hcju/isabelleps huggingface dataset:
closure_invariant_sound: fixes r :: "'b rexp"   and s :: "'b rexp"   and ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" assumes "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = Some (ws, ps, N)"   and "local.wf n r"   and "local.wf n s"   and "ws = []" shows "lang n r = lang n s"
rr: shows "secure_refinement \<R>\<^sub>A \<R> P"
le_or_fail_def: fixes m :: "'c nres"   and m' :: "'c nres" shows "m \<le>\<^sub>n m' \<equiv> nofail m \<longrightarrow> m \<le> m'"
Ch_irc_idem: fixes h :: "'c"   and B :: "'a set" shows "Ch h (Ch h B) = Ch h B"
Ints_def: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.Ints one plus zero minus = range (ring_1.of_int one plus zero minus)"
msg_distinct_92: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg" shows "Enc X Y \<noteq> Hash X'"
encode_problem_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "\<Phi> \<Pi> t \<equiv> \<Phi>\<^sub>I \<Pi> \<^bold>\<and> (encode_operators \<Pi> t \<^bold>\<and> (encode_all_frame_axioms \<Pi> t \<^bold>\<and> (\<Phi>\<^sub>G \<Pi>) t))"
ine_ins_neg1: fixes P :: "specID"   and m :: "Expression"   and x :: "chanID" assumes "\<not> ine P m"   and "exprChannel x m" shows "x \<notin> ins P"
s_ns_mul_ext_union_compat: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and C :: "'a multiset"   and D :: "'a multiset" assumes "(A, B) \<in> s_mul_ext ns s"   and "(C, D) \<in> ns_mul_ext ns s" shows "(A + C, B + D) \<in> s_mul_ext ns s"
Lambert_W_leI: fixes x :: "real"   and y :: "real" assumes "x \<le> y * exp y"   and "- 1 \<le> y"   and "- exp (- 1) \<le> x" shows "Lambert_W x \<le> y"
dc2c_def: fixes \<chi> :: "nat \<Rightarrow> complex" shows "m.dc2c \<chi> = (\<lambda>x. if x < m then \<chi> x else 0)"
nextl_app: fixes q :: "hf"   and xs :: "'a list"   and ys :: "'a list" shows "nextl q (xs @ ys) = nextl (nextl q xs) ys"
list_comb: fixes f :: "'a"   and xs :: "'a list" shows "pre_strong_term_class.wellformed (list_comb f xs) = (pre_strong_term_class.wellformed f \<and> list_all pre_strong_term_class.wellformed xs)"
active_subset_def: fixes M :: "('f \<times> 'l) set" shows "active_subset M = {CL \<in> M. snd CL = active}"
llist_all2_inf_llist: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'd" shows "llist_all2 P (inf_llist f) (inf_llist g) = (\<forall>n. P (f n) (g n))"
fin: shows "finite {v. p v \<noteq> (0::'b)}"
Base_foo_permits_acc: fixes S :: "qtname" shows "tprg \<turnstile> (Base, mdecl Base_foo) in Base permits_acc_from S"
simps: fixes stop :: "'c \<Rightarrow> bool"   and "next" :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "enat_unfold stop next a = (if stop a then 0 else eSuc (enat_unfold stop next (next a)))"
Z_good_fun_alt_sum: fixes g :: "nat \<Rightarrow> nat"   and P :: "(nat \<Rightarrow> nat) \<Rightarrow> 'b" assumes "g \<in> local.F_inj" shows "(\<Sum>f | f \<in> {0..<n} \<rightarrow> {0..<m} \<and> (\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i) \<and> inj_on f {0..<n} \<and> f ` {0..<n} = g ` {0..<n}. P f) = (\<Sum>\<pi> | \<pi> permutes {0..<n}. P (g \<circ> \<pi>))"
PMC_SC_axioms: shows "PMC_SC Cn \<gamma>\<^sub>T\<^sub>R \<top> \<bottom> (.\<longrightarrow>.) .\<not> (.\<and>.) (.\<or>.) (.\<longleftrightarrow>.)"
OUTfromChCorrect_data23: shows "OUTfromChCorrect data23"
DSourcesEmptySources: fixes i :: "AbstrLevelsID"   and C :: "CSet" assumes "DSources i C = {}" shows "Sources i C = {}"
H_simps_1: shows "H 0 = (0, {0<..}, 0)"
HVZK_unfold1: assumes "\<Sigma>0.\<Sigma>_protocol" shows "\<forall>h w e. (h, w) \<in> Rel0 \<longrightarrow> e \<in> carrier L \<longrightarrow> \<Sigma>0.R h w e = \<Sigma>0.S h e"
bot_least: fixes a :: "('a \<times> ('b \<times> 'a) list) set" shows "t_zero \<subseteq> a"
xor_evaluate_def: fixes A :: "bool"   and B :: "bool" shows "xor_evaluate A B = return_spmf (A \<oplus> B)"
suminf_nonneg: fixes f :: "nat \<Rightarrow> 'c" assumes "summable f"   and "\<And>n. (0::'c) \<le> f n" shows "(0::'c) \<le> suminf f"
pmf_embed_pmf: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV = 1" shows "pmf (embed_pmf f) x = f x"
type_subst_fresh: fixes X :: "tyvrs"   and T :: "ty"   and P :: "ty"   and Y :: "tyvrs" assumes "X \<sharp> T"   and "X \<sharp> P" shows "X \<sharp> T[Y \<mapsto> P]\<^sub>\<tau>"
stutter_extend_def: fixes G :: "('b, 'd) graph_rec_scheme" shows "stutter_extend G \<equiv> \<lparr>g_V = g_V G, g_E = stutter_extend_edges (g_V G) (g_E G), g_V0 = g_V0 G, \<dots> = graph_rec.more G\<rparr>"
assms: fixes i :: "'a" assumes "i \<in> I" shows "comm_group (G i)"
hyps: fixes y :: "'a list"   and ts :: "'a tree list"   and k :: "nat" assumes "length y < length xs"   and "length ts = 2 ^ k"   and "\<forall>i<2 ^ k. braun_list (ts ! i) (take_nths i k y)" shows "list_fast_rec ts = y"
C_verifies_PRes_lemma: fixes M :: "agent"   and MsgPRes :: "msg"   and evs :: "event list"   and C :: "agent"   and LID_M :: "nat"   and Trans :: "msg"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and XID :: "nat"   and Chall_C :: "nat" assumes "Crypt (priSK M) (Hash MsgPRes) \<in> parts (knows Spy evs)"   and "Notes C \<lbrace>Number LID_M, Trans\<rbrace> \<in> set evs"   and "Trans = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "MsgPRes = \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Hash (Number PurchAmt)\<rbrace>"   and "evs \<in> set_pur"   and "M \<notin> bad" shows "\<exists>j KP. Notes M \<lbrace>Number LID_M, Agent (PG j), Trans\<rbrace> \<in> set evs \<and> Gets M (EncB (priSK (PG j)) KP (pubEK M) \<lbrace>Number LID_M, Number XID, Number PurchAmt\<rbrace> authCode) \<in> set evs \<and> Says M C (sign (priSK M) MsgPRes) \<in> set evs"
assms_2: fixes X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and x :: "'var" assumes "isInBar (xs, s)"   and "wls s X" shows "phiAbs (xs, s) (Abs xs x X)"
distinct_13: fixes x2 :: "'a \<times> nat"   and x5 :: "'a \<times> nat" shows "Formal_in x2 \<noteq> Actual_out x5"
Un_subset_iff: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" shows "(A \<union> B \<subseteq> C) = (A \<subseteq> C \<and> B \<subseteq> C)"
induced_hom_equality: fixes S :: "'b set"   and P :: "('b \<times> bool) list set"   and f :: "'b \<Rightarrow> 'c"   and w :: "'b freeword" assumes "GroupByPresentationInducedFun S P f"   and "w \<in> FreeGroup S" shows "GroupByPresentationInducedFun.induced_hom S P f (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S) `` {w})) = freeword_funlift f w"
mmulti_join_link: fixes A_pos :: "nat set list"   and n :: "nat"   and A_neg :: "nat set list"   and L :: "'a option list set list" assumes "A_pos \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L" shows "mmulti_join n A_pos A_neg L = mmulti_join' A_pos A_neg L"
sorted_sorted_list_of_set: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
Collect_mono_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "(Collect (P::'b \<Rightarrow> bool) \<subseteq> Collect (Q::'b \<Rightarrow> bool)) = (\<forall>x::'b. P x \<longrightarrow> Q x)"
all_resolvents_upon_is_finite: fixes S :: "'at Literal set set"   and A :: "'at" assumes "all_fulfill finite S" shows "all_fulfill finite (S \<union> all_resolvents_upon S A)"
eventually_at_topological: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a"   and s :: "'a set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
is_first_def: fixes x :: "'a"   and u :: "'a list" shows "is_first x u = (\<exists>v. u = [x] @ v)"
spec\<^sub>5_def: fixes p :: "prog" shows "spec\<^sub>5 p \<equiv> vars p = vars\<^sub>0 \<and> body_split p s\<^sub>L\<^sub>0 s\<^sub>H\<^sub>0"
meets_atrans: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and p :: "'a"   and q :: "'a"   and r :: "'a" assumes "class.interval meets \<I>"   and "meets p q"   and "meets q r" shows "\<not> meets p r"
iso_map\<^sub>1_ide: fixes f :: "'a" assumes "C.ide f" shows "D.iso (\<tau>\<^sub>1 f)"
simps_3: fixes x :: "name"   and \<theta> :: "(name \<times> lam) list"   and e :: "lam" assumes "x \<sharp> \<theta>" shows "\<theta><Lam [x].e> = Lam [x].\<theta><e>"
hyps_3: assumes "\<And>x. x \<in> S \<Longrightarrow> set (coeffs (f x)) \<subseteq> \<real>" shows "set (coeffs (sum f S)) \<subseteq> \<real>"
EQ_Or: fixes f :: "atom fm"   and g :: "atom fm"   and xs :: "'a list" shows "EQ (Or f g) xs = EQ f xs \<union> EQ g xs"
senior_states_cases_subset: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and p :: "'d"   and n :: "nat"   and q :: "'d" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" obtains "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n \<subset> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n"   | "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n = semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n"   | "semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w q n \<subset> semi_mojmir_def.senior_states \<Sigma> \<delta> q\<^sub>0 w p n"
sem_Exi_explicit: fixes p :: "pvar"   and \<phi> :: "'aprop dfmla"   and env :: "pvar \<Rightarrow> nat"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "sem (Exi p \<phi>) env \<pi>l = (\<exists>\<pi>. wfp AP' \<pi> \<and> stateOf \<pi> = (if \<pi>l \<noteq> [] then stateOf (last \<pi>l) else s0) \<and> sem \<phi> (env(p := length \<pi>l)) (\<pi>l @ [\<pi>]))"
IH: shows "x = y \<or> (\<exists>s. s \<noteq> [] \<and> tl s \<noteq> [] \<and> s ! 0 = x \<and> s ! (length s - 1) = y \<and> (\<forall>i<length s - 1. s ! i \<rightarrow>\<^sub>i s ! Suc i))"
val_eq_Some_sum_if_valid_neq_Leaf: fixes t :: "'a tree" assumes "local.valid t"   and "t \<noteq> Leaf" shows "val t = Some (local.sum (l t) (r t))"
simps_2: fixes l :: "'b tree234"   and a :: "'b"   and r :: "'b tree234" shows "height (Up\<^sub>i l a r) = height l"
oghoare_step_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and x :: "'e"   and c' :: "('e, 'd, 'f) com"   and f :: "'f"   and P :: "('e, 'd, 'f) ann"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>\<turnstile> (c, Normal x) \<rightarrow> (c', Fault f)"   and "x \<in> pre P"   and "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P c Q, A" shows "f \<in> F"
small_insert_iff: fixes a :: "'c"   and X :: "'c set" shows "small (insert a X) = small X"
nts: shows "wf_ts_regex \<sigma> P j r nts"
lm_to_list_Assoc_List_set: fixes l :: "('b, 'c) assoc_list" shows "set (lm.to_list l) = Assoc_List.set l"
\<R>p_\<R>_eq: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "\<R>p \<Gamma> = \<R>_abv \<Gamma>"
set_tree_induct: fixes x :: "'a"   and t :: "'a tree"   and P :: "'a \<Rightarrow> 'a tree \<Rightarrow> bool" assumes "x \<in> set_tree t"   and "\<And>t. P (root t) t"   and "\<And>x t. \<lbrakk>x \<in> set_tree (left t); P x (left t)\<rbrakk> \<Longrightarrow> P x t"   and "\<And>x t. \<lbrakk>x \<in> set_tree (right t); P x (right t)\<rbrakk> \<Longrightarrow> P x t" shows "P x t"
val_Zp_ultrametric_eq: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp y < val_Zp x" shows "val_Zp (x \<oplus> y) = val_Zp y"
smooth_on_compose: fixes k :: "enat"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'b" assumes "k-smooth_on T f"   and "k-smooth_on S g"   and "open S"   and "open T"   and "g ` S \<subseteq> T" shows "k-smooth_on S (f \<circ> g)"
one_im_sheaf_def: shows "one_im_sheaf \<equiv> \<lambda>V. \<one>\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
row_conjugate: fixes i :: "nat"   and m :: "'a mat" assumes "i < dim_row m" shows "row (conjugate m) i = conjugate (row m i)"
inject: fixes x :: "'a"   and ya :: "'a" shows "(cofinite.cofinite x = cofinite.cofinite ya) = (x = ya)"
simps_1: fixes x :: "quat" shows "quat.Re (- x) = - quat.Re x"
distinct_5: fixes x2 :: "'d"   and x31 :: "'c"   and x32 :: "'d" shows "Skip x2 \<noteq> Yield x31 x32"
4: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p (unlabel \<A>)"
enum_alt_surj: shows "range enum_alt \<union> {None} = UNIV"
proj_P2R_raw: fixes \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "proj \<phi> = (\<lambda>x. P2R \<phi> `` {x})"
f: shows "(\<lambda>x. f (g x)) \<in> N \<rightarrow>\<^sub>M M"
a_empty: fixes G :: "nat set \<times> nat set set" assumes "short_cycles G k = {}" shows "P G k"
matrix_axis_0: fixes k :: "'b"   and A :: "(('b, 'c) vec, 'd) vec" assumes "k \<noteq> (0::'b)"   and "\<forall>i. A *v axis i k = 0" shows "A = 0"
homotopic_with_def: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "homotopic_with P X Y f g \<equiv> \<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x. h (0, x) = f x) \<and> (\<forall>x. h (1, x) = g x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x)))"
simps_3: fixes x1 :: "'d" shows "\<star> \<noteq> ln x1"
exp2: fixes B :: "nat" shows "Suc 0 < 3 * exp 3 B"
arith_simps_7: fixes m :: "num" shows "num.Bit1 m + num.One = num.Bit0 (m + num.One)"
le_less_Suc_eq: fixes x :: "nat"   and i :: "nat" assumes "x - Suc 0 \<le> i"   and "i < x" shows "x - Suc 0 = i"
h2: shows "poly (qs ! a) x < 0 \<and> poly (prod_list (retrieve_polys qs xa)) x < 0 \<longrightarrow> prod_list (map ((!) sign) (a # xa)) = 1"
Ramsey: fixes Z :: "'b set"   and r :: "nat"   and f :: "'b set \<Rightarrow> nat"   and s :: "nat" assumes "infinite Z"   and "\<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = r \<longrightarrow> f X < s" shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X = t)"
timpl_closure'_timpls_trancl_supset: fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' c \<subseteq> timpl_closure' (c\<^sup>+)"
wf_J_mdecl'_def: fixes P :: "'a J_prog"   and C :: "String.literal" shows "wf_J_mdecl' P C \<equiv> \<lambda>(M, Ts, T, pns, body). length Ts = length pns \<and> distinct pns \<and> this \<notin> set pns \<and> typeable_with P [this \<mapsto> Class C, pns [\<mapsto>] Ts] body T \<and> \<D> body \<lfloor>{this} \<union> set pns\<rfloor>"
orthogonal_matrix_def: fixes Q :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat (1::'d) \<and> Q ** Finite_Cartesian_Product.transpose Q = mat (1::'d))"
jnf: shows "jordan_nf (of_real_hom.mat_hom A) n_as"
cube_minus: shows "\<forall>p q. (p - q) ^ 3 = rat_poly_plus (p ^ 3 - rat_poly_times (rat_poly_times 3 (p\<^sup>2)) q) (rat_poly_times (rat_poly_times 3 p) (q\<^sup>2)) - q ^ 3"
eq: shows "X #[y \<and> x]_xs = X' #[y \<and> x']_xs"
reduce_rulesD_1: fixes r :: "('a, 'b) ta_rule"   and \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a set" assumes "r \<in> reduce_rules \<delta> P" shows "r \<in> \<delta>"
shd_sset: fixes a :: "'a stream" shows "shd a \<in> sset a"
monoE: fixes g :: "'b"   and f :: "'b"   and f' :: "'b" assumes "D.mono g"   and "D.seq g f"   and "D.seq g f'"   and "g \<cdot>\<^sub>D f = g \<cdot>\<^sub>D f'" shows "f' = f"
simple_cons: fixes e :: "'b \<times> 'b"   and p :: "('b \<times> 'b) list" shows "simple (e # p) = (uedge e \<notin> uedge ` set p \<and> simple p)"
assms: shows "a \<in> allAllocations'' N \<Omega>"
sep_empty_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_empty zero \<equiv> \<lambda>h. h = zero"
ret_le_down_conv: fixes m :: "'a nres"   and c :: "'b"   and R :: "('b \<times> 'a) set" assumes "nofail m" shows "(RETURN c \<le> \<Down> R m) = (\<exists>a. (c, a) \<in> R \<and> RETURN a \<le> m)"
Suc_as_int: shows "Suc = (\<lambda>a. nat (int a + 1))"
float_distinct_3: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_nan a \<and> is_denormal a)"
dim_image_eq: fixes f :: "(('a \<Rightarrow> real) \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real"   and S :: "(('a \<Rightarrow> real) \<Rightarrow> real) set" assumes "\<forall>x. x \<in> tangent_space p \<longrightarrow> f x \<in> sub.tangent_space p"   and "S \<subseteq> tangent_space p"   and "linear_on (tangent_space p) (sub.tangent_space p) (*\<^sub>R) (*\<^sub>R) f"   and "inj_on f (module_on.span (*\<^sub>R) S)" shows "sub.tangent_space.dim p (f ` S) = tangent_space.dim p S"
sets_Collect_finite_All': fixes S :: "'c set"   and P :: "'c \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>i. i \<in> S \<Longrightarrow> {x \<in> \<Omega>. P i x} \<in> M"   and "finite S"   and "S \<noteq> {}" shows "{x \<in> \<Omega>. \<forall>i\<in>S. P i x} \<in> M"
exE_some: fixes P :: "'b \<Rightarrow> bool"   and c :: "'b" assumes "Ex P"   and "c \<equiv> Eps P" shows "P c"
infinite_times_eqpoll_self: fixes A :: "'a set" assumes "infinite A" shows "A \<times> A \<approx> A"
irrefl_L_Prec: fixes l :: "label" shows "\<not> l \<sqsubset>l l"
eval_in_carrier: fixes p :: "'a list list"   and x :: "'a list" assumes "set p \<subseteq> carrier (K [X])"   and "x \<in> carrier (K [X])" shows "UP.eval p x \<in> carrier (K [X])"
atLeastAtMost_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "ord.atLeastAtMost less_eq l u \<equiv> ord.atLeast less_eq l \<inter> ord.atMost less_eq u"
max_Var_floatarith_le_max_Var_floatariths: fixes fa :: "floatarith"   and fas :: "floatarith list" assumes "fa \<in> set fas" shows "max_Var_floatarith fa \<le> max_Var_floatariths fas"
expands_to_MSLCons_0_asymp_equiv_hd: fixes f :: "real \<Rightarrow> real"   and C :: "'a"   and xs :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "(f expands_to MS (MSLCons (C, 0) xs) g) basis"   and "trimmed (MS (MSLCons (C, 0) xs) f)"   and "basis_wf basis" shows "f \<sim>[at_top] eval C"
consistency_def: fixes f :: "'a set \<Rightarrow> 'a set" shows "consistency f = (\<forall>B C. C \<subseteq> B \<longrightarrow> f B \<subseteq> C \<longrightarrow> f B = f C)"
space_measure_pmf: fixes p :: "'b pmf" shows "space (measure_pmf p) = UNIV"
powr_numeral: fixes x :: "real"   and n :: "num" assumes "0 \<le> x" shows "x powr numeral n = x ^ numeral n"
sgn_root: fixes n :: "nat"   and x :: "real" assumes "0 < n" shows "sgn (root n x) = sgn x"
inner_eq_zero_iff: fixes x :: "'b" shows "(inner x x = (0::'a)) = (x = (0::'b))"
field_hom_0'_axioms: shows "field_hom_0' of_rat"
approx_floatariths_def: fixes p :: "nat"   and ea :: "floatarith list"   and as :: "(real \<times> real pdevs) list" shows "approx_floatariths p ea as = (let da = degree_aforms as; aes = map (\<lambda>x. (x, 0)) as in approx_floatariths_aformerr p ea aes \<bind> (\<lambda>rs. let d = max da (degree_aforms_err rs) in Some (aforms_err_to_aforms d rs)))"
ap_conv_bind_state: fixes f :: "('c, 'e \<Rightarrow> 'd) state"   and x :: "('c, 'e) state" shows "f \<diamondop> x = f \<bind> (\<lambda>f. x \<bind> pure \<circ> f)"
minus_def: fixes x :: "'a" shows "- x = Abs (- Rep x mod int CARD('a))"
CHAR_eqI: fixes c :: "nat" assumes "of_nat c = (0::'a)"   and "\<And>x. of_nat x = (0::'a) \<Longrightarrow> c dvd x" shows "CHAR = c"
sorted_remove1: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and a :: "'c" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (remove1 a xs)"
set_bind_pmf: fixes M :: "'b pmf"   and N :: "'b \<Rightarrow> 'a pmf" shows "set_pmf (M \<bind> N) = (\<Union>M\<in>set_pmf M. set_pmf (N M))"
maxsimpD_maximal: fixes X :: "'b set set"   and x :: "'b set"   and z :: "'b set" assumes "SimplicialComplex X"   and "SimplicialComplex.maxsimp X x"   and "z \<in> X"   and "order.greater_eq z x" shows "z = x"
dvd_2: shows "r = [] \<or> degree r < degree (Irr K x)"
singleton_iff_atom: fixes p :: "'a" shows "singleton p = (p \<noteq> 0 \<and> (\<forall>x\<le>p. x = 0 \<or> x = p))"
symp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "symp r = (\<forall>x y. r x y \<longrightarrow> r y x)"
repeNeg_simps: fixes r :: "real" shows "(NEG_INF \<equiv>\<^sub>E r) = (r \<le> - 2147483647)"
Ici_subset_Ioi_iff: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.linorder less_eq less" shows "(ord.atLeast less_eq a \<subseteq> ord.greaterThan less b) = less b a"
hyps_2: shows "P' \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P''"
closed_Un: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "topological_space.closed open T" shows "topological_space.closed open (S \<union> T)"
partial_fraction_decomposition_def: fixes x :: "'b"   and ys :: "('b \<times> nat) list" shows "partial_fraction_decomposition x ys = (if ys = [] then (x, []) else let zs = map (\<lambda>i. let (y, n) = ys ! i in normalise_decomp (decompose x (map (\<lambda>(y, n). y ^ Suc n) ys) ! i) y (Suc n)) [0..<length ys] in (sum_list (map fst zs), map snd zs))"
fpairs_vinsert: fixes a :: "V"   and b :: "V"   and A :: "V" shows "fpairs (vinsert [a, b]\<^sub>\<circ> A) = set {[a, b]\<^sub>\<circ>} \<union>\<^sub>\<circ> fpairs A"
asubg_subset1: fixes H :: "'a set"   and x :: "'a" assumes "A +> H"   and "x \<in> H" shows "x \<in> carrier A"
map_bset_abs_rep_Tree\<^sub>\<alpha>: fixes tset\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha> set['e]" shows "map_bset abs_Tree\<^sub>\<alpha> (map_bset rep_Tree\<^sub>\<alpha> tset\<^sub>\<alpha>) = tset\<^sub>\<alpha>"
mbisim_simulation1: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tl1 :: "'t \<times> ('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "\<not> m\<tau>move1 s1 tl1 s1'"   and "r1.redT s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. \<tau>mRed2 s2 s2' \<and> r2.redT s2' tl2 s2'' \<and> \<not> m\<tau>move2 s2' tl2 s2'' \<and> s1' \<approx>m s2'' \<and> tl1 \<sim>T tl2"
shadows_consts_def: fixes C_info :: "(name, dt_def) fmap"   and heads :: "name fset"   and t :: "'b" shows "pre_constants.shadows_consts C_info heads t = (\<not> fdisjnt (pre_constants.all_consts C_info heads) (all_frees t))"