hcju/isabelleps huggingface dataset:
fresh_star_supp_conv: fixes x :: "'e"   and y :: "'d" assumes "supp x \<sharp>* y" shows "supp y \<sharp>* x"
multiple_incomplete: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and n :: "nat" assumes "incomplete_design point_set block_collection u_block_size"   and "0 < n" shows "incomplete_design point_set (repeat_mset n block_collection) u_block_size"
ActualBoxE: fixes v :: "i"   and \<phi> :: "\<o>" assumes "[\<^bold>\<A>(\<^bold>\<box>\<phi>) in v]" shows "[\<^bold>\<box>\<phi> in v]"
closed_eq: fixes f1 :: "'form"   and f2 :: "'form"   and s1 :: "'subst"   and a1 :: "nat"   and s2 :: "'subst"   and a2 :: "nat" assumes "closed f1"   and "closed f2" shows "(subst s1 (freshen a1 f1) = subst s2 (freshen a2 f2)) = (f1 = f2)"
lunit_closed: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and f :: "'b" assumes "subbicategory V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B Arr"   and "Arr f"   and "partial_magma.ide V f" shows "Arr (transformation_by_components.map V V (\<lambda>\<mu>. if partial_magma.arr V \<mu> then H (trg\<^sub>B \<mu>) \<mu> else partial_magma.null V) (bicategory.lunit V H \<a>\<^sub>B \<i> src\<^sub>B trg\<^sub>B) f)"
wfp_onI_min: fixes A :: "'e set"   and r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x Q. \<lbrakk>x \<in> Q; Q \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>z\<in>Q. \<forall>y\<in>A. r y z \<longrightarrow> y \<notin> Q" shows "wfp_on r A"
fz_eq: fixes z :: "complex" assumes "cmod z < 1" shows "f z = z * h z"
order_one_triv_iff: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "(order G = 1) = (carrier G = {\<one>\<^bsub>G\<^esub>})"
GodNecExists: shows "universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v)))"
simps_1: fixes f :: "'c" shows "Dom \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>C.dom f\<^bold>\<rangle>"
f\<^sub>\<alpha>_def: shows "f\<^sub>\<alpha> \<equiv> \<lambda>x. \<Sum>p\<in>sparsegrid dm lm. \<alpha> p * \<Phi> p x"
fps_unit_factor_decompose: fixes f :: "'b fps" shows "f = unit_factor f * fps_X ^ subdegree f"
is_sig_GB_upt_def: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" shows "is_sig_GB_upt d G u = (G \<subseteq> dgrad_sig_set' (length fs) d \<and> (\<forall>v. v \<prec>\<^sub>t u \<longrightarrow> d (pp_of_term v) \<le> dgrad_max d \<longrightarrow> component_of_term v < length fs \<longrightarrow> is_sig_GB_in d G v))"
int_pow_closed: fixes x :: "'c"   and i :: "int" assumes "x \<in> carrier (H\<lparr>carrier := generate H {b}\<rparr>)" shows "x [^]\<^bsub>H\<lparr>carrier := generate H {b}\<rparr>\<^esub> i \<in> carrier (H\<lparr>carrier := generate H {b}\<rparr>)"
hausdorff_distance_infdist_triangle: fixes B :: "'c set"   and C :: "'c set"   and x :: "'c" assumes "B \<noteq> {}"   and "bounded B"   and "bounded C" shows "infdist x C \<le> infdist x B + hausdorff_distance B C"
geodesic_subsetD: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "geodesic_subset S"   and "x \<in> S"   and "y \<in> S" shows "geodesic_segment_between {x--S--y} x y"
simps: fixes n :: "int" shows "dec_10 n = (if n < 10 then n else dec_10 (n - 10))"
equation: shows "rat_poly_plus (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A)) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A) = 0"
inext_neq_imp_less: fixes n :: "nat"   and I :: "nat set" assumes "n \<noteq> inext n I" shows "n < inext n I"
simps_1: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and p :: "'a"   and c' :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.next channel trans send recv c (Snapshot p) c' = (distributed_system.can_occur channel trans send recv (Snapshot p) c \<and> ps c' p = Some (states c p) \<and> (\<forall>p'. states c p' = states c' p') \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> ps c' p' = ps c p') \<and> (\<forall>i. (\<exists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i @ [Marker]) \<and> (\<forall>i. (\<exists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = (fst (cs c i), Recording)) \<and> (\<forall>i. (\<nexists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i) \<and> (\<forall>i. (\<nexists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = cs c i))"
one_le_power: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq one a" shows "less_eq one (power.power one times a n)"
Collect_case_prod_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<le> B" shows "{(x, y). A x y} \<subseteq> {(x, y). B x y}"
MsgFunction: shows "msgs cfg2 = sends p (states cfg1 p) (unpackMessage m) \<union># msgs cfg1 -# m"
finite_S: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and S :: "'c set" assumes "Separation G v0 v1 S" shows "finite S"
simps_2: fixes v :: "'b" shows "is_cnf (Atom v) = is_disj (Atom v)"
range_ex1_eq: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b" assumes "inj f" shows "(b \<in> range f) = (\<exists>!x. b = f x)"
conjugE: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v" obtains r :: "'b list" and s :: "'b list" where "r \<cdot> s = u" and "s \<cdot> r = v"
subg_sg_sg: fixes K :: "'a set"   and H :: "'a set" assumes "G \<guillemotright> K "   and "\<natural>K \<guillemotright> H " shows "G \<guillemotright> H "
le_inf_eq_inf_transp: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "w\<^sub>2 =\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>3"
dgrad_set_leI: fixes S :: "'b set"   and T :: "'b set"   and d :: "'b \<Rightarrow> nat" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. d s \<le> d t" shows "dgrad_set_le d S T"
le_numeral_extra_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
Discr_undiscr: fixes y :: "'b discr" shows "Discr (undiscr y) = y"
simps_1: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and uu :: "'Var ModeUpd" shows "has_annotated_stop (c@[uu]) = (if c = Stop then True else has_annotated_stop c)"
create_graph_def: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<equiv> expand (\<lparr>name = expand_new_name expand_init, incoming = {expand_init}, new = {\<phi>}, old = {}, next = {}\<rparr>, {}) \<bind> (\<lambda>(uu_, nds). RETURN nds)"
retraction_def: fixes g :: "'b" shows "D.retraction g \<equiv> \<exists>f. D.ide (g \<cdot>\<^sub>D f)"
insort_is_Cons: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "\<forall>x\<in>set xs. f a \<le> f x" shows "insort_key f a xs = a # xs"
starrel_def: shows "starrel = {(X, Y). \<forall>\<^sub>F n in \<U>. X n = Y n}"
subtype_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and P :: "POPLmark.type"   and Q :: "POPLmark.type"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile> P <: Q"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B" shows "\<up>\<^sub>e 1 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile> \<up>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> P <: \<up>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> Q"
normalize_poly_def: fixes p :: "(char list list \<times> int) list" shows "PAC_Polynomials_Operations.normalize_poly p = sort_poly_spec p \<bind> (\<lambda>p. RETURN (merge_coeffs p))"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com"   and t :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com" assumes "\<not> bval b s"   and "(c\<^sub>2, s) \<Rightarrow> t" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
assoc: fixes a :: "'b hrel"   and b :: "'b hrel"   and c :: "'b hrel" shows "(a ;; b) ;; c = a ;; b ;; c"
every_qo_extension_wf_imp_af: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>Q. (\<forall>x\<in>A. \<forall>y\<in>A. P x y \<longrightarrow> Q x y) \<and> qo_on Q A \<longrightarrow> wfp_on (strict Q) A"   and "qo_on P A" shows "almost_full_on P A"
set_intros_2: fixes y :: "'a"   and x22 :: "'a list"   and x21 :: "'a" assumes "y \<in> list.set x22" shows "y \<in> list.set (x21 # x22)"
0: shows "(sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs))\<^sup>*\<^sup>* (poly_of_pair p) (sig_trd bs (poly_of_pair p))"
that: fixes Us :: "LambdaType.type list"   and U :: "LambdaType.type" assumes "Ts = Us @ [U]"   and "e \<tturnstile> ts : Us"   and "e \<turnstile> t : U" shows "thesis"
m_s_anti_mono: fixes S1 :: "'av st"   and S2 :: "'av st"   and X :: "char list set" assumes "S1 \<le> S2" shows "m\<^sub>s S2 X \<le> m\<^sub>s S1 X"
select_convs_1: fixes lens_get :: "'c \<Rightarrow> 'd"   and lens_put :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and more :: "'e" shows "get\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_get"
2: assumes "\<forall>H\<in>set (tseytin_tran1 (tseytin_toatom F) F). \<A> \<Turnstile> H" shows "\<forall>H\<in>set (tseytin_tran1 (tseytin_toatom F) F). tseytin_update \<A> F \<Turnstile> H"
inv_inj_on_permutes: fixes S :: "'b set" shows "inj_on inv {p. p permutes S}"
gso_inj: fixes i :: "nat" assumes "i < length fs" shows "inj_on gso {0..<i}"
abs_eq: fixes x :: "('b \<times> real) nonzero" shows "chart_last (Proj x) = chart_last_nonzero x"
mset_subset_eqD: fixes A :: "'c multiset"   and B :: "'c multiset"   and x :: "'c" assumes "A \<subseteq># B"   and "x \<in># A" shows "x \<in># B"
normalize_idem: fixes a :: "'a"   and b :: "'a" shows "normalize (lcm a b) = lcm a b"
well_ordered_subset: fixes B :: "'a set" assumes "B \<subseteq> P" shows "well_ordered_set B (\<sqsubseteq>)"
Refl_Field_Restr2: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Refl r"   and "A \<subseteq> Field r" shows "Field (Restr r A) = A"
fpxs_nth_uminus: fixes f :: "'b fpxs"   and r :: "rat" shows "fpxs_nth (- f) r = - fpxs_nth f r"
hullp_hull_eq: fixes G :: "'b list set" shows "hullp (\<lambda>x. x \<in> G) = (\<lambda>x. x \<in> \<langle>G\<rangle>)"
fmfilter_empty: fixes P :: "'a \<Rightarrow> bool" shows "fmfilter P fmempty = fmempty"
pbij_lens_weak: fixes x :: "'c \<Longrightarrow> 'd" assumes "pbij_lens x" shows "weak_lens x"
load_word_mem_10_unchanged: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addra :: "word32"   and addr :: "word36"   and val :: "word8" assumes "low_equal s1 s2 \<and> load_word_mem s1 addra 10 = load_word_mem s2 addra 10" shows "load_word_mem (s1\<lparr>mem := (mem s1)(10 := mem s1 10(addr \<mapsto> val), 11 := (mem s1 11)(addr := None))\<rparr>) addra 10 = load_word_mem (s2\<lparr>mem := (mem s2)(10 := mem s2 10(addr \<mapsto> val), 11 := (mem s2 11)(addr := None))\<rparr>) addra 10"
invfun_l1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and a :: "'c" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "a \<in> A" shows "(f\<inverse>\<^bsub>B,A\<^esub>) (f a) = a"
intro: fixes is_empty :: "'a \<Rightarrow> bool"   and empty :: "'a"   and multiset :: "'a \<Rightarrow> 'b multiset"   and of_list :: "'b list \<Rightarrow> 'a" assumes "\<And>e. is_empty e \<Longrightarrow> e = empty"   and "is_empty empty"   and "multiset empty = {#}"   and "\<And>i. multiset (of_list i) = mset i" shows "Collection empty is_empty of_list multiset"
transpD: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
list_of_eucl_eq_iff: fixes x :: "'b"   and y :: "'c" shows "(list_of_eucl x = list_of_eucl y) = (DIM('b) = DIM('c) \<and> (\<forall>i<DIM('c). x \<bullet> Basis_list ! i = y \<bullet> Basis_list ! i))"
safe_distance_no_collsion_delta_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_no_collsion_delta a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> \<equiv> safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> \<and> safe_distance_no_collsion_delta_axioms a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
pair_measure_def: fixes A :: "'b measure"   and B :: "'c measure" shows "A \<Otimes>\<^sub>M B = measure_of (space A \<times> space B) {a \<times> b |a b. a \<in> sets A \<and> b \<in> sets B} (\<lambda>X. \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. indicator X (x, y) \<partial>B \<partial>A)"
pos_le_neg: fixes A :: "'a" shows "Pos A \<le> Neg A"
bind_sample2: fixes m :: "'m"   and p :: "'p pmf"   and f :: "'a \<Rightarrow> 'p \<Rightarrow> 'm" shows "bind m (\<lambda>y. sample p (f y)) = sample p (\<lambda>x. bind m (\<lambda>y. f y x))"
relpow_Suc: fixes n :: "nat"   and r :: "('b \<times> 'b) set" shows "r ^^ Suc n = r O r ^^ n"
opeV: fixes y :: "'c" assumes "y \<in> U" shows "openin (top_of_set U) (V y)"
SUP_least: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'h set"   and f :: "'h \<Rightarrow> 'g"   and u :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
incl_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "incl f = (arr f \<and> Dom f \<subseteq> Cod f \<and> f = incl_of (local.dom f) (cod f))"
size_subst: fixes D :: "'a literal multiset"   and \<sigma> :: "'s" shows "size (D \<cdot> \<sigma>) = size D"
AI_Some_measure: fixes m :: "parity \<Rightarrow> nat"   and c :: "com" assumes "(strict {(x, y). x \<sqsubseteq> y})\<inverse> \<subseteq> measure m"   and "\<forall>x y. x \<sqsubseteq> y \<and> y \<sqsubseteq> x \<longrightarrow> m x = m y" shows "\<exists>c'. AI_parity c = Some c'"
assms: shows "length (xvec1 @ xvec2) = length yvec"
deg_shifts_deg_shifts_pp: fixes xs :: "'x list"   and d :: "nat"   and fs :: "(('x, nat) pp \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.deg_shifts (set xs) d (map (Poly_Mapping.map_key PP) fs) = map (Poly_Mapping.map_key PP) (deg_shifts_pp xs d fs)"
smult_one: fixes c :: "'b" shows "smult c 1 = [:c:]"
f\<^sub>a: shows "ide f\<^sub>a \<and> \<guillemotleft>\<epsilon> : f \<star> f\<^sub>a \<Rightarrow> src r\<guillemotright> \<and> \<guillemotleft>\<nu> : r \<Rightarrow> g \<star> f\<^sub>a\<guillemotright> \<and> local.iso \<nu> \<and> ((r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sub>a] \<cdot> (\<rho> \<star> f\<^sub>a)) \<cdot> \<nu> = \<r>\<^sup>-\<^sup>1[r]"
isometry_on_closure: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "isometry_on X f"   and "continuous_on (closure X) f" shows "isometry_on (closure X) f"
empty_def: shows "DList_Set.empty \<equiv> Abs_dlist []"
similar_mat_witI: fixes P :: "'c mat"   and Q :: "'c mat"   and n :: "nat"   and A :: "'c mat"   and B :: "'c mat" assumes "P * Q = 1\<^sub>m n"   and "Q * P = 1\<^sub>m n"   and "A = P * B * Q"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "P \<in> carrier_mat n n"   and "Q \<in> carrier_mat n n" shows "similar_mat_wit A B P Q"
closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S = topspace X \<inter> (S \<union> X derived_set_of S)"
extremum_unique: fixes a :: "'b multiset" shows "(a \<subseteq># {#}) = (a = {#})"
distr_bind: fixes N :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and K :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and R :: "'d measure" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}"   and "f \<in> K \<rightarrow>\<^sub>M R" shows "distr (M \<bind> N) R f = M \<bind> (\<lambda>x. distr (N x) R f)"
double_lebesgue_sets: fixes S :: "'a set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue"   and "T \<in> sets lebesgue"   and "f ` S \<subseteq> T" shows "(\<forall>U. U \<in> sets lebesgue \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue) = (f \<in> borel_measurable (lebesgue_on S) \<and> (\<forall>U. negligible U \<and> U \<subseteq> T \<longrightarrow> {x \<in> S. f x \<in> U} \<in> sets lebesgue))"
slice_kind_Call: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and S :: "'b SDG_node set"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "sourcenode a \<notin> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "SDG.slice_kind sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S a = \<lambda>cf. False:r\<hookrightarrow>\<^bsub>p\<^esub>fs"
convex_hull_carrier: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n" shows "m.convex_hull Vs \<subseteq> carrier_vec n"
gs_carr_2: shows "gs1.gso i \<in> Rn"
alpha_qSwap_preserves_qGood: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('a, 'b, 'varSort, 'var, 'c) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'" shows "qGood X = qGood X'"
geometric_deriv_sums: fixes z :: "'a" assumes "norm z < 1" shows "(\<lambda>n. of_nat (Suc n) * z ^ n) sums ((1::'a) / ((1::'a) - z)\<^sup>2)"
sheaf_spec_is_sheaf: shows "sheaf_of_rings local.local.spectrum local.local.is_zariski_open local.local.sheaf_spec local.local.sheaf_spec_morphisms local.local.\<O>b local.local.add_sheaf_spec local.local.mult_sheaf_spec local.local.zero_sheaf_spec local.local.one_sheaf_spec"
append_simpl: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "i < length xs \<longrightarrow> (xs @ ys) ! i = xs ! i"
s1: fixes q :: "nat"   and s2 :: "nat"   and x :: "nat"   and y :: "nat"   and s1 :: "nat" assumes "secure_mult q"   and "s2 = (x * y + q - s1) mod q"   and "x < q"   and "y < q"   and "s1 < q" shows "s1 = (x * y + q - s2) mod q"
proves_eq_abstract_rule_pre: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')" shows "\<Theta>,{} \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (mk_eq' \<tau>' (f $ Bv 0) (g $ Bv 0)) \<longmapsto> mk_eq (Abs \<tau> (f $ Bv 0)) (Abs \<tau> (g $ Bv 0))"
rec: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "rec_optionT f (OptionT x) = f x"
span_add: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> span S"   and "y \<in> span S" shows "x + y \<in> span S"
assms_1: shows "X \<inter> range_vars \<theta> = {}"
assms_2: shows "divides_ff (to_fract n) (coeff p i)"
c: shows "Leq x c \<in> snd ` insert ia as"
wf_trm_subterms: fixes t :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m t" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subterms t)"
less_eq_Agent_def: fixes x :: "Agent"   and y :: "Agent" shows "x \<le> y \<equiv> x = y"
taq_def: fixes s :: "'a set"   and q :: "'a poly" shows "taq s q \<equiv> \<Sum>x\<in>s. Sturm_Tarski.sign (poly q x)"
to_hma\<^sub>m_def: fixes a :: "'a mat" shows "Mod_Type_Connect.to_hma\<^sub>m a = (\<chi>i j. a $$ (mod_type_class.to_nat i, mod_type_class.to_nat j))"
Assign': fixes s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and \<pi> :: "char list \<Rightarrow> com option" assumes "s' = s(x := (s x)(aval i s := aval a s))" shows "\<pi>: (x[i] ::= a, s) \<Rightarrow> s'"
strictly_redundant_def: fixes atom_ordering :: "('a \<times> 'a) set"   and C :: "'b Literal set"   and S :: "'b Literal set set" assumes "propositional_atoms atom_ordering" shows "strictly_redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> D \<subset> C))"
preserves_reflects_arr: fixes f :: "'s" shows "S'.arr (\<Phi> f) = S.arr f"
vimage_Int: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" shows "f -` (A \<inter> B) = f -` A \<inter> f -` B"
setmonotone_mk_regular1: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" shows "setmonotone (mk_regular1 P F)"
cont_on_right: shows "continuous_on (path_image (\<lambda>y. twoC (y, 1))) F"
measure\<^sub>s\<^sub>t_wellfounded: shows "wf measure\<^sub>s\<^sub>t"
abs_fresh_1: fixes b :: "name"   and a :: "name"   and x :: "'a" shows "b \<sharp> [a].x = (b = a \<or> b \<sharp> x)"
M_subterms_cl: shows "has_all_wt_instances_of \<Gamma> (subterms\<^sub>s\<^sub>e\<^sub>t (set M)) (set M)"
T_meas: shows "I.random_variable natural_extension_measure natural_extension_map"
create_character_data_preserves_wellformedness_2: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> create_character_data document_ptr text \<rightarrow>\<^sub>h h'"   and "type_wf h"   and "known_ptrs h" shows "type_wf h'"