> R"
assms_2: fixes i :: "nat" shows "0 \<le> snd (pdevs_apply (snd X) i)"
cos_arctan: fixes x :: "real" shows "cos (arctan x) = 1 / sqrt (1 + x\<^sup>2)"
finite_elements: fixes A :: "'a set"   and P :: "'a set set" assumes "finite A"   and "partition_on A P" shows "finite P"
vs_zero_lin_dep: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n"   and "lin_indpt S" shows "0\<^sub>v n \<notin> S"
iso_runit: fixes f :: "'c" assumes "C.ide f" shows "C.iso \<r>\<^sub>C[f]"
inf_sup_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
op_sub_is_pr: shows "(-) \<in> PrimRec2"
Closed_Kernel_axioms: shows "Closed_Kernel \<K>' (set \<K>\<^sub>s)"
simps_1: fixes fv :: "'b \<Rightarrow> 'a set"   and n :: "nat" shows "Regex.fv_regex fv (regex.Skip n) = {}"
snd_snd_splitFace_edges_incr: fixes g :: "graph"   and v\<^sub>1 :: "nat"   and v\<^sub>2 :: "nat"   and f :: "face"   and vs :: "nat list" assumes "pre_splitFace g v\<^sub>1 v\<^sub>2 f vs" shows "\<E> g \<subseteq> \<E> (snd (snd (splitFace g v\<^sub>1 v\<^sub>2 f vs)))"
hyp_ind: shows "\<forall>y. (y, x) \<in> trm_ord \<longrightarrow> trm_rep y S \<noteq> y \<longrightarrow> st_red S y"
len_fppvs: shows "length fppvs = length vs - length us"
Suc_2: shows "enat (Suc i) < llength (lselect s w)"
clinear_imp_scaleC: fixes D :: "complex \<Rightarrow> 'b" assumes "clinear D" obtains d :: "'b" where "D = (\<lambda>x. x *\<^sub>C d)"
sublens'_prop2: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and s\<^sub>1 :: "'e"   and s\<^sub>2 :: "'e" assumes "vwb_lens X"   and "X \<subseteq>\<^sub>L' Y" shows "get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> s\<^sub>1 (get\<^bsub>Y\<^esub> s\<^sub>2)) = get\<^bsub>X\<^esub> s\<^sub>2"
assms_2: shows "init_exp_of e\<^sub>0 = \<lfloor>unit\<rfloor>"
local_triangle1_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and \<beta> :: "'a \<times> ('b \<times> 'a) list"   and \<alpha> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>1 :: "'a \<times> (bool \<times> 'b \<times> 'a) list"   and \<sigma>2 :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>3 :: "'a \<times> (bool \<times> 'b \<times> 'a) list" shows "local_triangle1 ars \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (local_peak ars (\<beta>, \<alpha>) \<and> \<sigma>2 \<in> seq ars \<and> {\<sigma>1, \<sigma>3} \<subseteq> conv ars \<and> lst \<beta> = fst \<sigma>1 \<and> lst_conv \<sigma>1 = fst \<sigma>2 \<and> lst \<sigma>2 = fst \<sigma>3)"
compl_open_closed1: fixes m :: "'a set" assumes "m \<subseteq> carrier"   and "carrier - m closed" shows "m open"
self_evaluation_map_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "self_evaluation_map V H \<a> \<i> src trg \<equiv> bicategory V H \<a> \<i> src trg"
length_remove_nth: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "length (remove_nth i xs) = length xs - 1"
prems_2: shows "v \<notin> subst_domain \<delta>"
SeqHRP_unique: fixes x :: "tm"   and y :: "tm"   and s :: "tm"   and u :: "tm"   and y' :: "tm"   and s' :: "tm"   and u' :: "tm" shows "{SeqHRP x y s u, SeqHRP x y' s' u'} \<turnstile> y' EQ y"
hn_refine_ref: fixes m :: "'a nres"   and m' :: "'a nres"   and P :: "assn"   and c :: "'b Heap"   and Q :: "assn"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "m \<le> m'"   and "hn_refine P c Q R m" shows "hn_refine P c Q R m'"
moebius_mu_formula_real: fixes n :: "nat" shows "moebius_mu n = dirichlet_inverse (\<lambda>_. 1) 1 n"
rep_eq: fixes x :: "'b \<Rightarrow> 'c word" shows "target_size x = LENGTH('c)"
integral_FTC_Icc_real: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_real_derivative f x) (at x)"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> isCont f x" shows "LBINT x. f x * indicat_real {a..b} x = F b - F a"
\<Delta>_simps_4: shows "local.dom ((\<pi> \<cdot> \<mu> \<star> r\<^sub>0) \<cdot> \<rho>) = r\<^sub>1"
eq_inf_le_inf_transp: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream"   and w\<^sub>3 :: "'action stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"
b: shows "\<forall>z\<in>set_pmf (init \<bind> (\<lambda>s. snd (BIT_init, BIT_step) s r \<bind> (\<lambda>(a, is'). return_pmf (step (fst s) r a, is')))). snd (snd z) = s0"
bigo_fix: fixes f :: "nat \<Rightarrow> 'a"   and h :: "nat \<Rightarrow> 'a" assumes "(\<lambda>x. f (x + 1)) \<in> O(\<lambda>x. h (x + 1))"   and "f 0 = (0::'a)" shows "f \<in> O(h)"
remove_preserves_known_ptrs: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.remove child \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.known_ptrs h'"
DynCastFail: fixes 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 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 a :: "nat"   and Cs :: "char list list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>ref (a, Cs),(h, l)\<rangle>"   and "h a = \<lfloor>(D, S)\<rfloor>"   and "\<not> P \<turnstile> Path D to C unique"   and "\<not> P \<turnstile> Path last Cs to C unique"   and "C \<notin> set Cs" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,(h, l)\<rangle>"
start_unstable_cond: fixes p :: "nat" assumes "succs p \<noteq> {}"   and "p < length (g_V G)" shows "\<not> stable r step start p"