hcju/isabelleps huggingface dataset:
flatten_normalize: fixes c :: "('d, 'e, 'f) com"   and x :: "('d, 'e, 'f) com"   and xs :: "('d, 'e, 'f) com list" assumes "flatten (Language.normalize c) = x # xs" shows "case xs of [] \<Rightarrow> Language.normalize c = x | x' # xs' \<Rightarrow> Language.normalize c = Seq x (sequence Seq xs)"
exec1_pres_termis: fixes cs :: "com list"   and s :: "state"   and cs' :: "com list"   and s' :: "state" assumes "(cs, s) \<rightarrow> (cs', s')" shows "cs \<Down> s \<longrightarrow> cs' \<Down> s'"
eoutKnowsECorrect_def: fixes C :: "specID"   and e :: "Expression" shows "eoutKnowsECorrect C e \<equiv> eout C e = ((\<exists>k. e = kE k \<and> k \<in> specKeys C) \<or> (\<exists>s. e = sE s \<and> s \<in> specSecrets C) \<or> knows C [e])"
SC: shows "nths (xs @ [x]) (A - B) = nths xs (A - B) @ nths [x] {j. j + length xs \<in> A - B}"
strict_trans2: fixes b :: "'a Word.word"   and a :: "'a Word.word"   and c :: "'a Word.word" assumes "b <s a"   and "c \<le>s b" shows "c <s a"
combine_with_key_Mapping: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and t1 :: "('c, 'd) RBT.rbt"   and t2 :: "('c, 'd) RBT.rbt" shows "Mapping.combine_with_key f (RBT_Mapping.Mapping t1) (RBT_Mapping.Mapping t2) = RBT_Mapping.Mapping (RBT.combine_with_key f t1 t2)"
compl_sets: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set" assumes "algebra \<Omega> M"   and "a \<in> M" shows "\<Omega> - a \<in> M"
simps_8: fixes v :: "Ferrack.num" shows "not (Eq v) = NOT (Eq v)"
right_invariant_eq_app_right: fixes L :: "'b list set" shows "right_invariant (eq_app_right L)"
add_cancel_end: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "(x + (y + z) = y) = (x = - z)"
hyps: shows "count (image_mset f X) y = (\<Sum>x | x \<in># X \<and> y = f x. count X x)"
Diff_insert0: fixes x :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "x \<notin> A" shows "A - insert x B = A - B"
has_derivative_prod: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f' :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'b"   and S :: "'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within S)" shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within S)"
ta_bisim_init_fin_bisim_convert: fixes final1 :: "'b \<Rightarrow> bool"   and final2 :: "'d \<Rightarrow> bool"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta1 :: "('f, 'a, 'b, 'c, 'g, 'h) thread_action"   and ta2 :: "('f, 'a, 'd, 'e, 'g, 'h) thread_action" shows "ta_bisim (FWbisimulation_base.init_fin_bisim final1 final2 bisim) (convert_TA_initial (convert_obs_initial ta1)) (convert_TA_initial (convert_obs_initial ta2)) = ta_bisim bisim ta1 ta2"
finite_UNIV_char_def: shows "finite_UNIV = Phantom(char) True"
coeffs_times_linterm: fixes a :: "'a"   and p :: "'a poly"   and b :: "'a" shows "coeffs (pCons (0::'a) (Polynomial.smult a p) + Polynomial.smult b p) = strip_while ((=) (0::'a)) (map2 (\<lambda>c d. b * d + c * a) ((0::'a) # coeffs p) (coeffs p @ [0::'a]))"
fdia_star_unfoldr_var: fixes y :: "'a"   and x :: "'a" shows "d y +  |x\<^sup>\<star>\<rangle>  |x\<rangle> y =  |x\<^sup>\<star>\<rangle> y"
ideD_1: fixes a :: "'b \<times> 'b" assumes "B.VV.ide a" shows "B.VV.arr a"
not_mayDiverge_Atm: fixes atm :: "'atom"   and s :: "'state" shows "\<not> mayDiverge (Atm atm) s"
convol_o: fixes f :: "'e \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and h :: "'b \<Rightarrow> 'e" shows "BNF_Def.convol f g \<circ> h = BNF_Def.convol (f \<circ> h) (g \<circ> h)"
Pow_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "Pow A \<subseteq> Pow B"
assms_3: shows "vars F \<union> vars F' \<subseteq> vars valuation"
idx_sequence_less: fixes idx :: "nat \<Rightarrow> nat"   and n :: "nat"   and k :: "nat" assumes "idx_sequence idx" shows "idx n < idx (Suc (n + k))"
gb_graph_axioms_def: fixes G :: "('a, 'b) gb_graph_rec_scheme" shows "gb_graph_axioms G \<equiv> finite (gbg_F G) \<and> gbg_F G \<subseteq> Pow (g_V G)"
map_of_constant_assignments_dom: fixes m :: "'c \<Rightarrow> 'd option"   and d :: "'d"   and vs :: "'c list" assumes "m = map_of (map (\<lambda>v. (v, d)) vs)" shows "dom m = set vs"
Goal: fixes P :: "pi"   and s :: "(name \<times> name) list"   and Q :: "pi"   and x :: "name" assumes "P[
] \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q[
]"   and "x \<sharp> s" shows "<\<nu>x>P[
] \<leadsto>\<guillemotleft>weakBisim\<guillemotright> <\<nu>x>Q[
]"
ground_subst_dom_iff_img: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a" assumes "ground (subst_range \<sigma>)" shows "(x \<in> subst_domain \<sigma>) = (\<sigma> x \<in> subst_range \<sigma>)"
inj_image_mem_iff: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and A :: "'a set" assumes "inj f" shows "(f a \<in> f ` A) = (a \<in> A)"
assms: fixes x :: "'a" assumes "x \<in> sset xs" shows "p\<cdot>x \<noteq> \<bottom>"
atLeastAtMost_singleton_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.order less_eq less" shows "(ord.atLeastAtMost less_eq a b = {c}) = (a = b \<and> b = c)"
alw_HLD_iff_sconst: fixes x :: "'c"   and \<omega> :: "'c stream" shows "alw (HLD {x}) \<omega> = (\<omega> = sconst x)"
matrix_def: shows "matrix = {f. finite (nonzero_positions f)}"
fin: fixes i :: "'a" assumes "i \<in> I" shows "emeasure M (A i) \<noteq> \<infinity>"
1: assumes "Ide t" shows "t \<in> HHom (Src t) (Trg t) \<and> t \<in> VHom t t"
poincare_map_def: fixes P :: "'a set"   and x :: "'a" shows "Y.poincare_map P x = Y.flow0 x (Y.return_time P x)"
ipv4addr_of_nat_def: fixes n :: "nat" shows "ipv4addr_of_nat n = word_of_nat n"
unity_root_eq_1: fixes k :: "nat"   and n :: "int" assumes "int k dvd n" shows "unity_root k n = 1"
mat_mul_conjugate_transpose_vec_eq_0: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nr"   and "A *\<^sub>v (A\<^sup>H *\<^sub>v v) = 0\<^sub>v nr" shows "A\<^sup>H *\<^sub>v v = 0\<^sub>v nc"
powr_less_eq_inverse_iff': fixes x :: "real"   and y :: "real"   and z :: "real" assumes "0 < x"   and "0 < y"   and "0 < z" shows "(z \<le> x powr y) = (z powr inverse y \<le> x)"
par: assumes "n \<noteq> 0" shows "eval (Cn 1 r_parallel [at0, at1, c]) [f \<triangleright> n] = parallel (the (f 0)) (the (f 1)) (the (inconsist (amalg01 f) (f \<triangleright> n)))"
divide_fds_def: fixes f :: "'b fds"   and g :: "'b fds" shows "f / g = f * inverse g"
sup_ereal_def: fixes x :: "ereal"   and y :: "ereal" shows "sup x y = max x y"
nz: shows "prob (g -` {g w}) \<noteq> 0"
upper_semilattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "upper_semilattice L \<equiv> partial_order L \<and> upper_semilattice_axioms L"
col_space_eq: fixes A :: "'a mat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc" shows "col_space A = {y \<in> carrier_vec (dim_row A). \<exists>x\<in>carrier_vec (dim_col A). A *\<^sub>v x = y}"
simps_2: fixes m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" shows "cut_off_after_match_any (Rule m a # rs) = (if m = MatchAny \<and> (a = Accept \<or> a = Drop \<or> a = Reject) then [Rule m a] else Rule m a # cut_off_after_match_any rs)"
dom_simp: fixes f :: "'a" assumes "hhom\<^sub>C.arr f" shows "hhom\<^sub>C.dom f = C.dom f"
theNFTBoth_length_finished_3: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list"   and dl :: "('d, 'e, 'f) cmd list" assumes "n \<in> theNFTBoth cl dl" shows "n < length dl"
to_map_def: fixes A :: "('a \<times> 'a) set"   and f :: "'a \<times> 'a \<Rightarrow> 'a \<times> 'a" shows "to_map A f = \<lparr>edge_rev = swap_in A, edge_succ = f\<rparr>"
PInfty_neq_ereal_2: fixes r :: "real" shows "\<infinity> \<noteq> ereal r"
\<phi>\<psi>: shows "internal_equivalence f g \<phi> \<psi>"
da_elim_cases_33: fixes Env :: "env"   and B :: "lname set"   and unop :: "unop"   and e :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>UnOp unop e\<rangle>\<guillemotright> A" obtains "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> A"
sorted_filter_less_subset_take: fixes xs :: "'b list"   and i :: "nat" assumes "sorted xs"   and "i < length xs" shows "{#x \<in># mset xs. x < xs ! i#} \<subseteq># mset (take i xs)"
eval_power: fixes x :: "'b"   and n :: "num" shows "x ^ numeral n = x * x ^ pred_numeral n"
fold_keys_min_top_eq: fixes t :: "('b, unit) RBT.rbt" assumes "\<not> RBT.is_empty t" shows "fold_keys min t top = fold1_keys min t"
mono_star_fun: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f" shows "mono (star_fun f)"
get_elements_by_tag_name_def: shows "Shadow_DOM.get_elements_by_tag_name \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_elements_by_tag_name Shadow_DOM.to_tree_order"
cinner_sum_left: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'c" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner (comm_monoid_add.sum plus zero f A) y = (\<Sum>x\<in>A. cinner (f x) y)"
while: fixes b :: "'BExp"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simplep (While b c) mem (Stmt.If b (c ;; While b c) Stop) mem"
intro: fixes A :: "('c, 'd) aGroup_scheme" assumes "(\<plusminus>\<^bsub>A\<^esub>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A"   and "\<And>a b c. \<lbrakk>a \<in> carrier A; b \<in> carrier A; c \<in> carrier A\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)"   and "\<And>a b. \<lbrakk>a \<in> carrier A; b \<in> carrier A\<rbrakk> \<Longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b = b \<plusminus>\<^bsub>A\<^esub> a"   and "mop A \<in> carrier A \<rightarrow> carrier A"   and "\<And>a. a \<in> carrier A \<Longrightarrow> -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>"   and "\<zero>\<^bsub>A\<^esub> \<in> carrier A"   and "\<And>a. a \<in> carrier A \<Longrightarrow> \<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a" shows "aGroup A"
rfd_def: shows "fd\<^sub>\<R> = fd\<^sub>\<F> \<circ> \<F>"
append_child_def: shows "append_child \<equiv> i_append_child.a_append_child"
knows_decomposition_1_k: fixes a :: "Keys"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "kKS a \<notin> specKeysSecrets P"   and "kKS a \<notin> specKeysSecrets Q"   and "subcomponents PQ = {P, Q}"   and "knows PQ [kE a]"   and "correctCompositionIn PQ"   and "correctCompositionLoc PQ" shows "knows P [kE a] \<or> knows Q [kE a]"
assms_2: fixes i :: "'a" assumes "i \<in> J" shows "sets (M i) = sets (N i)"
br: fixes i :: "nat"   and \<C> :: "t_context"   and ts :: "t list"   and t1s :: "t list"   and t2s :: "t list" assumes "i < length (label \<C>)"   and "label \<C> ! i = ts" shows "\<C> \<turnstile> [Br i] : t1s @ ts _> t2s"
sturm_adjacent_root_aux: fixes i :: "nat"   and ps :: "real poly list"   and x :: "real" assumes "i < length ps - 1"   and "poly (ps ! i) x = 0"   and "poly (ps ! (i + 1)) x = 0"   and "\<And>i x. \<lbrakk>i < length ps - 2; poly (ps ! (i + 1)) x = 0\<rbrakk> \<Longrightarrow> sgn (poly (ps ! (i + 2)) x) = - sgn (poly (ps ! i) x)" shows "\<forall>j\<le>i + 1. poly (ps ! j) x = 0"
indicator_times_eq_if_1: fixes S :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "indicator S x * f x = (if x \<in> S then f x else (0::'a))"
assms: fixes xs :: "nat list" assumes "length xs = arity f" shows "eval f xs \<down>"
Limit_vproduct_vsubset_Vset_succI: fixes \<alpha> :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "I \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> A i \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "vproduct I A \<subseteq>\<^sub>\<circ> Vset (ZFC_in_HOL.succ \<alpha>)"
lipschitz_on_minus: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "C-lipschitz_on U f" shows "C-lipschitz_on U (\<lambda>x. - f x)"
perp_up_non_degenerate_in_S_at_end_2: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l"   and "\<not> proj2_incident a l" shows "B\<^sub>\<real> cart2_pt (perp_foot a l) cart2_pt a cart2_pt (perp_up a l)"
range: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "range T \<subseteq> V"
inv_image_eq': shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = (\<lambda>b. {x \<in> A. f' (inv p\<^sub>A x) = b}) ` B - {{}}"
filter_insort_triv: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less"   and "\<not> P x" shows "filter P (linorder.insort_key less_eq f x xs) = filter P xs"
rewrite_negated_primitives_preserves_not_has_disc: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and disc2 :: "'b common_primitive \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<not> has_disc disc2 m"   and "\<not> has_disc_negated disc False m"   and "\<forall>a. \<not> disc2 (C a)" shows "\<not> has_disc disc2 (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m)"
FunctorComp: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme"   and f :: "'f"   and g :: "'f" assumes "Ftor F : A \<longrightarrow> B"   and "f \<approx>>\<^bsub>A\<^esub> g" shows "F ## f ;;\<^bsub>A\<^esub> g = F ## f ;;\<^bsub>B\<^esub> (F ## g)"
1: fixes x :: "real"   and y :: "real"   and r :: "ennreal" assumes "ennreal x = ennreal y * r"   and "0 \<le> x"   and "0 < y" shows "ennreal (1 / y * x) = r"
seqp_deliverTE: fixes \<xi> :: "'e"   and l :: "'h"   and s\<^sub>d\<^sub>a\<^sub>t\<^sub>a :: "'e \<Rightarrow> nat"   and p :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p), a, \<xi>', q) \<in> seqp_sos \<Gamma>" obtains "a = deliver (s\<^sub>d\<^sub>a\<^sub>t\<^sub>a \<xi>)" and "\<xi>' = \<xi>" and "q = p"
lset_repeat: fixes x :: "'c" shows "lset (repeat x) = {x}"
elims_2: fixes x :: "'v"   and xa :: "enat list"   and xb :: "'v" assumes "path x xa xb" obtains u :: "'v" and v :: "'v" where "x = ua__" and "xa = []" and "xb = va__" and "ua__ = va__"   | u :: "'v" and l :: "enat" and ls :: "enat list" and v :: "'v" where "x = u" and "xa = l # ls" and "xb = v" and "\<exists>uh. l = w (u, uh) \<and> path uh ls v"
list_all2_nthD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'d list"   and p :: "nat" assumes "list_all2 P xs ys"   and "p < length xs" shows "P (xs ! p) (ys ! p)"
assms_2: shows "F\<lparr>ArrCod\<rparr> = A"
fi_f: fixes x :: "'u" assumes "x \<in> V" shows "fi (f x) = x"
partial_deliverTE: fixes s :: "'c net_state"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set" assumes "(s, i:deliver(d), s') \<in> pnet_sos S T" obtains sa :: "'c net_state" and s'a :: "'c net_state" and t :: "'c net_state" where "s = SubnetS sa__ t__" and "s' = SubnetS s'a t__" and "(sa__, i:deliver(d), s'a) \<in> S"   | t :: "'c net_state" and t' :: "'c net_state" and sa :: "'c net_state" where "s = SubnetS sa t" and "s' = SubnetS sa t'" and "(t, i:deliver(d), t') \<in> T"
clop_iso: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "cl_op x \<le> cl_op y"
bounded_linear_inner_left: fixes b :: "'a" shows "bounded_linear (\<lambda>a. a \<bullet> b)"
is_sublist_y_hd: fixes vs :: "'a list"   and y :: "'a"   and x :: "'a" assumes "distinct vs"   and "y = hd vs" shows "\<not> is_sublist [x, y] vs"
monoid_vec_simps_3: fixes ty :: "'a itself"   and n :: "nat" shows "\<one>\<^bsub>monoid_vec ty n\<^esub> = 0\<^sub>v n"
oalist_eq_raw_alt: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ys :: "('a \<times> 'c) list \<times> 'o" assumes "oalist_inv xs"   and "oalist_inv ys" shows "oalist_eq_raw xs ys = (lookup_raw xs = lookup_raw ys)"
exhaust: fixes y :: "Index" obtains "y = none"   | "y = some"
assms_2: shows "r \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
hom_char: fixes a :: "'a"   and b :: "'a" shows "Right.hom a b = hom a b \<inter> Collect (right (src \<mu>))"
norm_blinfun: fixes f :: "'d \<Rightarrow>\<^sub>L 'c"   and x :: "'d" shows "norm (blinfun_apply f x) \<le> norm f * norm x"
assms_1: shows "\<bar>x\<bar> \<noteq> \<infinity>"
finprod_insert: fixes F :: "'d set"   and a :: "'d"   and f :: "'d \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finprod G f (insert a F) = f a \<otimes> finprod G f F"
l: shows "l = matrix_to_lpolies (two_block_non_interfering A A\<^sup>T)"
FreeUltrafilterNat_inverse_real_of_posnat: fixes u :: "real" assumes "0 < u" shows "\<forall>\<^sub>F n in \<U>. inverse (real (Suc n)) < u"
open_right: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "open S"   and "x \<in> S"   and "x < y" shows "\<exists>b>x. {x..<b} \<subseteq> S"
unique: shows "q.unique_factorization_m (Polynomial.smult ilc f) (1, hs)"
inc_support_trading_strat: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and B :: "'c set" assumes "trading_strategy pf1" shows "\<forall>asset\<in>support_set pf1 \<union> B. borel_adapt_stoch_proc F (pf1 asset)"
strategyproofness: shows "(pmf (sds R14) d + pmf (sds R14) a < pmf (sds R9) d + pmf (sds R9) a \<or> pmf (sds R14) d + (pmf (sds R14) a + pmf (sds R14) c) < pmf (sds R9) d + (pmf (sds R9) a + pmf (sds R9) c)) \<or> pmf (sds R14) d + pmf (sds R14) a = pmf (sds R9) d + pmf (sds R9) a \<and> pmf (sds R14) d + (pmf (sds R14) a + pmf (sds R14) c) = pmf (sds R9) d + (pmf (sds R9) a + pmf (sds R9) c)"
tauStepChainPar1: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P" shows "\<Psi> \<rhd> P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P' \<parallel> Q"
assms: shows "almost_everywhere (M1 \<Otimes>\<^sub>M M2) Q"
emeasure_pair_measure_alt: fixes X :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M M2)" shows "emeasure (N \<Otimes>\<^sub>M M2) X = \<integral>\<^sup>+ x. emeasure M2 (Pair x -` X) \<partial>N"
sel_10: fixes x61 :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and x62 :: "'a expr list" shows "call_expr.args (Operation x61 x62) = x62"
f_neg: fixes v :: "'c" assumes "v \<in> carrier M" shows "f (\<ominus>\<^bsub>M\<^esub> v) = \<ominus>\<^bsub>N\<^esub> f v"
no_1_2: fixes c :: "nat \<Rightarrow>f value option"   and r :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" shows "\<not> obtains 1 c drinks2 2 r t"
w_def: shows "w = (\<lambda>n. u n * v n)"
Cons_in_shuffles_rightI: fixes zs :: "'a list"   and xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "zs \<in> shuffles xs ys" shows "z # zs \<in> shuffles xs (z # ys)"
reachable_reaches: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s\<^sub>0 :: "'c"   and a :: "'c"   and b :: "'c" assumes "Graph_Start_Defs.reachable E s\<^sub>0 a"   and "E\<^sup>*\<^sup>* a b" shows "Graph_Start_Defs.reachable E s\<^sub>0 b"
lzip_trans_fusion2: fixes g :: "('c, 'e) lgenerator'"   and h :: "'f \<Rightarrow> ('d, 'f) step"   and sg :: "'e"   and sh :: "'f" shows "lunstream (zip_raw (lgenerator g) h) (sg, sh, None) = lzip (lunstream' g sg) (lunstream h sh)"
pseudo_graph_def: fixes G :: "('c, 'd) pre_digraph" shows "pseudo_graph G \<equiv> fin_digraph G \<and> sym_digraph G"
MOST_rev_mp: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<forall>\<^sub>\<infinity>x. P x"   and "\<forall>\<^sub>\<infinity>x. P x \<longrightarrow> Q x" shows "\<forall>\<^sub>\<infinity>x. Q x"
that: fixes ab :: "'a set" assumes "path ab a b \<and> ray a b \<subseteq> ab" shows "thesis"
valid_rchild_if_well_valued0: fixes t :: "'a tree" assumes "well_shaped t"   and "local.well_valued0 t" shows "local.valid (rchild t)"
null_is_empty: fixes xq :: "'a seq" shows "null xq = is_empty (pred_of_seq xq)"
subst_fmla_cnj: fixes A :: "fmla"   and B :: "fmla"   and i :: "name"   and x :: "trm" shows "(A AND B)(i::=x) = A(i::=x) AND B(i::=x)"
A_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and QA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and QB :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Bisimulation_Invariants A B sim PA QA PB QB"   and "PA a"   and "A a b" shows "QA b"
harry_meas_True_start: fixes xs :: "bool list" shows "harry_meas (True # xs) = 1 + harry_meas xs"
smult_sg_nsg: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "G \<guillemotright> H \<diamondop> N "
assms: shows "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
dom_char: fixes f :: "'a \<times> 'a" shows "H.VV.dom f = (if H.VV.arr f then H.VxV.dom f else H.VxV.null)"
insert_delete_index: fixes i' :: "nat"   and i :: "nat" assumes "i' \<noteq> i" shows "insert_index i (delete_index i i') = i'"
dtor2_ctor2: fixes x :: "('d, 'd IF1, 'd IF2) F2" shows "dtor2 (ctor2 x) = x"
flat_below_iff: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.flat below" shows "below x y = (x = pcpo.bottom below \<or> x = y)"
pdouble_residue_eq: fixes a :: "int"   and p :: "int \<times> int \<times> int" shows "pdouble a p = mpdouble m a p"
nn_integral_count_compose_bij: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "bij_betw g A B" shows "\<integral>\<^sup>+x\<in>A. f (g x)\<partial>count_space UNIV = set_nn_integral (count_space UNIV) B f"
st_minus: fixes x :: "real star" assumes "x \<in> HFinite" shows "st (- x) = - st x"