hcju/isabelleps huggingface dataset:
summable_iff_shift: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" shows "summable (\<lambda>n. f (n + k)) = summable f"
hyp4: shows "fst (snd (foldl Gauss_Jordan_column_k_det_P_iarrays (1::'a, 0, matrix_to_iarray A) [0..<Suc k])) = fst (snd (foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k]))"
l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs"
Nom': fixes b :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "p at b in' set ((ps, a) # branch)"   and "Nom a at b in' set ((ps, a) # branch)"   and "\<forall>i. p = Nom i \<or> p = (\<^bold>\<diamond> Nom i) \<longrightarrow> i \<in> A"   and "A, n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
remdups_adj_adjacent: fixes i :: "nat"   and xs :: "'c list" assumes "Suc i < length (remdups_adj xs)" shows "remdups_adj xs ! i \<noteq> remdups_adj xs ! Suc i"
vdw_assms: fixes k' :: "nat" assumes "0 < k'" shows "\<exists>n_k'. vdw k' l n_k'"
tendsto_min: fixes X :: "'a \<Rightarrow> 'b"   and x :: "'b"   and net :: "'a filter"   and Y :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "(X \<longlongrightarrow> x) net"   and "(Y \<longlongrightarrow> y) net" shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
Inf_trans_greatest: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "S \<noteq> {}"   and "\<forall>t\<in>S. \<forall>P. le_trans u t" shows "le_trans u (Inf_trans S)"
zero_pdivides_zero: shows "[] pdivides\<^bsub>poly_ring R\<^esub> []"
diff_self: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and uminus :: "'g \<Rightarrow> 'g"   and a :: "'g" assumes "class.group_add minus plus zero uminus" shows "minus a a = zero"
responses_\<I>_full: fixes x :: "'b" shows "responses_\<I> \<I>_full x = UNIV"
irreflclpD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "r\<^sup>\<noteq>\<^sup>\<noteq> a b" shows "r a b \<and> a \<noteq> b"
DirProd_list_def: fixes Gs :: "('a, 'b) monoid_scheme list" shows "DirProd_list Gs = foldr (\<lambda>G H. image_group (\<lambda>(x, xs). x # xs) (G \<times>\<times> H)) Gs nil_monoid"
cauchy_filter_metric_filtermapI: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>P. eventually P F \<and> (\<forall>x y. P x \<and> P y \<longrightarrow> dist (f x) (f y) < e)" shows "cauchy_filter (filtermap f F)"
set_antichain2: fixes x :: "'a"   and X :: "'a antichain" assumes "x \<in>\<^sub>A X" shows "x \<in> set_antichain X"
dWP: shows "wlp (While M S) P \<in> carrier_mat d d"
xy_le_uxx_vyy: fixes u :: "real"   and v :: "real"   and x :: "real"   and y :: "real" assumes "0 < u"   and "u * v = 1" shows "x * y \<le> u * x\<^sup>2 / 2 + v * y\<^sup>2 / 2"
D1: shows "D2 r (\<sigma>, \<tau>, \<tau>', \<sigma>')"
set_custom_empty_axioms: shows "set_custom_empty op_hs_empty"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', P \<oplus>\<^sub>\<top> \<tau>.P, \<tau>.P) \<in> Rel"
sp_fundef_def: fixes F :: "'l \<Rightarrow> 'm fundef option"   and fd :: "'n fundef"   and xs :: "('o, 'p, 'l, 'd, 'e, 'g, 'q, 'r) Ubx.instr list" shows "sp_fundef F fd xs \<equiv> Subx.sp F xs (replicate (arity fd) None)"
hyps_3: fixes i :: "nat" assumes "i < length qs" shows "accs \<delta>' (ts ! i) (qs ! i)"
assms_3: shows "h = g \<ominus>\<^bsub>UP R\<^esub> monom (UP R) a 0"
unat_lt2p: fixes x :: "'a word" shows "unat x < 2 ^ LENGTH('a)"
cts_wp_Seq: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "bd_cts (wp a)"   and "bd_cts (wp b)"   and "healthy (wp b)" shows "bd_cts (wp (a ;; b))"
good_prod_inj: fixes A :: "(val \<times> val) fset" shows "inj_on good_prod (fset A)"
t_ins_subset: fixes Y :: "'b t_type"   and X :: "'b t_type" assumes "Y \<in> t_ins_set X" shows "t_ins_set Y \<subseteq> t_ins_set X"
NF: shows "NF ((Lam [x].t) \<degree>\<degree> ts)"
awalk_to_apath_subset: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "G.awalk u p v" shows "set (G.awalk_to_apath p) \<subseteq> set p"
dj_perm_perm_forget: fixes pi2 :: "('b \<times> 'b) list"   and pi1 :: "('a \<times> 'a) list" assumes "disjoint TYPE('a) TYPE('b)" shows "pi2 \<bullet> pi1 = pi1"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "dim_vec (Matrix.vec n f) = n"
CallNull: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and ps :: "char list exp list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>ps,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
reachable_eq_imp: fixes m :: "'b"   and r :: "'a" shows "eq_imp (\<lambda>r' s. (roots (s (mutator m)), ghost_honorary_root (s (mutator m)), \<Union> (ran ` obj_fields ` set_option (sys_heap s r')), filter (\<lambda>s. is_mw_Mutate s \<or> is_mw_Mutate_Payload s) (sys_mem_store_buffers (mutator m) s))) (mut_m.reachable m r)"
simps_1: fixes p :: "proc"   and v :: "val messageValue" shows "sends p s0 v = {# <p0, v1> }"
staysuntouched': fixes init :: "nat list"   and x :: "nat"   and y :: "nat"   and qs :: "nat list" assumes "distinct init"   and "x \<in> set init"   and "y \<in> set init"   and "set qs \<subseteq> set init"   and "x \<notin> set qs"   and "y \<notin> set qs" shows "x < y in fst (config\<^sub>p (rTS []) init qs) = x < y in init"
insert_Nil_None: fixes e1 :: "'c \<times> 'd \<times> bool"   and e2 :: "'c \<times> 'd \<times> bool"   and i :: "'c"   and i2 :: "'c option" assumes "fst e1 \<noteq> fst e2"   and "i \<noteq> fst e2"   and "i2 \<noteq> Some (fst e1)" shows "insert [] e2 i2 \<bind> (\<lambda>ys. insert ys e1 (Some i)) = None"
set_thread_value_invariant: fixes s :: "state_t"   and thrst :: "thread_id_t \<Rightarrow> thread_t" shows "atomic_step_invariant s = atomic_step_invariant (s\<lparr>thread := thrst\<rparr>)"
fps_to_fls_base_factor_to_fps: fixes f :: "'a fls" shows "fls_regpart_as_fls (fls_base_factor f) = fls_base_factor f"
insertion_fun_irrelevant_vars: fixes p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>m v. \<lbrakk>p m \<noteq> (0::'b); lookup m v \<noteq> 0\<rbrakk> \<Longrightarrow> f v = g v" shows "insertion_fun f p = insertion_fun g p"
npow_suc: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and n :: "nat" shows "x^\<^bsup>R Suc n\<^esup> = x^\<^bsup>R n\<^esup> \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
88: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - y \<sqinter> x = x"
unit_disc_fix_iff_ounit_circle: fixes M :: "moebius" shows "unit_disc_fix M = (moebius_ocircline M ounit_circle = ounit_circle)"
constant_rep_alt: fixes x1 :: "'a"   and r2 :: "int"   and x :: "'a" assumes "x1 \<in> \<V>"   and "\<B> rep x1 = r2"   and "x \<in> \<V>" shows "\<B> rep x = r2"
complement_sum_in_product_complement: fixes x :: "i"   and y :: "i" assumes "CEMC"   and "O x y"   and "x \<noteq> u"   and "y \<noteq> u" shows "P (\<midarrow> x \<oplus> \<midarrow> y) (\<midarrow> (x \<otimes> y))"
col_mat: fixes j :: "nat"   and nc :: "nat"   and nr :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "j < nc" shows "col (Matrix.mat nr nc f) j = Matrix.vec nr (\<lambda>i. f (i, j))"
ccw'_not_coll_2: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "det3 a c b \<noteq> 0"
prems_6: shows "getWatch2 stateA = getWatch2 stateB"
bezout_matrix_def: fixes A :: "(('d, 'f) vec, 'e) vec"   and a :: "'e"   and b :: "'e"   and j :: "'f"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "bezout_matrix A a b j bezout = (\<chi>x y. let (p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j) in if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then 1::'d else (0::'d))"
scc_root_scc_tree_trancl: fixes v :: "'v"   and scc :: "'v set"   and x :: "'v" assumes "scc_root s v scc"   and "x \<in> scc"   and "x \<in> dom (discovered s)"   and "x \<noteq> v" shows "(v, x) \<in> (tree_edges s)\<^sup>+"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (lowerPowers x xa xb) = (\<lambda>x. if x \<in> {x..<x + xa} then 0::'b else lookup xb (dec_above x xa x))"
disjointed_mono: fixes A :: "nat \<Rightarrow> 'b set"   and n :: "nat" assumes "incseq A" shows "disjointed A (Suc n) = A (Suc n) - A n"
b: fixes x :: "'a" assumes "x \<in> A" shows "bij_betw (b x) (F x) B"
assoc'_is_natural_1: 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 \<mu> :: "'c"   and \<nu> :: "'c"   and \<tau> :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "bicategory.\<a>' V H \<a> src trg \<mu> \<nu> \<tau> = V (H (H \<mu> \<nu>) \<tau>) (bicategory.\<a>' V H \<a> src trg (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
add_add_bot: fixes x :: "'a"   and y :: "'a" shows "x + y + bot = x + y"
assms: shows "lift n xs \<Sigma>1 = Ok (ys, \<Sigma>2)"
relative_to_def: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "P relative_to S \<equiv> \<lambda>T. \<exists>U. P U \<and> S \<inter> U = T"
2: shows "v1 \<noteq> v \<or> v2 \<noteq> v'"
except_keys: fixes p :: "'a \<Rightarrow>\<^sub>0 'b" shows "except p (keys p) = 0"
combine_rabin_pairs_UNIV: fixes \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and fin :: "('b \<times> 'c \<times> 'b) set"   and w :: "nat \<Rightarrow> 'c"   and fin' :: "('b \<times> 'c \<times> 'b) set"   and inf' :: "('b \<times> 'c \<times> 'b) set set" assumes "accepting_pair\<^sub>R \<delta> q\<^sub>0 (fin, UNIV) w"   and "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (fin', inf') w" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (fin \<union> fin', inf') w"
good_item_simps_78: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and y1 :: "'var" assumes "goodEnv rhoa"   and "freshEnv ys y rhoa" shows "rhoa &[y1 // y]_ys = rhoa [y \<leftarrow> Var ys y1]_ys"
diff_nat_eq_if: fixes z :: "int"   and z' :: "int" shows "nat z - nat z' = (if z' < 0 then nat z else let d = z - z' in if d < 0 then 0 else nat d)"
assms: shows "g = (\<lambda>i. cblinfun_apply (f i))"
existence_ivl_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "t0 \<in> existence_ivl t0 x0"
P1: shows "Col P C D \<and> \<not> Col P A B"
range_conc: fixes w\<^sub>1 :: "'c list"   and w\<^sub>2 :: "nat \<Rightarrow> 'c" shows "range (w\<^sub>1 \<frown> w\<^sub>2) = set w\<^sub>1 \<union> range w\<^sub>2"
simps_19: fixes f1 :: "ipc_direction_t \<Rightarrow> ipc_stage_t \<Rightarrow> thread_id_t \<Rightarrow> page_t \<Rightarrow> 'a"   and f2 :: "ev_wait_stage_t \<Rightarrow> ev_consume_t \<Rightarrow> 'a"   and f3 :: "ev_signal_stage_t \<Rightarrow> thread_id_t \<Rightarrow> 'a"   and f4 :: "'a" shows "(case NONE of SK_IPC x xa xb xc \<Rightarrow> f1 x xa xb xc | SK_EV_WAIT x xa \<Rightarrow> f2 x xa | SK_EV_SIGNAL x xa \<Rightarrow> f3 x xa | NONE \<Rightarrow> f4) = f4"
rel_generat_continuationD: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Out :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and x :: "('g, 'i, 'k) generat"   and y :: "('h, 'j, 'l) generat" assumes "rel_generat A Out R x y"   and "\<not> is_Pure x \<or> \<not> is_Pure y" shows "R (continuation x) (continuation y)"
Cod_comp: fixes f :: "('o, 'a) arr"   and g :: "('o, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Cod (COMP g f) = Cod g"
r_lifz: fixes xs :: "nat list"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and c :: "nat" assumes "length xs = n"   and "recfn n f"   and "recfn n g" shows "eval (r_lifz f g) (c # xs) = (if c = 0 then eval f xs else eval g xs)"
card_bind_constant: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set"   and k :: "nat" assumes "finite S"   and "\<forall>X\<in>S. finite (f X)"   and "disjoint_family_on f S"   and "\<And>x. x \<in> S \<Longrightarrow> card (f x) = k" shows "card (S \<bind> f) = card S * k"
simps_5: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Exists \<phi>) j = Monitor.progress \<sigma> \<phi> j"
card_of_leq: fixes A :: "'s set" assumes "A \<subseteq> S.Univ" shows "|A| <o cardSuc (cmax |UNIV| natLeq)"
M: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"
typed: shows "\<Gamma> \<turnstile>\<^sub>b e \<in> Low "
th': shows "real_of_int (fst (zsplit0 t)) * real_of_int x = real_of_int (fst (zsplit0 t) * x)"
R: shows "continuous_map X X' f \<and> f ` topspace X \<subseteq> S"
diff_le_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (minus a b) c = less_eq a (plus c b)"
hoare_pre: fixes R :: "'e"   and P :: "'e"   and S :: "'f \<Rightarrow> 'e"   and Q :: "'f" assumes "R \<sqsubseteq> P"   and "\<Turnstile> P{| S |}Q" shows "\<Turnstile> R{| S |}Q"
function_ring_car_eqI: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F"   and "g \<in> carrier F"   and "\<And>a. a \<in> S \<Longrightarrow> f a = g a" shows "f = g"
sd_pos: fixes V :: "'a set"   and x :: "'a" assumes "V \<in> \<C>"   and "x \<in> S"   and "x \<in> V" shows "0 < setdist {x} (S - V)"
f21: shows "f21 = Face (u # between (vertices f) u v @ v # ws) Nonfinal"
path_inf_priorities_ltl: fixes P :: "'a llist" shows "path_inf_priorities P = path_inf_priorities (ltl P)"
regular_closed_bot: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "bot = uminus (uminus bot)"
subst_frees: fixes env :: "(name, 'a) fmap"   and t :: "'a" assumes "closed_env env" shows "frees (subst t env) = frees t |-| fmdom env"
hyps_3: shows "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> Abrupt s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> Normal s' \<and> (\<Gamma>\<turnstile> \<langle>c2,Normal s'\<rangle> \<Rightarrow>\<notin>Fault ` F \<longrightarrow> strip F \<Gamma>\<turnstile>c2 \<down> Normal s')"
set: fixes x :: "'a option cdecl list" shows "set_prog (Program x) = (\<Union>uu\<in>set x. \<Union>uu\<in>Basic_BNFs.snds uu. \<Union>uu\<in>Basic_BNFs.snds uu. \<Union>uu\<in>Basic_BNFs.snds uu. \<Union>uu\<in>set uu. \<Union>uu\<in>Basic_BNFs.snds uu. \<Union>uu\<in>Basic_BNFs.snds uu. \<Union> (set_option ` Basic_BNFs.snds uu))"
i_set0_cut_closed_disj: fixes I :: "nat set"   and cut_op :: "nat set \<Rightarrow> nat \<Rightarrow> nat set"   and t :: "nat" assumes "I \<in> i_set0"   and "cut_op = (\<down>\<le>) \<or> cut_op = (\<down>\<ge>) \<or> cut_op = (\<down><) \<or> cut_op = (\<down>>)" shows "cut_op I t \<in> i_set0"
cutX: shows "S \<inter> (insert e X \<union> Y) = {}"
mem_affine: fixes S :: "'c set"   and x :: "'c"   and y :: "'c"   and u :: "real"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
assms_1: shows "b \<in>\<^sub>\<circ> A"
equivpartp_sympartp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equivpartp (sympartp r) = equivpartp r"
61103: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- - x \<sqinter> y) \<squnion> z = - x \<squnion> (- y \<squnion> z)"
exhaust: fixes y :: "'l \<times> 'm" obtains x1 :: "'l" and x2 :: "'m" where "y = (x1, x2)"
finvimageD: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)"
simply_connected_eq_contractible_loop_some: fixes S :: "'a set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> (\<exists>a. a \<in> S \<and> homotopic_loops S p (linepath a a))))"
uthird_gt0: shows "0 < real_of_float (rapprox_rat (max prec 1) 1 3) + 1"
INF_top: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf ((\<lambda>x. top) ` A) = top"
assms: shows "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> t"
simps_6: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Neg \<phi>) j = Monitor.progress \<sigma> P \<phi> j"
Cauchy_inequality: fixes f :: "complex \<Rightarrow> complex"   and \<xi> :: "complex"   and r :: "real"   and B :: "real"   and n :: "nat" assumes "f holomorphic_on ball \<xi> r"   and "continuous_on (cball \<xi> r) f"   and "0 < r"   and "\<And>x. cmod (\<xi> - x) = r \<Longrightarrow> cmod (f x) \<le> B" shows "cmod ((deriv ^^ n) f \<xi>) \<le> fact n * B / r ^ n"
assms_2: shows "\<forall>G. \<not> c_sinvar m G \<longrightarrow> (\<exists>OFF. c_offending_flows m G = {OFF})"
finite_DF: fixes k :: "nat"   and i :: "nat" shows "finite (DF k i)"
vals_nonequiv: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and i :: "nat"   and j :: "nat" assumes "vals_nonequiv K (Suc (Suc n)) vv"   and "i \<le> Suc (Suc n)"   and "j \<le> Suc (Suc n)"   and "i \<noteq> j" shows "\<not> v_equiv K (vv i) (vv j)"
triangular_columnD: fixes j :: "nat"   and A :: "'a mat"   and i :: "nat" assumes "triangular_column j A"   and "j < i"   and "i < dim_row A" shows "A $$ (i, j) = (0::'a)"
hs_remove_max_multiset: fixes m :: "'c"   and t' :: "'c Tree"   and t :: "'c Tree" assumes "(m, t') = hs_remove_max t"   and "t \<noteq> E" shows "{#m#} + Heap.multiset t' = Heap.multiset t"
ce_union: fixes A :: "nat set"   and B :: "nat set" assumes "A \<in> ce_sets"   and "B \<in> ce_sets" shows "A \<union> B \<in> ce_sets"
not_inf_implies_wf: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate" assumes "\<not> AlternativeSmallStep.inf \<Gamma> cs css s" shows "wf {(c2, c1). \<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* c1 \<and> \<Gamma>\<turnstile> c1 \<rightarrow> c2}"
max_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" shows "ord.max less_eq a b = (if less_eq a b then b else a)"
inside_def: fixes S :: "'b set" shows "inside S \<equiv> {x. x \<notin> S \<and> bounded (connected_component_set (- S) x)}"
refusals_def: fixes P :: "'e process"   and xs :: "'e list" shows "refusals P xs \<equiv> failures P `` {xs}"
simps_1: fixes bl :: "('b \<times> 'b trm) list" shows "[] \<lozenge> bl = bl"
\<mu>_ninf: fixes v :: "'a" assumes "v \<in> verts G"   and "dist v = - \<infinity>" shows "\<mu> c s v = - \<infinity>"
i_shrink_eq_Msg_iEx_iAll_cut_greater_conv: fixes k :: "nat"   and m :: "'b message_af"   and s :: "nat \<Rightarrow> 'b message_af"   and t :: "nat" assumes "0 < k"   and "m \<noteq> NoMsg" shows "((s \<div> k) t = m) = (\<diamond> t1 [t * k\<dots>,k - Suc 0]. s t1 = m \<and> (\<box> t2 [t * k\<dots>,k - Suc 0] \<down>> t1. s t2 = NoMsg))"
landau_symbol_ge_3_cong: fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "\<And>x. (3::'a) \<le> x \<Longrightarrow> f x = g x" shows "L at_top f = L at_top g"
conv_radius_cong: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in sequentially. norm (f x) = norm (g x)" shows "conv_radius f = conv_radius g"
xs_dec: shows "a @ [xs ! index xs x] @ b @ [xs ! index xs y] @ c = xs"
lset_ltake_ltl: fixes n :: "nat"   and xs :: "'c llist" shows "lset (ltake (enat n) (ltl xs)) \<subseteq> lset (ltake (enat (Suc n)) xs)"
hyps_2: shows "(p \<circ>\<^sub>p q = 0) = (p = 0)"
surj_def: fixes f :: "'c \<Rightarrow> 'b" shows "top_pres ((`) f) = (\<forall>y. \<exists>x. y = f x)"
simps_2: fixes e :: "'b"   and h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "prefixBeforeElement e (h # t) = (if element h = e then [] else h # prefixBeforeElement e t)"
fset_mp: fixes A :: "'c fset"   and B :: "'c fset"   and c :: "'c" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
from_hma_to_hma\<^sub>v: fixes v :: "'d Matrix.vec" assumes "v \<in> carrier_vec CARD('e)" shows "HMA_Connect.from_hma\<^sub>v (HMA_Connect.to_hma\<^sub>v v) = v"