hcju/isabelleps huggingface dataset:
list_slice2_def: fixes xs :: "'b list"   and k :: "nat" shows "list_slice2 xs k \<equiv> list_slice xs k @ (if length xs mod k = 0 then [] else [xs \<up> (length xs div k * k)])"
surjective_pairing: fixes t :: "'r \<times> 's" shows "t = (prod.fst t, prod.snd t)"
comm_ring_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "comm_ring_hom hom \<equiv> ring_hom hom"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "inv.B.dom (\<sigma> f) = F (A.dom f)"
A: shows "Esigma 69615 = Esigma 87633"
h1_def: shows "h1 = pr_conv_2_to_1 h"
extend_stable: fixes F :: "'a program"   and A :: "'a set" shows "(extend h F \<in> stable (extend_set h A)) = (F \<in> stable A)"
elimVar_atom_convert: fixes var :: "nat"   and L :: "atom list"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and L' :: "atomUni list"   and L2 :: "atom list"   and L2' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "convert_atom_list var L2 (xs' @ x # xs) = Some L2'"   and "length xs' = var" shows "(\<exists>f\<in>set L. eval (elimVar var L2 [] f) (xs' @ x # xs)) = (\<exists>f\<in>set L'. evalUni (elimVarUni_atom L2' f) x)"
delete_edges_edges_empty: fixes G :: "'a graph" shows "delete_edges G (edges G) = G\<lparr>edges := {}\<rparr>"
valid_def: fixes X :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<upsilon> X \<equiv> \<lambda>\<tau>. if X \<tau> = \<bottom> \<tau> then false \<tau> else true \<tau>"
borel_compact: fixes A :: "'c set" assumes "compact A" shows "A \<in> sets borel"
SucFbd_limit: fixes x1 :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set"   and x2 :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set" assumes "x1 \<in> Field SucFbd \<and> x2 \<in> Field SucFbd" shows "\<exists>y\<in>Field SucFbd. (x1 \<noteq> y \<and> (x1, y) \<in> SucFbd) \<and> x2 \<noteq> y \<and> (x2, y) \<in> SucFbd"
setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
absorb_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](g)" shows "\<Theta>[F](\<lambda>x. f x + g x) = \<Theta>[F](g)"
that: fixes T :: "'b set" assumes "subspace T"   and "T \<subseteq> span S"   and "dim T = n" shows "thesis"
discrete_subset_disconnected: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" shows "f ` S \<subseteq> {y. connected_component_set (f ` S) y = {y}}"
Rep_inverse: fixes x :: "'c stream" shows "stream_of_llist (llist_of_stream x) = x"
simps_12: fixes e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and A :: "char list set option" shows "\<D> (e\<^sub>1;; e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
path_component_of_refl: fixes X :: "'c topology"   and x :: "'c" shows "path_component_of X x x = (x \<in> topspace X)"
path_component_in_topspace: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" assumes "path_component_of X x y" shows "x \<in> topspace X \<and> y \<in> topspace X"
TermP_Eats: fixes t1 :: "tm"   and t2 :: "tm" shows "{TermP t1, TermP t2} \<turnstile> TermP (Q_Eats t1 t2)"
add_increasing: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "(0::'a) \<le> a"   and "b \<le> c" shows "b \<le> a + c"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_before_double ires rs (l, r) = (\<exists>ln rn. l = Bk # Bk # Bk \<up> ln @ Oc # ires \<and> r = Oc \<up> Suc (Suc rs) @ Bk \<up> rn)"
drop_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list" shows "drop n (take m xs) = take (m - n) (drop n xs)"
axioms_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "two_composable_arrows_of_spans C prj0 prj1 \<mu> \<nu>" shows "arrow_of_spans C \<nu>"
axioms_2: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and C :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and F :: "'d \<Rightarrow> 'e"   and G :: "'e \<Rightarrow> 'f" assumes "composite_functor A B C F G" shows "functor B C G"
Cons_1: fixes v :: "'item list"   and w :: "'item list"   and thesis :: "bool" assumes "\<And>p r s q. \<lbrakk>t' =\<^sub>F p @ r; u =\<^sub>F s @ q; v =\<^sub>F p @ s; w =\<^sub>F r @ q; Ind (set r) (set s)\<rbrakk> \<Longrightarrow> thesis"   and "t' @ u =\<^sub>F v @ w" shows "thesis"
clop_ext: fixes x :: "'a" shows "x \<le> cl_op x"
get_M_Mdocument_preserved3: fixes document_ptr :: "(_) document_ptr"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and setter :: "('l \<Rightarrow> 'l) \<Rightarrow> ('c, 'a, 'b, 'j option) RDocument_scheme \<Rightarrow> ('c, 'a, 'b, 'j option) RDocument_scheme"   and v :: "'l"   and h' :: "(_) heap"   and getter :: "(_) RObject_ext \<Rightarrow> 'm" assumes "cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 document_ptr \<noteq> object_ptr"   and "h \<turnstile> put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr setter v \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t object_ptr getter) h h'"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "(f a = f b) = (a \<ominus>\<^bsub>R\<^esub> b \<in> I)"
truth_provable: fixes H :: "fm set" shows "H \<turnstile> Neg Fls"
start_conc: fixes L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" shows "start (RegExp2NAe.conc L R) = True # start L"
mask_eq_0_eq_x: fixes x :: "'b word"   and w :: "'b word" shows "(x AND w = 0) = (x AND NOT w = x)"
overrides_commonD: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides old" shows "declclass new \<noteq> Object \<and> \<not> is_static new \<and> \<not> is_static old \<and> accmodi new \<noteq> Private \<and> msig new = msig old \<and> G\<turnstile>declclass new\<prec>\<^sub>C declclass old \<and> G\<turnstile>Method new declared_in declclass new \<and> G\<turnstile>Method old declared_in declclass old"
successively_append_iff: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'g list" shows "successively P (xs @ ys) = (successively P xs \<and> successively P ys \<and> (xs = [] \<or> ys = [] \<or> P (last xs) (hd ys)))"
continuous_compose_quotient_map_eq: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "quotient_map X X' f" shows "continuous_map X X'' (g \<circ> f) = continuous_map X' X'' g"
topological_basis_prod: fixes A :: "'b set set"   and B :: "'c set set" assumes "topological_basis A"   and "topological_basis B" shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
SC_siso_imp_SC_WbisT: fixes c :: "('test, 'atom) com" assumes "SC_siso c" shows "SC_WbisT c"
nonpos_Reals_divide_of_nat_iff: fixes a :: "'a"   and n :: "nat" shows "(a / of_nat n \<in> \<real>\<^sub>\<le>\<^sub>0) = (a \<in> \<real>\<^sub>\<le>\<^sub>0 \<or> n = 0)"
simps_5: fixes p :: "'a pfm"   and q :: "'a pfm" shows "reduce (p \<^bold>\<longrightarrow>\<^sub>! q) = (reduce p \<^bold>\<longrightarrow>\<^sub>! reduce q)"
p1: fixes k :: "nat" assumes "k < R" shows "exM1 * tensor_P (proj_psi_l k) (proj_k k) * exM1 = tensor_P (proj_psi_l k) (proj_k k)"
inj_on_apply_range: fixes f :: "'a perm"   and a :: "'a" shows "inj_on (\<lambda>m. (f ^ m) \<langle>$\<rangle> a) {..<order f a}"
INS: shows "GEN_OP ins2 insert (Rk' \<rightarrow> \<langle>Rk'\<rangle>Rs2 \<rightarrow> \<langle>Rk'\<rangle>Rs2)"
relcompp_assoc: fixes r :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(r OO s) OO t = r OO s OO t"
ldrop_enat: fixes n :: "nat"   and xs :: "'a llist" shows "ldrop (enat n) xs = ldropn n xs"
fin: fixes x :: "real"   and Q :: "nat \<Rightarrow> bool" shows "finite {p. prime p \<and> real p \<le> x \<and> Q p}"
propex\<^sub>3: fixes F :: "\<Pi>\<^sub>3" shows "\<exists>r. Some r = d\<^sub>3 F"
isomorphic_relative_homology_groups_Euclidean_complements: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and T :: "(nat \<Rightarrow> real) set"   and p :: "int" assumes "closedin (Euclidean_space n) S"   and "closedin (Euclidean_space n) T"   and "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T" shows "relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - S) \<cong> relative_homology_group p (Euclidean_space n) (topspace (Euclidean_space n) - T)"
not_coprimeI: 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c a"   and "dvd.dvd times c b"   and "\<not> dvd.dvd times c one" shows "\<not> algebraic_semidom.coprime times one a b"
lc2aux: shows "{l. length l = 2 \<and> valid l} = {[B, B]}"
all_forward: fixes P' :: "'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<forall>x. P x"
F2set3_Lev1: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xe :: "'d"   and xb :: "'e"   and x :: "'e" assumes "xf \<in> Lev1 s1 s2 n xe"   and "rv1 s1 s2 xf xe = Inr xb"   and "x \<in> F2set3 (s2 xb)" shows "xf @ [Inr (tobd_F23 s2 xb x)] \<in> Lev1 s1 s2 (Suc n) xe"
word_bw_same_1: fixes x :: "'a word" shows "x AND x = x"
INT_D: fixes b :: "'e"   and B :: "'f \<Rightarrow> 'e set"   and A :: "'f set"   and a :: "'f" assumes "b \<in> \<Inter> (B ` A)"   and "a \<in> A" shows "b \<in> B a"
face_distance_def: fixes F :: "'a set"   and D :: "'a set" shows "domain.face_distance F D \<equiv> domain.chamber_distance (domain.closest_supchamber F D) D"
power_even_eq: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (2 * n) = power.power one times (power.power one times a n) 2"
lcm_mono: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b d" shows "dvd.dvd times (lcm a b) (lcm c d)"
latestAct_less: fixes n' :: "nat"   and n :: "nat"   and nid :: "'nid"   and t :: "nat \<Rightarrow> cnf" assumes "latestAct_cond nid t n n'" shows "n' \<le> \<langle>nid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>"
update_unalive_val: fixes x :: "Value"   and s :: "Store"   and l :: "Location" assumes "\<not> alive x s" shows "s\<langle>l := x\<rangle> = s"
updEnv_preserves_good: fixes X :: "('e, 'f, 'varSort, 'var, 'g) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm set option"   and x :: "'var"   and xs :: "'varSort" assumes "good X"   and "goodEnv rho" shows "goodEnv (rho [x \<leftarrow> X]_xs)"
cSup_singleton: 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 x :: "'b::type" assumes "class.conditionally_complete_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)" shows "(Sup::'b::type set \<Rightarrow> 'b::type) {x::'b::type} = x"
opt_mru_guard_def: fixes mru_votes :: "process \<Rightarrow> (nat \<times> val) option"   and Q :: "process set"   and v :: "val" shows "opt_mru_guard mru_votes Q v \<equiv> Q \<in> Quorum \<and> (let mru = opt_mru_vote mru_votes Q in mru = None \<or> (\<exists>r. mru = Some (r, v)))"
closed_subset_contains_Inf: fixes C :: "real set"   and A :: "real set" assumes "closed C"   and "A \<subseteq> C"   and "A \<noteq> {}"   and "bdd_below A" shows "Inf A \<in> C"
tagged_fun_relD_none: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and A :: "('c \<times> 'e) set"   and B :: "('d \<times> 'f) set"   and x :: "'c"   and x' :: "'e" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
annil_def: fixes x :: "'a" shows "annil x = (\<forall>y. D x y \<and> x \<oplus> y = x)"
Says_Server_not_range: fixes A :: "agent"   and B :: "agent"   and K :: "nat"   and na :: "msg"   and nb :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Server A \<lbrace>Crypt (shrK A) \<lbrace>Agent B, Key K, na, nb\<rbrace>, X\<rbrace> \<in> set evs"   and "evs \<in> yahalom" shows "K \<notin> range shrK"
all_2: shows "\<forall>x\<in>X'. h x = h g'"
Collect_all_eq: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "{x. \<forall>y. P x y} = (\<Inter>y. {x. P x y})"
bounded_induct: fixes r :: "('c \<times> 'c) set"   and I :: "'c set"   and F :: "'d program"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set" assumes "wf r"   and "\<forall>m\<in>I. F \<in> A \<inter> f -` {m} \<longmapsto> A \<inter> f -` r\<inverse> `` {m} \<union> B" shows "F \<in> A \<longmapsto> A - f -` I \<union> B"
paths_Card_above: fixes p :: "Vars.var list"   and ps :: "Vars.var list set" assumes "p \<in> ps" shows "pathCard p \<sqsubseteq> pathsCard ps"
coerce_cfun': shows "COERCE('a \<rightarrow> 'b, 'c \<rightarrow> 'd) = cfun_map\<cdot>COERCE('c, 'a)\<cdot>COERCE('b, 'd)"
inf_not_lt: fixes x :: "'b DBMEntry" shows "(\<infinity> \<prec> x) = False"
sharing_consistent_all_shared: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "sharing_consistent \<S> \<O> sb" shows "all_shared sb \<subseteq> dom \<S> \<union> \<O>"
FunctorCodDom_2: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "cod\<^bsub>CatCod F\<^esub> (F ## f) = F @@ cod\<^bsub>CatDom F\<^esub> f"
subset_antisym: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
intros: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and DA :: "'a literal multiset"   and As :: "'a list"   and \<sigma> :: "'s" assumes "S DA = negs (mset As) \<or> S DA = {#} \<and> length As = 1 \<and> maximal_wrt (As ! 0 \<cdot>a \<sigma>) (DA \<cdot> \<sigma>)" shows "eligible S \<sigma> As DA"
assms_3: fixes y :: "'a"   and ys :: "[:'a:]" assumes "y \<noteq> \<bottom>"   and "ys \<noteq> \<bottom>"   and "xs = ys :@ [:y:]" shows "P"
exec_step_ind_New_Done: fixes sh :: "sheap"   and C :: "char list"   and obj :: "sfields"   and h :: "heap"   and a :: "nat"   and ics :: "init_call_status"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "sh C = \<lfloor>(obj, Done)\<rfloor>"   and "new_Addr h = \<lfloor>a\<rfloor>"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (New C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h(a \<mapsto> blank P C), (Addr a # stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
assms: shows "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
lemma_2_5_12_b: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c l\<rightarrow> a \<le> c l\<rightarrow> b"
deriv_const: fixes c :: "'b" shows "deriv (\<lambda>w. c) = (\<lambda>z. 0::'b)"
VLambda_set_def: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "VLambda A f = set {\<langle>a, f a\<rangle> |a. a \<in>\<^sub>\<circ> A}"
diff_left_mono: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a" shows "c - a \<le> c - b"
IH: fixes I :: "'a \<Rightarrow> bool option" shows "length (trace_parallel_plan_strips I \<pi>) \<le> length \<pi> + 1"
mono_trace_hide: fixes a :: "'a event list"   and b :: "'a event list"   and A :: "'a set" assumes "a \<le> b" shows "trace_hide a (ev ` A) \<le> trace_hide b (ev ` A)"
add: fixes x :: "real"   and y :: "real"   and xa :: "'c" shows "(x + y) *\<^sub>R xa = x *\<^sub>R xa + y *\<^sub>R xa"
S_def: shows "S \<equiv> \<lambda>n. bernoulli (2 * Suc N) / real n ^ (2 * N + 3)"
f_def: shows "f = (\<lambda>p a. if a \<in> A then b_to_a (p (a_to_b a)) else a)"
compile_env_entry_def: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'m \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and x :: "'m \<times> ('b, 'n, 'fun, 'd, 'e) Inca.instr fundef" shows "compile_env_entry \<A> \<O> x = map_result id (Pair (fst x)) (compile_fundef \<A> (\<O> (fst x)) (snd x))"
bind_gpv_sel: fixes r :: "('h, 'f, 'g) gpv"   and f :: "'h \<Rightarrow> ('e, 'f, 'g) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> the_gpv (f x) | IO out c \<Rightarrow> return_spmf (IO out (\<lambda>input. c input \<bind> f)))"
hyps: assumes "Subobjs P C (Cs @ C' # rev Ds)" shows "\<forall>C''\<in>set Ds. (C', C'') \<in> (subcls1 P)\<^sup>+"
in_dom: fixes i :: "'a" assumes "i \<in> dom" shows "Order_Predicates.preorder_on carrier (R i)"
GHOST_def: shows "GHOST \<equiv> \<lambda>x. x"
AG_mono: fixes p :: "'b set"   and q :: "'b set" assumes "p \<subseteq> q" shows "\<^bold>A\<^bold>G p \<subseteq> \<^bold>A\<^bold>G q"
group_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "Group.group_axioms G \<equiv> carrier G \<subseteq> Units G"
image_cong: fixes M :: "'b set"   and N :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
finite_cmax: fixes r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" assumes "Card_order r"   and "Card_order s" shows "finite (Field (cmax r s)) = (finite (Field r) \<and> finite (Field s))"
has_codomain_iff_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "(partial_magma.codomains C f \<noteq> {}) = partial_magma.arr C f"
th1: shows "s @ t setinterleaves ((ta, u), A) \<longrightarrow> [] \<le> ta \<and> u1 \<le> u"
LIMSEQ_imp_Cauchy: fixes "open" :: "'a set \<Rightarrow> bool"   and uniformity :: "('a \<times> 'a) filter"   and X :: "nat \<Rightarrow> 'a"   and x :: "'a" assumes "class.uniform_space open uniformity"   and "filterlim X (topological_space.nhds open x) sequentially" shows "uniform_space.Cauchy uniformity X"
iFROM_pred_insert_conv: fixes n :: "nat" shows "insert (n - Suc 0) [n\<dots>] = [n - Suc 0\<dots>]"
fresh_at_path_def: fixes t :: "('form, 'rule, 'a, 'b) itnode rose_tree"   and "is" :: "nat list" shows "fresh_at_path t is = \<Union> (fresh_at t ` set (prefixes is))"
xtrans_5: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x"   and "x \<le> y" shows "x = y"
complex_is_Real_iff: fixes z :: "complex" shows "(z \<in> \<real>) = (Im z = 0)"
restrict0_add: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "restrict0 A (f + g) = restrict0 A f + restrict0 A g"
prod_filter_principal_singleton: fixes x :: "'d"   and F :: "'e filter" shows "principal {x} \<times>\<^sub>F F = filtermap (Pair x) F"
intros: fixes r :: "('d, 'e) ta_rule"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d set" assumes "r \<in> \<delta>"   and "set (rhsq r) \<subseteq> Q" shows "lhs r \<in> bacc_step \<delta> Q"
rep_eq: fixes x :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and xb :: "('d, 'f) \<I>"   and xc :: "('e, 'g) \<I>" shows "rel_\<I> x xa xb xc = (rel_set x {out. Rep_\<I> xb out \<noteq> {}} {out. Rep_\<I> xc out \<noteq> {}} \<and> rel_fun x (rel_set xa) (Rep_\<I> xb) (Rep_\<I> xc))"
PCF_lr_POR_delta: shows "PCF_lr (unlr POR.delta)"
lfp_ordinal_induct: fixes f :: "'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "mono f"   and "\<And>S. \<lbrakk>P S; S \<le> lfp f\<rbrakk> \<Longrightarrow> P (f S)"   and "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)" shows "P (lfp f)"
not: fixes vs :: "'b list"   and bs :: "'b list"   and p :: "fm" shows "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
conforms_def: fixes xs :: "state"   and E :: "prog \<times> lenv" shows "xs\<Colon>\<preceq>E = (let (G, L) = E; s = snd xs; l = locals s in (\<forall>r. ! obj:globs s r: G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r) \<and> G,s\<turnstile>l[\<sim>\<Colon>\<preceq>]L \<and> (\<forall>a. fst xs = Some (Xcpt (Loc a)) \<longrightarrow> G,s\<turnstile>Addr a\<Colon>\<preceq>Class (SXcpt Throwable)) \<and> (fst xs = Some (Jump Ret) \<longrightarrow> l Result \<noteq> None))"
subst_fm_trans_commute_eq: fixes du :: "dbtm"   and u :: "tm"   and i :: "name"   and A :: "fm" assumes "du = trans_tm [] u" shows "subst_dbfm du i (trans_fm [] A) = trans_fm [] (A(i::=u))"
arr_iff_in_hom: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.arr f = Maps.in_hom f (Maps.dom f) (Maps.cod f)"
tendsto_powr_complex': fixes a :: "complex"   and b :: "complex"   and f :: "'a \<Rightarrow> complex"   and F :: "'a filter"   and g :: "'a \<Rightarrow> complex" assumes "a \<notin> \<real>\<^sub>\<le>\<^sub>0 \<or> a = 0 \<and> 0 < Re b"   and "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
finfun_const_False_conv_bot: shows "finfun_apply (finfun_const False) = bot"
IH: shows "eval r_leap [t, i, x] \<down>= encode_config (iterate t step ([(f, Partial_Recursive.list_decode x, [])], None))"
higher_differentiable_on_compose: fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "higher_differentiable_on T f n"   and "higher_differentiable_on S g n"   and "g ` S \<subseteq> T"   and "open S"   and "open T" shows "higher_differentiable_on S (f \<circ> g) n"
Unifier_comp': fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and u :: "('c, 'd) Term.term" assumes "Unifier \<delta> (t \<cdot> \<theta>) (u \<cdot> \<theta>)" shows "Unifier (\<theta> \<circ>\<^sub>s \<delta>) t u"
f_carrier: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs" shows "fs ! i \<in> carrier_vec n"
FinGroupRepresentation_axioms: shows "FinGroupRepresentation G (\<cdot>) V"