hcju/isabelleps huggingface dataset:
HSem_def': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "\<lbrace>\<Gamma>\<rbrace>\<rho> = (\<mu> x. \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>x\<^esub>)"
additive_def: fixes f :: "'d \<Rightarrow> 'e" shows "Modules.additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
distinct_25: fixes x2 :: "nat"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Bound x2 \<noteq> CN x81 x82 x83"
indexlt_subset_\<P>: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" assumes "indexlt k' u' k u" shows "\<P> k' (Suc u') \<subseteq> \<P> k u"
dual_equality: fixes x :: "'c"   and y :: "'c" shows "(dual x = dual y) = (x = y)"
add_edge_def: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "add_edge v e v' g \<equiv> \<lparr>nodes = {v, v'} \<union> nodes g, edges = insert (v, e, v') (edges g)\<rparr>"
length_two_unique: assumes "length (take 2 \<GG>) = 2" shows "take 2 \<GG> = [{\<one>\<^bsub>G\<lparr>carrier := \<GG> ! 1\<rparr>\<^esub>}, carrier (G\<lparr>carrier := \<GG> ! 1\<rparr>)]"
HPhase0Read_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and d :: "Disk"   and b :: "nat"   and v :: "InputsOrNi" assumes "HPhase0Read s s' q d"   and "maxBalInp s b v" shows "maxBalInp s' b v"
confTs_Err: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and n :: "nat"   and v :: "val" shows "P,h \<turnstile> replicate n v [:\<le>\<^sub>\<top>] replicate n Err"
map_eq_append_conv: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'b list"   and zs :: "'b list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
simps_13: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b msllist" shows "mslmap f (MSLCons x21 x22) = MSLCons (f x21) (mslmap f x22)"
sat_def: fixes S :: "'b formula set" shows "sat (S::'b formula set) \<equiv> \<exists>\<A>::'b \<Rightarrow> bool. \<forall>F::'b formula\<in>S. \<A> \<Turnstile> F"
subset_scaleR2_fstD: fixes X :: "('c \<times> 'd) set"   and l :: "ereal"   and u :: "ereal"   and Y :: "('c \<times> 'd) set" assumes "X \<subseteq> scaleR2 l u Y" shows "fst ` X \<subseteq> fst ` Y"
monotonicity: shows "monotone (\<lambda>x y. y < x) (\<lambda>x y. y < x) f"
sel_1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "shd (siterate f x) = x"
SN_on_relto_relcomp: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_on (S\<^sup>* O R O S\<^sup>*) T = SN_on (S\<^sup>* O R) T"
fmapU_error_simps_2: fixes f :: "udom \<rightarrow> udom"   and e :: "'a" shows "fmapU\<cdot>f\<cdot>(Err\<cdot>e) = Err\<cdot>e"
composeES_def: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec" shows "ES1 \<parallel> ES2 \<equiv> \<lparr>E_ES = E\<^bsub>ES1\<^esub> \<union> E\<^bsub>ES2\<^esub>, I_ES = I\<^bsub>ES1\<^esub> - O\<^bsub>ES2\<^esub> \<union> (I\<^bsub>ES2\<^esub> - O\<^bsub>ES1\<^esub>), O_ES = O\<^bsub>ES1\<^esub> - I\<^bsub>ES2\<^esub> \<union> (O\<^bsub>ES2\<^esub> - I\<^bsub>ES1\<^esub>), Tr_ES = {\<tau>. \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> \<in> Tr\<^bsub>ES2\<^esub> \<and> set \<tau> \<subseteq> E\<^bsub>ES1\<^esub> \<union> E\<^bsub>ES2\<^esub>}\<rparr>"
arr_Rel_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T \<in>\<^sub>\<circ> Vset \<alpha>"
sum_div: fixes A :: "'a set"   and b :: "'b"   and f :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<And>a. a \<in> A \<Longrightarrow> b dvd f a" shows "sum f A div b = (\<Sum>a\<in>A. f a div b)"
read_only_mono: fixes S :: "nat \<Rightarrow> bool option"   and S' :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "S \<subseteq>\<^sub>m S'"   and "a \<in> read_only S" shows "a \<in> read_only S'"
contour_integral_sum: fixes s :: "'a set"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and p :: "real \<Rightarrow> complex" assumes "finite s"   and "\<And>a. a \<in> s \<Longrightarrow> f a contour_integrable_on p" shows "contour_integral p (\<lambda>x. \<Sum>a\<in>s. f a x) = (\<Sum>a\<in>s. contour_integral p (f a))"
remdups_rev_def: fixes rs :: "'b list" shows "remdups_rev rs \<equiv> rev (remdups (rev rs))"
star_of_diff: fixes x :: "'b"   and y :: "'b" shows "star_of (x - y) = star_of x - star_of y"
singleton_def: fixes default :: "unit \<Rightarrow> 'a"   and A :: "'a pred" shows "singleton default A = (if \<exists>!x. eval A x then THE x. eval A x else default ())"
vpeq_transitive: shows "\<forall>a b c u. vpeq u a b \<and> vpeq u b c \<longrightarrow> vpeq u a c"
left_adjoint_is_ide: fixes f :: "'a" assumes "B.is_left_adjoint f" shows "B.ide f"
sel_4: fixes x21 :: "'b"   and x22 :: "'c"   and x23 :: "'a" shows "getGMCom (State x21 x22 x23) = x23"
pred_True: shows "list_all (\<lambda>_. True) = (\<lambda>_. True)"
discrete_topology_unique: fixes U :: "'d set"   and X :: "'d topology" shows "(discrete_topology U = X) = (topspace X = U \<and> (\<forall>x\<in>U. openin X {x}))"
add_zeror: fixes x :: "'c" shows "x \<squnion> \<bottom> = x"
INF_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'a"   and D :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Inf (C ` A) = Inf (D ` B)"
UP_zero_closed: shows "\<zero>\<^bsub>P\<^esub> \<in> carrier P"
add_canc2: fixes x :: "'c"   and y :: "'c" assumes "x \<preceq>\<^sub>R y" shows "x \<oplus> rquot y x = y"
inf_dual: fixes x :: "'a"   and y :: "'a" shows "\<partial> (x \<sqinter> y) = \<partial> x \<squnion> \<partial> y"
iso_refl: fixes G :: "('a, 'b) monoid_scheme" shows "G \<cong> G"
repr_independence: fixes y :: "'a"   and H :: "'a set"   and x :: "'a" assumes "y \<in> H #>\<^bsub>add_monoid R\<^esub> x"   and "x \<in> carrier R"   and "subgroup H (add_monoid R)" shows "H #>\<^bsub>add_monoid R\<^esub> x = H #>\<^bsub>add_monoid R\<^esub> y"
connected_openin: fixes S :: "'c set" shows "connected S = (\<nexists>E1 E2. openin (top_of_set S) E1 \<and> openin (top_of_set S) E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
mult_implies_mult2_s: fixes ns :: "('a \<times> 'a) set"   and X :: "'a multiset"   and Y :: "'a multiset"   and s :: "('a \<times> 'a) set" assumes "refl ns"   and "(X, Y) \<in> mult s" shows "(X, Y) \<in> mult2_s ns s"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> rev.existence_ivl0 x0" shows "x0 \<in> X"
less_imp_minus_plus_hmset: fixes m :: "hmultiset"   and n :: "hmultiset"   and k :: "hmultiset" assumes "m < n" shows "k < k - m + n"
sinit_append_split: fixes i :: "nat"   and j :: "nat"   and xs :: "'b seq" assumes "i < j" shows "sinit j xs = sinit i xs @ ssubstr i j xs"
eq_span_insert_eq: fixes x :: "'b"   and y :: "'b"   and S :: "'b set" assumes "x - y \<in> cspan S" shows "cspan (insert x S) = cspan (insert y S)"
pwd_spy_mult_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_mult n)"
liftn_absn: fixes n :: "nat"   and m :: "nat"   and t :: "dB"   and k :: "nat" shows "liftn n ((Abs ^^ m) t) k = (Abs ^^ m) (liftn n t (k + m))"
lt_plus_lessE: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.lt p \<prec> punit.lt (p + q)" shows "punit.lt p \<prec> punit.lt q"
prob0_fst_qubits_leq_2: shows "(cmod (\<Sum>k<2 ^ n. (- 1) ^ (f k + 1)))\<^sup>2 \<le> real (2 ^ (2 * n))"
complement_def: fixes A :: "('c, 'd) nba" shows "complement A \<equiv> nba (alphabet A) ({(\<lambda>_. Some (2 * card (nodes A))) |` initial A} \<times> {{}}) (complement_succ A) (\<lambda>(f, P). P = {})"
path_image_linepath: fixes a :: "'b"   and b :: "'b" shows "path_image (linepath a b) = closed_segment a b"
res_lim_Zp_cauchy_0_1: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "int"   and k :: "nat" assumes "is_Zp_cauchy s"   and "f = res_lim s k" shows "\<exists>N. \<forall>m>N. s m k = f"
cases: fixes a1 :: "nat"   and a2 :: "Lholed"   and a3 :: "e list"   and a4 :: "e list" assumes "Lfilled a1 a2 a3 a4" obtains vs :: "e list" and lholed :: "Lholed" and es' :: "e list" and es :: "e list" where "a1 = 0" and "a2 = lholeda__" and "a3 = esb__" and "a4 = vsa__ @ esb__ @ es'a__" and "const_list vsa__" and "lholeda__ = LBase vsa__ es'a__"   | vs :: "e list" and lholed :: "Lholed" and n :: "nat" and es' :: "e list" and l :: "Lholed" and es'' :: "e list" and k :: "nat" and es :: "e list" and lfilledk :: "e list" where "a1 = k + 1" and "a2 = lholed" and "a3 = es" and "a4 = vs @ [Label n es' lfilledk] @ es''" and "const_list vs" and "lholed = LRec vs n es' l es''" and "Lfilled k l es lfilledk"
assms: shows "\<not> rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1"
interp_rga_tail_unfold: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" shows "interp_rga (xs @ [x]) = insert_rga (interp_rga xs) x"
right_zero: fixes a :: "'a" assumes "a \<in> R" shows "a \<cdot> \<zero> = \<zero>"
last_acc_lengths: fixes ls :: "'a list list"   and acc :: "nat" assumes "ls \<noteq> []" shows "last (acc_lengths acc ls) = acc + sum_list (map length ls)"
collision_react_strict: assumes "s\<^sub>o \<le> u_max"   and "u_max < other.s_stop" shows "collision_react {\<delta>..} = collision_react {\<delta><..}"
prod_insert_eqpoll: fixes a :: "'a"   and A :: "'a set"   and B :: "'b set" assumes "a \<notin> A" shows "insert a A \<times> B \<approx> B <+> A \<times> B"
simps_1: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and k :: "'b" assumes "Set2_Join join inv" shows "Set2_Join.split join \<langle>\<rangle> k = (\<langle>\<rangle>, False, \<langle>\<rangle>)"
atMost_iff: fixes i :: "'b word"   and k :: "'b word" shows "(i \<in> signed.atMost k) = (i \<le>s k)"
equation_13: fixes h :: "'b mod_ring poly"   and u :: "'b mod_ring poly" assumes "degree h < degree u" shows "[h ^ CARD('b) = h] (mod u) = ((berlekamp_mat u)\<^sup>T *\<^sub>v vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) 0) = vec_of_list (coeffs h @ replicate (degree u - length (coeffs h)) 0))"
eventuallyI: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter" assumes "\<And>x. P x" shows "eventually P F"
size_3: fixes x31 :: "cmd"   and x32 :: "cmd" shows "size_cmd (x31;; x32) = size_cmd x31 + size_cmd x32 + Suc 0"
s_lam: fixes t :: "lam"   and t' :: "lam"   and x :: "name" assumes "t \<rightarrow> t'" shows "Lam [x].t \<rightarrow> t'"
XXY: shows "simple_function M (\<lambda>x. (X x, X x, Y x))"
eq: shows "\<Union> (B ` R) = (\<Union>i. if i \<in> R then B i else {})"
le_add_iff1: fixes a :: "'a"   and e :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" shows "(a * e + c \<le> b * e + d) = ((a - b) * e + c \<le> d)"
head_del_vert: fixes u :: "'a" shows "head (del_vert u) = head G"
conv_abscissa_MInfI: fixes f :: "'a fds" assumes "\<And>s. fds_converges f s" shows "conv_abscissa f = - \<infinity>"
right_impl_one: fixes a :: "'a" shows "a r\<rightarrow> a = (1::'a)"
SeqThrow: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile> (Seq Throw c\<^sub>2, Normal s) \<rightarrow> (Throw, Normal s)"
Aenc_synth: fixes X :: "msg"   and K :: "msg"   and H :: "msg set" assumes "Aenc X K \<in> synth H" obtains "Aenc X K \<in> H"   | "X \<in> synth H" and "K \<in> synth H"
S_intersections_distinct: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" assumes "p \<noteq> 0"   and "q \<noteq> 0"   and "proj2_abs p \<noteq> proj2_abs q"   and "proj2_abs p \<in> hyp2" shows "S_intersection1 p q \<noteq> S_intersection2 p q"
append_self_conv: fixes xs :: "'g list"   and ys :: "'g list" shows "(xs @ ys = xs) = (ys = [])"
"0_3": assumes "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> ((the_Abs f, the_Abs g) \<in> set TI \<or> (the_Abs g, the_Abs f) \<in> set TI \<or> (\<exists>ti\<in>set TI. (the_Abs f, snd ti) \<in> set TI \<and> (the_Abs g, snd ti) \<in> set TI))"
minus_q_mult_cancel: fixes a :: "nat"   and e :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" assumes "[a = e + b - q * c - d] (mod q)"   and "0 < e + b - d"   and "0 < e + b - q * c - d" shows "[a = e + b - d] (mod q)"
append_Nil2: fixes xs :: "'a list" shows "xs @ [] = xs"
errMOD_igSubstIGAbsSTR: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igAbsIPresIGWls Delta MOD"   and "igWlsDisj MOD"   and "FixSyn.igSubstIPresIGWls Delta MOD"   and "FixSyn.igSubstIGAbs Delta MOD" shows "igSubstIGAbsSTR (FixSyn.errMOD Delta MOD)"
appE: fixes A :: "'b ptrm"   and B :: "'b ptrm"   and Y :: "'b ptrm" assumes "PApp A B \<approx> Y" obtains B :: "'b ptrm" and D :: "'b ptrm" where "Y = PApp B D" and "A \<approx> B" and "Ba__ \<approx> D"
inclusion: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>" shows "arr f"
NotNull: shows "\<not> (a * e - b * d = 0 \<and> a * f - c * d = 0 \<and> e * c = b * f)"
simps: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "char list exp list"   and a3 :: "ty list" shows "(P,a1,h \<turnstile> a2 [:] a3) = ((\<exists>E. a1 = E \<and> a2 = [] \<and> a3 = []) \<or> (\<exists>E e T es Ts. a1 = E \<and> a2 = e # es \<and> a3 = T # Ts \<and> P,E,h \<turnstile> e : T \<and> P,E,h \<turnstile> es [:] Ts))"
integrable_on_subcbox: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and a :: "'c"   and b :: "'c" assumes "f integrable_on S"   and "cbox a b \<subseteq> S" shows "f integrable_on cbox a b"
UINF_image_eq: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'c \<Rightarrow> ('a, 'b) uexpr" shows "(\<Sqinter> i \<in> f ` A \<bullet> g i) = (\<Sqinter> i \<in> A \<bullet> g (f i))"
simps_43: fixes l :: "formula list" shows "safe_formula (Ands l) = (let (pos, neg) = partition safe_formula l in pos \<noteq> [] \<and> list_all safe_formula (map remove_neg neg) \<and> \<Union> (set (map fv neg)) \<subseteq> \<Union> (set (map fv pos)))"
assms_3: shows "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops'"
simps_2: fixes P :: "'b"   and Cal :: "'b processCalculus"   and n :: "nat"   and Q :: "'b" shows "P \<longmapsto>Cal\<^bsup>Suc n\<^esup> Q = (\<exists>P'. P \<longmapsto>Cal\<^bsup>n\<^esup> P' \<and> P' \<longmapsto>Cal Q)"
cut_greater_Int_right: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<inter> B) \<down>> t = A \<inter> B \<down>> t"
hm_empty_impl: shows "imp_map_empty is_hashmap hm_new"
star_sim2: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<cdot> x \<le> y \<cdot> z" shows "z \<cdot> x\<^sup>\<star> \<le> y\<^sup>\<star> \<cdot> z"
in_keys_less_degree: fixes n :: "nat"   and f :: "nat \<Rightarrow>\<^sub>0 'b" assumes "n \<in> keys f" shows "n < degree f"
norm_eq_iff: fixes x :: "nat"   and y :: "nat" shows "(x mod 2 ^ LENGTH('b) = y mod 2 ^ LENGTH('b)) = (word_of_nat x = word_of_nat y)"
two_not_zero: shows "\<guillemotleft>2\<guillemotright> \<noteq> \<zero>"
hyps_1: assumes "j + 2 < card X"   and "0 < j + 1"   and "j + 1 + 1 \<le> card X" shows "aa (j + 1) + bb (j + 1) \<le> Dube_aux (card X) (poly_deg f) (j + 1)"
face_of_eq: fixes T :: "'b set"   and S :: "'b set"   and U :: "'b set" assumes "T face_of S"   and "U face_of S"   and "rel_interior T \<inter> rel_interior U \<noteq> {}" shows "T = U"
Entry_Exit_edge: fixes prog :: "cmd" shows "\<exists>a. valid_edge prog a \<and> sourcenode a = (_Entry_) \<and> targetnode a = (_Exit_) \<and> kind a = (\<lambda>s. False)\<^sub>\<surd>"
pnop_vdomain: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "pnop A n f" shows "\<D>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A ^\<^sub>\<times> n"
sp_append_one: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and g :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and e :: "('d \<times> nat) edge"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel subs" shows "SubRel.subpath g v1 (es @ [e]) v2 subs = (SubRel.subpath g v1 es (src e) subs \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs))"
fls_const_mult_const: fixes x :: "'b"   and y :: "'b" shows "fls_const x * fls_const y = fls_const (x * y)"
commute: fixes a :: "'a"   and b :: "'a" shows "a XOR b = b XOR a"
poss2: shows "squash (rpoly q x1) = 1 \<and> squash (rpoly q x2) = - 1 \<longrightarrow> (\<exists>r<x1. x2 \<le> r \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
matrix_to_iarray_snd_Gauss_Jordan_upt_k_det_P: fixes k :: "nat"   and A :: "(('d, 'e) vec, 'f) vec" assumes "k < ncols A" shows "matrix_to_iarray (snd (Gauss_Jordan_upt_k_det_P A k)) = snd (Gauss_Jordan_upt_k_det_P_iarrays (matrix_to_iarray A) k)"
arr_iff_in_hom: fixes f :: "'a" shows "B.arr f = \<guillemotleft>f : B.dom f \<rightarrow>\<^sub>B B.cod f\<guillemotright>"
valid_combine_if_valid: fixes a :: "'b tree"   and as :: "'b list"   and z :: "'b tree" assumes "0 < l a"   and "valid as a"   and "valid as z"   and "a \<noteq> Leaf"   and "z \<noteq> Leaf"   and "l z = Suc (r a)" shows "valid as (combine a z)"
le: shows "\<forall>y. y < length vs \<and> vs ! y \<noteq> 0 \<longrightarrow> y \<le> length vs"
\<gamma>1': shows "\<gamma>1' \<in> seq ars"
iwlsFSbSwTR_imp_iwlsFSb: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" assumes "iwlsFSbSwTR MOD" shows "iwlsFSb MOD"
adds_satisfied: fixes P :: "ast_problem"   and f :: "object atom formula"   and a :: "object atom formula list"   and d :: "object atom formula list"   and M :: "object atom formula set" assumes "wf_ast_problem P"   and "f \<in> set a" shows "f \<in> apply_effect (Effect a d) M"
poly_monom: fixes a :: "'b"   and n :: "nat"   and x :: "'b" shows "poly (monom a n) x = a * x ^ n"
IH: assumes "\<forall>a\<in>ran \<Gamma>. wf_chain \<Gamma> a"   and "\<forall>rsg\<in>ran \<Gamma>. \<forall>r\<in>set rsg. (\<forall>chain. get_action r \<noteq> Goto chain) \<and> get_action r \<noteq> Unknown"   and "\<forall>y m. \<forall>r\<in>set rs. r = Rule m (Call y) \<longrightarrow> (\<exists>a. \<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m (Call y)], Undecided\<rangle> \<Rightarrow> a)"   and "wf_chain \<Gamma> rs"   and "\<forall>r\<in>set rs. (\<forall>chain. get_action r \<noteq> Goto chain) \<and> get_action r \<noteq> Unknown" shows "(\<exists>a. \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> a) \<or> (\<exists>rs_called1 rs_called2 m'. rs = rs_called1 @ [Rule m' Return] @ rs_called2 \<and> matches \<gamma> m' p \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs_called1, Undecided\<rangle> \<Rightarrow> Undecided)"
path_weight_singleton: fixes G :: "graph"   and x :: "nat" shows "path_weight G [x] = 0"
mini_maxi_neq: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd" assumes "Suc 0 < length xs" shows "mini xs key \<noteq> maxi xs key"
restrict_inj: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and A1 :: "'d set" assumes "inj_on f A"   and "A1 \<subseteq> A" shows "inj_on f A1"
code: fixes a :: "'d" shows "Generative_Probabilistic_Value.Done a = GPV (return_spmf (Pure a))"
norm_minus_commute: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus a b) = norm (minus b a)"
read_only_reads_antimono: fixes \<O> :: "nat set"   and \<O>' :: "nat set"   and sb :: "'b memref list" assumes "\<O> \<subseteq> \<O>'" shows "read_only_reads \<O>' sb \<subseteq> read_only_reads \<O> sb"
nth_Abs_fls_lower_bound: fixes N :: "int"   and f :: "int \<Rightarrow> 'a"   and n :: "int" assumes "\<forall>n<N. f n = (0::'a)" shows "fls_nth (Abs_fls f) n = f n"
filter_map: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
simps: fixes v :: "'a"   and n :: "nat" shows "Time_Monad.val (TM v n) = v"
modular_lattice_def: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "class.modular_lattice inf less_eq less sup \<equiv> class.lattice inf less_eq less sup \<and> class.modular_lattice_axioms inf less_eq sup"
map_le_same_upd: fixes f :: "'g \<Rightarrow> 'f option"   and x :: "'g"   and y :: "'f" assumes "f x = None" shows "f \<subseteq>\<^sub>m f(x \<mapsto> y)"
seq_sim_def: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and A :: "('b \<Rightarrow> bool) set"   and \<tau> :: "nat \<Rightarrow> 'b" shows "(\<sigma> \<simeq>A\<simeq> \<tau>) = (\<forall>n. \<sigma> n ~A~ \<tau> n)"
validRules: shows "\<forall>(N, \<alpha>)\<in>\<RR>. N \<in> \<NN> \<and> (\<forall>s\<in>set \<alpha>. s \<in> \<NN> \<union> \<TT>)"
o_id: fixes f :: "'a \<Rightarrow> 'b" shows "f \<cdot> id = f"
ev_Stream: fixes P :: "'a stream \<Rightarrow> bool"   and x :: "'a"   and s :: "'a stream" shows "ev P (x ## s) = (P (x ## s) \<or> ev P s)"
iarray_simps_1: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and i :: "nat"   and x :: "'a" shows "iarray_update (IArray.of_fun f n) i x = IArray.of_fun (f(i := x)) n"
Optimal_BST2_axioms: shows "Optimal_BST2 w"
sdrop_siterate: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "sdrop n (siterate f x) = siterate f ((f ^^ n) x)"
assms_3: shows "p \<sim>\<^bsub>poly_ring R\<^esub> q"
bij_betw_empty2: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "bij_betw f A {}" shows "A = {}"
filter_ASSERT_def: fixes c :: "'b \<Rightarrow> bool"   and m :: "'b nres" shows "filter_ASSERT c m \<equiv> m \<bind> (\<lambda>x. ASSERT (c x) \<bind> (\<lambda>_. RETURN x))"
induct: fixes x :: "'node SDG_node"   and n :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> bool" assumes "x \<in> sum_SDG_slice2 n"   and "valid_SDG_node n \<Longrightarrow> P n"   and "\<And>n'' n'. \<lbrakk>n'' s\<longrightarrow>\<^bsub>cd\<^esub> n'; n' \<in> sum_SDG_slice2 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' V n'. \<lbrakk>n'' s-V\<rightarrow>\<^sub>d\<^sub>d n'; n' \<in> sum_SDG_slice2 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p n'. \<lbrakk>n'' s-p\<rightarrow>\<^bsub>ret\<^esub> n'; n' \<in> sum_SDG_slice2 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p V n'. \<lbrakk>n'' s-p:V\<rightarrow>\<^bsub>out\<^esub> n'; n' \<in> sum_SDG_slice2 n; P n'\<rbrakk> \<Longrightarrow> P n''"   and "\<And>n'' p n'. \<lbrakk>n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'; n' \<in> sum_SDG_slice2 n; P n'\<rbrakk> \<Longrightarrow> P n''" shows "P x"
fls_eqI: fixes x :: "'b fls"   and y :: "'b fls" assumes "\<And>xa. fls_nth x xa = fls_nth y xa" shows "x = y"
comp_ide_self: fixes a :: "'a" assumes "A.ide a" shows "a \<cdot>\<^sub>A a = a"
legs_form_cospan_2: shows "C.cospan \<mu>.cod.leg0 \<nu>.cod.leg1"