hcju/isabelleps huggingface dataset:
SDG_edge_valid_SDG_node_2: 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 n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n'"
filter_remove1: fixes Q :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
Upd_eqvt: fixes \<pi> :: "perm"   and v :: "Vars.var" shows "\<pi> \<bullet> Upd v = Upd (\<pi> \<bullet> v)"
silent_move_call: fixes f :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and a' :: "'edge"   and ms :: "'node list"   and S :: "'node SDG_node set"   and ms' :: "'node list" assumes "pred (f a) s"   and "transfer (f a) s = s'"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "valid_edge a'"   and "a' \<in> get_return_edges a"   and "(\<exists>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>) \<or> hd ms \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "Ball (set (tl ms)) return_node"   and "length ms = length s"   and "length s' = Suc (length s)"   and "hd ms = sourcenode a"   and "ms' = targetnode a # targetnode a' # tl ms" shows "S,f \<turnstile> (ms,s) -a\<rightarrow>\<^sub>\<tau> (ms',s')"
atLeastatMost_empty_iff2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "({} = subset_mset.atLeastAtMost a b) = (\<not> a \<subseteq># b)"
integrable_\<phi>2: fixes p :: "nat \<times> int"   and q :: "nat \<times> int" shows "integrable lborel (\<lambda>x. \<phi> p x * \<phi> q x)"
fin_set_imp_fin_orbits: assumes "finite M" shows "finite orbits"
insert_DiffM2: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "M \<ominus> x \<oplus> x = M"
iso_cong: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "ChamberComplexIsomorphism X Y f"   and "fun_eq_on g f (\<Union> X)" shows "ChamberComplexIsomorphism X Y g"
mod_add_left_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus (modulo a c) b) c = modulo (plus a b) c"
path_path_shift: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and m :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>" shows "IFC_def.is_path prob (\<pi> \<guillemotleft> m)"
total_on_refl_on_consistent_into: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "total_on A r"   and "refl_on A r"   and "order_consistent r s"   and "x \<in> A"   and "y \<in> A"   and "(x, y) \<in> s" shows "(x, y) \<in> r"
next_good_set_def: shows "next_good_set = (\<lambda>K. (T ^^ good_time K) -` K \<inter> A)"
unitary_monom_eq_var_pow: fixes K :: "'a set"   and n :: "nat" assumes "subring K R" shows "local.monom \<one> n = X [^]\<^bsub>K [X]\<^esub> n"
suf_comp_last_eq: fixes u :: "'b list"   and v :: "'b list" assumes "u \<bowtie>\<^sub>s v"   and "u \<noteq> \<epsilon>"   and "v \<noteq> \<epsilon>" shows "last u = last v"
integral_product_swap: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable (lborel \<Otimes>\<^sub>M lborel)" shows "LINT (x, y)|(lborel \<Otimes>\<^sub>M lborel). f (y, x) = integral\<^sup>L (lborel \<Otimes>\<^sub>M lborel) f"
assert_all_state_def: fixes t :: "(nat \<times> linear_poly) list"   and ats :: "('i \<times> 'a atom) list" shows "assert_all_state t ats \<equiv> check (assert_bound_loop ats (init t))"
interpret_Unop_1: shows "interpret_floatarith Unop xs = unop xs"
plus_of_nat: fixes x :: "real"   and n :: "nat" shows "fds_perzeta (x + real n) = fds_perzeta x"
openin_topspace: fixes U :: "'b topology" shows "openin U (topspace U)"
spant: shows "orthogonal_complement (f -` {0}) = cspan {t}"
eq: fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> V' \<Longrightarrow> \<sigma> x = \<rho> x" shows "expr_sem_rf \<sigma> e' = expr_sem_rf \<rho> e'"
run_writer_def: fixes writerT :: "('b, 'c, 'a) writerT" shows "run_writer writerT = (case writerT of WriterT x \<Rightarrow> x)"
deg_pm_plus: fixes s :: "'c \<Rightarrow>\<^sub>0 'b"   and t :: "'c \<Rightarrow>\<^sub>0 'b" shows "deg_pm (s + t) = deg_pm s + deg_pm t"
isomorphism_expand: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" shows "(f \<circ> g = id \<and> g \<circ> f = id) = ((\<forall>x. f (g x) = x) \<and> (\<forall>x. g (f x) = x))"
cases: fixes x :: "'b sort_key extended \<times> 'c sort_key extended" obtains l :: "'b sort_key extended" and u :: "'c sort_key extended" where "x = (l, u)"
coeff_def: shows "coeff \<equiv> map_fun Rep_linear_poly id fun_coeff"
asymptotics4: shows "\<forall>\<^sub>F x in at_top. 1 \<le> (1 - ln x powr (- e / 2)) * 2"
is_class_xcpt: fixes C :: "char list"   and wf_md :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "C \<in> sys_xcpts"   and "wf_prog wf_md P" shows "is_class P C"
not_eq_order_implies_strict: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a \<noteq> b"   and "b \<preceq>\<^sub>p a" shows "b \<prec>\<^sub>p a"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>(\<tau>.P), \<alpha>\<cdot>Q) \<in> Rel"
residue_of_sum: fixes p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p" shows "(a \<oplus>\<^bsub>padic_int p\<^esub> b) k = a k \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> b k"
index_map_mat_2: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat" shows "dim_row (map_mat f A) = dim_row A"
notin_outstanding_non_volatile_takeWhile_Un_lem': fixes a :: "nat"   and ls :: "('b \<times> 'c \<times> 'd \<times> 'e memref list \<times> 'f \<times> 'g \<times> 'h) list" assumes "a \<notin> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ls. outstanding_refs (Not \<circ> is_volatile) sb)" shows "a \<notin> (\<Union>(uu_, uu_, uu_, sb, uu_, uu_, uu_)\<in>set ls. outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb))"
fold_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) rbt" shows "RBT_Impl.fold f t = List.fold (\<lambda>(x, y). f x y) (entries t)"
stut_before: fixes F :: "'b \<Rightarrow> 'c" shows "STUTINV $F"
gcd_add1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and m :: "'b"   and n :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (plus m n) n = gcd m n"
Step1_5_3b: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> \<box>(ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p)) \<and> \<box>(\<forall>l. $MemInv mm l) \<and> \<box>$ImpInv rmhist p \<and> ImpLive p \<longrightarrow> WF(RNext memCh mm (resbar rmhist) p)_(rtrner memCh!p, resbar rmhist!p)"
simps: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and n :: "nat"   and \<rho> :: "complex mat" shows "denote_while_n_comp M0 M1 DS n \<rho> = M1 * denote_while_n_iter M0 M1 DS n \<rho> * adjoint M1"
assms: shows "Fun c [] \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
valid_empty_edges_iff_exists_offending_flows: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" assumes "SecurityInvariant_withOffendingFlows.sinvar_mono sinvar"   and "wf_graph G"   and "\<not> sinvar G nP" shows "sinvar \<lparr>nodes = nodes G, edges = {}\<rparr> nP = (SecurityInvariant_withOffendingFlows.set_offending_flows sinvar G nP \<noteq> {})"
k1: shows "distinct k1 \<and> fset (fmdom P1) = set k1"
obtain_orbit_list_func: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> orbit f s"   and "\<And>x. x \<notin> orbit f s \<Longrightarrow> f x = x" obtains xs :: "'b list" where "f = list_succ xs" and "set xs = orbit f s" and "distinct xs" and "hd xs = s"
length_ls_insls: fixes l :: "('a \<times> 'b) list list"   and ls :: "('a \<times> 'b) list" assumes "1 < length l" shows "1 < length (ls_insls ls l)"
distinct_29: fixes x51 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and x52 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and x53 :: "((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set"   and x54 :: "acom"   and x61 :: "(((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat) \<times> char list set \<times> (char list \<Rightarrow> char list set)"   and x62 :: "bexp"   and x63 :: "acom" shows "{x51/x52/x53} CONSEQ x54 \<noteq> {x61} WHILE x62 DO x63"
Var_right: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "x \<notin> vars_term t" shows "UNIF1 (subst x t) (add_mset (t, Var x) E) (subst_mset (subst x t) E)"
assms_2: fixes x :: "'a"   and s :: "'s" assumes "x \<in> S s" shows "healthy (wp (a x))"
CD_on_XD_smallest: fixes ds :: "'b set"   and X :: "'a set" assumes "Singleton_for_d.stable_on ds X" shows "Singleton_for_d.CD_on ds (Singleton_for_d.XD_smallest ds X) = X"
i_eq_least: shows "i = (LEAST n. A $ n $ j \<noteq> (0::'a))"
Step1_4_4c: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MemReturn rmCh ires p \<and> $S4 rmhist p \<and> S5 rmhist p$ \<and> unchanged (e p, c p, r p) \<longrightarrow> unchanged (rtrner memCh!p, resbar rmhist!p)"
rev_morph_arg: fixes f :: "'c list \<Rightarrow> 'b list"   and u :: "'c list" shows "rev_morph f u = rev (f (rev u))"
oFix_def: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" shows "oFix F a = oLimit (\<lambda>n. iter n F a)"
step: shows "\<Gamma>\<turnstile> (ccs, css, s) \<rightarrow> (cs', css', t)"
det_row_operation: fixes i :: "'b"   and j :: "'b"   and A :: "(('c, 'b) vec, 'b) vec"   and c :: "'c" assumes "i \<noteq> j" shows "det (\<chi>k. if k = i then row i A + c *s row j A else row k A) = det A"
Gromov_product_at_def: fixes e :: "'a"   and x :: "'a"   and y :: "'a" shows "Gromov_product_at e x y = (dist e x + dist e y - dist x y) / 2"
unique_connecting_path_tl: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and u :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "u \<in> set (w \<leadsto>\<^bsub>G\<^esub> v)"   and "u \<rightarrow>\<^bsub>G\<^esub> w" shows "tl (w \<leadsto>\<^bsub>G\<^esub> v) = u \<leadsto>\<^bsub>G\<^esub> v"
Rep_unat_inverse: fixes x :: "unat" shows "Abs_unat (Rep_unat x) = x"
Pause_neq_ABORT: fixes out :: "'g"   and c :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.Pause out c \<noteq> ABORT"
pred_mono: fixes P :: "'b \<Rightarrow> bool"   and Pa :: "'b \<Rightarrow> bool" assumes "P \<le> Pa" shows "list_all P \<le> list_all Pa"
env: fixes tm :: "taylor_model" assumes "tm \<in> set env" shows "num_params (tm_poly tm) \<le> length I"
finrankI: fixes M :: "'v set"   and n :: "nat" assumes "\<And>ms. \<lbrakk>set ms \<subseteq> M; R_lin_independent ms\<rbrakk> \<Longrightarrow> length ms \<le> n" shows "R_finrank M"
DCaxiom_def: shows "DCaxiom = (([[EvolveODE (OVar vid1) (Pc pid1)]]Pc pid3) \<rightarrow> [[EvolveODE (OVar vid1) (Pc pid1)]]Pc pid2 \<leftrightarrow> ([[EvolveODE (OVar vid1) (Pc pid1 && Pc pid3)]]Pc pid2))"
rewrite_X_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (rewrite_X \<phi>) \<subseteq> atoms_ltln \<phi>"
Composition': fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b"   and \<Psi>''' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'"   and "\<Psi>'' \<simeq> \<Psi>'''" shows "\<Psi> \<otimes> \<Psi>'' \<simeq> \<Psi>' \<otimes> \<Psi>'''"
inf_over_nats: fixes c :: "real"   and a :: "real" assumes "c < 0" shows "(INF x. ereal (a + real x * c)) = - \<infinity>"
new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'ai" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
GuardFault: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and f :: "'f"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
a_repr_independence': fixes G :: "('e, 'f) ring_scheme"   and y :: "'e"   and H :: "'e set"   and x :: "'e" assumes "abelian_group G"   and "y \<in> H #>\<^bsub>add_monoid G\<^esub> x"   and "x \<in> carrier G"   and "subgroup H (add_monoid G)" shows "H #>\<^bsub>add_monoid G\<^esub> x = H #>\<^bsub>add_monoid G\<^esub> y"
adjoined_in_adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "a \<in> adjoin (G\<lparr>carrier := adjoin G H a\<rparr>) H a"
image_chamber_system: shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> domain.\<C>"
stirling_limit_aux2: fixes z :: "complex" shows "((\<lambda>y. complex_of_real y * Ln (1 + z / complex_of_real y)) \<longlongrightarrow> z) at_top"
assms_2: fixes x :: "real" assumes "x \<in> {a..b}" shows "l \<le> f' x"
exhaust: fixes y :: "'b infty" obtains "y = Infty"   | x2 :: "'b" where "y = Num x2"
simps_2: fixes p :: "char list"   and ts :: "hterm list" shows "flit_of_hlit (Neg p ts) = Neg p (fterms_of_hterms ts)"
eq_\<I>_gpv_1: shows "\<I> \<turnstile>g gpv \<surd>"
composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> A"   and "b \<in> A" shows "multA a b \<in> A"
2: shows "{i. enat i < llength w \<and> (w \<bar>\<bar> iterates Suc 0) ?! i \<in> UNIV \<times> s} = {i \<in> s. enat i < llength w}"
intros_12: fixes \<tau> :: "'a OCL_Types.type" shows "Bag \<tau> \<sqsubset> Collection \<tau>"
hiding_game_ind_cpa_def: shows "hiding_game_ind_cpa \<equiv> \<lambda>uu. case uu of (\<A>1, \<A>2) \<Rightarrow> TRY key_gen \<bind> (\<lambda>(ck, vk). \<A>1 vk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_spmf (valid_msg m0 \<and> valid_msg m1) \<bind> (\<lambda>_. coin_spmf \<bind> (\<lambda>b. commit ck (if b then m0 else m1) \<bind> (\<lambda>(c, d). \<A>2 c \<sigma> \<bind> (\<lambda>b'. return_spmf (b' = b))))))) ELSE coin_spmf"
cosh_real_nonpos_le_iff: fixes x :: "real"   and y :: "real" assumes "x \<le> 0"   and "y \<le> 0" shows "(cosh x \<le> cosh y) = (y \<le> x)"
u_simps_5: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>" shows "partial_magma.cod V u = u"
rep_eq: fixes x :: "'c \<Longrightarrow> 'b" shows "\<lbrakk>x\<rbrakk>\<^sub>e = get\<^bsub>x\<^esub>"
eq_fold: fixes A :: "'a set" shows "card A = Finite_Set.fold (\<lambda>_. Suc) 0 A"
goodsteinC_def': fixes c :: "nat"   and n :: "C" shows "goodsteinC c n = goodsteinO c (C2O n)"
restrict_ext: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "restrict f A = restrict g A"
infinite_le_lepoll: fixes A :: "'a set" shows "infinite A = (UNIV \<lesssim> A)"
Id_O_R: fixes R :: "('b \<times> 'c) set" shows "Id O R = R"
rconjby_uminus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.group_add minus plus zero uminus" shows "plus (plus (uminus x) (uminus y)) x = uminus (plus (plus (uminus x) y) x)"
elims_3: fixes x :: "'b"   and xa :: "'b \<times> 'b" assumes "\<not> elem x xa" obtains xb :: "'b" and l :: "'b" and u :: "'b" where "x = xb" and "xa = (l, u)" and "\<not> (l < xb \<and> xb < u)"
covering_space_lift_unique_gen: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and g1 :: "'e \<Rightarrow> 'c"   and a :: "'e"   and g2 :: "'e \<Rightarrow> 'c"   and T :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and U :: "'e set"   and x :: "'e" assumes "covering_space c p S"   and "g1 a = g2 a"   and "continuous_on T f"   and "f ` T \<subseteq> S"   and "continuous_on T g1"   and "g1 ` T \<subseteq> c"   and "\<And>x. x \<in> T \<Longrightarrow> f x = p (g1 x)"   and "continuous_on T g2"   and "g2 ` T \<subseteq> c"   and "\<And>x. x \<in> T \<Longrightarrow> f x = p (g2 x)"   and "U \<in> components T"   and "a \<in> U"   and "x \<in> U" shows "g1 x = g2 x"
E: shows "\<not> 0 < index key x (length ns) mi ma"
f_def: fixes s0 :: "'a list"   and rs :: "'a list" shows "f s0 rs \<equiv> real (T_on Aon s0 rs)"
P5: shows "X' Out C' Q \<and> Cong X' C' X C"
\<tau>_continuous: fixes T :: "real set" shows "continuous_on T \<tau>"
eq_zero_or_degree_less: fixes p :: "'a poly"   and n :: "nat" assumes "degree p \<le> n"   and "coeff p n = (0::'a)" shows "p = 0 \<or> degree p < n"
ide_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.ide f = (f = J.Zero \<or> f = J.One)"
x_y_prop: fixes x :: "'a"   and y :: "'a" shows "1\<^sub>\<sigma> \<sqinter> (x \<sqinter> nc) \<cdot> y = 1\<^sub>\<sigma> \<sqinter> x \<cdot> y"
acyclic_def: fixes r :: "('ap \<times> 'ap) set" shows "acyclic r = (\<forall>x. (x, x) \<notin> r\<^sup>+)"
zero_in_PPs: fixes X :: "'a set" shows "0 \<in> .[X]"
child_sc: shows "h \<turnstile> get_scdom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr) \<rightarrow>\<^sub>r child_sc"
length_selsort: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (linorder_list0.selsort le xs) = length xs"
assms_2: shows "strictly_subsumes (fst Dl) (fst Cl) \<or> subsumes (fst Dl) (fst Cl) \<and> snd Dl \<sqsubset>l snd Cl"
linear_inj_iff_eq_0: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "inj f = (\<forall>x. f x = 0 \<longrightarrow> x = 0)"
coeff_sym_mpoly: fixes X :: "nat set"   and k :: "nat"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (sym_mpoly X k) mon = (if finite X \<and> (\<exists>Y\<subseteq>X. card Y = k \<and> mon = monom_of_set Y) then 1::'b else (0::'b))"
C: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
suffixE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>s ys" obtains zs :: "'b list" where "ys = zs \<cdot> xs"
continuous_on_ego_other2: fixes T :: "real set" shows "continuous_on T ego_other2"
product_complement_character: fixes x :: "i"   and y :: "i" assumes "CMC"   and "\<exists>w. P w x \<and> \<not> O w y" shows "\<forall>w. P w (x \<otimes> \<midarrow> y) = (P w x \<and> \<not> O w y)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires = (\<exists>rn. l = ires \<and> r = [Bk, Bk] @ <lm> @ Bk \<up> rn)"
exhaust_sel: fixes complex :: "complex" obtains "complex = Complex (Re complex) (Im complex)"
partition_on_le_set_elements: fixes A :: "'c set"   and P :: "'c set set" assumes "finite A"   and "partition_on A P" shows "card P \<le> card A"
triangle_in_hom_2: shows "\<guillemotleft>(f\<^sup>* \<star> \<rho>.\<epsilon>) \<cdot> \<a>[f\<^sup>*, f, f\<^sup>*] \<cdot> (\<rho>.\<eta> \<star> f\<^sup>*) : trg f\<^sup>* \<star> f\<^sup>* \<Rightarrow> f\<^sup>* \<star> src f\<^sup>*\<guillemotright>"
prob_space_P: fixes x :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<i} M)" shows "prob_space (P i x)"
p_fun_inj: fixes x :: "'a" shows "is_p_fun x = is_inj (x\<^sup>\<smile>)"
indep_setsD: fixes F :: "'b \<Rightarrow> 'a set set"   and I :: "'b set"   and J :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "indep_sets F I"   and "J \<subseteq> I"   and "J \<noteq> {}"   and "finite J"   and "\<forall>j\<in>J. A j \<in> F j" shows "prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))"
Ri_effective: fixes \<gamma> :: "'a inference"   and N :: "'a literal multiset set" assumes "\<gamma> \<in> gr.ord_\<Gamma>"   and "concl_of \<gamma> \<in> N \<union> sr.Rf N" shows "\<gamma> \<in> sr.Ri N"
EF_step_step: fixes x :: "'a"   and y :: "'a"   and f :: "'a set" assumes "x \<rightarrow>\<^sub>i y"   and "y \<in> EF f" shows "x \<in> EF f"
PCallE: fixes \<pi> :: "char list \<Rightarrow> com option"   and p :: "char list"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (PCall p, s) \<Rightarrow> t" obtains c :: "com" where "\<pi> p = Some c" and "\<pi>: (c, s) \<Rightarrow> t"
iso_is_retraction: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.retraction f"
star_def: fixes A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" shows "RegExp2NA.star A = or epsilon (RegExp2NA.plus A)"
rPar1: fixes \<alpha> :: "act"   and P' :: "ccs"   and \<C> :: "'a" assumes "P \<longmapsto>\<alpha> \<prec> P'" shows "Prop \<C> (P \<parallel> !P) \<alpha> (P' \<parallel> !P)"
matrix_mult_sum: fixes A :: "(('d, 'f) vec, 'e) vec"   and x :: "('d, 'f) vec" shows "A *v x = (\<Sum>i\<in>UNIV. x $ i *s column i A)"
map_1: fixes f :: "'f \<Rightarrow> 'e" shows "map f [] = []"