hcju/isabelleps huggingface dataset:
matrix'2: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'c) vec, 'c) vec"   and f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" assumes "is_basis (set_of_vector X)"   and "is_basis (set_of_vector Y)"   and "\<forall>i. f (X $ i) = (\<Sum>j\<in>UNIV. A $ j $ i *s Y $ j)" shows "matrix' X Y f = A"
pp_pp_inf_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> y = bot) = (- - x \<sqinter> - - y = bot)"
div_gcd_dvd1: fixes a :: "'b"   and b :: "'b" shows "a div gcd a b dvd a"
rel_def: fixes m :: "'b \<Rightarrow> 'b ref" shows "rel m = {(x, y). m x = Ref y}"
nlcons_list: fixes a :: "('a, 'b) Node"   and t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.toList (FingerTreeStruc.nlcons a t) = FingerTreeStruc.nodeToList a @ FingerTreeStruc.toList t"
preserves_reflects_arr: fixes f :: "'c" shows "D.arr (local.map f) = C.arr f"
mn: shows "nat \<bar>a\<bar> = m\<^sup>2 - n\<^sup>2 \<and> nat \<bar>b\<bar> = 2 * m * n \<and> nat \<bar>c\<bar> = m\<^sup>2 + n\<^sup>2 \<and> coprime m n"
simps: fixes a :: "lam" shows "NF a = ((\<exists>ts x. a = Var x \<degree>\<degree> ts \<and> listsp NF ts) \<or> (\<exists>t x. a = (Lam [x].t) \<and> NF t))"
simps_4: fixes p :: "'a pfm"   and q :: "'a pfm" shows "reduce (p \<^bold>\<and>\<^sub>! q) = (reduce p \<^bold>\<and>\<^sub>! reduce q)"
rcosets_H_funcset_M: shows "(\<lambda>C\<in>rcosets H. M1 #> (SOME g. g \<in> carrier G \<and> H #> g = C)) \<in> rcosets H \<rightarrow> M"
bounded_subset_ballI: fixes S :: "'b set"   and x :: "'b"   and r :: "real" assumes "S \<subseteq> ball x r" shows "bounded S"
2: shows "\<exists>i<length (annos C2). m\<^sub>o (annos C2 ! i) (vars (strip C2)) < m\<^sub>o (annos C1 ! i) (vars (strip C2))"
card_eq_dim_span_indep: fixes A :: "('a, 'b) vec set" assumes "vec.dim (vec.span A) = card A"   and "finite A" shows "vec.independent A"
set_less_aux_conv_set_less_eq_aux: fixes A :: "'a set"   and B :: "'a set" shows "(A \<sqsubset>' B) = (A \<sqsubseteq>' B \<and> \<not> B \<sqsubseteq>' A)"
prefix_le_fininf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<le> w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
dBox_choice: fixes P :: "'b hrel"   and Q :: "'b hrel"   and \<Phi> :: "'b upred" shows "\<^bold>[P \<sqinter> Q\<^bold>]\<Phi> = (\<^bold>[P\<^bold>]\<Phi> \<and> \<^bold>[Q\<^bold>]\<Phi>)"
kruskal3_def: shows "kruskal3 \<equiv> obtain_sorted_carrier''' \<bind> (\<lambda>(sl, mn). let initial_union_find = per_init' (mn + 1) in nfoldli sl (\<lambda>_. True) (\<lambda>ce (uf, T). ASSERT (\<alpha> ce \<in> E) \<bind> (\<lambda>_. endpoints ce \<bind> (\<lambda>(a, b). ASSERT (a \<in> Domain uf \<and> b \<in> Domain uf) \<bind> (\<lambda>_. if \<not> per_compare uf a b then let uf = per_union uf a b in ASSERT (ce \<notin> set T) \<bind> (\<lambda>_. RETURN (uf, T @ [ce])) else RETURN (uf, T))))) (initial_union_find, []) \<bind> (\<lambda>(per, spanning_forest). RETURN spanning_forest))"
inf2I: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "A x y"   and "B x y" shows "(A \<sqinter> B) x y"
reductive_inference_system_axioms: shows "reductive_inference_system gd.ord_\<Gamma>"
assms_2: fixes k :: "nat" shows "w k \<preceq>\<^sub>F\<^sub>I v"
MapsToObj_1: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "X \<in> obj\<^bsub>C\<^esub>"
rank_eq_dim_col_space: fixes A :: "(('b, 'c) vec, 'd) vec" shows "rank A = vec.dim (col_space A)"
bmod_auxr: fixes f :: "'d \<Rightarrow> 'c"   and y :: "'d"   and z :: "'e"   and G :: "('e \<Rightarrow> 'c) set" shows "{f y \<cdot> x |x. \<exists>g. x = g z \<and> g \<in> G} = {f y \<cdot> g z |g. g \<in> G}"
num_ni_nz_if: fixes t :: "nat"   and ts :: "nat \<Rightarrow> thread" assumes "t < N"   and "ts t \<noteq> INIT" shows "num_ni ts \<noteq> 0"
simps_3: fixes ux :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f"   and uy :: "'g" shows "ap_sum ux (Inr e) (pure uy) = Inr e"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "P 0"   and "\<And>n. \<lbrakk>\<And>x. x \<in> set_pmf (pmf_of_set {..n}) \<Longrightarrow> P x; \<And>x xa. \<lbrakk>x \<in> set_pmf (pmf_of_set {..n}); xa \<in> set_pmf (rqs_cost x)\<rbrakk> \<Longrightarrow> P (n - x)\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P a0"
seq_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>\<^sub>SC.seq g f = (\<F>\<^sub>SC.arr f \<and> \<F>\<^sub>SC.arr g \<and> \<F>C.seq g f)"
fds_nth_fds': fixes f :: "nat \<Rightarrow> 'b" assumes "f 0 = (0::'b)" shows "fds_nth (fds f) = f"
det3_switch': fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "det3 a b c = - det3 b a c"
connected_def: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Graph G" shows "Graph.connected G v w \<equiv> \<exists>xs. Graph.walk G xs \<and> xs \<noteq> [] \<and> hd xs = v \<and> last xs = w"
hyperpow_zero: fixes n :: "nat star" shows "0 pow (n + 1) = 0"
Not_Rex: fixes F :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "\<turnstile> \<not> (\<exists>x. F x) = (\<forall>x. \<not> F x)"
power2_less_eq_zero_iff: 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 a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times a 2) zero = (a = zero)"
product_is_first_summand: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "\<not> O y z" shows "(x \<oplus> y) \<otimes> (x \<oplus> z) = x"
xs2_def: shows "xs2 = take (Suc hi - lo) (drop lo xs)"
cexpr_typing_cexpr_comp: fixes t1 :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and g :: "cexpr"   and t2 :: "pdf_type"   and f :: "cexpr"   and t3 :: "pdf_type" assumes "case_nat t1 \<Gamma> \<turnstile>\<^sub>c g : t2"   and "case_nat t2 \<Gamma> \<turnstile>\<^sub>c f : t3" shows "case_nat t1 \<Gamma> \<turnstile>\<^sub>c f \<circ>\<^sub>c g : t3"
map_sq_matrix_compose: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'd"   and M :: "('e, 'c) sq_matrix" shows "map_sq_matrix f (map_sq_matrix g M) = map_sq_matrix (\<lambda>x. f (g x)) M"
frrestriction_vsubset_vpairs: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<bullet> A \<subseteq>\<^sub>\<circ> fpairs r"
hd: shows "hd (take 2 \<GG>) = {\<one>\<^bsub>G\<lparr>carrier := \<GG> ! 1\<rparr>\<^esub>}"
conc_def: shows "RegExp2NAe.conc = (\<lambda>(ql, dl, fl) (qr, dr, fr). (True # ql, \<lambda>a s. case s of [] \<Rightarrow> {} | left # s \<Rightarrow> if left then True ## dl a s \<union> (if fl s \<and> a = None then {False # qr} else {}) else False ## dr a s, \<lambda>s. case s of [] \<Rightarrow> False | left # s \<Rightarrow> \<not> left \<and> fr s))"
hyps_1: shows "P \<turnstile> Path last Cs to C via Cs' "
to_fun_X_pow: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and n :: "nat" assumes "UP_cring R"   and "x \<in> carrier R" shows "UP_cring.to_fun R (X_poly R [^]\<^bsub>UP R\<^esub> n) x = x [^]\<^bsub>R\<^esub> n"
ptyping_length_1: fixes p :: "pat"   and T :: "POPLmarkRecord.type"   and \<Delta> :: "binding list" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>" shows "\<parallel>p\<parallel>\<^sub>p = \<parallel>\<Delta>\<parallel>"
thm_cont_e_2: fixes v :: "i" shows "[Contingent E! in v]"
SUCC_Subset_Ord_lemma: fixes k' :: "tm"   and k :: "tm" shows "{k' IN k, OrdP k} \<turnstile> SUCC k' SUBS k"
intro: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'e set"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'f set" assumes "finite_dimensional_vector_space s1 B1"   and "finite_dimensional_vector_space s2 B2" shows "finite_dimensional_vector_space_pair s1 B1 s2 B2"
constrains_Int: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A co A'"   and "F \<in> B co B'" shows "F \<in> A \<inter> B co A' \<inter> B'"
prfOf_Pf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" shows "let PPf = \<lambda>t1 t2. psubst Pf [(t1, yy), (t2, xx)] in prf \<in> proof \<and> \<phi> \<in> fmla \<and> Fvars \<phi> = {} \<longrightarrow> prfOf prf \<phi> \<longrightarrow> bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
P2_OT_12_inf_theoretic: fixes funct_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "OT_14_sim.perfect_sec_P2 funct_14_OT M C"
hyps: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "length u = n"   and "length v = n"   and "length w = n" shows "foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u (map (\<lambda>xy. fst xy + snd xy) (zip v w))) (0::'a) = foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u v) (0::'a) + foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u w) (0::'a)"
compactin_euclidean_iff: fixes S :: "'b set" shows "compactin euclidean S = compact S"
purgeIdle_set_isState: fixes s :: "('St, 'U, 'C) state"   and sl :: "('St, 'U, 'C) state list" assumes "s \<in> set (purgeIdle sl)" shows "isState s"
simps_1: fixes ys :: "'g list" shows "[] @ ys = ys"
abs_summable_on_cong_neutral: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'b)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'b)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "(f abs_summable_on A) = (g abs_summable_on B)"
redundancy_criterion_def: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set" shows "redundancy_criterion \<Gamma> Rf Ri \<equiv> ((\<forall>N. Ri N \<subseteq> \<Gamma>) \<and> (\<forall>N N'. N \<subseteq> N' \<longrightarrow> Rf N \<subseteq> Rf N') \<and> (\<forall>N N'. N \<subseteq> N' \<longrightarrow> Ri N \<subseteq> Ri N')) \<and> (\<forall>N' N. N' \<subseteq> Rf N \<longrightarrow> Rf N \<subseteq> Rf (N - N')) \<and> (\<forall>N' N. N' \<subseteq> Rf N \<longrightarrow> Ri N \<subseteq> Ri (N - N')) \<and> (\<forall>N. satisfiable (N - Rf N) \<longrightarrow> satisfiable N)"
\<ll>_agreement: shows "MC.\<ll> = (\<lambda>f. if arr f then f \<cdot> \<l>[local.dom f] else null)"
constant_rules_2: fixes C :: "'b set"   and G :: "('b Standard_Constant, 'b + 'c) labeled_graph"   and c :: "'b" assumes "standard' C G"   and "c \<in> C" shows "maintained (const_exists_rev c) G"
inject_3: fixes x3 :: "real \<times> real \<times> real"   and y3 :: "real \<times> real \<times> real" shows "(LeqUni x3 = LeqUni y3) = (x3 = y3)"
prems_1: shows "S \<subseteq> \<gamma>\<^sub>o S'"
valid_states_def: shows "valid_states \<equiv> {s. dom s = Dom \<and> (\<forall>x\<in>Dom. the (s x) \<in> range_of_var x)}"
UP_subring_taylor_appr: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"   and "a \<in> S"   and "b \<in> S" shows "\<exists>c\<in>S. to_fun g a = to_fun g b \<oplus> deriv g b \<otimes> (a \<ominus> b) \<oplus> c \<otimes> (a \<ominus> b) [^] 2"
assms_1: shows "rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
subset_emptyI: fixes A :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "order.greater_eq {} A"
Pow_top: fixes A :: "'a set" shows "A \<in> Pow A"
max_list_Cons: fixes x :: "'a"   and xs :: "'a list" shows "Missing_Lemmas.max_list (x # xs) = (if xs = [] then x else max x (Missing_Lemmas.max_list xs))"
class_add_is_relevant_entry: fixes P :: "jvm_prog"   and i :: "instr"   and pc :: "nat"   and e :: "ex_entry"   and C :: "char list"   and cdec :: "jvm_method class" assumes "is_relevant_entry P i pc e"   and "\<not> is_class P C" shows "is_relevant_entry (class_add P (C, cdec)) i pc e"
mono: fixes x :: "'a option acom"   and y :: "'a option acom" assumes "x \<sqsubseteq> y" shows "f x \<sqsubseteq> f y"
Methd: fixes E :: "env"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and dt :: "loc \<Rightarrow> ty option"   and T :: "ty" assumes "is_class (prg E) C"   and "methd (prg E) C sig = Some m"   and "E,dt\<Turnstile>Body (declclass m) (stmt (mbody (mthd m)))\<Colon>-T" shows "E,dt\<Turnstile>Methd C sig\<Colon>-T"
rep_eq: fixes x :: "nat \<Rightarrow> bool" shows "Rep_uint64 (set_bits x) = set_bits x"
mono_union_L: fixes X :: "char list set"   and c :: "com" shows "mono (\<lambda>Y. X \<union> L c Y)"
source_stability: fixes x :: "'a \<Longrightarrow> 'b"   and \<sigma> :: "'b" assumes "wb_lens x" shows "\<exists>v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
no_ofail_wp_comb1: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and P' :: "'c \<Rightarrow> bool" assumes "no_ofail P f"   and "\<And>s. P' s \<Longrightarrow> P s" shows "no_ofail P' f"
PROTECT2_def: fixes x :: "'i"   and y :: "prop" shows "PROTECT2 x PROP y \<equiv> x"
R_G_cfg_on_div_def: fixes st :: "'s \<times> ('c \<Rightarrow> real) set" shows "R_G_cfg_on_div st \<equiv> R_G.cfg_on st \<inter> Collect R_G_div_cfg"
SourcesS14_L2: shows "Sources level2 sS14 = {sS9, sS11}"
vector_covector: fixes v :: "'a"   and w :: "'a" assumes "v \<sqinter> top = v"   and "w \<sqinter> top = w" shows "v \<sqinter> id w = v \<sqinter> id w"
S_def: shows "S = {k. [[a\<^sub>n f k b]] \<and> k < card Y}"
UP_membI: fixes x :: "hf"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "x \<^bold>\<in> ide_to_hf a" shows "UP x \<in> local.set a"
vfinite_induct: fixes F :: "V"   and P :: "V \<Rightarrow> bool" assumes "vfinite F"   and "P 0"   and "\<And>x F. \<lbrakk>vfinite F; x \<notin>\<^sub>\<circ> F; P F\<rbrakk> \<Longrightarrow> P (vinsert x F)" shows "P F"
Receive: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c (update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
reads_consistent_drop_volatile_writes_no_volatile_reads: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list" assumes "reads_consistent pending_write \<O> m sb" shows "outstanding_refs is_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) = {}"
N_stl: fixes \<omega> :: "'s stream" assumes "N \<omega>" shows "N (stl \<omega>)"
exp_integer_2pi: fixes n :: "real" assumes "n \<in> \<int>" shows "exp (complex_of_real (2 * n * pi) * \<i>) = 1"
usedI: fixes c :: "msg"   and evs :: "event list" assumes "c \<in> parts (knows Spy evs)" shows "c \<in> used evs"
join_self_below_2: fixes x :: "'a"   and y :: "'a" shows "(x = y \<squnion> x) = (y \<sqsubseteq> x)"
swap_valid_boundaries: fixes C :: "real \<times> real \<Rightarrow> real \<times> real"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "\<forall>(k, \<gamma>)\<in>boundary C. valid_path \<gamma>"   and "(k, \<gamma>) \<in> boundary (prod.swap \<circ> C \<circ> prod.swap)" shows "valid_path \<gamma>"
has_vector_derivative_add: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and net :: "real filter"   and g :: "real \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_vector_derivative f') net"   and "(g has_vector_derivative g') net" shows "((\<lambda>x. f x + g x) has_vector_derivative f' + g') net"
arr_char: fixes f :: "'a" shows "S.arr f = left b f"
distinct_137: shows "GreaterThan \<noteq> ShiftRightZeros"
Diff_eq_empty_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A - B = {}) = (A \<subseteq> B)"
upset_anti_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.preorder less_eq less" shows "(ord.upset less_eq y \<subseteq> ord.upset less_eq x) = less_eq x y"
fps_divide_subdegree: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "g \<noteq> 0"   and "subdegree g \<le> subdegree f" shows "subdegree (f / g) = subdegree f - subdegree g"
A4: shows "Esigma (2 ^ k * r) = Esigma (2 ^ k) * Esigma r"
B: shows "B = {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}"
cod_simps_3: shows "\<nu>\<pi>.cod.leg1 = \<nu>\<pi>.leg1"
arsinh_real_at_top: shows "filterlim arsinh at_top at_top"
minus_float_round_down_eq: fixes prec :: "nat"   and x :: "float" shows "- float_round_down prec x = float_round_up prec (- x)"
assms_5: shows "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"
sincreasing_def: fixes s :: "'b stream" shows "sincreasing s = (\<forall>x. \<exists>i. x < s !! i)"
bn_inv_intro: fixes ns :: "nat list"   and p :: "nat"   and u :: "nat"   and xs :: "'c list" assumes "foldl (+) 0 ns \<le> p" shows "bn_inv (p - bn_count ns) (foldl (+) 0 ns - bn_count ns) (u, ns, xs)"
Units_l_cancel: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Group.monoid G"   and "x \<in> Units G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> y = x \<otimes>\<^bsub>G\<^esub> z) = (y = z)"
congruence_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" shows "congruence M H \<equiv> mat_adj M *\<^sub>m\<^sub>m H *\<^sub>m\<^sub>m M"
zmset_frontier_empty: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c" assumes "dataflow_topology summary results_in" shows "zmset_of (mset_set (set_antichain (frontier {#}\<^sub>z))) = {#}\<^sub>z"
vconsD: fixes a :: "V"   and xs :: "V"   and x :: "V" assumes "a \<in>\<^sub>\<circ> xs #\<^sub>\<circ> x" shows "a \<in>\<^sub>\<circ> vinsert \<langle>vcard xs, x\<rangle> xs"
id_outside_supp: fixes x :: "'a"   and cs :: "'a list" assumes "x \<notin> set cs" shows "cycle_of_list cs x = x"
simps_1: fixes i :: "nat"   and \<Gamma> :: "nat \<Rightarrow> nat"   and c :: "nat"   and k :: "nat" shows "transdb_inv (DBVar i) \<Gamma> c k = Var (\<Gamma> i)"
satisfies_bounded_rderiv0: fixes \<AA> :: "'i"   and a :: "'a"   and x :: "'x" assumes "wf0 (#\<^sub>V \<AA>) a"   and "#\<^sub>V \<AA> = size x" shows "(\<AA> \<Turnstile>\<^sub>b rderiv0 x a) = (SNOC x \<AA> \<Turnstile>\<^sub>0 a)"
size_bal_list: fixes n :: "nat"   and xs :: "'a list" assumes "n \<le> length xs" shows "size (bal_list n xs) = n"
UN_simps_10: fixes B :: "'c \<Rightarrow> 'b set"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "\<Union> (B ` f ` A) = (\<Union>a\<in>A. B (f a))"
max_lin_independent_set_in_Span: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and vs :: "'b list"   and us :: "'b list" assumes "VectorSpace smult V"   and "set vs \<subseteq> V"   and "set us \<subseteq> R_scalar_mult.RSpan UNIV smult vs"   and "R_scalar_mult.R_lin_independent UNIV smult us" shows "length us \<le> length vs"
dom_add_closed: fixes x :: "'a"   and y :: "'a" shows "d (d x + d y) = d x + d y"
ResSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" shows "<\<nu>x>(P \<oplus> Q) \<equiv>\<^sub>s <\<nu>x>P \<oplus> <\<nu>x>Q"
iterate_add_eq_add_iff2: fixes i :: "nat"   and j :: "nat"   and u :: "'k"   and m :: "'k"   and n :: "'k" assumes "i \<le> j" shows "(iterate_add i u + m = iterate_add j u + n) = (m = iterate_add (j - i) u + n)"
unique_keys_setD: fixes S :: "('b \<times> 'c) set"   and i :: "'b"   and x :: "'c"   and y :: "'c" assumes "unique_keys_set S"   and "(i, x) \<in> S"   and "(i, y) \<in> S" shows "x = y"
cyclic: fixes cs :: "'a list" assumes "cs \<in> set css" shows "cyclic_on f (set cs)"
holF: fixes h :: "complex \<Rightarrow> complex" assumes "h \<in> F" shows "h holomorphic_on S"
no_better_in_s: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "x \<in> arg_max_set f S" shows "\<nexists>y. y \<in> S \<and> f x < f y"
below_fun_def: shows "(\<sqsubseteq>) \<equiv> \<lambda>f g. \<forall>x. f x \<sqsubseteq> g x"
simps_2: fixes i :: "nat"   and xs :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_aux_PQ (i # xs) bezout (P, A, Q) = diagonal_to_Smith_aux_PQ xs bezout (Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q))"
pw_le_iff: fixes S :: "'b nres"   and S' :: "'b nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
intros_1: fixes m :: "'a" shows "lformula0 (Fo m)"
that: fixes a' :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r' :: "'ret"   and fs' :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" assumes "a \<in> get_return_edges a'"   and "valid_edge a'"   and "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'"   and "CFG_node (sourcenode a') s-p\<rightarrow>\<^bsub>sum\<^esub> CFG_node (targetnode a)" shows "thesis"
longest_Lyndon_suf_ext: fixes a :: "'a"   and w :: "'a list" assumes "\<not> Lyndon (a # w)" shows "LynSuf w = LynSuf (a # w)"
substCases: fixes Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and xvec :: "name list"   and Tvec :: "'a list" shows "subs (Cases Cs) xvec Tvec = Cases caseListSubst Cs xvec Tvec"
axioms_1: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" assumes "ring_isomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'" shows "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit'"
t_op_def: fixes x :: "('a \<times> 'a) set" shows "rel_kat.t_op x = rel_at (rel_at x)"
mem_mod_privilege: fixes s' :: "'b sparc_state"   and a1 :: "word8"   and a2 :: "word36"   and v :: "word8"   and s :: "'b sparc_state" assumes "s' = mem_mod a1 a2 v s \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
7: shows "regular (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)"
integrate_insert_commute_delete_def: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and m :: "'a delete_message" shows "integrate_insert_commute_delete M a s m \<equiv> integrate_insert_commute M a s \<and> integrate_insert_commute_delete_axioms M m"
msb_shiftl: fixes x :: "int"   and n :: "nat" shows "msb (x << n) = msb x"
mag_times: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "mag (x \<cdot> y) = mag x \<cdot> mag y"
scalar_prod_def: fixes v :: "'a vec"   and w :: "'a vec" shows "v \<bullet> w \<equiv> \<Sum>i = 0..<dim_vec w. v $ i * w $ i"
B: fixes s :: "char list \<Rightarrow> int" assumes "P s < \<infinity>" shows "\<exists>t p. (c, s) \<Rightarrow> p \<Down> t \<and> enat p + enat k * Q t \<le> enat k * P s"