hcju/isabelleps huggingface dataset:
strip_3: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> P x" shows "\<forall>x\<in>A. P x"
fst_eqvt: fixes p :: "perm"   and x :: "'c \<times> 'd" shows "p \<bullet> fst x = fst (p \<bullet> x)"
d_sub_id_ax: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d x \<subseteq> 1\<^sub>\<sigma>"
Crypt_valid_pubEK: fixes i :: "nat"   and EKi :: "nat"   and evs :: "event list" assumes "Crypt (priSK RCA) \<lbrace>Agent (CA i), Key EKi, onlyEnc\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_mr" shows "EKi = pubEK (CA i)"
awalk_conv: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "G.awalk u p v = (set (pawalk_verts u p) \<subseteq> pverts G \<and> set p \<subseteq> parcs G \<and> hd (pawalk_verts u p) = u \<and> last (pawalk_verts u p) = v \<and> pcas u p v)"
pfa_delete_def: shows "pfa_delete \<equiv> \<lambda>(a, n) i. array_shl a (i + 1) 1 \<bind> (\<lambda>_. return (a, n - (1::'c)))"
total_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "total_on A r = (\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r)"
neq_Leaf_iff: fixes t :: "'a tree" shows "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
both_comm: fixes t :: "'b ttree"   and t' :: "'b ttree" shows "t \<otimes>\<otimes> t' = t' \<otimes>\<otimes> t"
ipoly_of_real: fixes p :: "int poly"   and x :: "real" shows "ipoly p (of_real x) = of_real (poly (real_of_int_poly p) x)"
simps_6: fixes a :: "Cooper.num" shows "bound0 (Ge a) = numbound0 a"
euler_imp_connected: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.euler_trail G u p v" shows "Digraph_Component.connected G"
\<tau>move2_LAssD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (V:=e) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
to_n: fixes k :: "nat" shows "real (n - 1 choose k) \<le> real (n - 1 choose (n - 1) div 2)"
4: shows "[m] \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> [m] \<langle>g\<rangle> [l] \<and> ms \<langle>g\<rangle> [l] \<odot> [l] \<langle>f\<rangle> [l] \<preceq> ms \<langle>g\<rangle> [l]"
sup_continuous_const: fixes c :: "'e" shows "sup_continuous (\<lambda>x. c)"
limit2: shows "((\<lambda>x. sqrt (4 / x - 1) * x) \<longlongrightarrow> 0) (at_right 0)"
sums_infsetsum_nat: fixes f :: "nat \<Rightarrow> 'd"   and A :: "nat set" assumes "f abs_summable_on A" shows "(\<lambda>n. if n \<in> A then f n else (0::'d)) sums infsetsum f A"
effect_bindE: fixes f :: "'b Heap"   and g :: "'b \<Rightarrow> 'a Heap"   and h :: "heap"   and h'' :: "heap"   and r' :: "'a" assumes "effect (f \<bind> g) h h'' r'" obtains h' :: "heap" and r :: "'b" where "effect f h h' r" and "effect (g r) h' h'' r'"
value_array_inhabitants: fixes typeId :: "Arraytype"   and P :: "Value \<Rightarrow> bool" shows "(\<forall>x. \<tau> x = ArrT typeId \<longrightarrow> P x) = (\<forall>a. P (arrV typeId a))"
aux_lemma: fixes p1a :: "'a list"   and p2a :: "'a list"   and n :: "nat" assumes "set p1a \<subseteq> carrier R"   and "set p2a \<subseteq> carrier R" shows "poly_add p1a p2a = poly_add (replicate n \<zero> @ p1a) p2a"
morphism_def: fixes f :: "'b list \<Rightarrow> 'c list" shows "morphism f \<equiv> \<forall>u v. f (u \<cdot> v) = f u \<cdot> f v"
free_tv_TVar: fixes m :: "nat" shows "free_tv (TVar m) = {m}"
mult_sm_ex_id1: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "\<exists>k. k \<noteq> 0 \<and> k *\<^sub>s\<^sub>m M = M"
card_of_cong: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r =o r'" shows "|Field r| =o |Field r'|"
is_volatile_simps_3: fixes p :: "'b"   and p' :: "'b"   and "is" :: "instr list" shows "is_volatile (Prog\<^sub>s\<^sub>b p p' is) = False"
has_extend_witness: fixes n :: "nat"   and S :: "'a set"   and xs :: "'a list" assumes "has n S"   and "set xs \<subseteq> S"   and "length xs < n" shows "set xs \<subset> S"
con_comp_sinks_aux_range: fixes U :: "'e option set"   and xs :: "'f list"   and p :: "'g \<Rightarrow> 'f"   and q :: "'h \<Rightarrow> 'f"   and I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e" assumes "U \<subseteq> range Some"   and "set xs \<subseteq> range p \<union> range q" shows "sinks_aux (con_comp_pol I) (con_comp_map D E p q) U xs \<subseteq> range Some"
diff_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.cancel_comm_monoid_add plus minus zero" shows "minus a zero = a"
sup_inf_distrib1: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.distrib_lattice inf less_eq less sup" shows "sup x (inf y z) = inf (sup x y) (sup x z)"
range_vfsequence_of_vlist: shows "range vfsequence_of_vlist = {xs. vfsequence xs}"
Gromov_hyperbolic_invariant_under_quasi_isometry_explicit: fixes lambda :: "real"   and C :: "real"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry f" shows "Gromov_hyperbolic_subset (752 * lambda ^ 3 * (C + deltaG TYPE('d))) UNIV"
disconnected_tree_orders: shows "h \<turnstile> map_M (to_tree_order \<circ> 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) disc_nodes \<rightarrow>\<^sub>r disconnected_tree_orders"
adapt_stoch_proc_borel_measurable: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and X :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and N :: "'d measure" assumes "filtrated_prob_space M F"   and "adapt_stoch_proc F X N" shows "\<forall>n. X n \<in> M \<rightarrow>\<^sub>M N"
add_uminus_conv_diff: fixes a :: "'a"   and b :: "'a" shows "a + - b = a - b"
R_lin_independent_Cons: fixes m :: "'v"   and ms :: "'v list" shows "R_lin_independent (m # ms) = (R_lin_independent ms \<and> (\<forall>r rs. set (r # rs) \<subseteq> UNIV \<and> (r # rs) \<bullet>\<cdot> (m # ms) = (0::'v) \<longrightarrow> r = (0::'f)))"
that_2: shows "set_of B \<subseteq> set_of B'"
successE: fixes f :: "'a Heap"   and h :: "heap" assumes "success f h" obtains r :: "'a" and h' :: "heap" where "execute f h = Some (r, h')"
valid_reads_axioms: fixes m :: "nat \<Rightarrow> nat" shows "valid_reads m ts\<^sub>s\<^sub>b"
F_UU: shows "F \<bottom> = {(s, X). front_tickFree s}"
dvd_field_iff: fixes a :: "'a"   and b :: "'a" shows "(a dvd b) = (a = (0::'a) \<longrightarrow> b = (0::'a))"
collect_def: fixes F :: "('b \<Rightarrow> 'a set) set"   and x :: "'b" shows "BNF_Def.collect F x = (\<Union>f\<in>F. f x)"
neighbors_edges: fixes g :: "graph"   and a :: "nat"   and b :: "nat" assumes "minGraphProps g"   and "a \<in> \<V> g" shows "(b \<in> set (neighbors g a)) = ((a, b) \<in> \<E> g)"
is_expansion_powr: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and basis :: "(real \<Rightarrow> real) list"   and F :: "'a"   and abort :: "bool"   and p :: "real" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term"   and "basis_wf basis"   and "trimmed F"   and "0 < fst (dominant_term F)"   and "is_expansion F basis" shows "is_expansion (powr_expansion abort F p) basis"
simps_4: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "future_reach (Or \<phi> \<psi>) = max (future_reach \<phi>) (future_reach \<psi>)"
apply_rsp': fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'c"   and y :: "'d" assumes "rel_fun R1 R2 f g"   and "R1 x y" shows "R2 (f x) (g y)"
the_min_min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and the_min :: "'a"   and x :: "'a" assumes "class.the_min less_eq less the_min" shows "less_eq the_min x"
a: shows "1\<^sub>\<sigma> + x \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
id_2: shows "[3..<Suc (Suc i)] = [3..<Suc i] @ [Suc i]"
gallery_snocI: fixes xs :: "'a set list"   and x :: "'a set"   and y :: "'a set" assumes "gallery (xs @ [x])"   and "chamber y"   and "x \<sim> y" shows "gallery (xs @ [x, y])"
class_add_is_relevant_class: fixes i :: "instr"   and P :: "jvm_prog"   and C\<^sub>0 :: "char list"   and C :: "char list"   and cdec :: "jvm_method class" assumes "is_relevant_class i P C\<^sub>0"   and "\<not> is_class P C" shows "is_relevant_class i (class_add P (C, cdec)) C\<^sub>0"
assms: shows "h \<turnstile> ok adopt_node document_ptr child"
mult_im_sheaf_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and S' :: "'f set"   and is_open' :: "'f set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f" assumes "im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.mult_im_sheaf S mult_str f \<equiv> \<lambda>V. mult_str (f \<^sup>\<inverse> S V)"
mresolution_rule: fixes C\<^sub>1 :: "fterm literal set"   and Cs :: "fterm literal set set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "C\<^sub>1 \<in> Cs"   and "C\<^sub>2 \<in> Cs"   and "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>" shows "mresolution_step Cs (Cs \<union> {mresolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>})"
assms_3: shows "\<forall>x. norm (f x) \<le> norm x * K"
eq_set_inc: fixes a :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "a \<in> A"   and "A = B" shows "a \<in> B"
take_bit_word_eq_self: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "take_bit n w = w"
comb_def: fixes l :: "'b \<Rightarrow> real" shows "comb l = (\<Sum>b\<in>B. l b *\<^sub>R b)"
fun_upd_idem_iff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'b" shows "(f(x := y) = f) = (f x = y)"
induced_ghom_someTr1: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and a :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "a \<in> carrier F" shows "f (SOME xa. xa \<in> gker\<^bsub>F,G\<^esub> f \<bullet>\<^bsub>F\<^esub> a) = f a"
CompFunCarAudioSystem_the2: shows "the (CompFun CarAudioSystem ''CarAudioSystem'') = {AudioPlayer_CTRL, CDPlayer_CTRL}"
preserves_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg (UP \<mu>) = UP.map\<^sub>0 (trg\<^sub>B \<mu>)"
countableI_bij: fixes f :: "nat \<Rightarrow> 'b"   and C :: "nat set"   and S :: "'b set" assumes "bij_betw f C S" shows "countable S"
E: shows "map (inv p) (filter (\<lambda>x. x \<in> range p) xs') \<in> divergences P"
tt_sem: fixes I :: "('b, 'c, 'a) interp" shows "fml_sem I TT = UNIV"
prems_6: shows "(N, \<alpha> @ \<beta>) \<in> \<RR>"
simps: fixes n :: "nat"   and l :: "cell list"   and r :: "cell list" shows "inv_end1 n (l, r) = (0 < n \<and> (l, r) = ([Bk], Oc # Bk \<up> n @ Oc \<up> n))"
pos_inf_h: assumes "sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} \<noteq> []"   and "sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! (length (sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0}) - 1) < x1" shows "sga = sign_vec qs (real_of_int (crb (prod_list (cast_rat_list qs))))"
sel_2: fixes xs :: "'b list" shows "stl (cycle xs) = cycle (tl xs @ [hd xs])"
induct_game: fixes P :: "game list \<Rightarrow> bool"   and a :: "game list" assumes "\<And>x. \<forall>y. (y, x) \<in> lprod option_of \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
force_inverse: fixes xs :: "'a llist" shows "Lazy_llist (\<lambda>_. force xs) = xs"
that: fixes n :: "nat" assumes "p ^^ n = id"   and "0 < n" shows "thesis"
create_subseqs: fixes base :: "'b"   and elements :: "'a list"   and n :: "nat"   and out :: "'b list"   and state :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list" assumes "local.create_subseqs base elements n = (out, state)" shows "invariant base elements n state \<and> set out = S base elements n"
has_integral_twiddle: fixes r :: "real"   and h :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and a :: "'b"   and b :: "'b" assumes "0 < r"   and "\<And>x. h (g x) = x"   and "\<And>x. g (h x) = x"   and "\<And>x. isCont g x"   and "\<And>u v. \<exists>w z. g ` cbox u v = cbox w z"   and "\<And>u v. \<exists>w z. h ` cbox u v = cbox w z"   and "\<And>u v. content (g ` cbox u v) = r * content (cbox u v)"   and "(f has_integral i) (cbox a b)" shows "((\<lambda>x. f (g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
Reals_inverse: fixes a :: "'a" assumes "a \<in> \<real>" shows "inverse a \<in> \<real>"
face_im: fixes w :: "'b set"   and y :: "'c set"   and f :: "'b \<Rightarrow> 'c" assumes "w \<in> Y"   and "order.greater_eq (f ` w) y" shows "y \<in> f \<turnstile> Y"
divide_eq_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and b :: "'e"   and c :: "'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "(divide b c = a) = (if c \<noteq> zero then b = times a c else a = zero)"
SUP_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'a"   and D :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "\<Squnion> (C ` A) = \<Squnion> (D ` B)"
connected_component_refl: fixes x :: "'b"   and S :: "'b set" assumes "x \<in> S" shows "connected_component S x x"
suffix_commute: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "s |\<^sub>s n |\<^sub>s m = s |\<^sub>s m |\<^sub>s n"
RestrictedP_cong: fixes H :: "fm set"   and f :: "tm"   and f' :: "tm"   and k :: "tm"   and A' :: "tm"   and g :: "tm"   and g' :: "tm" assumes "H \<turnstile> f EQ f'"   and "H \<turnstile> k EQ A'"   and "H \<turnstile> g EQ g'" shows "H \<turnstile> RestrictedP f k g IFF RestrictedP f' A' g'"
expand: fixes list :: "'c list"   and list' :: "'c list" assumes "(list = []) = (list' = [])"   and "\<lbrakk>list \<noteq> []; list' \<noteq> []\<rbrakk> \<Longrightarrow> hd list = hd list' \<and> tl list = tl list'" shows "list = list'"
induct: fixes P :: "char list \<Rightarrow> cmd \<Rightarrow> bool"   and a0 :: "char list"   and a1 :: "cmd" assumes "\<And>uu_. P uu_ Skip"   and "\<And>V' V e. P V' (V:=e)"   and "\<And>V' c\<^sub>1 c\<^sub>2. \<lbrakk>P V' c\<^sub>1; P V' c\<^sub>2\<rbrakk> \<Longrightarrow> P V' (c\<^sub>1;; c\<^sub>2)"   and "\<And>V' b c\<^sub>1 c\<^sub>2. \<lbrakk>P V' c\<^sub>1; P V' c\<^sub>2\<rbrakk> \<Longrightarrow> P V' (if (b) c\<^sub>1 else c\<^sub>2)"   and "\<And>V' b c. P V' c \<Longrightarrow> P V' (while (b) c)" shows "P a0 a1"
gallery_Cons_reduce: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X (x # xs)" shows "SimplicialComplex.maxsimpchain X xs"
fds_abs_converges_norm_iff': fixes f :: "'a fds"   and s :: "real" shows "fds_abs_converges (fds_norm f) s = fds_abs_converges f (of_real s)"
comp: fixes f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec"   and g :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec" assumes "linear f"   and "linear g"   and "P f"   and "P g" shows "P (f \<circ> g)"
intros_3: fixes A :: "'b set"   and x :: "'b"   and k :: "nat" assumes "finite A"   and "x \<notin> A"   and "R A (Suc k)"   and "R (insert x A) k" shows "R (insert x A) (Suc k)"
is_natural_1: fixes f :: "'c" assumes "C.arr f" shows "C (C.map f) (C.map (C.dom f)) = C.map f"
those_eq_Some_lengthD: fixes xs :: "'a option list"   and ys :: "'a list" assumes "those xs = Some ys" shows "length xs = length ys"
cases: fixes x :: "'b rexp" obtains r :: "'b rexp" and s :: "'b rexp" where "x = Plus r s"   | "x = Zero"   | "x = One"   | v :: "'b" where "x = Atom v__"   | v :: "'b rexp" and va :: "'b rexp" where "x = Times va__ va"   | v :: "'b rexp" where "x = Star v"
that: assumes "\<Z>_vfsequence \<alpha> \<SS>"   and "vcard \<SS> = 3\<^sub>\<nat>"   and "\<Z>_csgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vplus\<rparr>]\<^sub>\<circ>"   and "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> (a +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b) *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c"   and "\<And>a b c. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c) = a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b +\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a *\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c" shows "thesis"
divide_def: fixes x :: "complex_homo"   and y :: "complex_homo" shows "x :\<^sub>h y = x *\<^sub>h reciprocal y"
countable_Int2: fixes B :: "'b set"   and A :: "'b set" assumes "countable B" shows "countable (A \<inter> B)"
help_rtrancl_Range: fixes a :: "'d"   and b :: "'d"   and R :: "('d \<times> 'd) set" assumes "a \<noteq> b"   and "(a, b) \<in> R\<^sup>*" shows "b \<in> Range R"
intros: fixes trsys :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'c"   and tl :: "'d"   and s' :: "'c"   and \<tau>move :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool" assumes "trsys s tl s'"   and "\<tau>move s tl s'" shows "\<tau>trsys.silent_move trsys \<tau>move s s'"
OUTfromChCorrect_data20: shows "OUTfromChCorrect data20"
sum_ereal: fixes f :: "'c \<Rightarrow> real"   and A :: "'c set" shows "(\<Sum>x\<in>A. ereal (f x)) = ereal (sum f A)"
seq_sim_trans: fixes \<rho> :: "nat \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<rho> \<simeq>A\<simeq> \<sigma>"   and "\<sigma> \<simeq>A\<simeq> \<tau>" shows "\<rho> \<simeq>A\<simeq> \<tau>"
pgeq: fixes y :: "'c" assumes "y \<in> U" shows "p (g y) = f y"
assms_2: shows "Z \<subseteq>\<^sub>\<circ> X"
in_set_segments_of_aform_aform_valE: fixes x2 :: "real \<times> real"   and y2 :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(x2, y2) \<in> set (segments_of_aform X)" obtains e :: "nat \<Rightarrow> real" where "y2 = aform_val e X" and "e \<in> UNIV \<rightarrow> {- 1..1}"
word_of_int_uint: fixes w :: "'a word" shows "word_of_int (uint w) = w"
linear_Poly_list_of_vec: shows "Poly \<circ> list_of_vec \<in> module_hom class_ring local.V (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u)})"
intros_14: shows "Collection OclSuper \<sqsubset> OclSuper"
continuous_map_from_subtopology_mono: fixes X :: "'a topology"   and T :: "'a set"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "continuous_map (subtopology X T) X' f"   and "S \<subseteq> T" shows "continuous_map (subtopology X S) X' f"
orthogonal_complement_orthoI': fixes x :: "'b"   and M :: "'b set"   and y :: "'b" assumes "x \<in> M"   and "y \<in> orthogonal_complement M" shows "is_orthogonal x y"
farkas_coefficients_preprocess': fixes cs :: "('a \<times> QDelta ns_constraint) list"   and S :: "'a istate"   and C :: "(rat \<times> QDelta atom) list" assumes "preprocess' cs (start_fresh_variable cs) = S"   and "farkas_coefficients_atoms_tableau (snd ` set (Atoms S)) (Tableau S) C" shows "\<exists>C. farkas_coefficients_ns (snd ` set cs) C"
exp_conj_nneg: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" shows "\<lambda>s. 0 \<tturnstile> P && Q"
isDiagonal_diagonal_to_Smith_aux_general: fixes xs :: "nat list"   and A :: "(('a, 'b) vec, 'c) vec"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "isDiagonal A" shows "isDiagonal (diagonal_to_Smith_aux A xs bezout)"
Pair_inject: fixes a :: "'e"   and b :: "'f"   and a' :: "'e"   and b' :: "'f" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
two_comparisons_into_case_order_3: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and Q :: "'c"   and R :: "'c" shows "(if le x y then if le y x then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
emeasure_CI': fixes J :: "nat set"   and X :: "(nat \<Rightarrow> 'a) set" assumes "finite J"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (CI J) X = emeasure (C 0 (up_to J) (\<lambda>_. undefined)) (PF.emb {0..<up_to J} J X)"
pos_le_divideR_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and a :: "'a"   and b :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq a (scaleR (inverse c) b) = less_eq (scaleR c a) b"
InitializePhase_rdBy: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and pp :: "Proc"   and qq :: "Proc"   and dd :: "Disk" assumes "InitializePhase s s' p" shows "rdBy s' pp qq dd \<subseteq> rdBy s pp qq dd"
simps: fixes a1 :: "meth_def"   and a2 :: "char list" shows "method_name a1 a2 = (\<exists>cl meth vds meth_body. a1 = meth_def_def (meth_sig_def cl meth vds) meth_body \<and> a2 = meth)"