hcju/isabelleps huggingface dataset:
pmf_nonneg: fixes p :: "'h pmf"   and x :: "'h" shows "0 \<le> pmf p x"
hyps: fixes y :: "nat" assumes "y < n"   and "0 < m" shows "gcd (fib m) (fib (y mod m)) = gcd (fib m) (fib y)"
simps: fixes a :: "nat" shows "ya_odd a = (\<exists>x. a = x + 1 \<and> ya_even x)"
front_tickFree_mono: fixes t :: "'c event list"   and r :: "'c event list" shows "front_tickFree (t @ r) \<and> r \<noteq> [] \<longrightarrow> tickFree t \<and> front_tickFree r"
tensor_in_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set"   and g :: "'c local.term set"   and c :: "'c local.term set"   and d :: "'c local.term set" assumes "\<F>C.in_hom f a b"   and "\<F>C.in_hom g c d" shows "\<F>C.in_hom (f \<otimes> g) (a \<otimes> c) (b \<otimes> d)"
nnfis_rv: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> nnfis f M" shows "f \<in> rv M"
contractible_imp_Borsukian: fixes S :: "'a set" assumes "contractible S" shows "Borsukian S"
distinct_1: fixes x21 :: "color"   and x22 :: "('b, 'c) RBT_Impl.rbt"   and x23 :: "'b"   and x24 :: "'c"   and x25 :: "('b, 'c) RBT_Impl.rbt" shows "rbt.Empty \<noteq> Branch x21 x22 x23 x24 x25"
bv_add_def: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_add w1 w2 = nat_to_bv (bv_to_nat w1 + bv_to_nat w2)"
extensional'_def: fixes A :: "'b set" shows "extensional' A \<equiv> {f. \<forall>x. x \<notin> A \<longrightarrow> f x = unity}"
residual_flow_attainability: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "RES.wf_residual_network"   and "flow (residual_network f) f" shows "flow_attainability (RES.residual_network f)"
ent_disjI1: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" assumes "P \<or>\<^sub>A Q \<Longrightarrow>\<^sub>A R" shows "P \<Longrightarrow>\<^sub>A R"
De: shows "\<D> e \<lfloor>dom l\<rfloor>"
unfold_llist_eq_LCons: fixes IS_LNIL :: "'c \<Rightarrow> bool"   and LHD :: "'c \<Rightarrow> 'b"   and LTL :: "'c \<Rightarrow> 'c"   and b :: "'c"   and x :: "'b"   and xs :: "'b llist" shows "(unfold_llist IS_LNIL LHD LTL b = LCons x xs) = (\<not> IS_LNIL b \<and> x = LHD b \<and> xs = unfold_llist IS_LNIL LHD LTL (LTL b))"
valid_eqvt: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and x :: "('e, 'c, 'd) formula"   and p :: "perm" assumes "nominal_ts satisfies transition"   and "nominal_ts.valid satisfies transition P x" shows "nominal_ts.valid satisfies transition (p \<bullet> P) (p \<bullet> x)"
move_left_pos: fixes b :: "'a textbuffer" shows "get_pos (move_left b) = get_pos b - 1"
top_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and T :: "'a"   and A :: "'a set" shows "Semilat.top r T A = (\<forall>x\<in>A. x \<sqsubseteq>\<^bsub>r\<^esub> T)"
two_conj: fixes c :: "one\<^sub>\<bottom>" shows "c = many \<or> c = once \<or> c = none"
coeff_eq_0: fixes p :: "'c poly"   and n :: "nat" assumes "degree p < n" shows "coeff p n = (0::'c)"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "finite {y. P x y}"
conj_subset_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "((A::'a set) \<subseteq> _Collect (x::'a) ((P::'a \<Rightarrow> bool) x \<and> (Q::'a \<Rightarrow> bool) x)) = (A \<subseteq> _Collect (x::'a) (P x) \<and> A \<subseteq> _Collect (x::'a) (Q x))"
simps_22: fixes g1 :: "'b gexp"   and g2 :: "'b gexp"   and v :: "bool" shows "GExp_Lexorder.less_gexp_aux (Nor g1 g2) (Bc v) = False"
simps: fixes n :: "nat"   and j :: "nat"   and "is" :: "nat list" shows "prime_nat_main n j is = (case is of [] \<Rightarrow> case next_candidates j of (j, is) \<Rightarrow> prime_nat_main n j is | i # is \<Rightarrow> if i dvd n then n \<le> i else if i * i \<le> n then prime_nat_main n j is else True)"
weakBisimE_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<approx> Q" shows "\<Psi> \<rhd> P \<leadsto><weakBisim> Q"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" shows "efficient_power f y x 0 = y"
right_unique_iff: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "right_unique R = (\<forall>z. \<exists>\<^sub>\<le>\<^sub>1 x. R z x)"
admits_SNF_HA_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "admits_SNF_HA A = (isDiagonal A \<longrightarrow> (\<exists>P Q. invertible P \<and> invertible Q \<and> Smith_normal_form (P ** A ** Q)))"
g_loop: fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "flow_network \<Delta> g" shows "g (x, x) = 0"
Mxs_2: shows "sum_weights (map (\<lambda>(s, l, t). l) xs) = s"
prems_2: shows "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel S)"
map_graph_def: fixes f :: "'c \<Rightarrow> 'd option" shows "map_graph f = {(x, y) |x y. f x = Some y}"
simps_1: fixes d :: "nat \<Rightarrow> nat" shows "from_digits 0 d = 0"
fmlookup_default_if_2: fixes ys :: "('b, 'a) fmap"   and a :: "'b"   and d :: "'a" assumes "fmlookup ys a = None" shows "fmlookup_default d ys a = d"
rel_intros_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x21 :: "'c"   and y21 :: "'d"   and x22 :: "'c llist"   and y22 :: "'d llist" assumes "R x21 y21"   and "llist_all2 R x22 y22" shows "llist_all2 R (LCons x21 x22) (LCons y21 y22)"
leg1_simps_6: shows "cod p\<^sub>1 = p\<^sub>1"
map_reindex: fixes l :: "'d list"   and g :: "'d \<Rightarrow> 'e"   and f :: "nat \<Rightarrow> 'e" assumes "\<forall>i<length l. g (l ! i) = f i" shows "map f [0..<length l] = map g l"
bind_returns_heap_E: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and h'' :: "'l" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'o" and h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
c0: fixes t :: "'a" assumes "t \<in> insert a T" shows "0 \<le> c t"
pow_one_id: fixes u :: "'b list" shows "u \<^sup>@ 1 = u"
ControlSystem_def: fixes s :: "nat \<Rightarrow> nat list" shows "ControlSystem s \<equiv> ts s \<and> (\<forall>j. 200 \<le> hd (s j) \<and> hd (s j) \<le> 800)"
intro: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>"   and "natural_isomorphism_axioms A B \<tau>" shows "natural_isomorphism A B F G \<tau>"
eq_on_subset: fixes arr1 :: "'a \<Rightarrow> 'b"   and arr2 :: "'a \<Rightarrow> 'b"   and R :: "'a set"   and S :: "'a set" assumes "arr1 = arr2 on R"   and "S \<subseteq> R" shows "arr1 = arr2 on S"
assms_1: shows "approx_slp_outer p n slp XS = Some RS"
less_eq_top_def: fixes x :: "'b top"   and y :: "'b top" shows "(x \<le> y) = (case y of top.Value y \<Rightarrow> case x of top.Value x \<Rightarrow> x \<le> y | top.Top \<Rightarrow> False | top.Top \<Rightarrow> True)"
mId_def: fixes M :: "('e, 'f, 'g) Module_scheme" shows "mId\<^bsub>M\<^esub>  = (\<lambda>m\<in>carrier M. m)"
\<sigma>'_ds: shows "(fst \<sigma>1', snd \<sigma>1' @ snd \<sigma>2' @ snd \<sigma>3') \<in> seq ars"
less_b_least_impl_zero: fixes y :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "y < b_least f x" shows "f x y = 0"
conditional_mutual_information_nonneg: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e"   and Z :: "'a \<Rightarrow> 'f" assumes "simple_function M X"   and "simple_function M Y"   and "simple_function M Z" shows "0 \<le> \<I>( X ; Y | Z )"
a: assumes "u < v" shows "f u \<subseteq> f v"
iMin_le_iff: fixes I :: "'a set"   and n :: "'a" assumes "I \<noteq> {}" shows "(iMin I \<le> n) = (\<exists>a\<in>I. a \<le> n)"
std_normal_moment_even: fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k)) (fact (2 * k) / (2 ^ k * fact k))"
ell2_norm_smult_1: fixes x :: "'c \<Rightarrow> complex"   and c :: "complex" assumes "has_ell2_norm x" shows "has_ell2_norm (\<lambda>i. c * x i)"
swap_events: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and i :: "nat"   and j :: "nat"   and t :: "('d, 'e, 'f) event list" assumes "computation channel trans send recv init final"   and "i < j"   and "j < length t"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "computation.postrecording_event channel trans send recv init t i"   and "computation.prerecording_event channel trans send recv init t j"   and "distributed_system.trace channel trans send recv init t final" shows "distributed_system.trace channel trans send recv init (swap_events i j t) final \<and> (\<forall>k\<ge>j + 1. distributed_system.s channel trans send recv init (swap_events i j t) k = distributed_system.s channel trans send recv init t k) \<and> (\<forall>k\<le>i. distributed_system.s channel trans send recv init (swap_events i j t) k = distributed_system.s channel trans send recv init t k) \<and> computation.prerecording_event channel trans send recv init (swap_events i j t) i \<and> computation.postrecording_event channel trans send recv init (swap_events i j t) (i + 1) \<and> (\<forall>k. i + 1 < k \<and> k < j + 1 \<longrightarrow> \<not> regular_event (swap_events i j t ! k))"
coeff_add: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "coeff (UP R) (p \<oplus>\<^bsub>UP R\<^esub> q) n = coeff (UP R) p n \<oplus>\<^bsub>R\<^esub> coeff (UP R) q n"
simps_6: fixes r :: "'a rexp" shows "lang (Star r) = lang r\<star>"
smap_stoch_proc_length: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and k :: "nat"   and w :: "'c" shows "length (smap_stoch_proc n f k w) = n"
map_option_eq_Some: fixes f :: "'c \<Rightarrow> 'b"   and xo :: "'c option"   and y :: "'b" shows "(map_option f xo = \<lfloor>y\<rfloor>) = (\<exists>z. xo = \<lfloor>z\<rfloor> \<and> f z = y)"
hyps: fixes X :: "'a nmultiset multiset"   and Y :: "'a nmultiset multiset"   and k :: "'a nmultiset"   and a :: "'a nmultiset" assumes "X \<noteq> {#}"   and "X \<subseteq># N"   and "M = N - X + Y"   and "k \<in># Y"   and "k < a" shows "|k| \<le> |a|"
inverse_arrows_lunit: fixes a :: "'a" assumes "ide a" shows "inverse_arrows \<l>[a] \<l>\<^sup>-\<^sup>1[a]"
init_triv_filt_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" shows "init_triv_filt M F = (Filtration.filtration M F \<and> sets (F \<bottom>) = {{}, space M})"
assms_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "d x \<cdot> y \<cdot> z = d x \<cdot> (y \<cdot> z)"
pathstart_rectpath: fixes a1 :: "complex"   and a3 :: "complex" shows "pathstart (rectpath a1 a3) = a1"
orbit_def: fixes s :: "'a" shows "\<gamma>\<^sup>\<phi> s = g_orbital (\<lambda>t. f) (\<lambda>s. True) (\<lambda>s. T) S 0 s"
real_of_nat_indicator: fixes A :: "'b set"   and x :: "'b" shows "real (indicator A x) = indicat_real A x"
those_eq_Some_lengthD: fixes xs :: "'b option list"   and ys :: "'b list" assumes "those xs = Some ys" shows "length xs = length ys"
is_glbD2: fixes S :: "'a set"   and x :: "'a"   and u :: "'a" assumes "S >>| x"   and "S >| u" shows "u \<sqsubseteq> x"
universe_difference: fixes x :: "i" assumes "\<exists>w. \<not> O w x" shows "\<forall>w. P w (u \<ominus> x) = (\<not> O w x)"
Var: fixes rs :: "dB list"   and n :: "nat" assumes "listsp IT rs" shows "IT (Var n \<degree>\<degree> rs)"
enc_Inr: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list"   and M :: "nat"   and P :: "nat set"   and p :: "nat" assumes "x \<in> enc (w, I)"   and "\<forall>x\<in>set I. case x of Inl a \<Rightarrow> True | Inr P \<Rightarrow> finite P"   and "M < length I"   and "I ! M = Inr P" shows "(p \<in> P) = (p < length x \<and> snd (x ! p) ! M)"
abs: fixes rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list" shows "rs, \<Gamma> \<turnstile>\<^sub>v Sabs cs \<down> Vabs cs \<Gamma>"
removeNones_none_hd: fixes ls :: "'a option list" shows "removeNones (None # ls) = removeNones ls"
simps_1: fixes m :: "nat" shows "free_tv (FVar m) = {m}"
subst_lsst_cons: fixes a :: "'d strand_label \<times> ('e, 'f) stateful_strand_step"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "a # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = (a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) # (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
assms_3: shows "density_context V V' \<Gamma> \<delta>"
alpha_refl: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "qGood X" shows "X #= X"
card_matching_vertex_cover: fixes C :: "'b set"   and M :: "'b set set"   and E :: "'b set set" assumes "finite C"   and "matching M"   and "M \<subseteq> E"   and "vertex_cover E C" shows "card M \<le> card C"
le_pow2: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "x \<le> 2 ^ y"
rad_of_strict_mono: fixes x :: "real"   and y :: "real" shows "(rad_of x < rad_of y) = (x < y)"
primary_ideal_proper1: fixes q :: "'a set" assumes "primary_ideal R q" shows "1\<^sub>r \<notin> q"
cases: fixes x :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" obtains \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" and axs :: "Core.term set" where "x = (\<Sigma>, axs)"
eucl_inf: fixes x :: "'c"   and y :: "'c" shows "inf x y = (\<Sum>i\<in>Basis. inf (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
summandTransition_3: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "hnf P" shows "P \<longmapsto> a[b] \<prec> P' = (a{b}.P' \<in> summands P)"
abs_conv_abscissa_def: fixes f :: "'b fds" shows "abs_conv_abscissa f = (INF s\<in>{s. fds_abs_converges f s}. ereal (s \<bullet> (1::'b)))"
is_empty_empty_simp: shows "PQ.is_empty PQ.empty"
matchers_vars_term_eq: fixes \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term"   and P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) set"   and \<tau> :: "'d \<Rightarrow> ('e, 'f) Term.term"   and s :: "('e, 'd) Term.term"   and t :: "('e, 'f) Term.term" assumes "\<sigma> \<in> matchers P"   and "\<tau> \<in> matchers P"   and "(s, t) \<in> P" shows "\<forall>x\<in>vars_term s. \<sigma> x = \<tau> x"
sorted_iff_nth_mono_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a 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))"
disjoint_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
SPN_r_SPN_s_leg0_eq: shows "SPN_r_SPN_s.leg0 = rs.S\<^sub>0 \<odot> Maps.PRJ\<^sub>0 rs.R\<^sub>0 rs.S\<^sub>1"
\<Gamma>: shows "\<Gamma> = \<Gamma>1 @ \<bottom> # \<Gamma>2"
prems: shows "steps0 (1, tp) A (Suc n) = (0, tp')"
multpw_cancelL: fixes ns :: "('a \<times> 'a) set"   and Z :: "'a multiset"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "refl ns"   and "trans ns" shows "((Z + X, Z + Y) \<in> multpw ns) = ((X, Y) \<in> multpw ns)"
assms_1: shows "is1 \<lhd> nths ds A"
ctxt_subst_mem_VarB: fixes x :: "vrs"   and T :: "ty"   and \<Gamma> :: "binding list"   and Y :: "tyvrs"   and U :: "ty" assumes "VarB x T \<in> set \<Gamma>" shows "VarB x (T[Y \<mapsto> U]\<^sub>\<tau>) \<in> set (\<Gamma>[Y \<mapsto> U]\<^sub>e)"
diff_inv_def: shows "diff_inv = (\<pi> - initial_value pf) / constant_image (prices Mkt pos_stock 0)"
fsubset_finsertI2: fixes A :: "'b fset"   and B :: "'b fset"   and b :: "'b" assumes "A |\<subseteq>| B" shows "A |\<subseteq>| finsert b B"
fl: fixes j :: "'a"   and x :: "'b" assumes "j \<in> I"   and "a < j"   and "j \<le> b" shows "(f(i := x) \<longlongrightarrow> l j) (at_left j)"
is_Abs_def: fixes "term" :: "Core.term" shows "is_Abs term = (\<exists>x41 x42. term = Abs x41 x42)"
ennreal_escape_rate: fixes x :: "'a" shows "ennreal (escape_rate x) = \<integral>\<^sup>+ x. ennreal (R x x) \<partial>count_space UNIV"
list_disj_def: fixes ps :: "fm list" shows "list_disj ps \<equiv> foldr Parametric_Ferrante_Rackoff.disj ps F"
rel_pmf_reflI: fixes p :: "'e pmf"   and P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> P x x" shows "rel_pmf P p p"
conjugation_multl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<sqinter> z = \<bottom>) = (x \<sqinter> (z \<lhd> y) = \<bottom>)"
repr_bad: fixes \<psi> :: "'b" assumes "\<psi> \<notin> span B" shows "repr \<psi> = (\<lambda>_. 0)"
case_distrib: fixes h :: "'i \<Rightarrow> 'h"   and f1 :: "'j \<Rightarrow> 'i"   and f2 :: "'k \<Rightarrow> 'i"   and sum :: "'j + 'k" shows "h (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum of Inl x \<Rightarrow> h (f1 x) | Inr x \<Rightarrow> h (f2 x))"
associative: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M"   and "b \<in> M"   and "c \<in> M" shows "composition (composition a b) c = composition a (composition b c)"
him: shows "h ` sphere 0 1 \<subseteq> S"
e: shows "edges (map_graph (R O on_graph B f) A) \<subseteq> edges (map_graph_fn B f)"
measurable_compose_countable': fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'b \<Rightarrow> 'a" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space I"   and "countable I" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
rsA: shows "recurrent_subset A \<in> sets (restrict_space M A)"
ActualForallI: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" assumes "[\<^bold>\<forall>x. \<^bold>\<A>\<phi> x in v]" shows "[\<^bold>\<A>(\<^bold>\<forall>x. \<phi> x) in v]"
add_col_sub_row_carrier_2: fixes a :: "'b"   and k :: "nat"   and l :: "nat"   and A :: "'b mat" shows "dim_col (add_col_sub_row a k l A) = dim_col A"
simps_4: fixes p :: "'a pfm"   and q :: "'a pfm" shows "static (p \<^bold>\<and>\<^sub>! q) = (static p \<and> static q)"
emeasure_real: fixes A :: "(int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) set" shows "\<exists>r\<ge>0. emeasure MI A = ennreal r"
distinct_2: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.MkArr x21 x22 x23 \<noteq> concrete_category.Null"
inv_equality: fixes y :: "'c"   and x :: "'c" assumes "y \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"   and "x \<in> carrier H"   and "y \<in> carrier H" shows "inv\<^bsub>H\<^esub> x = y"
card_subset_not_gt_card: fixes A :: "'b set"   and ps :: "'b set" assumes "finite A"   and "card A < card ps" shows "\<not> ps \<subseteq> A"
hom_kernel_dim: fixes A :: "'a mat" shows "kernel_dim (mat\<^sub>h A) = kernel_dim A"
simps: fixes M1 :: "('e, 'f, 'g) FSM"   and M2 :: "('e, 'f, 'h) FSM"   and iseq :: "'e list"   and \<Omega> :: "('e, 'f) ATC set" shows "atc_io_reduction_on M1 M2 iseq \<Omega> = (M1 \<preceq>\<lbrakk>{iseq}\<rbrakk> M2 \<and> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {iseq}. B M1 io \<Omega> \<subseteq> B M2 io \<Omega>))"
assms: shows "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'"
elims: fixes x :: "\<langle>'a\<rangle>\<^sub>\<bottom>"   and y :: "'a" assumes "\<lceil>x\<rceil> = y" obtains v :: "'a" where "x = \<lfloor>v\<rfloor>" and "y = v"   | "x = \<bottom>" and "y = undefined"
disjoint_family_on_def: fixes A :: "'a \<Rightarrow> 'b set"   and S :: "'a set" shows "disjoint_family_on A S = (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
reduced_IF_depends: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool"   and t :: "'f ifex"   and X :: "'f set" assumes "reduced_bdt_checkers ifex_of val"   and "reduced t X"   and "t \<noteq> Trueif"   and "t \<noteq> Falseif" shows "\<exists>s1 s2. val_ifex t s1 \<noteq> val_ifex t s2"
exception_of_option_option_of_exception: fixes x :: "'h exception" shows "exception_of_option (option_of_exception x) = x"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'c" assumes "i < nr"   and "j < nc" shows "mat nr nc f $$ (i, j) = f (i, j)"
not_bot_dense: fixes uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.linorder_stone_algebra_expansion uminus less_eq less bot top inf sup"   and "x \<noteq> bot" shows "uminus (uminus x) = top"
unat_mod: fixes x :: "'b word"   and y :: "'b word" shows "unat (x mod y) = unat x mod unat y"
bind_return'': fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "M \<bind> return N = M"
pre_subdivFace_face_def: fixes f :: "face"   and v' :: "nat"   and vOptionList :: "nat option list" shows "pre_subdivFace_face f v' vOptionList \<equiv> filter (\<lambda>v. v \<in> set (removeNones vOptionList)) (verticesFrom f v') = removeNones vOptionList \<and> \<not> final f \<and> distinct (vertices f) \<and> hd vOptionList = Some v' \<and> v' \<in> \<V> f \<and> last vOptionList = Some (last (verticesFrom f v')) \<and> hd (tl vOptionList) \<noteq> last vOptionList \<and> 2 < |vOptionList| \<and> vOptionList \<noteq> [] \<and> tl vOptionList \<noteq> []"
qtm_conc: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and q' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.queue_to_multiset (q @ q') = SkewBinomialHeapStruc.queue_to_multiset q + SkewBinomialHeapStruc.queue_to_multiset q'"
tensor_relations4_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and MN :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "TR4 R M N MN = {x. \<exists>m\<in>carrier M. \<exists>n\<in>carrier N. \<exists>a\<in>carrier R. x = (m, a \<cdot>\<^sub>s\<^bsub>N\<^esub> n) \<plusminus>\<^bsub>MN\<^esub> -\<^sub>a\<^bsub>MN\<^esub> a \<cdot>\<^sub>s\<^bsub>MN\<^esub> (m, n)}"
index_mult_mat_1: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and B :: "'b mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = row A i \<bullet> col B j"