hcju/isabelleps huggingface dataset:
l8_14_2_1a: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "A B Perp C D"
bidir_1: shows "\<exists>rev. bidirected_digraph G rev"
push_down_state_rank_token_run: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = Some i" shows "\<exists>x. semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n = q \<and> x \<le> n"
Ipoly_num_params_cong: fixes p :: "'a poly"   and xs :: "'a list"   and ys :: "'a list" assumes "\<And>i. i < num_params p \<Longrightarrow> xs ! i = ys ! i" shows "Ipoly xs p = Ipoly ys p"
measurable_trace_at': shows "(\<lambda>((s, j), \<omega>). trace_at s \<omega> j) \<in> (count_space UNIV \<Otimes>\<^sub>M borel) \<Otimes>\<^sub>M stream_space S \<rightarrow>\<^sub>M count_space UNIV"
split_part: fixes P :: "bool"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<lambda>(a, b). P \<and> Q a b) = (\<lambda>ab. P \<and> (case ab of (x, xa) \<Rightarrow> Q x xa))"
interpret_Unop_2: shows "interpret_floatarith Unop' (interpret_slp slp xs) = unop xs"
add_One: fixes x :: "num" shows "x + num.One = Num.inc x"
simps_1: fixes ys :: "'c list" shows "append [] ys = ys"
phi_complemented: fixes x :: "'c regular" shows "complement (stone_phi x) (stone_phi (- x))"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<exists>F G. inverse_functors A B F G" shows "isomorphic_categories_axioms A B"
Rep_real_int: fixes x :: "real_int" shows "Rep_real_int x \<in> {r. fst r \<le> snd r}"
poincare_mapsto_congI1: fixes A :: "'a set"   and B :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and C :: "'a set"   and D :: "'a set"   and E :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" assumes "poincare_mapsto A B C D E"   and "closed A" shows "c1_on_open_euclidean.poincare_mapsto g Y A B C D E"
bij_\<phi>: shows "bij_betw \<psi> S'.Univ S.Univ"
tendsto_0_le: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'c"   and K :: "real" assumes "(f \<longlongrightarrow> (0::'b)) F"   and "\<forall>\<^sub>F x in F. norm (g x) \<le> norm (f x) * K" shows "(g \<longlongrightarrow> (0::'c)) F"
exhaust_sel: fixes tllist :: "('a, 'b) tllist" obtains "tllist = TNil (terminal tllist)"   | "tllist = TCons (thd tllist) (ttl tllist)"
set_takeWhileD: fixes x :: "'d"   and P :: "'d \<Rightarrow> bool"   and xs :: "'d list" assumes "x \<in> set (takeWhile P xs)" shows "x \<in> set xs \<and> P x"
order_iff_strict: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b \<subseteq># a) = (b \<subset># a \<or> a = b)"
log2_of_power_less: fixes m :: "nat"   and n :: "nat" assumes "m < 2 ^ n"   and "0 < m" shows "log 2 (real m) < real n"
zero_le_floor: fixes x :: "'d" shows "(0 \<le> \<lfloor>x\<rfloor>) = ((0::'d) \<le> x)"
eval_red_Tensor: fixes a :: "'c local.term"   and b :: "'c local.term" assumes "Ide a"   and "Ide b" shows "\<lbrace>(a \<^bold>\<otimes> b)\<^bold>\<down>\<rbrace> = \<lbrace>\<^bold>\<lfloor>a\<^bold>\<rfloor> \<^bold>\<Down> \<^bold>\<lfloor>b\<^bold>\<rfloor>\<rbrace> \<cdot> (\<lbrace>a\<^bold>\<down>\<rbrace> \<otimes> \<lbrace>b\<^bold>\<down>\<rbrace>)"
build_succ_impl_def: shows "build_succ_impl \<equiv> \<lambda>x. LIST_FOREACH' (RETURN (id x)) Impl_Array_Map.iam_invar (\<lambda>xa. LIST_FOREACH' (RETURN (it_to_list ((map_iterator_dom \<circ>\<circ>\<circ> (\<circ>)) foldli rbt_to_list) (incoming_impl xa))) Impl_Array_Map.iam_invar (\<lambda>xc xd. if xc = 0 then RETURN xd else RETURN (Impl_Array_Map.iam_update xc (glist_insert (=) (name_impl xa) (the_default [] (Impl_Array_Map.iam_lookup xc xd))) xd))) (Impl_Array_Map.iam_empty ())"
elementsButlastTrailIsButlastElementsTrail: fixes M :: "('b \<times> bool) list" shows "elements (butlast M) = butlast (elements M)"
lin: fixes U :: "'a set" assumes "U \<in> \<U>" shows "Lindelof_space (subtopology X U)"
dist_norm: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "class.dist_norm minus dist norm" shows "dist x y = norm (minus x y)"
z: shows "\<forall>w. P w z = (\<not> O w x)"
choice_CBasis_iff: fixes P :: "'a \<Rightarrow> complex \<Rightarrow> bool" shows "(\<forall>i\<in>CBasis. \<exists>x. P i x) = (\<exists>x. \<forall>i\<in>CBasis. P i \<langle>x, i\<rangle>)"
up_a_inv_closed: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a" assumes "ring R"   and "p \<in> up R" shows "(\<lambda>i. \<ominus>\<^bsub>R\<^esub> p i) \<in> up R"
finite_keys: fixes f :: "'a \<Rightarrow>\<^sub>0 'b" shows "finite (keys f)"
map_poly_pCons: fixes c :: "'c"   and p :: "'c poly"   and f :: "'c \<Rightarrow> 'd" assumes "c \<noteq> (0::'c) \<or> p \<noteq> 0" shows "map_poly f (pCons c p) = pCons (f c) (map_poly f p)"
map_of_join_raw: fixes ys :: "('b \<times> 'c) list"   and f :: "'b \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list"   and x :: "'b" assumes "distinct (map fst ys)" shows "map_of (join_raw f xs ys) x = (case map_of xs x of None \<Rightarrow> map_of ys x | Some v \<Rightarrow> case map_of ys x of None \<Rightarrow> Some v | Some v' \<Rightarrow> Some (f x (v, v')))"
one_def: shows "1\<^sub>h \<equiv> abs_complex_homo 1\<^sub>h\<^sub>c"
must_sync_def2: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap" shows "execd_mthr.must_sync P t x m = (\<exists>taa x' m' s. execd_mthr.r_syntax P t x m taa x' m' \<and> exec_mthr.actions_ok s t taa)"
match_supp_fst: fixes p :: "pat"   and u :: "trm"   and \<theta> :: "(name \<times> trm) list" assumes "\<turnstile> p \<rhd> u \<Rightarrow> \<theta>" shows "supp (map fst \<theta>) = supp p"
f_Exec_Stream_expand_aggregate_map_nth_eq_i_nth: fixes k :: "nat"   and n :: "nat"   and n' :: "nat"   and f :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e"   and ag :: "'d list \<Rightarrow> 'd" assumes "0 < k"   and "n < n'" shows "f_aggregate (map f (f_Exec_Comp_Stream trans_fun (input \<Down> n' \<odot> k) c)) k ag ! n = i_aggregate (f \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) k ag n"
sum_subtractf: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Sum>x\<in>A. f x - g x) = sum f A - sum g A"
mult_less_cancel_right1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times b c) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
leq_gen_cong: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr" shows "R \<le> cong.gen_cong rel eval R"
assms_7: shows "InvariantCEntailed (getConflictFlag state) F0 (getC state)"
Local: fixes P :: "state \<Rightarrow> bool"   and P' :: "state \<Rightarrow> state \<Rightarrow> bool"   and f :: "state \<Rightarrow> state"   and c :: "com"   and Q :: "state \<Rightarrow> bool"   and g :: "state \<Rightarrow> state \<Rightarrow> state" assumes "\<And>s. P s \<Longrightarrow> P' s (f s)"   and "\<forall>p. \<turnstile>\<^sub>t {P' p} c {Q \<circ> g p}" shows "\<turnstile>\<^sub>t {P} LOCAL f; c; g {Q}"
carrier_mat_triv: fixes m :: "'a Matrix.mat" shows "m \<in> carrier_mat (dim_row m) (dim_col m)"
pirreducibleE_3: fixes K :: "'a list set"   and p :: "'a list list"   and q :: "'a list list"   and r :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "pirreducible\<^bsub>K [X]\<^esub> K p"   and "q \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "r \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)"   and "p = q \<otimes>\<^bsub>K [X]\<^bsub>K [X]\<^esub>\<^esub> r" shows "q \<in> Units (K [X]\<^bsub>K [X]\<^esub>) \<or> r \<in> Units (K [X]\<^bsub>K [X]\<^esub>)"
gf: shows "quotient_map X X'' (g \<circ> f)"
rBang: fixes \<alpha> :: "act"   and P' :: "ccs"   and \<C> :: "'a" assumes "P \<parallel> !P \<longmapsto>\<alpha> \<prec> P'"   and "\<And>\<C>. Prop \<C> (P \<parallel> !P) \<alpha> P'" shows "Prop \<C> !P \<alpha> P'"
rep_eq: fixes x :: "int"   and xa :: "float" shows "real_of_float (float_up x xa) = round_up x (real_of_float xa)"
integrable_Un: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible (A \<inter> B)"   and "f integrable_on A"   and "f integrable_on B" shows "f integrable_on A \<union> B"
lm061: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G \<subseteq> allocationsUniverse \<and> allAllocations N G \<subseteq> {a. Domain a \<subseteq> N \<and> \<Union> (Range a) = G}"
check_plan_def: fixes \<pi>s :: "char list list" shows "check_plan \<pi>s = (case simulate_plan \<pi>s I of None \<Rightarrow> False | Some s' \<Rightarrow> s' \<in> G)"
berlekamp_mat_def: fixes u :: "'a mod_ring poly" shows "berlekamp_mat u = (let n = degree u; mul_p = power_poly_f_mod u [:0, 1:] CARD('a); xks = power_polys mul_p u 1 n in mat_of_rows_list n (map (\<lambda>cs. let coeffs_cs = coeffs cs; k = n - length (coeffs cs) in coeffs cs @ replicate k 0) xks))"
induced_subgraphI_arc_mono: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and H :: "('a, 'b) pre_digraph" assumes "max_subgraph P H"   and "arc_mono P" shows "induced_subgraph H G"
less_top_Value_Top_code: fixes x :: "'a" shows "(top.Value x < Top) = True"
\<xi>_def: shows "\<xi> = (\<Sum>k. bb k / aa k)"
subgroupE_1: fixes H :: "'a set" assumes "subgroup H B" shows "H \<subseteq> carrier B"
Subseqs_def: fixes A :: "'a list set" shows "Subseqs A = (\<Union>w\<in>A. set (subseqs w))"
integral_fst': fixes f :: "'a \<times> 'b \<Rightarrow> 'd" assumes "integrable (M1 \<Otimes>\<^sub>M M2) f" shows "LINT x|M1. LINT y|M2. f (x, y) = integral\<^sup>L (M1 \<Otimes>\<^sub>M M2) f"
finite_induct: fixes F :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
elims_1: fixes x :: "'c \<Rightarrow> 'd list"   and xa :: "'d \<Rightarrow> 'c"   and xb :: "nat \<times> nat list \<times> 'd list"   and y :: "bool" assumes "stab_inv x xa xb = y" obtains f :: "'c \<Rightarrow> 'd list" and key :: "'d \<Rightarrow> 'c" and u :: "nat" and ns :: "nat list" and xs :: "'d list" where "x = f" and "xa = key" and "xb = (u, ns, xs)" and "y = (\<forall>k. filter (\<lambda>x. key x = k) xs = f k)"
nomulti_digraph_def: fixes G :: "('c, 'd) pre_digraph" shows "nomulti_digraph G \<equiv> wf_digraph G \<and> nomulti_digraph_axioms G"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_erase2 m rs (l, r) = (\<exists>ln rn. l = Bk \<up> ln @ Bk # Oc # Oc \<up> Suc rs @ Bk # Oc \<up> Suc m \<and> tl r = Bk \<up> rn)"
AB: shows "\<forall>x\<in>A. \<exists>y\<in>B. x \<le> y"
funpow_add: fixes m :: "nat"   and n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
Constrains_UNIV: fixes F :: "'b program"   and A :: "'b set" shows "F \<in> A Co UNIV"
hom: fixes g :: "'v"   and g' :: "'v" assumes "g \<in> V"   and "g' \<in> V" shows "T (g + g') = T g + T g'"
unat_div: fixes x :: "'b word"   and y :: "'b word" shows "unat (x div y) = unat x div unat y"
tyenv_eq_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.tyenv_eq ev\<^sub>B dma_type \<Gamma> mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. sifum_types.type_max ev\<^sub>B (sifum_types.to_total dma_type \<Gamma> x) mem\<^sub>1 = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
insert_list_def: fixes x :: "'b"   and xs :: "'b list" shows "insert_list x xs = (if x \<in> set xs then xs else x # xs)"
Jordan_Brouwer_frontier: fixes S :: "'a set"   and a :: "'a"   and r :: "real"   and T :: "'a set" assumes "S homeomorphic sphere a r"   and "T \<in> components (- S)"   and "2 \<le> DIM('a)" shows "frontier T = S"
iter_dist_nodes: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list"   and v :: "nat"   and m :: "nat"   and q :: "nat"   and p :: "nat" assumes "wf_tr M T"   and "wf_dfa M v"   and "\<forall>p q. dfa_is_node M q \<and> p < q \<longrightarrow> tr_lookup T q p = (\<exists>n<m. dist_nodes M n v p q)"   and "0 < m"   and "dfa_is_node M q"   and "p < q" shows "tr_lookup (snd (iter M T)) q p = (\<exists>n<Suc m. dist_nodes M n v p q)"
if_image_distrib: fixes P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd"   and S :: "'e set" shows "(\<lambda>x. if P x then f x else g x) ` S = f ` (S \<inter> {x. P x}) \<union> g ` (S \<inter> {x. \<not> P x})"
r_prod_encode: fixes m :: "nat"   and n :: "nat" shows "eval r_prod_encode [m, n] \<down>= prod_encode (m, n)"
simps_2: fixes e :: "('a, 'b, 'c) exp"   and es :: "('a, 'b, 'c) exp list" shows "max_varss (e # es) = max (max_vars e) (max_varss es)"
wf_rexp_of': fixes \<Sigma> :: "'b list"   and n :: "nat"   and \<phi> :: "'b formula" assumes "formula \<Sigma>"   and "formula.wf_formula \<Sigma> n \<phi>" shows "alphabet.wf (wf_atom \<Sigma>) n (formula.rexp_of' \<Sigma> n \<phi>)"
vproduct_vdoubletonD_5: fixes f :: "V"   and a :: "V"   and b :: "V"   and A :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> vproduct (set {a, b}) A" shows "f = set {\<langle>a, f\<lparr>a\<rparr>\<rangle>, \<langle>b, f\<lparr>b\<rparr>\<rangle>}"
lookup_row_to_poly_not_zeroE: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ts :: "'g list"   and r :: "'f vec"   and t :: "'g" assumes "term_powerprod pair_of_term term_of_pair"   and "lookup (row_to_poly ts r) t \<noteq> (0::'f)" obtains i :: "nat" where "i < length ts" and "t = ts ! i"
subdistl: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "z ; x \<le> z ; (x + y)"
add_rat: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
steps_imp_seq: fixes a :: "'a"   and b :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(a, b) \<in> (unlabel ars)\<^sup>*" shows "\<exists>ss\<in>seq ars. fst ss = a \<and> lst ss = b"
desargues_config_2D_non_collinear_1: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, A'} = 3"
jkbpC_prefix_closed: fixes t :: "'s Trace"   and s :: "'s" assumes "t \<leadsto> s \<in> jkbpC" shows "t \<in> jkbpC"
inducts: fixes A :: "'c ring"   and B :: "'c ring" assumes "A \<lesssim> B" obtains "id \<in> ring_hom A B"
emeasure_Diff: fixes M :: "'a measure"   and B :: "'a set"   and A :: "'a set" assumes "emeasure M B \<noteq> \<infinity>"   and "A \<in> sets M"   and "B \<in> sets M"   and "B \<subseteq> A" shows "emeasure M (A - B) = emeasure M A - emeasure M B"
wp_mono: fixes A :: "'b set"   and B :: "'b set"   and act :: "('b \<times> 'b) set" assumes "A \<subseteq> B" shows "wp act A \<subseteq> wp act B"
ft_dghm_v11_on_Hom: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "v11 (\<FF>\<lparr>ArrMap\<rparr> \<restriction>\<^sup>l\<^sub>\<circ> Hom \<AA> a b)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vrat_mult"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vrat_mult" shows "b = c"
demand_suffices': fixes f :: "C \<rightarrow> 'a"   and n :: "nat" assumes "demand f = C\<^bsup>n\<^esup>" shows "f\<cdot>(demand f) \<noteq> \<bottom>"
ord_le_eq_trans: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
lfilter_repeat: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "lfilter P (repeat x) = (if P x then repeat x else LNil)"
arrE: fixes f :: "'a" assumes "Right_a.arr f" shows "right a f"
simps_2: fixes x :: "'b"   and AB :: "'b tree"   and CD :: "'b tree" shows "splay x \<langle>AB, x, CD\<rangle> = \<langle>AB, x, CD\<rangle>"
wf_TVarB: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" shows "TVarB T \<Colon> \<Gamma> \<turnstile>\<^sub>w\<^sub>f"
smap_siterate: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "smap f (siterate f x) = siterate f (f x)"
blinding_of_hash_R1: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "rel_list_R1 bo \<le> vimage2p (map_list_R1 h) (map_list_R1 h) (=)"
set_integrable_bigo: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "f \<in> O(g)"   and "set_integrable lborel {a..} g"   and "\<And>b. a \<le> b \<Longrightarrow> set_integrable lborel {a..<b} f"   and "set_borel_measurable borel {a..} f" shows "set_integrable lborel {a..} f"
sum_nonneg_leq_bound: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'c"   and i :: "'d" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite s"   and "\<And>i. i \<in> s \<Longrightarrow> less_eq zero (f i)"   and "comm_monoid_add.sum plus zero f s = B"   and "i \<in> s" shows "less_eq (f i) B"
execute_trap_result: fixes s :: "'b sparc_state" assumes "\<not> (reset_trap_val s = False \<and> get_ET (cpu_reg_val PSR s) = 0)" shows "snd (execute_trap () s) = False"
exp_integer_2pi_plus1: fixes n :: "real" assumes "n \<in> \<int>" shows "exp (complex_of_real ((2 * n + 1) * pi) * \<i>) = - 1"
ats_val_process_adapted: fixes p :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" assumes "trading_strategy p"   and "support_adapt Mkt p" shows "borel_adapt_stoch_proc G (val_process Mkt p)"
a_factorgroup_is_comm_group: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "abelian_subgroup H G" shows "comm_group (G A_Mod H)"
count_single: fixes b :: "'b"   and a :: "'b" shows "count {#b#} a = (if b = a then 1 else 0)"
bool_equiv_test_def: shows "bool_equiv_test \<equiv> equiv_test ifex_of"
rt_strictly_fresher_singleI: fixes rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and i :: "nat"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "the (rt1 i) \<sqsubseteq> the (rt2 i)"   and "\<not> rt1 \<approx>\<^bsub>i\<^esub> rt2" shows "rt1 \<sqsubset>\<^bsub>i\<^esub> rt2"
has_integral_iff_nn_integral_lebesgue: fixes f :: "'a \<Rightarrow> real"   and r :: "real" assumes "\<And>x. 0 \<le> f x" shows "(f has_integral r) UNIV = (f \<in> borel_measurable lebesgue \<and> \<integral>\<^sup>+ x. ennreal (f x) \<partial>lebesgue = ennreal r \<and> 0 \<le> r)"
\<phi>_\<psi>: fixes b :: "'d"   and a :: "'d"   and x :: "(('d + ('j \<Rightarrow> 'd)) set, 'd + ('j \<Rightarrow> 'd) \<Rightarrow> 'd + ('j \<Rightarrow> 'd)) concrete_category.arr" assumes "D.ide b"   and "D.ide a"   and "x \<in> G\<chi>.Hom.set (b, a)" shows "(replete_setcat.UP \<circ> G\<chi>.inl) (G\<chi>.\<psi> (b, a) x) = x"
ins_height_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Unbal t'" shows "height t \<le> height t' \<and> height t' \<le> height t + 1"
A: shows "\<forall>\<^sub>F x in at x\<^sub>0. sgn (poly (p * pderiv p) x) = (if x\<^sub>0 < x then 1 else - 1)"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree"   and xs :: "'b list" shows "braun_list \<langle>l, x, r\<rangle> xs = (xs \<noteq> [] \<and> x = hd xs \<and> braun_list l (take_nths 1 1 xs) \<and> braun_list r (take_nths 2 1 xs))"
span_def: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and S :: "'c set" assumes "Module.module R M" shows "LinearCombinations.module.span R M S = {module.lincomb M a A |a A. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R}"
restrict_top_ifex_minimal_invar: fixes i :: "'a ifex"   and a :: "'a"   and val :: "bool" assumes "ifex_minimal i" shows "ifex_minimal (restrict_top i a val)"
mset_eq_length: fixes xs :: "'k list"   and ys :: "'k list" assumes "mset xs = mset ys" shows "length xs = length ys"
deg_fun_def: fixes s :: "'d \<Rightarrow> 'c" shows "deg_fun s \<equiv> sum s (supp_fun s)"
Int_Diff: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b"   and B :: "'a set" assumes "finite A" shows "prod g A = prod g (A \<inter> B) * prod g (A - B)"
simps_2: fixes n :: "nat"   and A :: "'b list set" shows "A ^^ Suc n = A @@ A ^^ n"
has_subprob_densityI: fixes f :: "'b \<Rightarrow> ennreal"   and N :: "'b measure"   and M :: "'b measure" assumes "f \<in> borel_measurable N"   and "M = density N f"   and "subprob_space M" shows "has_subprob_density M N f"
no_isolated_singularity': fixes K :: "complex set"   and f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "\<And>z. z \<in> K \<Longrightarrow> (f \<longlongrightarrow> f z) (at z within S)"   and "f holomorphic_on S - K"   and "open S"   and "finite K" shows "f holomorphic_on S"
stat_def: fixes C :: "'s option set" shows "K2.stat C = point_measure UNIV (\<lambda>x. indicator C x / K2.U' x x)"
o_inv_o_cancel: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "inj f" shows "g \<circ> inv f \<circ> f = g"
\<mu>\<nu>: shows "VV.arr \<mu>\<nu>"
nth_prime_upper_bound: fixes n :: "nat" assumes "0 < n" shows "real (nth_prime (n - 1)) < 12 * (real n * ln (real n) + real n * ln (12 / exp 1))"
obj_char: fixes a :: "'a" shows "S.obj a = (S.arr a \<and> B.obj a)"
mk_eqcl_fixpt_fst_bound: fixes M :: "nat bdd list \<times> bool list"   and i :: "nat" assumes "dfa_is_node M i" shows "fst (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))) ! i < length (snd (mk_eqcl (replicate (length (fst M)) None) [] 0 (fixpt M (init_tr M))))"
prime_product: fixes p :: "nat"   and q :: "nat" assumes "prime (p * q)" shows "p = 1 \<or> q = 1"