hcju/isabelleps huggingface dataset:
intro: fixes set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog" assumes "l_set_disconnected_nodes_get_parent set_disconnected_nodes_locs get_parent_locs"   and "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"   and "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs" shows "l_set_disconnected_nodes_get_ancestors\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_disconnected_nodes set_disconnected_nodes_locs get_child_nodes get_child_nodes_locs get_parent get_parent_locs type_wf known_ptrs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
valid_8: shows "load_tmps_fresh ts\<^sub>s\<^sub>b'"
ev: shows "\<forall>\<^sub>F s in at 1. s \<in> {s. 0 < Re s} - {1}"
dom_delete_aux: fixes ps :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst ps)" shows "fst ` list.set (delete_aux k ps) = fst ` list.set ps - {k}"
finite_lderivs: fixes r :: "'b rexp" shows "finite {\<guillemotleft>lderivs xs r\<guillemotright> |xs. True}"
extend_indets_ideal_subset: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "extend_indets ` ideal F \<subseteq> ideal (extend_indets ` F)"
Entry_Exit_edge: 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" assumes "CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit" shows "\<exists>a. valid_edge a \<and> sourcenode a = Entry \<and> targetnode a = Exit \<and> kind a = (\<lambda>s. False)\<^sub>\<surd>"
filter_leI: fixes F' :: "'c filter"   and F :: "'c filter" assumes "\<And>P. eventually P F' \<Longrightarrow> eventually P F" shows "F \<le> F'"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<gamma>.F.map \<gamma>.G.map \<gamma>.map"
set_iterator_abs_genord_I2: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and S0 :: "'f set"   and Rc :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and invar :: "'f \<Rightarrow> bool"   and Ra :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and \<alpha> :: "'f \<Rightarrow> 'g"   and S0' :: "'g set" assumes "set_iterator_genord iti S0 Rc"   and "\<And>xc1 xc2. \<lbrakk>invar xc1; invar xc2; Rc xc1 xc2\<rbrakk> \<Longrightarrow> Ra (\<alpha> xc1) (\<alpha> xc2)"   and "\<And>xc1 xc2. \<lbrakk>invar xc1; invar xc2; xc1 \<in> S0; xc2 \<in> S0; \<alpha> xc1 = \<alpha> xc2\<rbrakk> \<Longrightarrow> xc1 = xc2"   and "\<And>xc. xc \<in> S0 \<Longrightarrow> invar xc"   and "S0' = \<alpha> ` S0" shows "set_iterator_abs_genord \<alpha> invar iti S0' Ra"
induct: fixes P :: "'a prog"   and x1 :: "char list"   and x2 :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and Pa :: "char list \<Rightarrow> (char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option) \<Rightarrow> bool" assumes "P \<turnstile> x1 sees_methods x2"   and "\<And>D fs ms Mm. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms\<rbrakk> \<Longrightarrow> Pa Object Mm"   and "\<And>C D fs ms Mm Mm'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D sees_methods Mm; Pa D Mm; Mm' = Mm ++ (map_option (\<lambda>m. (m, C)) \<circ> map_of ms)\<rbrakk> \<Longrightarrow> Pa C Mm'" shows "Pa x1 x2"
coprime_absorb_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and y :: "'b"   and x :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times y x" shows "algebraic_semidom.coprime times one x y = dvd.dvd times y one"
isomorphic_def: fixes a :: "'b"   and a' :: "'b" shows "D.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f)"
strategy_extends_backwards_VVpstar: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> VV p**"   and "strategy p \<sigma>"   and "\<And>w. v0 \<rightarrow> w \<Longrightarrow> winning_strategy p \<sigma> w" shows "winning_strategy p \<sigma> v0"
FAssStatic\<^sub>1: fixes P :: "nat exp prog"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and a :: "nat"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(C, fs)\<rfloor>"   and "P \<turnstile> C has F,Static:t in D" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2)\<rangle>"
\<beta>_is_revrev\<beta>: shows "\<beta> = rev rev\<beta>"
1: fixes i :: "'a" assumes "contractible_space (product_topology X I)"   and "i \<in> I" shows "contractible_space (X i)"
case_prod_beta: fixes f :: "'h \<Rightarrow> 'i \<Rightarrow> 'g"   and p :: "'h \<times> 'i" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (fst p) (snd p)"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "(poly_mapping.lookup f k \<noteq> (0::'b)) = (k \<in> Poly_Mapping.keys f)"
disc_2: fixes x21 :: "'a"   and x22 :: "nat"   and x23 :: "'a interval" shows "\<not> is_INS (DEL x21 x22 x23)"
Inf2_inf: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type"   and c :: "'c::type"   and d :: "'c::type" assumes "class.refinement_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) {(f::'c::type \<Rightarrow> 'b::type) x |x::'c::type. x \<in> {c::'c::type, d::'c::type}} = (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (f c) (f d)"
mono_Sup: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" assumes "mono f" shows "Sup (f ` A) \<le> f (Sup A)"
simps_1: fixes c :: "int" shows "zsplit0 (C c) = (0, C c)"
finsum_insert: fixes F :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F" shows "sum f (insert a F) = f a + sum f F"
valid_unMultigraph_axioms: shows "valid_unMultigraph kon_graph"
M2Zf: shows "m2z\<^bsub>C\<^esub>f |\<in>| ((YFtor C @@ Y) @@ X)"
lo: shows "left_over = remove1 \<beta> (remove1 \<alpha> xs)"
simps_2: fixes e :: "'b" shows "alist_and' [Pos e] = Match e"
inj_comm_ring_hom_axioms: shows "inj_comm_ring_hom (map_poly hom)"
r_cancel_one: fixes x :: "'a"   and a :: "'a" assumes "x \<in> carrier \<G>"   and "a \<in> carrier \<G>" shows "(a \<otimes> x = x) = (a = \<one>)"
inv_comp: fixes f :: "'d"   and g :: "'d" assumes "D.iso f"   and "D.iso g"   and "D.seq g f" shows "D.inv (g \<cdot>\<^sub>D f) = D.inv f \<cdot>\<^sub>D D.inv g"
ty_cmd_imp_ty_la_aux: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and loop :: "bool"   and c :: "cmd"   and a :: "action"   and c' :: "cmd" assumes "ty_cmd (\<Gamma>, loop) c"   and "cfg c (Inl a) c'" shows "ty_la \<Gamma> a"
mpi_less_Im_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "- pi < Im (ln z)"
truncate_up_le: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "x \<le> y" shows "x \<le> truncate_up prec y"
abs: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma>\<^sub>1 :: "(name, value) fmap"   and \<Gamma>\<^sub>2 :: "(name, value) fmap" assumes "fmrel_on_fset (frees (Sabs cs)) vrelated \<Gamma>\<^sub>1 \<Gamma>\<^sub>2"   and "fmrel_on_fset (consts (Sabs cs)) vrelated (fmap_of_list rs) \<Gamma>\<^sub>2" shows "\<turnstile>\<^sub>v Vabs cs \<Gamma>\<^sub>1 \<approx> Vabs cs \<Gamma>\<^sub>2"
vrange_def: fixes D :: "V" shows "vrange D = (\<lambda>r\<in>\<^sub>\<circ>D. set {b. \<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r})"
poly_mult_in_carrier: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "set (R.poly_mult p1 p2) \<subseteq> carrier R"
tell_state_def: fixes tell :: "'i \<Rightarrow> 'h \<Rightarrow> 'h"   and w :: "'i"   and m :: "('g, 'h) stateT" shows "tell_state tell w m = StateT (\<lambda>s. tell w (run_state m s))"
fpxs_nth_1'_1: shows "fpxs_nth 1 0 = (1::'b)"
Orbit_Relation_def: shows "Orbit_Relation = {(x, y). x \<in> G \<and> y \<in> G \<and> (\<exists>\<alpha>\<in>translation ` K. y = \<alpha> x)}"
ldropn_lmap: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b llist" shows "ldropn n (lmap f xs) = lmap f (ldropn n xs)"
fin_f: shows "finite {a. f a \<noteq> (0::'b)}"
carrier_both: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "carrier (t \<otimes>\<otimes> t') = carrier t \<union> carrier t'"
inf_demorgan: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) = - - x \<squnion> - - y"
assms_7: shows "\<not> ([[a b c d]] \<or> [[a b d c]] \<or> [[a c b d]] \<or> [[a c d b]] \<or> [[a d b c]] \<or> [[a d c b]])"
Domain_def: fixes f :: "ZF" shows "HOLZF.Domain f \<equiv> Replacement f (\<lambda>p. if isOpair p then Some (Fst p) else None)"
closure_injective_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear f"   and "inj f" shows "f ` closure S = closure (f ` S)"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P mark_guards f c Q,A"
NoActivityNoMessageLoss: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and Q :: "'a set"   and cfg2 :: "('a, 'c, 'b) configuration"   and p :: "'a"   and m' :: "('a, 'c) message" assumes "asynchronousSystem.qReachable trans sends cfg1 Q cfg2"   and "p \<notin> Q"   and "isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
B: fixes y :: "'b process" assumes "cont g" shows "cont (\<lambda>x. y `;` g x)"
lub_C: fixes S :: "nat \<Rightarrow> C" assumes "chain S" shows "Lub S = Abs_C (\<Squnion>i. Rep_C (S i))"
assms: fixes n :: "nat" shows "ltake (enat n) xs \<sqsubseteq> ltake (enat n) ys"
assms_2: fixes b\<^sub>\<sigma> :: "bool" shows "etp_rsa.HCP_adv etp.\<A> m2 b\<^sub>\<sigma> D \<le> HCP_ad"
nth_shiftr1: fixes w :: "'b word"   and n :: "nat" shows "shiftr1 w !! n = w !! Suc n"
polychain_of_ccw_convex: fixes e :: "real \<times> real \<Rightarrow> real"   and P :: "real \<times> real"   and Q :: "real \<times> real"   and Ps :: "(real \<times> real) list"   and Pc :: "real \<times> real" assumes "e \<in> UNIV \<rightarrow> {0<..<1}"   and "ccw'.sortedP 0 (P # Q # Ps)" shows "list_all (\<lambda>(xi, xj). ccw' xi xj (Pc + (\<Sum>P\<in>set (P # Q # Ps). e P *\<^sub>R P))) (polychain_of Pc (P # Q # Ps))"
emeasure_lborel_box: fixes l :: "'a"   and u :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "emeasure lborel (box l u) = ennreal (prod ((\<bullet>) (u - l)) Basis)"
INTRO_KD_def: fixes uu :: "'h"   and uv :: "'i" shows "INTRO_KD uu uv \<equiv> True"
Q'''Trans: shows "Q''' \<longmapsto> a<x> \<prec> Q''"
pmf_inverse_switch_eqals: fixes a :: "real"   and p :: "'a pmf"   and q :: "'a pmf" assumes "a \<in> {0..1}" shows "mix_pmf a p q = mix_pmf (1 - a) q p"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "option_le P (Some x) None = False"
gt_half_sum: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "(a + b) / ((1::'a) + (1::'a)) < b"
rell_F_compp: fixes L1 :: "'n \<Rightarrow> 'aa \<Rightarrow> bool"   and L1' :: "'aa \<Rightarrow> 'x \<Rightarrow> bool"   and L2 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and L2' :: "'ab \<Rightarrow> 'y \<Rightarrow> bool"   and L3 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and L3' :: "'ac \<Rightarrow> 'z \<Rightarrow> bool" shows "rell_F (L1 OO L1') (L2 OO L2') (L3 OO L3') = rell_F L1 L2 L3 OO rell_F L1' L2' L3'"
approximation_preproc_floatarith_8: fixes x :: "real"   and y :: "real" shows "x / y = x * inverse y"
incl: fixes h :: "'c"   and H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
Ms2_imp: shows "\<forall>M m D. Ms2 M = \<lfloor>(m, D)\<rfloor> \<longrightarrow> P \<turnstile> D \<preceq>\<^sup>* C"
ring_record_simps_5: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
pivotandupdate_check_precond: fixes dir :: "('i, 'a) Direction"   and s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat"   and l\<^sub>i :: "'a" assumes "dir = (if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative)"   and "min_lvar_not_in_bounds s = Some x\<^sub>i"   and "min_rvar_incdec dir s x\<^sub>i = Inr x\<^sub>j"   and "l\<^sub>i = the (LB dir s x\<^sub>i)"   and "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s" shows "\<triangle> (\<T> (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)) \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s) \<and> \<diamond> (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s) \<and> \<nabla> (pivot_and_update x\<^sub>i x\<^sub>j l\<^sub>i s)"
ring_simprules_6: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
prj_simps_4: shows "C.arr f_src.prj\<^sub>0"
bigoI_tendsto: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter" assumes "((\<lambda>x. f x / g x) \<longlongrightarrow> c) F"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'b)" shows "f \<in> O[F](g)"
one_chain_line_integral_point_paths: fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and basis :: "(real \<times> real) set"   and ps :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "finite one_chain"   and "finite basis"   and "\<forall>(k, \<gamma>)\<in>ps. point_path \<gamma>" shows "one_chain_line_integral F basis (one_chain - ps) = one_chain_line_integral F basis one_chain"
closedI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Semilat A r f" shows "closed A f"
fls_X_inv_power_nth: fixes n :: "nat"   and k :: "int" shows "fls_nth (fls_X_inv ^ n) k = (if k = - int n then 1::'a else (0::'a))"
OUTfromChCorrect_data5: shows "OUTfromChCorrect data5"
gb_schema_dummy_connectible: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and data :: "nat \<times> nat \<times> 'd"   and f :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c"   and g :: "('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c" assumes "struct_spec sel ap ab compl"   and "compl_conn compl"   and "dickson_grading d"   and "fst ` set gs \<subseteq> dgrad_p_set d m"   and "is_Groebner_basis (fst ` set gs)"   and "fst ` set bs \<subseteq> dgrad_p_set d m"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "D \<subseteq> (set gs \<union> set bs) \<times> (set gs \<union> set bs)"   and "set ps \<inter>\<^sub>p D = {}"   and "unique_idx (gs @ bs) (snd data)"   and "\<And>p q. \<lbrakk>processed (p, q) (gs @ bs) ps; (p, q) \<notin>\<^sub>p D; fst p \<noteq> 0; fst q \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs)) (fst p) (fst q)"   and "\<And>B a b. \<lbrakk>set gs \<union> set bs \<subseteq> B; fst ` B \<subseteq> dgrad_p_set d m; (a, b) \<in>\<^sub>p D; fst a \<noteq> 0; fst b \<noteq> 0; \<And>x y. \<lbrakk>x \<in> set gs \<union> set bs; y \<in> set gs \<union> set bs; (x, y) \<notin>\<^sub>p D; fst x \<noteq> 0; fst y \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst x) (fst y)\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)"   and "f \<in> set (fst (gb_schema_dummy data D bs ps))"   and "g \<in> set (fst (gb_schema_dummy data D bs ps))"   and "fst f \<noteq> 0"   and "fst g \<noteq> 0" shows "crit_pair_cbelow_on d m (fst ` set (fst (gb_schema_dummy data D bs ps))) (fst f) (fst g)"
pre_def: shows "pre = (\<lambda>(lo, hi, xs). quicksort_pre R h xs0 lo hi xs)"
facets': fixes w :: "'w"   and s :: "'w" assumes "w \<in> W"   and "s \<in> S" shows "dual_order.poset_simplex_map \<P> {w, w + s} \<lhd> dual_order.poset_simplex_map \<P> {w}"
Complete_subset_\<pi>: fixes k :: "nat"   and X :: "item set"   and T :: "(symbol \<times> character list) set" shows "Complete k X \<subseteq> \<pi> k T X"
simps_5: fixes x :: "'d"   and p :: "('e, 'f, 'd) formula" shows "FVF (Exists x p) = FVF p - {Inl x}"
Keys_insert: fixes a :: "'b \<Rightarrow>\<^sub>0 'c"   and A :: "('b \<Rightarrow>\<^sub>0 'c) set" shows "Keys (insert a A) = keys a \<union> Keys A"
T_A_nneg: fixes n :: "nat" shows "0 \<le> T_A n"
that: fixes c :: "'b"   and v :: "'t" assumes "c \<noteq> (0::'b)"   and "lc p = c"   and "lt p = v"   and "p = monomial c v" shows "thesis"
weakenL': fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and F :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
OUTfromChCorrect_data10: shows "OUTfromChCorrect data10"
gen_dfs_eq_assert: shows "gen_dfs = gen_dfs_assert"
assms_3: shows "is_rcoset_replist H (g # gs)"
gauss_sum_nat: fixes n :: "nat" shows "\<Sum> {0..n} = n * Suc n div 2"
distinct_24: fixes x4 :: "key"   and x3 :: "fresh_t" shows "Key x4 \<noteq> Nonce x3"
dirichlet_prod_moebius_commute: fixes f :: "nat \<Rightarrow> 'a" shows "dirichlet_prod f moebius_mu = dirichlet_prod moebius_mu f"
weakPsiCongScopeExtChain: fixes xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<parallel> Q \<doteq> P \<parallel> (\<lparr>\<nu>*xvec\<rparr>Q)"
map_cong: fixes x :: "'a \<Rightarrow> 'b"   and ya :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "x = ya"   and "\<And>z. z \<in> range ya \<Longrightarrow> f z = g z" shows "f \<circ> x = g \<circ> ya"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> (\<BB>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>)"
p_inf_pp: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus (inf x (uminus (uminus y))) = uminus (inf x y)"
tendsto_mono: fixes "open" :: "'c set \<Rightarrow> bool"   and F :: "'d filter"   and F' :: "'d filter"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c" assumes "class.topological_space open"   and "F \<le> F'"   and "filterlim f (topological_space.nhds open l) F'" shows "filterlim f (topological_space.nhds open l) F"
Int_Un_eq_2: fixes S :: "'b set"   and T :: "'b set" shows "S \<inter> T \<union> T = T"
awalk_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "awalk u p v"   and "awalk u' p v'" shows "p \<noteq> [] \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
subsumes_empty_left: fixes C :: "'a literal multiset" shows "subsumes {#} C"
d_delta_lnexp_cf2: fixes x :: "real" shows "((\<lambda>x. ln (exp_cf2 x) - x) has_real_derivative diff_delta_lnexp_cf2 x) (at x)"
ordermap_bij: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "wf r"   and "total_on A r"   and "small A" shows "bij_betw (ordermap A r) A (elts (ordertype A r))"
integrable_continuous_real: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
homotopic_loops_subset: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and t :: "'b set" assumes "homotopic_loops s p q"   and "s \<subseteq> t" shows "homotopic_loops t p q"
\<ll>_ide_simp: fixes a :: "'d" assumes "D.ide a" shows "D.\<ll> a = \<l>\<^sub>D[a]"
absorb1: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "b \<preceq> a" shows "linordered_monoid.max a b = a"
case_1: fixes f1 :: "'a"   and f2 :: "nat \<Rightarrow> 'a" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"