hcju/isabelleps huggingface dataset:
sorted_iff_nth_mono_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! i) (xs ! j))"
ST: fixes C :: "'a set" assumes "openin (top_of_set S) C" shows "openin (top_of_set T) (f ` C)"
subringE_2: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subring H R" shows "\<zero>\<^bsub>R\<^esub> \<in> H"
le_imp_less_or_eq: fixes x :: "'b list"   and y :: "'b list" assumes "x \<le>p y" shows "x <p y \<or> x = y"
lookup_one_zero: shows "lookup 1 (0::'c) = (1::'b)"
sharing_consis: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "sharing_consistent \<S>\<^sub>s\<^sub>b \<O> sb"
inf_comp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y * z = x * z \<sqinter> (y * z)"
Q_subst: fixes v :: "'v"   and t' :: "('f, 'v) Term.term"   and E :: "(('f, 'v) Term.term \<times> ('f, 'v) Term.term) list"   and B :: "('v \<times> ('f, 'v) Term.term) list" assumes "v \<notin> fv t'"   and "Q ((Var v, t') # E) (subst_of B)" shows "Q (subst_list (subst v t') E) (subst_of ((v, t') # B))"
hom_restrict0_freeword_funlift: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "GroupHom (FreeGroup S) (res_freeword_funlift f S)"
frac_Vr_is_K: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K" shows "\<exists>s\<in>carrier (Vr K v). \<exists>t\<in>carrier (Vr K v) - {\<zero>}. x = s \<cdot>\<^sub>r t\<^bsup>\<hyphen> K\<^esup>"
ground_replacement: fixes t :: "'b trm"   and p :: "indices list"   and v :: "'b trm"   and s :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "replace_subterm t p v s"   and "ground_term (t \<lhd> \<sigma>)"   and "ground_term (v \<lhd> \<sigma>)" shows "ground_term (s \<lhd> \<sigma>)"
nummul: fixes bs :: "int list"   and i :: "int"   and t :: "Cooper.num" shows "Inum bs (nummul i t) = Inum bs (Mul i t)"
select_trap_monad: fixes s :: "'b sparc_state" assumes "snd (select_trap () s) = False" shows "err_mode_val s = err_mode_val (snd (fst (select_trap () s)))"
map_add_def: fixes m1 :: "'b \<Rightarrow> 'c option"   and m2 :: "'b \<Rightarrow> 'c option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
simps_1: fixes c :: "int" shows "isnorm (Pc c) = True"
this_wall_betwI: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "C \<sim> D" shows "THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>ThinChamberComplex.foldpairs X. {{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}}) \<and> separated_by H C D) = {f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}"
d: shows "\<bar>\<Sum>i = M..N. a i / real i\<bar> < d"
f2r_def: fixes f :: "'b \<Rightarrow> 'c set" shows "\<R> f = {(x, y). s2p (f x) y}"
div_add_self2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a b) b = plus (divide a b) one"
prj_simps_12: shows "C.seq src_src.prj\<^sub>0 \<p>\<^sub>0[f.leg0, src.leg1 \<cdot> src_src.prj\<^sub>1]"
inducts_2: fixes P1 :: "exp \<Rightarrow> bool"   and P2 :: "assn \<Rightarrow> bool"   and assn :: "assn" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P1 x1 \<Longrightarrow> P1 (App x1 x2)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (LetA x1 x2)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (Lam [x1]. x2)"   and "\<And>x. P1 (Bool x)"   and "\<And>x1 x2 x3. \<lbrakk>P1 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (x1 ? x2 : x3)"   and "P2 ANil"   and "\<And>x1 x2 x3. \<lbrakk>P1 x2; P2 x3\<rbrakk> \<Longrightarrow> P2 (ACons x1 x2 x3)" shows "P2 assn"
kbo_stable: fixes s :: "('f, 'a) Term.term"   and t :: "('f, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('f, 'a) Term.term" assumes "NS s t" shows "(S s t \<longrightarrow> S (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)) \<and> NS (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
evContainsEx_containsEx0: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and A :: "formula" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))" shows "EV (contains f (i, FAll Neg A)) \<longrightarrow> EV (contains f (0, FAll Neg A))"
assoc_four_block_mat: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" shows "four_block_mat A (0\<^sub>m (dim_row A) (dim_col (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C))) (0\<^sub>m (dim_row (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C)) (dim_col A)) (four_block_mat B (0\<^sub>m (dim_row B) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col B)) C) = four_block_mat (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B) (0\<^sub>m (dim_row (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B)) (dim_col C)) (0\<^sub>m (dim_row C) (dim_col (four_block_mat A (0\<^sub>m (dim_row A) (dim_col B)) (0\<^sub>m (dim_row B) (dim_col A)) B))) C"
spr_repSim: fixes t :: "'s Trace" assumes "t \<in> SPR.jkbpC" shows "sim (spr_simMCt t) ((spr_repMC \<circ> fst) (spr_sim t)) spr_repSim"
abs_ast_plan_def: fixes \<pi>s :: "char list list" shows "abs_ast_plan \<pi>s \<equiv> map (abs_ast_operator \<circ> the \<circ> lookup_operator) \<pi>s"
convex_contains_segment: fixes S :: "'a set" shows "convex S = (\<forall>a\<in>S. \<forall>b\<in>S. closed_segment a b \<subseteq> S)"
unbounded_outside: fixes S :: "'a set" assumes "bounded S" shows "\<not> bounded (outside S)"
gen_cong_eval_rel_fun: 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 "rel_fun (rel (cong.gen_cong rel eval R)) (cong.gen_cong rel eval R) eval eval"
pref_profile_eqI: fixes agents :: "'a set"   and alts :: "'b set"   and R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pref_profile_wf agents alts R1"   and "pref_profile_wf agents alts R2"   and "\<And>x. x \<in> agents \<Longrightarrow> R1 x = R2 x" shows "R1 = R2"
commute: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "abel_semigroup f" shows "f a b = f b a"
card_partitions_with_k_parts: fixes n :: "nat"   and k :: "nat" shows "card {N. number_partition n N \<and> size N = k} = Partition n k"
verticesFrom_nth: fixes f :: "face"   and d :: "nat"   and v :: "nat" assumes "distinct (vertices f)"   and "d < |vertices f|"   and "v \<in> \<V> f" shows "verticesFrom f v ! d = f\<^bsup>d\<^esup> \<bullet> v"
edge'_Inner1: fixes x :: "'v"   and yo :: "'v vertex" assumes "edge' \<langle>x\<rangle> yo" obtains y :: "'v" where "yo = \<langle>y\<rangle>" and "edge \<Gamma> x y \<or> edge \<Gamma> y x"   | "yo = vertex.SINK" and "x \<in> A \<Gamma>"
unique: fixes f :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a msllist \<Rightarrow> 'a msllist" assumes "f = (\<lambda>fa xs. case xs of msllist.MSLNil \<Rightarrow> msllist.MSLNil | msllist.MSLCons x xs \<Rightarrow> msllist.MSLCons (fa x) (f fa xs))" shows "f = msllist.mslmap"
algebraic_diff_cancel_left: fixes x :: "'b"   and y :: "'b" assumes "algebraic x" shows "algebraic (x - y) = algebraic y"
ts_a': fixes x :: "nat"   and y :: "nat"   and qs :: "nat list"   and h :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [question (Atom x), Atom y, Atom y])"   and "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)" shows "TS_inv' (Partial_Cost_Model.config' (rTS h0) ([x, y], h) qs) (last qs) [x, y] \<and> Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) qs = 2"
f_h: shows "f h = (\<Sum>m<n. diff m 0 / fact m * h ^ m) + B * (h ^ n / fact n)"
elem_list_splits_elem: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and zs :: "'a list" assumes "(xs, y, ys) \<in> set (list_splits zs)" shows "y \<in> set zs"
wf_semiJoin_3: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and QQ :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" assumes "getIJ getIJ"   and "1 \<le> card J"   and "wf_query n J Q Qn"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = (\<lambda>tab. New_max.semiJoin tab (st, t)) ` Q" shows "covering J QQ"
fresh_star_insert: fixes a :: "atom"   and as :: "atom set"   and x :: "'a" shows "insert a as \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
indep_iff_rank_of: fixes X :: "'a set" assumes "X \<subseteq> carrier" shows "indep X = (rank_of X = card X)"
f'_def: fixes g :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" shows "f' g = f (g \<union> trap)"
incoming'_edges: fixes U :: "nat set" shows "incoming' U \<subseteq> E"
pdiv_pmod: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])" shows "p = q \<otimes>\<^bsub>K [X]\<^esub> (p pdiv q) \<oplus>\<^bsub>K [X]\<^esub> (p pmod q)"
simps_4: fixes vb :: "nat"   and va :: "Core.term list" shows "strip_comb_imp (Bv vb, va) = (Bv vb, va)"
ennreal_inj: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b" shows "(ennreal a = ennreal b) = (a = b)"
q01: fixes t :: "'a"   and n :: "nat" assumes "t \<in> S" shows "q n t \<in> {0..1}"
assms_3: shows "\<mu> \<star> \<sigma> \<noteq> null"
zero_pdivides: fixes p :: "'a list list" shows "[] pdivides\<^bsub>poly_ring R\<^esub> p = (p = [])"
lower_def: shows "lower \<equiv> map_fun bounds_of_interval id fst"
wbisim23: shows "delay_bisimulation_diverge_final trsys2 trsys3 bisim23 tlsim23 \<tau>move2 \<tau>move3 final2 final3"
map_upd_Some_unfold: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a"   and x :: "'b"   and y :: "'a" shows "((m(a \<mapsto> b)) x = \<lfloor>y\<rfloor>) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = \<lfloor>y\<rfloor>)"
dijkstra'_def: shows "dijkstra' \<equiv> dinit \<bind> (\<lambda>\<sigma>0. WHILE\<^sub>T\<^bsup>dinvar\<^esup> (\<lambda>(wl, uu_). wl \<noteq> {}) (\<lambda>\<sigma>. pop_min \<sigma> \<bind> (\<lambda>(x, y). update' x y)) \<sigma>0 \<bind> (\<lambda>(uu_, res). RETURN res))"
\<nu>'\<nu>_r: shows "\<guillemotleft>(\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g] : g \<star> src f \<Rightarrow> g \<star> w' \<star> w\<guillemotright>"
alpha1_def: shows "alpha1 \<equiv> $pc1 = #a \<and> pc1$ = #b \<and> $sem = Suc<sem$> \<and> unchanged (x, y, pc2)"
inf_sup_aci_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inf x (inf y z) = inf y (inf x z)"
Diagram_DataRefinement2: fixes D :: "I \<times> I \<Rightarrow> 'a \<Rightarrow> 'a"   and R :: "I \<Rightarrow> 'a \<Rightarrow> 'b"   and P :: "I \<Rightarrow> 'a"   and Q :: "I \<Rightarrow> 'a"   and D' :: "I \<times> I \<Rightarrow> 'b \<Rightarrow> 'b" assumes "dmono D"   and "Disjunctive_fun R"   and "Hoare_dgr P D Q"   and "DgrDataRefinement2 P D R D'" shows "Hoare_dgr (R .. P) D' ((R .. P) \<sqinter> - grd (step D'))"
map_relation_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and R :: "('c \<times> 'c) set" shows "map_relation A f R = {(x, y). (x, y) \<in> A \<times> A \<and> (f x, f y) \<in> R}"
older_seniors_tower': fixes y :: "nat"   and x :: "nat"   and n :: "nat" assumes "y \<in> older_seniors x n"   and "older_seniors x n = older_seniors x (Suc n)" shows "older_seniors y n = older_seniors y (Suc n)"
span_add_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "module scale"   and "x \<in> module.span scale S" shows "(x + y \<in> module.span scale S) = (y \<in> module.span scale S)"
sees_wf_mdecl: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b option"   and D :: "String.literal" assumes "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "wf_mdecl wf_md P D (M, Ts, T, m)"
exhaust: fixes y :: "'a Heap" obtains x :: "heap \<Rightarrow> ('a \<times> heap) option" where "y = Heap x"
antisym: fixes x :: "'e list"   and y :: "'e list" assumes "prefix x y"   and "prefix y x" shows "x = y"
module_def: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme" shows "module R M \<equiv> cring R \<and> abelian_group M \<and> module_axioms R M"
image_insert: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'d"   and B :: "'d set" shows "f ` insert a B = insert (f a) (f ` B)"
compress_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "compress w1 w2" shows "kauff_mat w1 = kauff_mat w2"
assms_3: shows "u \<cdot> v = v \<cdot> u"
split_list_last: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs"
degree_of_gcd: fixes q :: "int poly"   and r :: "int poly" shows "(degree (gcd q r) \<noteq> 0) = (degree (gcd (of_int_poly q) (of_int_poly r)) \<noteq> 0)"
continuous_map_alt: fixes T1 :: "'e topology"   and T2 :: "'f topology"   and f :: "'e \<Rightarrow> 'f" shows "continuous_map T1 T2 f = ((\<forall>U. openin T2 U \<longrightarrow> openin T1 (f -` U \<inter> topspace T1)) \<and> f ` topspace T1 \<subseteq> topspace T2)"
nat_pow_mult: fixes x :: "'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier \<G>" shows "x [^] n \<otimes> x [^] m = x [^] (n + m)"
strict_order_iff: fixes b :: "'a"   and a :: "'a" shows "(b < a) = (a = sup a b \<and> a \<noteq> b)"
times_vector_def: shows "times_vector = (\<lambda>((a, c), b, d) (a', c'). (a * a' + b * c', c * a' + d * c'))"
ThrowThrow: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
derives1_implies_Derives1: fixes u :: "symbol list"   and v :: "symbol list" assumes "derives1 u v" shows "\<exists>i r. Derives1 u i r v"
of_nat_less_0_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (semiring_1.of_nat one plus zero m) zero"
above_Field: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "above r a \<subseteq> Field r"
Idempotent_def: fixes H :: "'d upred \<Rightarrow> 'd upred" shows "Idempotent H = (\<forall>P. H (H P) = H P)"
sum_sqs_eq: fixes x :: "'b"   and y :: "'b" assumes "x * x + y * y = x * (y * (2::'b))" shows "y = x"
Int_insert_right_if0: fixes a :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "a \<notin> A" shows "A \<inter> insert a B = A \<inter> B"
f_def: shows "f \<equiv> \<lambda>(l, u). (l, SOME u. u \<in> R)"
MON_PU: shows "mon_env fg ww \<inter> (mon fg p \<union> mon_n fg u) = {}"
lub_eq: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'f \<Rightarrow> 'e"   and Y :: "'f \<Rightarrow> 'e" assumes "class.cpo below"   and "\<And>i. X i = Y i" shows "po.lub below (range X) = po.lub below (range Y)"
chop_fun: fixes t :: "('a, 'b) tm" assumes "is_App t"   and "is_App (fun t)" shows "App (chop (fun t)) (arg t) = chop t"
assms_2: shows "(l, x) \<in> set (transaction_strand T)"
set_def: fixes xs :: "['b]" shows "set xs = {x. listmem x xs}"
E_inf_iterate: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "f \<in> borel_measurable St" shows "E_inf s f = (\<Sqinter>D\<in>K s. \<integral>\<^sup>+ t. E_inf t (\<lambda>\<omega>. f (t ## \<omega>)) \<partial>measure_pmf D)"
hd_remdups_adj: fixes xs :: "'b list" shows "hd (remdups_adj xs) = hd xs"
distinct_27: fixes x41 :: "'b rexp"   and x42 :: "'b rexp"   and x6 :: "'b rexp" shows "Plus x41 x42 \<noteq> Star x6"
connected_compact_interval_1: fixes S :: "real set" shows "(connected S \<and> compact S) = (\<exists>a b. S = {a..b})"
has_derivative_transform_at: fixes d :: "real"   and x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "0 < d"   and "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'"   and "(f has_derivative f') (at x)" shows "(g has_derivative f') (at x)"
fundamental_theorem_algebra_factorized: fixes p :: "complex poly" shows "\<exists>as. smult (lead_coeff p) (\<Prod>a\<leftarrow>as. [:- a, 1:]) = p \<and> length as = degree p"
pr_joint_monic: fixes a :: "'a"   and b :: "'a"   and h :: "'a"   and h' :: "'a" assumes "seq \<pp>\<^sub>0[a, b] h"   and "\<pp>\<^sub>0[a, b] \<cdot> h = \<pp>\<^sub>0[a, b] \<cdot> h'"   and "\<pp>\<^sub>1[a, b] \<cdot> h = \<pp>\<^sub>1[a, b] \<cdot> h'" shows "h = h'"
\<alpha>nl_simps1_1: fixes fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" shows "\<alpha>nl fg \<circ> ENV = \<alpha>n fg"
binomial_fact': fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "n choose k = fact n div (fact k * fact (n - k))"
Ints_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "one \<in> ring_1.Ints one plus zero minus"
combine_pairs'_prop: fixes \<P> :: "((('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set \<times> (('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<times> 'b set \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set) set"   and \<delta> :: "('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option) \<Rightarrow> 'b set \<Rightarrow> 'b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option"   and q\<^sub>0 :: "'b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option"   and w :: "nat \<Rightarrow> 'b set" shows "(\<forall>P\<in>\<P>. accepting_pair\<^sub>R \<delta> q\<^sub>0 P w) = accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (combine_pairs' \<P>) w"
select_convs_2: fixes break_status :: "bool"   and return_status :: "bool"   and more :: "'b" shows "return_status \<lparr>break_status = break_status, return_status = return_status, \<dots> = more\<rparr> = return_status"
le_Suc_base: fixes a :: "nat \<Rightarrow> nat"   and j :: "nat" assumes "a \<in> s" shows "a j \<le> Suc (b j)"
right_neutral: fixes a :: "'b" shows "a + (0::'b) = a"
onorm_pos_lt: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "(0 < onorm f) = (\<not> (\<forall>x. f x = (0::'b)))"
arev_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(edge_rev GM x = edge_rev GM y) = (x = y)"
mult_right_mono_ge: fixes x :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "x \<le> a * c"   and "a \<le> b"   and "(0::'a) \<le> c" shows "x \<le> b * c"
simps_2: fixes xs :: "'b list" shows "minus_poly_rev_list xs [] = xs"
same_span_imp_card_eq_dim_span: fixes V :: "'a vec set"   and W :: "'a vec set" assumes "V \<subseteq> carrier_vec n"   and "W \<subseteq> carrier_vec n"   and "local.span V = local.span W"   and "lin_indpt V" shows "card V = dim_span W"
Class_in_Partition: fixes a :: "'a" assumes "a \<in> Ker" shows "L.Class a \<in> L.Partition"
p2_ident: shows "int (CARD('a) - 2) = p - 2"
correct1: fixes x1 :: "'a"   and w1 :: "'b"   and x0 :: "'g"   and e :: "'e" assumes "(x1, w1) \<in> Rel1"   and "x0 \<in> valid_pub0"   and "e \<in> carrier L" shows "\<Sigma>_OR.completeness_game (x0, x1) (Inr w1) e = return_spmf True"
pos: fixes A :: "'a set" assumes "A \<in> sets M" shows "emeasure N A \<le> emeasure M A"
idiff_0: fixes n :: "enat" shows "0 - n = 0"
intro: fixes x :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v" shows "pbij_lens_axioms x"
prod_assn_pure_conv: fixes R1 :: "('c \<times> 'a) set"   and R2 :: "('d \<times> 'b) set" shows "pure R1 \<times>\<^sub>a pure R2 = pure (R1 \<times>\<^sub>r R2)"
tranclp_into_rtranclp: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
red_supset_wf_on: fixes d :: "'a \<Rightarrow> nat"   and K :: "unit set"   and m :: "nat" assumes "dickson_grading d"   and "finite K" shows "wfp_on punit.red_supset (Pow (punit.dgrad_p_set d m) \<inter> {F. local.punit.component_of_term ` Keys F \<subseteq> K})"
cancels_to_1_at_def: fixes i :: "nat"   and l1 :: "(bool \<times> 'b) list"   and l2 :: "(bool \<times> 'b) list" shows "cancels_to_1_at i l1 l2 = (0 \<le> i \<and> 1 + i < length l1 \<and> canceling (l1 ! i) (l1 ! (1 + i)) \<and> l2 = cancel_at i l1)"
NIL_NOTIN_stateset: fixes X :: "'a list" shows "[] \<notin> state_set X"
continuous_map_from_discrete_topology: fixes U :: "'a set"   and X :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map (discrete_topology U) X f = (f ` U \<subseteq> topspace X)"
DecisionValuesExist: fixes c :: "('p, 'v, 's) configuration"   and p :: "'p" assumes "\<And>cc Q. terminationPseudo 1 cc Q"   and "initReachable c" shows "val[p,c] \<noteq> {}"
union_disjoint: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite A"   and "finite B"   and "A \<inter> B = {}" shows "abel_semigroup_set.F f z g (A \<union> B) = f (abel_semigroup_set.F f z g A) (abel_semigroup_set.F f z g B)"