hcju/isabelleps huggingface dataset:
Y_arr_ide_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and b :: "'a"   and f :: "'a"   and a :: "'a"   and a' :: "'a" assumes "yoneda_functor C S \<phi>"   and "partial_magma.ide C b"   and "partial_magma.in_hom C f a a'" shows "yoneda_functor.Y C S \<phi> f b = set_category_given_img.mkArr S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (cardSuc (cmax |UNIV| natLeq)) (hom_functor.set C \<phi> (b, a)) (hom_functor.set C \<phi> (b, a')) (\<phi> (b, a') \<circ> C f \<circ> hom_functor.\<psi> C \<phi> (b, a))"
algebra_subset_lambda_system: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "increasing M f"   and "Measure_Space.additive M f" shows "M \<subseteq> lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
\<K>_card: fixes K :: "nat set"   and j0 :: "nat"   and j :: "nat" assumes "K \<in> \<K> j0 j" shows "card K = j"
out_degree_eq_card_arc_ends: fixes v :: "'a" shows "out_degree G v = card (arc_to_ends G ` out_arcs G v)"
less_eq_fract_respect: fixes b :: "'b"   and b' :: "'b"   and d :: "'b"   and d' :: "'b"   and a :: "'b"   and a' :: "'b"   and c :: "'b"   and c' :: "'b" assumes "b \<noteq> (0::'b)"   and "b' \<noteq> (0::'b)"   and "d \<noteq> (0::'b)"   and "d' \<noteq> (0::'b)"   and "a * b' = a' * b"   and "c * d' = c' * d" shows "(a * d * (b * d) \<le> c * b * (b * d)) = (a' * d' * (b' * d') \<le> c' * b' * (b' * d'))"
auth_prikey_used: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "Auth_PriKey n \<in> used s"
sep_true_def: shows "sep_true \<equiv> \<lambda>s. True"
injphi: shows "\<forall>F\<in>set Fs\<^sub>1. \<forall>F'\<in>set Fs\<^sub>1. inj_on \<phi> (set F \<union> set F')"
fst_readVariableRecursive: fixes n :: "'node"   and g :: "'g"   and v :: "'var"   and phis :: "('node \<times> 'var, ('var \<times> 'node \<times> Def) list) mapping" assumes "n \<in> set (\<alpha>n g)" shows "fst (readVariableRecursive g v n phis) = lookupDef g n v"
Lim_Un: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and x :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "(f \<longlongrightarrow> l) (at x within S)"   and "(f \<longlongrightarrow> l) (at x within T)" shows "(f \<longlongrightarrow> l) (at x within S \<union> T)"
frag_split: fixes c :: "'c \<Rightarrow>\<^sub>0 int"   and S :: "'c set"   and T :: "'c set" assumes "Poly_Mapping.keys c \<subseteq> S \<union> T" obtains d :: "'c \<Rightarrow>\<^sub>0 int" and e :: "'c \<Rightarrow>\<^sub>0 int" where "Poly_Mapping.keys d \<subseteq> S" and "Poly_Mapping.keys e \<subseteq> T" and "d + e = c"
STA_bridge': fixes A :: "'b set"   and j :: "'b"   and n :: "nat"   and ps :: "('c, 'b) fm list"   and a :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and k :: "'b"   and i :: "'b"   and xs :: "(nat \<times> nat) set" assumes "infinite UNIV"   and "finite A"   and "j \<in> A"   and "A, n \<turnstile> (ps, a) # branch"   and "descendants k i ((ps, a) # branch) xs"   and "Nom k at j in' set branch" shows "A \<turnstile> mapi_branch (bridge k j xs) ((ps, a) # branch)"
vifunion_least: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and C :: "V" assumes "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f i \<subseteq>\<^sub>\<circ> C" shows "\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I) \<subseteq>\<^sub>\<circ> C"
hd_P_k_v0: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size" shows "hd (ProofStepInduct_NonTrivial.P_k G v0 v1 paths P_new sep_size) = v0"
returns_toI: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and t :: "real"   and x :: "'b"   and P :: "'b set" assumes "auto_ll_on_open f X"   and "0 < t"   and "t \<in> auto_ll_on_open.existence_ivl0 f X x"   and "auto_ll_on_open.flow0 f X x t \<in> P"   and "\<forall>\<^sub>F t in at_right 0. auto_ll_on_open.flow0 f X x t \<notin> P"   and "closed P" shows "auto_ll_on_open.returns_to f X P x"
sphere_cball: fixes z :: "'a"   and r :: "real" shows "sphere z r \<subseteq> cball z r"
intros_2: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" shows "\<turnstile>\<^sub>1 {\<lambda>l s. P l (s[a/x])} x ::= a { \<lambda>s. Suc 0 \<Down> P}"
inverse_norm: fixes n :: "int"   and w :: "'b word" shows "(word_of_int n = w) = (sint w = signed_take_bit (LENGTH('b) - 1) n)"
IH: assumes "i_b \<le> k'"   and "k' \<noteq> i_b \<longrightarrow> 0 < i_b \<longrightarrow> A $$ (i_b - 1, i_b) \<noteq> (1::'b)" shows "set (identify_blocks_main A i_b ((i_b, i_e) # list)) = set ((i_b, i_e) # list) \<union> {(i, j) |i j. i \<le> j \<and> j < i_b \<and> (\<forall>l\<ge>i. l < j \<longrightarrow> A $$ (l, Suc l) = (1::'b)) \<and> (Suc j \<noteq> k' \<longrightarrow> A $$ (j, Suc j) \<noteq> (1::'b)) \<and> (0 < i \<longrightarrow> A $$ (i - 1, i) \<noteq> (1::'b))}"
GreenThm_typeII_twoCube_2: shows "\<forall>(k, \<gamma>)\<in>boundary twoC. line_integral_exists F {j} \<gamma>"
entail_set_all_formulas: fixes N1 :: "'g set"   and N2 :: "'g set" shows "(N1 \<Turnstile>G N2) = (\<forall>C\<in>N2. N1 \<Turnstile>G {C})"
cases: fixes a :: "'c llist" assumes "lfinite a" obtains "a = LNil"   | x :: "'c" and xs :: "'c llist" where "a = LCons x xs" and "lfinite xs"
r'_r: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and x :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "perron_frobenius A"   and "x \<in> perron_frobenius.Y A" shows "perron_frobenius.r' A x = perron_frobenius.r A x"
ordLeq_total: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order r'" shows "ordLeq3 r r' \<or> ordLeq3 r' r"
e: shows "evalDdb e1\<cdot>env_empty_db = evalDdb e2\<cdot>env_empty_db"
midpoint__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A Midpoint B C" shows "Coplanar A B C D"
lookup_single_eq: fixes k :: "'b"   and v :: "'a" shows "lookup (Poly_Mapping.single k v) k = v"
node_deliver_messages_distinct: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and xs :: "'c event list"   and i :: "nat" assumes "network history msg_id"   and "node_histories.prefix_of_node_history history xs i" shows "distinct (trivial_network.node_deliver_messages xs)"
drop_Cons': fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "drop n (x # xs) = (if n = 0 then x # xs else drop (n - 1) xs)"
set_bits_conv_set_bits_aux: fixes f :: "nat \<Rightarrow> bool" shows "set_bits f = set_bits_aux f 0 LENGTH('a)"
assign_doesnt_modify: fixes x :: "'Var"   and y :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list" assumes "x \<noteq> y" shows "doesnt_modify ((x \<leftarrow> e) \<otimes> annos) y"
zero_le_sum_single: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b set"   and t :: "'a" shows "0 \<le> zcount (\<Sum>x\<in>M. {#f x#}\<^sub>z) t"
Un: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<union> b \<in> sets M"
mod_diff_cong: fixes a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "a mod c = a' mod c"   and "b mod c = b' mod c" shows "(a - b) mod c = (a' - b') mod c"
stack_passes_lower_bound_1: fixes c\<^sub>3 :: "'a"   and c\<^sub>4 :: "'a"   and c\<^sub>2 :: "'a" assumes "c\<^sub>3 \<Rightarrow> c\<^sub>4"   and "stack c\<^sub>2 \<lesssim> stack c\<^sub>3"   and "\<not> stack c\<^sub>2 \<lesssim> stack c\<^sub>4" shows "stack c\<^sub>3 = stack c\<^sub>2"
step: fixes A' :: "'a set"   and s :: "'b"   and x :: "'a" assumes "I A' s"   and "x \<notin> A'"   and "x \<in> A"   and "A' \<subseteq> A" shows "I (insert x A') (f x s)"
hyps: assumes "\<forall>a\<in>set As. (0::'a) \<in> a"   and "add_independentS As"   and "add_independentS Bs"   and "add_independentS [\<Sum>X\<leftarrow>As. X, \<Sum>X\<leftarrow>Bs. X]" shows "add_independentS (As @ Bs)"
card_insert_disjoint: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A" shows "card (x \<triangleright> A) = Suc (card A)"
no_multiple_local_consts: fixes r :: "'rule"   and i :: "nat"   and i' :: "nat" assumes "r \<in> i.R rules"   and "i < length (antecedent r)"   and "i' < length (antecedent r)" shows "a_fresh (antecedent r ! i) \<inter> a_fresh (antecedent r ! i') = {} \<or> i = i'"
liouville_irrational_algebraic: fixes x :: "real" assumes "x \<notin> \<rat>"   and "algebraic x" obtains c :: "real" and n :: "nat" where "0 < c" and "\<And>p q. 0 < q \<Longrightarrow> c / real_of_int q ^ n < \<bar>x - real_of_int p / real_of_int q\<bar>"
assms_2: shows "\<forall>a\<in>A. P a = Q (f a)"
fin_2: shows "finite {R \<in> Rs. (a, b) \<notin> R}"
oalist_inv_rawD1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "(0::'b) \<notin> snd ` set xs"
iso_some_unit_2: fixes a :: "'a" assumes "weak_unit a" shows "\<guillemotleft>some_unit a : a \<star> a \<Rightarrow> a\<guillemotright>"
dgrad_p_set_zero: fixes m :: "nat" shows "dgrad_p_set (\<lambda>_. 0) m = UNIV"
par_by_merge_mono: fixes P\<^sub>2 :: "('d, 'e) urel"   and P\<^sub>1 :: "('d, 'e) urel"   and Q\<^sub>2 :: "('d, 'f) urel"   and Q\<^sub>1 :: "('d, 'f) urel"   and M :: "(('d, 'e, 'f) mrg, 'g) urel" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 \<parallel>\<^bsub>M\<^esub> Q\<^sub>1 \<sqsubseteq> P\<^sub>2 \<parallel>\<^bsub>M\<^esub> Q\<^sub>2"
rrs1_1: fixes z :: "'b list" shows "rotater 0 z = z"
vimage_snd: fixes A :: "'d set" shows "snd -` A = UNIV \<times> A"
Icarr: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme"   and i :: "'a" assumes "ideal I R"   and "i \<in> I" shows "i \<in> carrier R"
simps: fixes a :: "real mat convnet" shows "shared_weight_net a = ((\<exists>M. a = Input M) \<or> (\<exists>m A. a = Conv A m \<and> shared_weight_net m) \<or> (\<exists>m1 m2. a = Pool m1 m2 \<and> shared_weight_net m1 \<and> shared_weight_net m2 \<and> count_weights True (remove_weights m1) = count_weights True (remove_weights m2) \<and> (\<forall>x<count_weights True (remove_weights m1). extract_weights True m1 x = extract_weights True m2 x)))"
r: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> Y" shows "cmod (h w) \<le> r"
cod_cod: fixes f :: "'a" assumes "S.arr f" shows "S.cod (S.cod f) = S.cod f"
strict_suffix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "strict_suffix xs ys = (suffix xs ys \<and> xs \<noteq> ys)"
VU: shows "\<forall>i. i \<in> I \<longrightarrow> V i \<subseteq> U \<and> s i \<in> local.im_sheaf (V i)"
right_unique_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
xs_1: shows "v0 \<leadsto>xs\<leadsto> w"
pivot_valuation_id: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)" shows "\<V> (pivot x\<^sub>i x\<^sub>j s) = \<V> s"
add_diff_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "minus (plus a b) b = a"
strict_prefixI': fixes ys :: "'c list"   and xs :: "'c list"   and z :: "'c"   and zs :: "'c list" assumes "ys = xs @ z # zs" shows "strict_prefix xs ys"
decompose_com_ctxt: fixes c' :: "('g, 'h, 'i, 'j) com"   and ctxts :: "(('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com) \<times> (('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list)"   and c :: "('g, 'h, 'i, 'j) com" assumes "(c', ctxts) \<in> decompose_com c" shows "ctxts \<in> ctxt"
while_option_commute_invariant: fixes P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and b' :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and c' :: "'c \<Rightarrow> 'c"   and s :: "'b" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. P s \<Longrightarrow> b s = b' (f s)"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> f (c s) = c' (f s)"   and "P s" shows "map_option f (while_option b c s) = while_option b' c' (f s)"
hyps: assumes "b \<in> set bs" shows "Inl (p, b) \<in> set (new_spairs bs p)"
brec_det_algo: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist"   and Qi :: "('a, unit) HashMap.hashmap" assumes "rqrm_invar rqrm"   and "rqrm_prop (ls.\<alpha> \<delta>) rqrm"   and "ls.invar \<delta>"   and "hs.invar Qi" shows "det_while_algo (brec_det_algo rqrm Qi \<delta>)"
T': shows "T' = poly_mod.Mp r (T - Polynomial.smult p B')"
extremum_strict: fixes a :: "'c list" shows "\<not> strict_prefix a []"
f_def: fixes x :: "(real, 'm) vec" shows "f x = (1 / norm x) *\<^sub>R x"
abs_eq: fixes x :: "'b set"   and xa :: "'b" assumes "eq_onp (\<lambda>A. finite A \<and> incomparable A) x x" shows "(xa \<in>\<^sub>A antichain x) = (xa \<in> x)"
is_category: shows "category (\<cdot>\<^sup>o\<^sup>p)"
llast_singleton: fixes x :: "'c" shows "llast (LCons x LNil) = x"
bind_game_alt_def: fixes \<A> :: "'a \<Rightarrow> ('a \<times> nat \<times> nat \<times> nat \<times> nat) spmf" shows "ped_commit.bind_game \<A> = TRY key_gen \<bind> (\<lambda>(ck, vk). \<A> ck \<bind> (\<lambda>(c, m, d, m', d'). assert_spmf (m \<noteq> m' \<and> valid_msg m \<and> valid_msg m') \<bind> (\<lambda>_. let b = verify vk m c d; b' = verify vk m' c d' in assert_spmf (b \<and> b') \<bind> (\<lambda>_. return_spmf True)))) ELSE return_spmf False"
line_integral_point_path: fixes \<gamma> :: "real \<Rightarrow> 'a"   and basis :: "'a set"   and F :: "'a \<Rightarrow> 'a" assumes "point_path \<gamma>"   and "finite basis" shows "line_integral F basis \<gamma> = 0"
Gr_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "Gr A f = {(a, f a) |a. a \<in> A}"
rotate5: fixes A5 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A5 (insert A1 (insert A2 (insert A3 (insert A4 H)))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 H)))) \<turnstile> B"
r2f_Inf_pres_var: fixes R :: "('a \<times> 'b) set set" shows "\<F> (\<Inter> R) = \<Sqinter> \<P> \<F> R"
char_power: fixes a :: "'a"   and k :: "nat" assumes "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "\<chi> (a [^]\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> k) = \<chi> a ^ k"
signed_mod_arith: fixes a :: "'b word"   and b :: "'b word" shows "sint (a smod b) = signed_take_bit (LENGTH('b) - 1) (sint a smod sint b)"
SubobjsSet_init_objSet: 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" shows "{Cs. Subobjs P C Cs} = {Cs. \<exists>vmap. init_obj P C (Cs, vmap)}"
select_result_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and P :: "'c \<Rightarrow> bool" assumes "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> P x" shows "P |h \<turnstile> f|\<^sub>r"
Mg: shows "g \<in> borel_measurable (embed_measure M f)"
balanced_alt_def_all: shows "\<forall>ps\<subseteq>\<V>. int (card ps) = \<t> \<longrightarrow> int (\<B>' index ps) = \<Lambda>\<^sub>t'"
cases: fixes a1 :: "'node SDG_node set"   and a2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a3 :: "'node list"   and a4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and a5 :: "'edge list"   and a6 :: "'node list"   and a7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "a1,a2 \<turnstile> (a3,a4) =a5\<Rightarrow> (a6,a7)" obtains S :: "'node SDG_node set" and f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind" and ms :: "'node list" and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and as :: "'edge list" and ms' :: "'node list" and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and a :: "'edge" and ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "a1 = S" and "a2 = f" and "a3 = ms" and "a4 = s" and "a5 = as @ [a]" and "a6 = ms''" and "a7 = s''" and "S,f \<turnstile> (ms,s) =as\<Rightarrow>\<^sub>\<tau> (ms',s')" and "S,f \<turnstile> (ms',s') -a\<rightarrow> (ms'',s'')"
list_assn_aux_ineq_len: fixes l :: "'a list"   and li :: "'b list"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> assn" assumes "length l \<noteq> length li" shows "list_assn A l li = false"
arsinh_sinh_real: fixes x :: "real" shows "arsinh (sinh x) = x"
mersenne_mod_def: fixes k :: "int"   and n :: "nat" shows "mersenne_mod k n = k mod 2 ^ n + k div 2 ^ n"
SUP_union: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and M :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (M ` (A \<union> B)) = sup (Sup (M ` A)) (Sup (M ` B))"
Step: shows "cfg1 \<turnstile> m \<mapsto> cfg2"
lsemimodule3: fixes s :: "'b"   and t :: "'b"   and a :: "'a" shows "(s \<odot> t) \<circ> a = s \<circ> t \<circ> a"
H_fcs_in_G_fcs: fixes b :: "'b" assumes "b \<in> arcs H - (G.face_cycle_set a \<union> G.face_cycle_set a')" shows "H.face_cycle_set b \<in> G.face_cycle_sets - {G.face_cycle_set a, G.face_cycle_set a'}"
Red_F_\<G>_empty_q_def: fixes q :: "'q"   and N :: "'f set" shows "no_labels.Red_F_\<G>_empty_q q N = {C. \<forall>D\<in>\<G>_F_q q C. D \<in> Red_F_q q (no_labels.\<G>_Fset_q q N)}"
T_Sub: fixes \<Gamma> :: "binding list"   and t :: "trm"   and S :: "POPLmark.type"   and T :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : S"   and "\<Gamma> \<turnstile> S <: T" shows "\<Gamma> \<turnstile> t : T"
arr_assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set"   and c :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b"   and "partial_magma.ide (free_monoidal_category.comp C) c" shows "partial_magma.arr (free_monoidal_category.comp C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C a b c)"
simps: fixes a :: "ind" shows "Nat.Nat a = (a = Zero_Rep \<or> (\<exists>i. a = Suc_Rep i \<and> Nat.Nat i))"
trace_length: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat" shows "length (trace n f g xs x) = Suc x"
eq_span_insert_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and S :: "'b set" assumes "module scale"   and "x - y \<in> module.span scale S" shows "module.span scale (insert x S) = module.span scale (insert y S)"
rep_eq: fixes x :: "'a list" shows "pdevs_apply (pdevs_of_list x) = (\<lambda>i. if i < length x then x ! i else (0::'a))"
r_pdec122_prim: shows "prim_recfn 1 r_pdec122"
assms: shows "a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
algebraic_of_rat: fixes x :: "rat" shows "algebraic (of_rat x)"
prog_clock_monotone: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and prog :: "top0 list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_prog ck env s prog res"   and "ck" shows "clock (fst res) \<le> clock s"
measurable_abs_powr: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and p :: "real" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
cancel_comm_semigroup_axioms: shows "cancel_comm_semigroup f\<^sub>a fi\<^sub>a"
f0_def: shows "PROP f0 \<equiv> (\<And>kvs. P 0 kvs)"
in_replicate_mset: fixes x :: "'a"   and n :: "nat"   and y :: "'a" shows "(x \<in># replicate_mset n y) = (0 < n \<and> x = y)"
outside: fixes S :: "'a set" shows "outside S = {x. \<not> bounded (connected_component_set (- S) x)}"
finprod_one_eqI: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<zero>\<^bsub>S\<^esub>" shows "finsum S f A = \<zero>\<^bsub>S\<^esub>"
tbisim_cases: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and bisim_wait :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and nw :: "bool"   and t :: "'a"   and ts1 :: "('b \<times> 'f \<Rightarrow>f nat) option"   and m1 :: "'c"   and ts2 :: "('d \<times> 'f \<Rightarrow>f nat) option"   and m2 :: "'e" assumes "FWbisimulation_base.tbisim bisim bisim_wait nw t ts1 m1 ts2 m2" obtains "ts1 = None" and "ts2 = None"   | x :: "'b" and ln :: "'f \<Rightarrow>f nat" and x' :: "'d" where "ts1 = \<lfloor>(x, ln)\<rfloor>" and "ts2 = \<lfloor>(x', ln)\<rfloor>" and "bisim t (x, m1) (x', m2)" and "nw \<or> bisim_wait x x'"
size_2: fixes x :: "'c" shows "size (dual x) = Suc 0"
Qeq: fixes h :: "'b \<Rightarrow> 'c"   and k :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> t \<Longrightarrow> h x = k x" shows "Q h = Q k"
Call_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and M :: "String.literal"   and es :: "'addr expr1 list" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (e\<bullet>M(es)) h s s'"
interior_std_simplex: shows "interior (convex hull insert (0::'a) Basis) = {x. (\<forall>i\<in>Basis. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) Basis < 1}"
MaxR_def: fixes r :: "('a \<times> 'a) set" shows "MaxR r \<equiv> Linear_order r"
IH: fixes g :: "graph"   and f :: "face"   and u :: "nat"   and n :: "nat" assumes "minGraphProps g"   and "pre_subdivFace' g f r u n ovs"   and "f \<in> \<F> g"   and "ovs = [] \<longrightarrow> n = 0 \<and> u = last (verticesFrom f r)" shows "\<exists>f'\<in>set (finals (subdivFace' g f u n ovs)) - set (finals g). (f\<^bsup>-1\<^esup> \<bullet> r, r) \<in> \<E> f' \<and> |vertices f'| = n + |ovs| + (if r = u then 1 else |between (vertices f) r u| + 2)"
edges_ins_nodes: fixes v :: "'b"   and g :: "'b ugraph" shows "edges (ins_node v g) = edges g"
low_mds_eq_sym: fixes dma :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and \<C>_vars :: "'c \<Rightarrow> 'c set"   and eval :: "((('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('e \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and mds :: "Mode \<Rightarrow> 'c set"   and mem\<^sub>1 :: "'c \<Rightarrow> 'd"   and mem\<^sub>2 :: "'c \<Rightarrow> 'd" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.low_mds_eq dma \<C>_vars mds mem\<^sub>1 mem\<^sub>2" shows "sifum_security_init.low_mds_eq dma \<C>_vars mds mem\<^sub>2 mem\<^sub>1"
rep_eq: fixes x :: "('a, 'b) \<I>" shows "responses_\<I> x = Rep_\<I> x"
exhaust: fixes y :: "dag" obtains "y = Tip"   | x21 :: "dag" and x22 :: "ref" and x23 :: "dag" where "y = Node x21 x22 x23"
lub_eqI: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and M :: "'d set"   and l :: "'d" assumes "class.po below"   and "po.is_lub below M l" shows "po.lub below M = l"
in_id_succg_rel_iff: fixes s :: "'a \<Rightarrow> 'a list"   and E :: "('a \<times> 'a) set" shows "((s, E) \<in> \<langle>Id\<rangle>succg_rel) = (\<forall>v. distinct (s v) \<and> set (s v) = E `` {v})"
of_nat_less_iff: fixes m :: "nat"   and n :: "nat" shows "(of_nat m < of_nat n) = (m < n)"