hcju/isabelleps huggingface dataset:
partitionsI: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "\<And>i. p i \<noteq> 0 \<Longrightarrow> 1 \<le> i \<and> i \<le> n"   and "(\<Sum>i\<le>n. p i * i) = n" shows "p partitions n"
map_us_map_IInf_iff: fixes f :: "'a upset \<Rightarrow> 'b upset"   and g :: "'a \<Rightarrow> 'b" shows "(f = map_us g) = (map_IInf f = g)"
sup_inf_distrib1: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<squnion> y \<sqinter> z = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
power4_eq_xxxx: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.monoid_mult one times" shows "power.power one times x 4 = times (times (times x x) x) x"
vars_term_scf_subset: fixes scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and s :: "('b, 'a) Term.term" shows "vars_term (scf_term scf s) \<subseteq> vars_term s"
finite_imp_ENR: fixes S :: "'a set" assumes "finite S" shows "ENR S"
composition_closed: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> Partition"   and "b \<in> Partition" shows "a [\<cdot>] b \<in> Partition"
components_pmdl: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "component_of_term ` Keys (pmdl B) = component_of_term ` Keys B"
least: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "class.pcpo below" shows "\<exists>x. \<forall>y. below x y"
is_partition_Cons: fixes x :: "'b set"   and xs :: "'b set list" shows "is_partition (x # xs) = (is_partition xs \<and> x \<inter> \<Union> (set xs) = {})"
simps_1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and xs :: "'varSort"   and x :: "'var" shows "qInit MOD (qVar xs x) = igVar MOD xs x"
simps_2: fixes S :: "Runs.var set" shows "option_inter S None = None"
fbox_antitone_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows " |y] z \<le>  |x] z"
cases: fixes a1 :: "('fenv_inca, 'c, 'fun) prog"   and a2 :: "('fenv_inca, 'c, ('fun, 'd) frame) state" assumes "Sinca.load a1 a2" obtains F :: "'fenv_inca" and main :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef" and H :: "'c" where "a1 = Prog F H main" and "a2 = State F H [Frame main 0 []]" and "Finca_get F main = Some fd" and "arity fd = 0"
le_refl: fixes x :: "'a" assumes "x \<in> carrier L" shows "x \<sqsubseteq> x"
simps_4: fixes I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and ux :: "char list set"   and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Es :: "char list set \<times> (char list \<Rightarrow> char list set)"   and b :: "bexp"   and C :: "acom" shows "varacom ({((I, ux), S, E, Es)} WHILE b DO C) = support I \<union> varacom C"
elapsed_ge0: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" shows "(0::'a) \<le> elapsed x y"
is_zero_row_upt_k_le: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" assumes "is_zero_row_upt_k i (Suc k) A" shows "is_zero_row_upt_k i k A"
aezfun_setspan_projD1: fixes a :: "'a"   and A :: "'a set"   and x :: "('b, 'a) aezfun" assumes "a \<in> A" shows "aezfun (aezfun_setspan_proj A x) a = aezfun x a"
sns_2: shows "g \<turnstile> Entry g-sns\<rightarrow>n"
altc_sample_writer_def: fixes altc_sample :: "'n \<Rightarrow> ('o \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'n"   and f :: "'o \<Rightarrow> ('k, 'l, 'm) writerT" shows "altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
stutter_sampler_build: fixes f :: "nat \<Rightarrow> nat"   and w :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "stutter_sampler f w" shows "stutter_sampler (0 ## (Suc \<circ> f)) (a ## w)"
nth_prime_at_top: shows "filterlim nth_prime sequentially sequentially"
acquired_reads_mono: fixes A :: "nat set"   and B :: "nat set"   and pending_write :: "bool"   and xs :: "'b memref list" assumes "A \<subseteq> B" shows "acquired_reads pending_write xs A \<subseteq> acquired_reads pending_write xs B"
Arity_up_zero_join: fixes x :: "Arity\<^sub>\<bottom>" shows "x \<squnion> up\<cdot>0 = up\<cdot>0"
summable_geometric: assumes "norm c < 1" shows "summable ((^) c)"
vimage_vempty: fixes A :: "V" shows "0 `\<^sub>\<circ> A = 0"
separable_def: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "dcharacter n \<chi>" shows "dcharacter.separable n \<chi> k = (dcharacter.gauss_sum n \<chi> k = cnj (\<chi> k) * dcharacter.gauss_sum n \<chi> 1)"
Gromov_converging_at_boundary_rel: fixes u :: "nat \<Rightarrow> 'a" assumes "Gromov_converging_at_boundary u" shows "Gromov_completion_rel u u"
wfP_def: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "wfP r = wf {(x, y). r x y}"
PiE_eq_empty_iff: fixes I :: "'a set"   and F :: "'a \<Rightarrow> 'b set" shows "(Pi\<^sub>E I F = {}) = (\<exists>i\<in>I. F i = {})"
sym: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "f x y = f y x"
N_unique: fixes \<phi> :: "'fmla" shows "let NN = \<lambda>t1 t2. psubst N [(t1, xx), (t2, yy)] in \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {} \<longrightarrow> bprv (all yy (all yy' (imp (cnj (NN \<langle>\<phi>\<rangle> (Var yy)) (NN \<langle>\<phi>\<rangle> (Var yy'))) (eql (Var yy) (Var yy')))))"
tendsto_min: fixes X :: "'d \<Rightarrow> 'a"   and x :: "'a"   and net :: "'d filter"   and Y :: "'d \<Rightarrow> 'a"   and y :: "'a" assumes "(X \<longlongrightarrow> x) net"   and "(Y \<longlongrightarrow> y) net" shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
emeasure_eq_measure2: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> fmeasurable M" shows "emeasure M A = ennreal (Sigma_Algebra.measure M A)"
snd_conv: fixes x1 :: "'m"   and x2 :: "'l" shows "snd (x1, x2) = x2"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
butlast_Subobjs_Rep: 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 Cs :: "char list list"   and D :: "char list" assumes "Subobjs\<^sub>R P (hd Cs) (Cs @ [D])"   and "Cs \<noteq> []" shows "Subobjs\<^sub>R P (hd Cs) Cs"
homeomorphic_affine_sets: fixes S :: "'b set"   and T :: "'c set" assumes "affine S"   and "affine T"   and "aff_dim S = aff_dim T" shows "S homeomorphic T"
vsv_axioms: shows "vsv (T\<lparr>ArrVal\<rparr>)"
trancl_into_trancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
RSD_pareto_eqclassI: fixes A :: "'a set"   and alts :: "'a set"   and agents :: "'b set"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<noteq> {}"   and "A \<subseteq> alts"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> alts\<rbrakk> \<Longrightarrow> (y \<in> A) = (\<forall>i\<in>agents. R i x y)" shows "RSD_pareto_eqclass agents alts R A"
positive: shows "positive (Pow A) (\<lambda>a. if finite a then of_nat (card a) else \<infinity>)"
2: shows "rawpsubstT (rawpsubstT t2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubstT (rawpsubstT t2 (zip (map Var vs2) (map snd txs))) (zip (map fst txs) vs2)"
simps_3: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "upd_lock l t UnlockFail = l"
graph_def: fixes \<alpha> :: "'a \<Rightarrow> ('b, 'c) graph"   and invar :: "'a \<Rightarrow> bool" shows "graph \<alpha> invar \<equiv> (\<forall>g. invar g \<longrightarrow> finite (nodes (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> finite (edges (\<alpha> g))) \<and> (\<forall>g. invar g \<longrightarrow> valid_graph (\<alpha> g))"
conc_fst: fixes n :: "nat"   and w :: "'a list"   and x :: "nat \<Rightarrow> 'a" assumes "n < length w" shows "(w \<frown> x) n = w ! n"
src: shows "\<R>\<^sub>G \<alpha> \<beta> s (\<alpha> r) = Some (\<R>\<^sub>S \<alpha> \<beta> \<sigma>, \<R>\<^sub>S \<alpha> \<beta> \<tau>, (\<R>\<^sub>C \<alpha> \<beta> \<E>) [Rfork (\<R>\<^sub>E \<alpha> \<beta> e)])"
existence_ivl_eq_rev: fixes t0 :: "real"   and x0 :: "'a" shows "(+) t0 ` existence_ivl0 x0 = preflect t0 ` ll_on_open.existence_ivl (range (preflect t0)) (\<lambda>t. - f) X t0 x0"
simps: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter" assumes "triple phi" shows "triple.pairs_uminus phi (x, y) = (- x, phi x)"
intTF_def: fixes \<sigma> :: "'c"   and a :: "'d" shows "intTF \<sigma> a \<equiv> True"
list_slice_nth: fixes m :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < length xs div k" shows "list_slice xs k ! m = xs \<up> (m * k) \<down> k"
take_induct: fixes P :: "'e\<cdot>('f, 'g) writerT \<Rightarrow> bool"   and x :: "'e\<cdot>('f, 'g) writerT" assumes "\<And>n. P (writerT_take n\<cdot>x)" shows "P x"
subst_sst_cons: fixes a :: "('a, 'b) stateful_strand_step"   and A :: "('a, 'b) stateful_strand_step list"   and \<delta> :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "a # A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta> = (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) # (A \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
finite_dimensionE: fixes K :: "'a set"   and E :: "'a set" assumes "subfield K R"   and "finite_dimension K E" shows "dimension ((local.dim over K) E) K E"
simps_11: fixes v :: "'b rexp" shows "le_rexp One (Star v) = True"
semiring_simprules_2: fixes R :: "('a, 'b) ring_scheme" assumes "semiring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
case_prod_conv: fixes f :: "'i \<Rightarrow> 'j \<Rightarrow> 'h"   and a :: "'i"   and b :: "'j" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
parallel_def: fixes i :: "nat"   and j :: "nat"   and x :: "nat" shows "parallel i j x \<equiv> eval r_parallel [i, j, x]"
integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> c"   and "c \<le> b"   and "f integrable_on {a..b}" shows "integral {a..c} f + integral {c..b} f = integral {a..b} f"
intro: fixes B :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and b :: "'ag" assumes "partial_magma.ide B b" shows "constant_functor_axioms B b"
blinfun_scaleR_def: fixes a :: "'a \<Rightarrow>\<^sub>L real"   and b :: "'b" shows "blinfun_scaleR a b = blinfun_scaleR_left b o\<^sub>L a"
supp_map_transform: fixes ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "supp (map_transform trans ae \<Gamma>) \<subseteq> supp \<Gamma>"
nxt_ev: fixes n :: "nat"   and \<phi> :: "'b stream \<Rightarrow> bool"   and xs :: "'b stream" assumes "(nxt ^^ n) \<phi> xs" shows "ev \<phi> xs"
Fiber_equality: fixes a :: "'a"   and b :: "'a" assumes "a \<in> S"   and "b \<in> S" shows "(Class a = Class b) = (\<alpha> a = \<alpha> b)"
elims: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and xa :: "'b list"   and y :: "'a" assumes "bisum x xa = y" obtains f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a" and a :: "'b" and as :: "'b list" where "x = fa__" and "xa = a # as" and "y = fa__ (0::'b) a + bisumTail fa__ (a # as)"   | f :: "'b \<Rightarrow> 'b \<Rightarrow> 'a" where "x = f" and "xa = []" and "y = f (0::'b) (0::'b)"
PORI_3: fixes n :: "nat" shows "(\<lambda>i. ValN\<cdot>n) \<in> unlr POR.delta"
tagged_def: fixes tag :: "String.literal"   and x :: "String.literal option"   and ts :: "xml_tree list" shows "xml.tagged tag x ts = xml.Elem tag (case x of None \<Rightarrow> [] | Some x' \<Rightarrow> [(STR ''0'', x')]) ts"
norm_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "free_monoidal_category.norm C (free_monoidal_category.rep C f) = free_monoidal_category.rep C f"
regular_locally_noteq: fixes x :: "'a" assumes "x \<in> X"   and "f x \<noteq> (0::'a)" shows "\<forall>\<^sub>F t in at 0. flow0 x t \<noteq> x"
assms_1: shows "A B C Cong3 A' B' C'"
rep_eq: shows "fls_nth fls_X = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
Suc: shows "\<Union> {pcp_seq C S n |n. n \<le> m} = pcp_seq C S m"
assms: shows "max_Var_floatarith v \<le> m"
g_bigtheta: shows "g \<in> \<Theta>(\<lambda>x. g' (real x))"
less_eq_extended_case: fixes x1a :: "'b extended"   and x2a :: "'b extended" shows "(x1a \<le> x2a) = (case x1a of Fin xda \<Rightarrow> case x2a of Fin x2da \<Rightarrow> xda \<le> x2da | \<infinity> \<Rightarrow> True | -\<infinity> \<Rightarrow> False | \<infinity> \<Rightarrow> case x2a of \<infinity> \<Rightarrow> True | _ \<Rightarrow> False | -\<infinity> \<Rightarrow> case x2a of Fin xba \<Rightarrow> True | _ \<Rightarrow> True)"
preserves_ide: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "C.ide (fst a \<star>\<^sub>C snd a)"
map_family_triangle4: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and f :: "'c \<Rightarrow> 'f \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. f j \<in> aHom S (A j)" shows "\<forall>j\<in>I. compos S (f j) aI\<^bsub>S\<^esub> = f j"
ereal_open_closed_aux: fixes S :: "ereal set" assumes "open S"   and "closed S"   and "- \<infinity> \<notin> S" shows "S = {}"
union_Bag: fixes xs :: "('b, nat) alist"   and ys :: "('b, nat) alist" shows "Bag xs + Bag ys = Bag (join (\<lambda>x (n1, n2). n1 + n2) xs ys)"
lst_def: fixes ss :: "'c \<times> ('d \<times> 'c) list" shows "lst ss = (if snd ss = [] then fst ss else snd (last (snd ss)))"
add_mono_neutr: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "Le (0::'c) \<preceq> b" shows "a \<preceq> dbm_add a b"
mod_0_imp_dvd: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m"   and b :: "'m" assumes "class.semiring_modulo plus minus zero times one divide modulo"   and "modulo a b = zero" shows "dvd.dvd times b a"
eq_iff: fixes a :: "'g multiset"   and b :: "'g multiset" shows "(a = b) = (b \<subseteq># a \<and> a \<subseteq># b)"
lexord_irreflexive: fixes r :: "('d \<times> 'd) set"   and xs :: "'d list" assumes "\<forall>x. (x, x) \<notin> r" shows "(xs, xs) \<notin> lexord r"
1: shows "p \<odot> monomial c (lt f) = local.punit.monom_mult (c / lc f) (0::'a) p \<odot> monomial (lc f) (lt f)"
laplace_integrand_absolutely_integrable_on_Ici_iff: fixes M :: "real"   and c :: "real"   and s :: "complex"   and a :: "real" assumes "exponential_order M c f"   and "c < Re s" shows "complex_set_integrable lebesgue {a..} (local.laplace_integrand s) = (\<forall>k>a. complex_set_integrable lebesgue {a..k} (local.laplace_integrand s))"
val_of_divide: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> nonzero Zp"   and "val_Zp y \<le> val_Zp x" shows "val_Zp (local.divide x y) = val_Zp x - val_Zp y"
t5: shows "typ_of s = typ_of (Abs_fv x \<tau>' s $ Fv x \<tau>')"
AT_fps_altdef: fixes f :: "nat \<Rightarrow> real"   and n :: "nat" shows "AT_fps f n = (\<Sum>m\<le>n. fps_const (real (Stirling n m)) * (fps_X - 1) ^ m * (fps_deriv ^^ m) (AT_fps f 0))"
wx_def: shows "wx = geodesic_segment_param {e--x} e t"
HInfinite_mult: fixes x :: "'b star"   and y :: "'b star" assumes "x \<in> HInfinite"   and "y \<in> HInfinite" shows "x * y \<in> HInfinite"
connectedD_const: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "connected S"   and "continuous_on S P" shows "\<exists>c. \<forall>s\<in>S. P s = c"
not_cd_3: shows "i' icd\<^bsup>\<pi>'\<^esup>\<rightarrow> 0"
dgrad_p_set_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat" shows "dgrad_p_set d m = {p. pp_of_term ` keys p \<subseteq> dgrad_set d m}"
not_prime_elem_zero: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.prime_elem times one zero zero"
iso_dghm_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<BB>\<lparr>Obj\<rparr>"
hom_zero: shows "h \<zero> = \<zero>\<^bsub>S\<^esub>"
silent_moves_Nil: fixes S :: "'node set"   and f :: "'edge \<Rightarrow> 'state edge_kind"   and n :: "'node"   and s :: "'state" shows "S,f \<turnstile> (n,s) =[]\<Rightarrow>\<^sub>\<tau> (n,s)"
transpose_transpose: fixes A :: "'a mat" shows "A\<^sup>T\<^sup>T = A"
nres_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
rightUniqueInjectiveOnFirstImplication: fixes P :: "('c \<times> 'd) set" assumes "runiq P" shows "inj_on fst P"
A_def: shows "A = {v \<in> natpermute k (m + 1). k \<in> set v}"
cancel1_append_same1: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "cancel1 xs ys" shows "cancel1 (zs @ xs) (zs @ ys)"
xin: shows "x \<in> set [a .e. 2 ^ n - 1]"
nterm_to_term_eq_closed: fixes t :: "nterm"   and \<Gamma> :: "name list"   and \<Gamma>' :: "name list" assumes "closed t" shows "nterm_to_term \<Gamma> t = nterm_to_term \<Gamma>' t"
fps_nth_fps_expansion: fixes f :: "complex \<Rightarrow> complex"   and F :: "complex fps"   and n :: "nat" assumes "f has_fps_expansion F" shows "fps_nth F n = (deriv ^^ n) f 0 / fact n"
extend_basis_def: fixes B :: "'b set" shows "extend_basis B = (SOME B'. B \<subseteq> B' \<and> independent B' \<and> span B' = UNIV)"
comp_tuple_arr: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "span f g"   and "arr h"   and "local.dom f = cod h" shows "\<langle>f, g\<rangle> \<cdot> h = \<langle>f \<cdot> h, g \<cdot> h\<rangle>"
t_LAM_inversion: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and t :: "lam"   and T :: "ty" assumes "\<Gamma> \<turnstile> LAM [x].t : T"   and "x \<sharp> \<Gamma>" shows "\<exists>T\<^sub>1 T\<^sub>2. T = T\<^sub>1 \<rightarrow> T\<^sub>2 \<and> (x, T\<^sub>1) # \<Gamma> \<turnstile> t : T\<^sub>2"
obj_ty_eq1: fixes obj :: "obj"   and obj' :: "obj" assumes "tag obj = tag obj'" shows "obj_ty obj = obj_ty obj'"
simps_6: fixes P :: "'a prog"   and h :: "'heap"   and v :: "'addr val" shows "\<tau>move1 P h (Val v) = False"
coef_s'_0_ge_s: shows "coef s' 0 \<ge>\<^sub>p coef s 0"
indep_setD_ev1: fixes A :: "'a set set"   and B :: "'a set set" assumes "indep_set A B" shows "A \<subseteq> events"
brD: fixes c :: "'d"   and a :: "'e"   and \<alpha> :: "'d \<Rightarrow> 'e"   and I :: "'d \<Rightarrow> bool" assumes "(c, a) \<in> br \<alpha> I" shows "a = \<alpha> c \<and> I c"
StaticCastThrow: 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)"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
subset_imageE: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "B \<subseteq> \<P> f A" obtains C :: "'c set" where "C \<subseteq> A" and "B = \<P> f C"
vrestr_image_3: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "x \<in> f --` A" shows "f x \<in> A"
le_less_linear: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "less_eq x y \<or> less y x"
dist: shows "\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
AnnBasic: fixes r :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and q :: "'a set" assumes "r \<subseteq> {s. f s \<in> q}" shows "\<turnstile> AnnBasic r f  q"
locally_faithful_iff_faithful: shows "G.locally_faithful = faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
ide_Ide: fixes a :: "'a \<times> 'a" assumes "V.ide (fst a) \<and> V.ide (snd a)" shows "H.VxV.ide a"
simps_3: fixes x21 :: "'b"   and x22 :: "'b heap list" shows "Hp x21 x22 \<noteq> Empty"
prv_exiI: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "prv (subst \<phi> t x)" shows "prv (exi x \<phi>)"
i0: shows "vec_norm (vs ! i) \<noteq> 0"
remdups_adj_singleton_iff: fixes xs :: "'b list" shows "(length (remdups_adj xs) = Suc 0) = (xs \<noteq> [] \<and> xs = replicate (length xs) (hd xs))"
cSup_eq: fixes X :: "'a set"   and a :: "'a" assumes "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y" shows "Sup X = a"
add_comm: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pas times D"   and "D x y" shows "D y x \<and> times x y = times y x"
x: shows "x \<in> eball (0::'a) (fps_conv_radius f)"
evnodd_empty: fixes b :: "nat" shows "evnodd {} b = {}"
REL_INTFI: fixes R :: "'a"   and I :: "'b" shows "REL_INTF R I"
less: fixes u :: "real" assumes "u \<in> {0..1}" shows "(1 - x * y) * u < 1"
CDERIV_pow: fixes n :: "nat"   and x :: "complex" shows "((\<lambda>x. x ^ n) has_field_derivative complex_of_real (real n) * x ^ (n - Suc 0)) (at x)"
lowlink_set_finite: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c" assumes "DFS_invar G param s" shows "finite (param_DFS_defs.lowlink_set G s v)"
hfst_def: fixes p :: "hf" shows "hfst p \<equiv> THE x. \<exists>y. p = \<langle>x, y\<rangle>"
f_consequent_def: fixes r :: "prop_rule" shows "predicate.f_consequent r = fset_from_list (consequent r)"
minus_1_div_exp_eq_int: fixes n :: "nat" shows "- 1 div 2 ^ n = - 1"
qml_act_1: fixes \<phi> :: "\<o>" shows "[[\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<A>\<phi>]]"
singleton_equal: fixes p :: "'a"   and q :: "'a" assumes "singleton p"   and "singleton q"   and "q \<le> p" shows "q = p"
A: shows "\<tau> \<Turnstile> H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(x)"
distinct_mset_mono: fixes D' :: "'b multiset"   and D :: "'b multiset" assumes "D' \<subseteq># D"   and "distinct_mset D" shows "distinct_mset D'"
try_spmf_return_None: fixes q :: "'h spmf" shows "TRY return_pmf None ELSE q = q"
trans: fixes R :: "('b \<times> 'b) set" assumes "R \<in> C" shows "trans R"
add_mset_commute: fixes x :: "'c"   and y :: "'c"   and M :: "'c multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
simps: fixes E :: "char list \<Rightarrow> hterm" shows "sub_of_denot E = fterm_of_hterm \<circ> E"
add_closed: fixes U :: "'c set"   and V :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Subspace.subspace U V"   and "x \<in> U"   and "y \<in> U" shows "x + y \<in> U"
bigtheta_inverse_eq_set_inverse: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "\<Theta>[F](\<lambda>x. inverse (f x)) = set_inverse \<Theta>[F](f)"
paint_color_of: fixes t :: "('a, 'b) rbt" shows "color_of (paint B t) = B"
cut_ge_Un: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A \<union> B) \<down>\<ge> t = A \<down>\<ge> t \<union> B \<down>\<ge> t"
assms_3: shows "lookup_raw (xs, ko) = lookup_raw (ys, ko)"