hcju/isabelleps huggingface dataset:
Inl_not_Inr: fixes a :: "'d"   and b :: "'e" shows "Inl a \<noteq> Inr b"
simps: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and z :: "'f"   and a1 :: "'e set"   and a2 :: "'f" shows "fold_graph f z a1 a2 = (a1 = {} \<and> a2 = z \<or> (\<exists>x A y. a1 = insert x A \<and> a2 = f x y \<and> x \<notin> A \<and> fold_graph f z A y))"
Rdef: fixes x :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "R x = (P x \<and> Q x)"
compute_\<ff>s_inner_bounds_1: fixes s :: "'a list"   and \<ff>s :: "nat list"   and ix :: "nat"   and j :: "nat"   and i :: "nat" assumes "I_out_cb s (\<ff>s, ix, j)"   and "j < length \<ff>s"   and "I_in_cb s j i" shows "i - 1 < length s"
A: shows "add_mset a M \<subseteq># {#b#}"
iff_eq: fixes x :: "'a"   and y :: "'a" shows "(proj x = proj y \<and> norm x = norm y) = (x = y)"
VEnd_polymap: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and p :: "'a poly" assumes "VectorSpaceEnd smult V T" shows "VectorSpaceEnd smult V (VectorSpaceEnd.polymap smult V T p)"
simps_2: fixes G :: "prog"   and rt :: "ref_ty" shows "is_type G (RefT rt) = isrtype G rt"
is_faithful: fixes f :: "'d"   and f' :: "'d" assumes "D.par f f'"   and "D.cod \<iota>\<^sub>1 \<otimes>\<^sub>D f = D.cod \<iota>\<^sub>1 \<otimes>\<^sub>D f'" shows "f = f'"
twiddleObj_betaExtend: fixes \<beta> :: "(Location \<times> Location) set"   and o1 :: "Class \<times> (Field \<times> Val) list"   and o2 :: "Class \<times> (Field \<times> Val) list"   and \<gamma> :: "(Location \<times> Location) set" assumes "twiddleObj \<beta> o1 o2"   and "Pbij_extends \<gamma> \<beta>" shows "twiddleObj \<gamma> o1 o2"
c: fixes a :: "complex"   and x :: "complex"   and y :: "complex" shows "(cmod (cnj a * x - y))\<^sup>2 = (cmod (a * x))\<^sup>2 - Re (a * y * cnj x + x * cnj (a * y)) + (cmod y)\<^sup>2"
rep_eq: fixes x :: "('b, 'a) aezfun"   and xa :: "('b, 'a) aezfun" shows "aezfun (x * xa) = convolution (aezfun x) (aezfun xa)"
tendsto_const: fixes "open" :: "'a set \<Rightarrow> bool"   and k :: "'a"   and F :: "'b filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
inf1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A \<equiv> inf k a" shows "inf A b \<equiv> inf k (inf a b)"
hurwitz_zeta_formula_aux: fixes a :: "real"   and s :: "complex" assumes "a \<in> {0<..1}"   and "1 < Re s" shows "rGamma s * hurwitz_zeta a (1 - s) = complex_of_real (2 * pi) powr - s * (\<i> powr - s * eval_fds (fds_perzeta a) s + \<i> powr s * eval_fds (fds_perzeta (- a)) s)"
chkchop_same_def: fixes gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" shows "chkchop_same gt t s = (if is_Var (head t) then is_Hd t \<or> chkchop gt (chop t) s else chkchop gt t s)"
diff_intersect_left_idem: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M - M \<inter># N = M - N"
witness_l0'_valid: shows "valid_net' (witness' r0 [M])"
iMOD_mult_div_right_inj_on: fixes k :: "nat"   and r :: "nat"   and m :: "nat" shows "inj_on (\<lambda>x. x div k) [ r, mod k * m ]"
suma_sym: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I" shows "D E F A B C SumA G H I"
normalize_1_iff: fixes a :: "'b poly" shows "(smult (inverse (lead_coeff a)) a = 1) = is_unit a"
hyp_static: assumes "dynmethd G statC dynC sig = Some statM" shows "thesis"
strict_weak_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubseteq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubset> z"
List_rev_drop_geqn: fixes n :: "nat"   and x :: "'a list" assumes "n \<le> length x" shows "take n (rev x) = rev (drop (length x - n) x)"
invertible_Gauss_Jordan_up_to_k: fixes A :: "(('b, 'd) vec, 'c) vec"   and k :: "nat" shows "\<exists>P. invertible P \<and> Gauss_Jordan_upt_k A k = P ** A"
nontriv_solution_def: shows "nontriv_solution = (\<lambda>(a, b). (a, b) \<noteq> (1::'b, 0::'b) \<and> a\<^sup>2 = (1::'b) + of_nat D * b\<^sup>2)"
subtopI: fixes t :: "'a set" assumes "t open\<^bsub>T\<^esub>" shows "t \<inter> carrier open"
resultant_def: fixes p :: "'a poly"   and q :: "'a poly" shows "resultant p q = det (sylvester_mat p q)"
has_derivative_transform: fixes x :: "'d"   and s :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e" assumes "x \<in> s"   and "\<And>x. x \<in> s \<Longrightarrow> g x = f x"   and "(f has_derivative f') (at x within s)" shows "(g has_derivative f') (at x within s)"
range_one: shows "Poly_Mapping.range 1 = {1::'a}"
tab_simps_3: shows "trg \<rho> = trg r"
set_shadow_root_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_shadow_root_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
Fourier_coefficient_add: fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and i :: "nat" assumes "f absolutely_integrable_on {- pi..pi}"   and "g absolutely_integrable_on {- pi..pi}" shows "Fourier_coefficient (\<lambda>x. f x + g x) i = Fourier_coefficient f i + Fourier_coefficient g i"
coeff_0_reflect_poly: fixes p :: "'a poly" shows "poly.coeff (reflect_poly p) 0 = lead_coeff p"
completion_ex_borel_measurable_real: fixes g :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "g \<in> borel_measurable (completion M)" shows "\<exists>g'\<in>borel_measurable M. AE x in M. g x = g' x"
expr_typing_cong': fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and t :: "pdf_type"   and \<Gamma>' :: "nat \<Rightarrow> pdf_type" assumes "\<Gamma> \<turnstile> e : t"   and "\<And>x. x \<in> free_vars e \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "\<Gamma>' \<turnstile> e : t"
PDG_path_cdep: fixes n :: "'node"   and n' :: "'node" assumes "PDG.PDG_edge sourcenode targetnode valid_edge Def Use weak_control_dependence n None n'" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n n'"
IH_3: assumes "list_all is_Update (unlabel S)" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t S = S"
subdistl_iso: 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 f :: "'b \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(f \<circ> Inf \<le> Inf \<circ> (`) f) = order.mono less_eq f"
uniq_inter: fixes l :: "Lines"   and m :: "Lines"   and P :: "Points" assumes "l \<noteq> m"   and "incid P l"   and "incid P m" shows "Pascal_Property.inter l m = P"
cast_and_box_imp_typeof: fixes \<tau> :: "Unboxed.type"   and blob :: "('a, 'ubx1, 'ubx2) unboxed"   and d :: "'dyn" assumes "cast_and_box \<tau> blob = Some d" shows "typeof blob = Some \<tau>"
lookup_Abs_poly_mapping_when_finite: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S" shows "lookup (Abs_poly_mapping (\<lambda>x. f x when x \<in> S)) = (\<lambda>x. f x when x \<in> S)"
carrier_nhd: fixes x :: "'b"   and T :: "'b set set" assumes "x \<in> carrier\<^bsub>T\<^esub>" shows "carrier\<^bsub>T\<^esub> \<in> nhds\<^bsub>T\<^esub> x"
ID_ceq: fixes eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ID CEQ('b) = Some eq" shows "equal eq"
max_add_distrib_right: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x + subset_mset.max y z = subset_mset.max (x + y) (x + z)"
hoare1_valid_def: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and q :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "(\<Turnstile>\<^sub>1 {P} c { q \<Down>Q}) = (\<exists>k>0. \<forall>l s. P l s \<longrightarrow> (\<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> p \<le> k * q s \<and> Q l t))"
f2': fixes f1 :: "real"   and f2 :: "real"   and f1' :: "real"   and f2' :: "real"   and e :: "real"   and c :: "real" assumes "Table0 f1 f2 f1' f2' e c" shows "0 < f2'"
sds_impossibility_4_4_axioms: shows "sds_impossibility_4_4 agents' alts' lowered A1 A2 A3 A4 a b c d"
ide_in_hom: fixes a :: "'s" shows "S.ide a = \<guillemotleft>a : a \<rightarrow>\<^sub>S a\<guillemotright>"
comp_fun_idem_sup: shows "comp_fun_idem (\<squnion>)"
endo: shows "VectorSpaceEnd (\<sharp>\<cdot>) V T"
Fvars_subst_in: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "x \<in> Fvars \<phi>" shows "Fvars (subst \<phi> t x) = Fvars \<phi> - {x} \<union> FvarsT t"
edges_subset: shows "edges_of_walk xs \<subseteq> \<Union> (edges_of_walk ` paths_with_new)"
p_res_ring_1_field: shows "field (residue_ring (p ^ 1))"
less_p_zero: fixes q :: "nat" assumes "q < p" shows "xs ! q = (0::'a)"
simps_3: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e" shows "sprCRel \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprCRel"
has_subprob_density_equal_on_space: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'c \<Rightarrow> ennreal" assumes "has_subprob_density M N f"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_subprob_density M N g"
parent_child_rel_acyclic: fixes h :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h" shows "acyclic (Shadow_DOM.parent_child_rel h)"
simps_9: fixes f :: "'d \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d list" shows "map f (x21 # x22) = f x21 # map f x22"
iT_Plus_neg_assoc: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" shows "I \<oplus>- a \<oplus>- b = I \<oplus>- a + b"
flat_lub_in_chain: fixes b :: "'b"   and A :: "'b set"   and a :: "'b" assumes "Complete_Partial_Order.chain (flat_ord b) A"   and "flat_lub b A = a" shows "a = b \<or> a \<in> A"
add_nonneg_pos: fixes a :: "'e multiset"   and b :: "'e multiset" assumes "{#} \<subseteq># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
P1: shows "Per A' B' C' \<and> D E F LtA A' B' C'"
elims_3: fixes x :: "object ast_effect" assumes "\<not> wf_effect_inst x" obtains a :: "object atom formula list" and d :: "object atom formula list" where "x = Effect a d" and "\<not> Ball (set a \<union> set d) (wf_fmla_atom objT)"
mat_mul_finsum_alt: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "A * B = mat\<^sub>r nr nc (\<lambda>i. finsum_vec TYPE('b) nc (\<lambda>k. A $$ (i, k) \<cdot>\<^sub>v row B k) {0..<n})"
Inf_superset_mono: fixes B :: "('c \<times> 'd) set"   and A :: "('c \<times> 'd) set" assumes "B \<subseteq> A" shows "less_eq_prod (Inf_prod A) (Inf_prod B)"
matcher_agree_on_exact_matches_gammaE: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and X :: "'c"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "\<beta> X p = TernaryTrue" shows "\<gamma> X p"
simps_4: fixes e :: "(name, nat) finfun"   and t :: "trm"   and u :: "trm" shows "eval_trm e (pls t u) = eval_trm e t + eval_trm e u"
finite_graph: fixes es :: "nat set set" shows "finite (uverts (edge_ugraph es))"
nsqn\<^sub>r_inv: fixes dsn :: "nat"   and dsk :: "k"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "nsqn\<^sub>r (dsn, dsk, Aodv_Basic.inv, hops, nhip, pre) = dsn - 1"
fold_congs_2: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and f :: "('b word \<Rightarrow> word5 \<Rightarrow> word32) \<Rightarrow> 'b word \<Rightarrow> word5 \<Rightarrow> word32"   and f' :: "('b word \<Rightarrow> word5 \<Rightarrow> word32) \<Rightarrow> 'b word \<Rightarrow> word5 \<Rightarrow> word32" assumes "r = r'"   and "user_reg r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "user_reg_update f r = user_reg_update f' r'"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" shows "fw m n (Suc k) = fwi (fw m n k) n (Suc k) n n"
wls_item_simps_4: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsBinp delta binp" shows "goodBinp binp"
ide_in_hom_1: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>f : src f \<rightarrow>\<^sub>B trg f\<guillemotright>"
simps_1: shows "atoms_pltl false\<^sub>p = {}"
simps_12: fixes x31 :: "'b"   and x32 :: "'b"   and x33 :: "instr list"   and x21 :: "bool"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat" shows "Prog\<^sub>s\<^sub>b x31 x32 x33 \<noteq> Read\<^sub>s\<^sub>b x21 x22 x23 x24"
REC_annot_def: fixes pre :: "'b \<Rightarrow> bool"   and post :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and body :: "('b \<Rightarrow> 'a nres) \<Rightarrow> 'b \<Rightarrow> 'a nres"   and x :: "'b" shows "REC_annot pre post body x \<equiv> REC (\<lambda>D x. assert (pre x) \<bind> (\<lambda>_. body D x \<bind> (\<lambda>r. assert (post x r) \<bind> (\<lambda>_. return r)))) x"
submodule_subset: fixes H :: "'a set" assumes "submodule R M H" shows "H \<subseteq> carrier M"
Bound: fixes a :: "subject"   and x :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<sharp> a"   and "x \<sharp> C" shows "\<exists>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> derivative P' Q' a x Rel"
resultsp_gpv_results_gpv_eq: fixes \<Gamma> :: "('d, 'e) \<I>"   and x :: "'f"   and gpv :: "('f, 'd, 'e) gpv" shows "resultsp_gpv \<Gamma> x gpv = (x \<in> results_gpv \<Gamma> gpv)"
vpa_no_slpa: 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 cs :: "'a list"   and as :: "'a list"   and n :: "'b"   and n' :: "'b" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_path_aux kind get_return_edges cs as"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "CFG.valid_call_list sourcenode kind valid_edge get_proc cs n"   and "cs \<noteq> []"   and "\<forall>xs ys. as = xs @ ys \<longrightarrow> \<not> CFG.same_level_path_aux kind get_return_edges cs xs \<or> CFG.upd_cs kind cs xs \<noteq> []" shows "\<exists>a Q r fs. valid_edge a \<and> kind a = Q:r\<hookrightarrow>\<^bsub>get_proc n'\<^esub>fs"
is_process5: fixes P :: "('a event list \<times> 'a event set) set \<times> 'a event list set" assumes "is_process P" shows "\<forall>sa X Y. (sa, X) \<in> FAILURES P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (sa @ [c], {}) \<notin> FAILURES P) \<longrightarrow> (sa, X \<union> Y) \<in> FAILURES P"
unitary_is_sq_length_preserving_bis: fixes U :: "complex mat"   and v :: "complex mat" assumes "unitary U"   and "dim_row v = dim_col U"   and "dim_col v = 1" shows "\<parallel>col (U * v) 0\<parallel>\<^sup>2 = \<parallel>col v 0\<parallel>\<^sup>2"
publicKey_neq_symKey: fixes K :: "nat"   and b :: "bool"   and A :: "agent" assumes "K \<in> symKeys" shows "publicKey b A \<noteq> K"
bigo_const2: fixes c :: "'b" shows "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1::'b)"
reflexive: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<leadsto>\<^sup>^<Rel> P"
assms_2: shows "l \<in> B \<rightarrow>\<^sub>M A"
admD: fixes P :: "'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b" assumes "adm P"   and "chain Y"   and "\<And>i. P (Y i)" shows "P (\<Squnion>i. Y i)"
least_eq_k: shows "(LEAST n. Gauss_Jordan_in_ij A (0::'rows) (mod_type_class.from_nat k) $ i $ n \<noteq> (0::'a)) = mod_type_class.from_nat k"
cp_StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b"   and \<tau> :: "'a state \<times> 'a state" shows "StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y \<tau> = StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t (\<lambda>_. x \<tau>) (\<lambda>_. y \<tau>) \<tau>"
is_lasso_pre_def: fixes m :: "'b"   and L :: "'Q lasso" shows "graph.is_lasso_pre (to_gbg_ext m) L \<equiv> lasso_v0 L \<in> g_V0 (to_gbg_ext m) \<and> path (g_E (to_gbg_ext m)) (lasso_v0 L) (lasso_reach L) (lasso_va L) \<and> path (g_E (to_gbg_ext m)) (lasso_va L) (lasso_cycle L) (lasso_va L)"
edges_subset: fixes g :: "'b ugraph" shows "edges g \<subseteq> nodes g \<times> nodes g"
wf_newQuery: fixes J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'a option list set) set"   and Qn0 :: "(nat set \<times> 'a option list set) set"   and QQ :: "(nat set \<times> 'a option list set) set"   and t :: "nat set \<times> 'a option list"   and QQn :: "(nat set \<times> 'a option list set) set"   and Qn :: "(nat set \<times> 'a option list set) set" assumes "1 \<le> card J"   and "wf_query n J Q Qn0"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = newQuery J Q t"   and "QQn = newQuery J Qn t"   and "non_empty_query Qn"   and "Qn = filterQuery J Qn0" shows "rwf_query n J QQ QQn"
HomCHomOp: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and g :: "'e"   and X :: "'d" shows "HomC\<^bsub>C\<^esub>[g,X] = Hom\<^bsub>Op C\<^esub>[X,g]"
e1e2notP: shows "\<not> P (nP e1) (nP e2)"
exec_Normal_elim_cases_13: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "'b"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> \<Rightarrow> t" obtains "t = Abrupt s"
diff_less_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less (minus a b) c = less a (plus c b)"
induct: fixes x :: "'c set"   and \<Omega> :: "'c set"   and M :: "'c set set"   and P :: "'c set \<Rightarrow> bool" assumes "x \<in> smallest_ccdi_sets \<Omega> M"   and "\<And>a. a \<in> M \<Longrightarrow> P a"   and "\<And>a. \<lbrakk>a \<in> smallest_ccdi_sets \<Omega> M; P a\<rbrakk> \<Longrightarrow> P (\<Omega> - a)"   and "\<And>A. \<lbrakk>range A \<in> Pow (smallest_ccdi_sets \<Omega> M \<inter> Collect P); A 0 = {}; \<And>n. A n \<subseteq> A (Suc n)\<rbrakk> \<Longrightarrow> P (\<Union> (range A))"   and "\<And>A. \<lbrakk>range A \<in> Pow (smallest_ccdi_sets \<Omega> M \<inter> Collect P); disjoint_family A\<rbrakk> \<Longrightarrow> P (\<Union> (range A))" shows "P x"
has_integral_interior: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "negligible (frontier S)" shows "(f has_integral y) (interior S) = (f has_integral y) S"
1: shows "ccpo.admissible lSup lprefix (\<lambda>v. lproject A v \<le> x)"
monotone_enat_minus1: fixes m :: "enat" shows "monotone (\<le>) (\<le>) (\<lambda>n. n - m)"
strict_monoidal_extension_to_free_monoidal_category_axioms: shows "strict_monoidal_extension_to_free_monoidal_category (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V E.map"
preserves_iso: fixes f :: "'c" assumes "C.iso f" shows "C.iso (C'.T (C.cod \<iota>\<^sub>C, f))"
pdevs_val_add_pdevs: fixes e :: "nat \<Rightarrow> real"   and X :: "'a pdevs"   and Y :: "'a pdevs" shows "pdevs_val e (add_pdevs X Y) = pdevs_val e X + pdevs_val e Y"
hom_add: fixes x :: "'c poly poly"   and y :: "'c poly poly" shows "poly_y_x (x + y) = poly_y_x x + poly_y_x y"
isoE: fixes f :: "'a1" assumes "C1.iso f" obtains g :: "'a1" where "C1.inverse_arrows f g"
\<phi>_biuniversal_prop_1: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "\<guillemotleft>\<mu> : r\<^sub>0 \<star> v \<Rightarrow> s\<^sub>1 \<star> u\<guillemotright>" shows "\<exists>w \<theta> \<nu>. ide w \<and> \<guillemotleft>\<theta> : p\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<nu> : v \<Rightarrow> p\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (s\<^sub>1 \<star> \<theta>) \<cdot> \<a>[s\<^sub>1, p\<^sub>0, w] \<cdot> (r\<^sub>0s\<^sub>1.\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[r\<^sub>0, p\<^sub>1, w] \<cdot> (r\<^sub>0 \<star> \<nu>) = \<mu>"
dgrad_p_set_exhaust: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" obtains m :: "nat" where "F \<subseteq> dgrad_p_set d m"
hindered_mono_web: fixes \<Gamma> :: "('k, 'l) web_scheme"   and \<Gamma>' :: "('k, 'l) web_scheme" assumes "edge \<Gamma> = edge \<Gamma>'"   and "A \<Gamma> = A \<Gamma>'"   and "B \<Gamma> = B \<Gamma>'"   and "\<And>x. x \<notin> A \<Gamma>' \<Longrightarrow> weight \<Gamma> x = weight \<Gamma>' x"   and "\<And>a. a \<in> A \<Gamma>' \<Longrightarrow> weight \<Gamma>' a \<le> weight \<Gamma> a"   and "hindered \<Gamma>'" shows "hindered \<Gamma>"
prv_eql_subst_trm_id_back: fixes y :: "'var"   and \<phi> :: "'fmla"   and n :: "'trm" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "n \<in> num" shows "prv (imp (eql (Var y) n) (imp (subst \<phi> n y) \<phi>))"
assms_3: shows "\<langle>v|v\<rangle> = 1"
phis_same_var: fixes g :: "'a"   and n :: "'b"   and v :: "'e"   and vs :: "'e list"   and v' :: "'e" assumes "Mapping.lookup (phis g) (n, v) = Some vs"   and "v' \<in> set vs" shows "var g v' = var g v"
smaller_sys_are_good: fixes p :: "real poly"   and qs1 :: "real poly list"   and qs2 :: "real poly list" assumes "p \<noteq> 0"   and "0 < length qs1"   and "\<forall>q. List.member qs1 q \<longrightarrow> coprime p q"   and "0 < length qs2"   and "\<forall>q. List.member qs2 q \<longrightarrow> coprime p q"   and "set (find_consistent_signs_at_roots p qs1) = set (characterize_consistent_signs_at_roots_copr p qs1)"   and "set (find_consistent_signs_at_roots p qs2) = set (characterize_consistent_signs_at_roots_copr p qs2)" shows "set (snd (snd (reduce_system p (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))) = set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2))"
assms: shows "\<langle>M; n\<rangle> \<turnstile>\<^sub>n t"
sel: fixes a :: "'a \<Rightarrow> 'e"   and b :: "'f \<Rightarrow> 'b"   and out :: "'g \<Rightarrow> 'c"   and inn :: "'d \<Rightarrow> 'h"   and conv :: "('e, 'f, 'g, 'h) converter" shows "run_converter (map_converter a b out inn conv) = map_gpv (map_prod b (map_converter a b out inn)) out \<circ> map_gpv' id id inn \<circ> run_converter conv \<circ> a"
spectrum_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "comm_ring R addition multiplication zero unit" shows "comm_ring.spectrum R addition multiplication zero unit \<equiv> {I. pr_ideal R I addition multiplication zero unit}"
sel_1: fixes x1 :: "'b match_expr"   and x2 :: "action" shows "get_match (Rule x1 x2) = x1"
spanE: fixes p :: "'a"   and B :: "'a set" assumes "p \<in> ideal B" obtains A :: "'a set" and q :: "'a \<Rightarrow> 'a" where "finite A" and "A \<subseteq> B" and "p = (\<Sum>b\<in>A. q b * b)"