hcju/isabelleps huggingface dataset:
add_left: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a"   and a' :: "'a"   and b :: "'b" assumes "bounded_cbilinear prod" shows "prod (a + a') b = prod a b + prod a' b"
case_5: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "char list \<Rightarrow> 'a"   and x5 :: "char list" shows "(case Class x5 of Void \<Rightarrow> f1 | Boolean \<Rightarrow> f2 | Integer \<Rightarrow> f3 | NT \<Rightarrow> f4 | Class x \<Rightarrow> f5 x) = f5 x5"
int_of_integer_symbolic_def: shows "int_of_integer_symbolic = int_of_integer"
ll_on_open_axioms_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" shows "ll_on_open_axioms T f X \<equiv> (local_lipschitz T X f \<and> (\<forall>x. x \<in> X \<longrightarrow> continuous_on T (\<lambda>t. f t x))) \<and> open T \<and> open X"
that: fixes g :: "'a \<Rightarrow> real" assumes "continuous_on UNIV g"   and "g \<in> lspace (lebesgue_on S) (ennreal p)"   and "lnorm (lebesgue_on S) p (f - g) < e" shows "thesis"
N_lt_imp_dvd: fixes q :: "real"   and a :: "int"   and n :: "nat" assumes "fbnorm q"   and "fbnorm.N q a < 1 / q ^ n"   and "0 < n" shows "int n dvd a"
l_cancel_one': fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and a :: "'b" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = x \<otimes>\<^bsub>G\<^esub> a) = (a = \<one>\<^bsub>G\<^esub>)"
lfp_wp_loop_unfold: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "healthy (wp body)"   and "sound P" shows "lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s) = (\<lambda>s. \<guillemotleft> G \<guillemotright> s * wp body (lfp_exp (\<lambda>Q s. \<guillemotleft> G \<guillemotright> s * wp body Q s + \<guillemotleft> \<N> G \<guillemotright> s * P s)) s + \<guillemotleft> \<N> G \<guillemotright> s * P s)"
is_prebicategory_with_homs: shows "prebicategory_with_homs (\<cdot>) (\<star>) \<a> src trg"
mult_strict_left_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
set_zip_leftD: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" shows "x \<in> set xs"
cut_greater_bound: fixes n :: "'a"   and i :: "'a set"   and t :: "'a" assumes "n \<in> i \<down>> t" shows "t < n"
analytic_Gamma: fixes A :: "complex set" assumes "A \<inter> \<int>\<^sub>\<le>\<^sub>0 = {}" shows "Gamma analytic_on A"
Bex_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "Bex A P = (\<exists>x. x \<in> A \<and> P x)"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelTEQ Enc TRel"
eq1: fixes a :: "(('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" shows "fst (fst (app_args spp_of a)) = map spp_of (fst (fst a))"
l5_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Bet A B C"   and "Bet A B D" shows "Bet B C D \<or> Bet B D C"
last_half_segments_eq_mirror_hd: fixes X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "half_segments_of_aform X \<noteq> []" shows "snd (last (half_segments_of_aform X)) = mirror_point (fst X) (fst (hd (half_segments_of_aform X)))"
tendsto: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'a"   and a :: "'a"   and F :: "'c filter" assumes "bounded_linear f"   and "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
solve_def: fixes cs :: "('i \<times> constraint) list" shows "solve cs \<equiv> case solve_exec cs of Inl c \<Rightarrow> Inl c | Inr v \<Rightarrow> Inr \<langle>v\<rangle>"
assms_5: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> dom (\<alpha> m)"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>dom (\<alpha> m) - it. k \<le> j" shows "P \<sigma>"
Ixx: shows "Gromov_product_at px x' x = M"
dcommutes: fixes act :: "('a \<times> 'a) set"   and s :: "'a"   and t :: "'a" assumes "act \<in> Acts F"   and "s \<in> T"   and "(s, t) \<in> relcl L" shows "s \<in> B \<or> t \<in> B \<or> (funof act s, funof act t) \<in> relcl L"
fcompare_double_eq: fixes x :: "double"   and y :: "double" assumes "\<not> Code_Double.is_zero x \<or> \<not> Code_Double.is_zero y"   and "fcompare_double x y = ccode.Eq" shows "x = y"
finite_UNIV: shows "finite_UNIV = Phantom('a) (finite UNIV)"
drop_eq_Nil: fixes n :: "nat"   and xs :: "'c list" shows "(drop n xs = []) = (length xs \<le> n)"
h_def: shows "h = (\<lambda>x. 1 / g x)"
map_value_image_filter_axioms: shows "map_value_image_filter hm.\<alpha> hm.invar hm.\<alpha> hm.invar hh_mvif.g_value_image_filter"
WT_restD_rfunc_adv: fixes \<I>_adv :: "('a, 'b) \<I>"   and \<I>_usr :: "('c, 'd) \<I>"   and I :: "'e \<Rightarrow> bool"   and rest :: "('e, 'f, 'a, 'c, 'b, 'd, 'e) rest_scheme"   and y :: "'b"   and es :: "'f list"   and s' :: "'e"   and s :: "'e"   and x :: "'a" assumes "WT_rest \<I>_adv \<I>_usr I rest"   and "((y, es), s') \<in> set_spmf (rfunc_adv rest s x)"   and "x \<in> outs_\<I> \<I>_adv"   and "I s" shows "y \<in> responses_\<I> \<I>_adv x \<and> I s'"
diff_diff_left: fixes i :: "nat"   and j :: "nat"   and k :: "nat" shows "i - j - k = i - (j + k)"
scaleC_1_apply: fixes x :: "complex"   and y :: "'a" shows "x *\<^sub>C 1 *\<^sub>V y = x *\<^sub>C y"
polytope_imp_polyhedron: fixes S :: "'b set" assumes "polytope S" shows "polyhedron S"
mem_val_alt_8_mem_equal_3: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 3) s1 = mem_val_alt 8 ((pa AND 68719476732) + 3) s2"
closed_approachable: fixes S :: "'a set"   and x :: "'a" assumes "closed S" shows "(\<forall>e>0. \<exists>y\<in>S. dist y x < e) = (x \<in> S)"
assms_2: shows "\<forall>\<^sub>F x in at_top. 0 \<le> l x"
coprime_cong_mult_poly: fixes a :: "'b poly"   and b :: "'b poly"   and m :: "'b poly"   and n :: "'b poly" assumes "[a = b] (mod m)"   and "[a = b] (mod n)"   and "algebraic_semidom_class.coprime m n" shows "[a = b] (mod m * n)"
F0: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>"   and "i \<noteq> 0" shows "(F i = 0) = (k < i)"
Rep_Abs_ND: fixes P :: "'c process"   and Q :: "'c process" shows "Rep_process (Abs_process (F P \<union> F Q, D P \<union> D Q)) = (F P \<union> F Q, D P \<union> D Q)"
simps_10: shows "pratt2_max \<top>\<^sub>2 \<infinity>\<^sub>2 = \<top>\<^sub>2"
older_seniors_subset: fixes y :: "nat"   and x :: "nat"   and n :: "nat" assumes "y \<in> older_seniors x n" shows "older_seniors y n \<subset> older_seniors x n"
SEQ_suc: fixes n :: "nat"   and s :: "'b sparc_state"   and t :: "'b sparc_state" assumes "SEQ n s = Some t" shows "SEQ (Suc n) s = NEXT t"
removeAll_notin: fixes a :: "'a"   and A :: "'a multiset" assumes "a \<notin># A" shows "removeAll_mset a A = A"
dual_iff: fixes x :: "'b"   and y :: "'b dual" shows "(\<partial> x = y) = (x = \<partial>\<^sup>- y)"
bitlen_nonneg: fixes x :: "int" shows "0 \<le> bitlen x"
finite_sequence_to_countable_set: fixes X :: "'a set" assumes "countable X" obtains F :: "nat \<Rightarrow> 'a set" where "\<And>i. F i \<subseteq> X" and "\<And>i. F i \<subseteq> F (Suc i)" and "\<And>i. finite (F i)" and "\<Union> (range F) = X"
to_g: fixes x :: "complex" assumes "x \<in> S" shows "\<exists>d h. 0 < d \<and> summable h \<and> range h \<subseteq> \<real>\<^sub>\<ge>\<^sub>0 \<and> (\<forall>\<^sub>F n in sequentially. \<forall>y\<in>ball x d \<inter> S. cmod (f n y) \<le> cmod (h n))"
prod_prime_factors': fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "completely_multiplicative_function f"   and "0 < n" shows "f n = (\<Prod>p\<in>prime_factors n. f p ^ multiplicity p n)"
inv\<^sub>2E_1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "inv\<^sub>2 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V" shows "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"
progress_Until_le: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and \<phi> :: "Formula.formula"   and I :: "\<I>"   and \<psi> :: "Formula.formula"   and j :: "nat" shows "Monitor.progress \<sigma> P (formula.Until \<phi> I \<psi>) j \<le> min (Monitor.progress \<sigma> P \<phi> j) (Monitor.progress \<sigma> P \<psi> j)"
partition_on_the_part_eq: fixes A :: "'a set"   and P :: "'a set set"   and x :: "'a"   and X :: "'a set" assumes "partition_on A P"   and "x \<in> X"   and "X \<in> P" shows "(THE X. x \<in> X \<and> X \<in> P) = X"
embed_implies_iso_Restr: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'c" assumes "Well_order r"   and "Well_order r'"   and "embed r' r f" shows "iso r' (Restr r (f ` Field r')) f"
lns_distI: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" assumes "\<And>x e. \<lbrakk>x \<in> B; 0 < e\<rbrakk> \<Longrightarrow> \<exists>y\<in>B. dist y x \<le> e \<and> y \<succeq>[P] x \<and> (x, y) \<notin> P" shows "local_nonsatiation B P"
fs_state: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)"   and "length fs = m" shows "fs_state state = fs"
Integer_arith_strict_6: fixes x :: "Integer" shows "x - \<bottom> = \<bottom>"
UNION_f_eqvt: fixes pi :: "('b \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b set"   and X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> \<Union> (f ` X) = \<Union> ((pi \<bullet> f) ` (pi \<bullet> X))"
swap_at_base_simps_2: fixes x :: "'a"   and y :: "'a" assumes "sort_of (atom x) = sort_of (atom y)" shows "(atom x \<rightleftharpoons> atom y) \<bullet> y = x"
closed_contains_Inf_cl: fixes S :: "'b set" assumes "closed S"   and "S \<noteq> {}" shows "Inf S \<in> S"
locally_path_connected_space_open_path_components: fixes X :: "'a topology" shows "locally_path_connected_space X = (\<forall>U c. openin X U \<and> c \<in> path_components_of (subtopology X U) \<longrightarrow> openin X c)"
valid0': shows "valid_unMultigraph (del_unEdge v w v' G)"
simps_2: fixes e :: "'g"   and f :: "'i \<Rightarrow> 'g +\<^sub>\<bottom> 'h" shows "sum_bot.Left e \<bind> f = sum_bot.Left e"
scalar_prod_right_mono: fixes u :: "'a vec"   and n :: "nat"   and v :: "'a vec"   and w :: "'a vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n"   and "vec_comp_all (\<succ>) v w" shows "u \<bullet> v \<succ> u \<bullet> w"
dual_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y) ^ o = x ^ o \<sqinter> y ^ o"
preserves_adjunction_data: fixes f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "adjunction_data_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>" shows "adjunction_data_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) (D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) (D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g))"
keys_plus_ninv_comm_monoid_add: fixes s :: "'b \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (s + t) = keys s \<union> keys t"
intros_2: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
blocks_induct: fixes P :: "char list list \<Rightarrow> ty list \<Rightarrow> val list \<Rightarrow> char list exp \<Rightarrow> bool"   and a0a :: "char list list"   and a0b :: "ty list"   and a0c :: "val list"   and a0d :: "char list exp" assumes "\<And>V Vs T Ts v vs e. P Vs Ts vs e \<Longrightarrow> P (V # Vs) (T # Ts) (v # vs) e"   and "\<And>e. P [] [] [] e"   and "\<And>vc vd a b. P [] (vc # vd) a b"   and "\<And>v ve vf vd. P [] v (ve # vf) vd"   and "\<And>va vb a b. P (va # vb) [] a b"   and "\<And>v vd ve vb. P v [] (vd # ve) vb"   and "\<And>va vc vb ve. P (va # vc) vb [] ve"   and "\<And>v va vc ve. P v (va # vc) [] ve" shows "P a0a a0b a0c a0d"
zero_reducibility_implies_lt_divisibility': fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "(red F)\<^sup>*\<^sup>* f 0"   and "f \<noteq> 0" shows "\<exists>h\<in>F. h \<noteq> 0 \<and> lt h adds\<^sub>t lt f"
coincides_cls_val_process': fixes Mkt :: "('c, 'd) discrete_market"   and Mkt2 :: "('c, 'd) discrete_market"   and A :: "'d set"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "coincides_on Mkt Mkt2 A"   and "support_set pf \<subseteq> A" shows "\<forall>n w. cls_val_process Mkt pf (Suc n) w = cls_val_process Mkt2 pf (Suc n) w"
set_dropWhileD: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "x \<in> list.set (dropWhile P xs)" shows "x \<in> list.set xs"
store_preserved1: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and arb_return :: "t list option"   and arb_label :: "t list list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "store_typing s \<S>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ map typeof vs, label := arb_label, return := arb_return\<rparr>"   and "i < length (s_inst \<S>)" shows "store_typing s' \<S>"
not_is_final: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n1 :: "nat"   and n2 :: "nat" assumes "\<not> is_final (steps c p n1)"   and "n2 \<le> n1" shows "\<not> is_final (steps c p n2)"
hidden_def: fixes xs :: "'c list"   and i :: "nat" shows "hidden xs i \<equiv> i < length xs \<and> xs ! i \<in> set (drop (i + 1) xs)"
intros: fixes F :: "'fenv_std"   and main :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op) Std.instr fundef"   and H :: "'c" assumes "Fstd_get F main = Some fd"   and "arity fd = 0" shows "Sstd.load (Prog F H main) (State F H [Frame main 0 []])"
assms_6: shows "negligible (\<lambda>\<eta>. PRF.advantage \<eta> (reduction_prf \<eta> (\<A> \<eta>)))"
del_maxD: fixes t :: "('b \<times> nat) tree"   and x :: "'b"   and t' :: "('b \<times> nat) tree" assumes "del_max t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "Tree2.inorder t' @ [x] = Tree2.inorder t"
vars_clause_def: fixes clause :: "Literal list" shows "vars clause \<equiv> varsClause clause"
ts_invD: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and I :: "'g \<Rightarrow> 'f option"   and ts :: "'g \<Rightarrow> ('h \<times> 'j \<Rightarrow>f nat) option"   and m :: "'i"   and t :: "'g"   and x :: "'h"   and ln :: "'j \<Rightarrow>f nat" assumes "ts_inv P I ts m"   and "ts t = \<lfloor>(x, ln)\<rfloor>" shows "\<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m"
sorted_list_of_set_3: fixes A :: "'a Word.word set" shows "distinct (signed.sorted_list_of_set A)"
csrp_same: fixes I :: "'a set"   and a :: "'a" assumes "ideal R I"   and "a \<in> carrier R" shows "csrp_fn R I (a \<uplus>\<^bsub>R\<^esub> I) \<uplus>\<^bsub>R\<^esub> I = a \<uplus>\<^bsub>R\<^esub> I"
match_result_elem: fixes env :: "v sem_env"   and s :: "'b state"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and e :: "exp0"   and env' :: "(char list \<times> v) list" assumes "match_result env s v0 pes err_v = Rval (e, env')" shows "\<exists>pat. (pat, e) \<in> set pes"
assms_4: shows "\<R>\<^sub>\<circ> f = A"
nn_integral_cong: fixes M :: "'d measure"   and u :: "'d \<Rightarrow> ennreal"   and v :: "'d \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
norm_le_l1: fixes x :: "'b" shows "norm x \<le> (\<Sum>b\<in>Basis. \<bar>x \<bullet> b\<bar>)"
hyps: assumes "xs = [] \<Longrightarrow> thesis"   and "\<And>ys x. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis" shows "thesis"
supp_eqvt_at: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "eqvt_at f x"   and "finite (supp x)" shows "supp (f x) \<subseteq> supp x"
Union_included_in_supp: fixes X :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite X" shows "\<Union> (supp ` X) \<subseteq> supp X"
hcomp_eqI': fixes f :: "'a"   and g :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "hseq f g"   and "f = f'"   and "g = g'" shows "f \<star>\<^sub>B g = f' \<star> g'"
scopeFresh: fixes x :: "name"   and P :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>P \<sim>\<^sup>s P"
Rp_2: shows "LINT x|M. fp x * real_cond_exp M F g x = LINT x|M. fp x * g x"
filter_mset_eq_conv: fixes P :: "'a \<Rightarrow> bool"   and M :: "'a multiset"   and N :: "'a multiset" shows "(filter_mset P M = N) = (N \<subseteq># M \<and> (\<forall>b\<in>#N. P b) \<and> (\<forall>a\<in>#M - N. \<not> P a))"
ZFfunIdRight: fixes f :: "ZF" assumes "isZFfun f" shows "f |o| ZFfun (|cod|f) (|cod|f) (\<lambda>x. x) = f"
bind_sample2: fixes m :: "'m"   and p :: "'p pmf"   and f :: "'a \<times> 'w list \<Rightarrow> 'p \<Rightarrow> 'm" shows "bind m (\<lambda>y. sample p (f y)) = sample p (\<lambda>x. bind m (\<lambda>y. f y x))"
pf_nonneg_mat_axioms: shows "pf_nonneg_mat A"
a_2: shows "f' (real_of_float u) \<le> real_of_float a"
atom_Sup_iff: fixes x :: "'a" shows "atom x = (x \<noteq> \<bottom> \<and> (\<forall>Y. (\<exists>y\<in>Y. x \<le> y) = (x \<le> \<Squnion> Y)))"
i1: shows "range ((*\<^sub>V) P) = (\<lambda>x. x - P *\<^sub>V x) -` {0::'a}"
assms_3: fixes u :: "'t" assumes "v \<prec>\<^sub>t u" shows "lookup p u = lookup q u"
MTimesR_def: fixes R :: "mregex set"   and s :: "mregex" shows "MTimesR R s = (\<lambda>r. MTimes r s) ` R"
simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term"   and u :: "'b local.term" assumes "monoidal_language C" shows "monoidal_language.Arr C (t \<^bold>\<otimes> u) = (monoidal_language.Arr C t \<and> monoidal_language.Arr C u)"
NatTransMapsTo: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and X :: "'d" assumes "NatTransP \<eta>"   and "X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub>" shows "\<eta> $$ X maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ X to NTCod \<eta> @@ X"
poly_mult_append_zero: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and q :: "'c list" assumes "ring R"   and "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R" shows "ring.poly_mult R (p @ [\<zero>\<^bsub>R\<^esub>]) q = ring.normalize R (ring.poly_mult R p q @ [\<zero>\<^bsub>R\<^esub>])"
sgn_bernoulli_even: fixes n :: "nat" assumes "0 < n" shows "sgn (bernoulli (2 * n)) = (- 1) ^ Suc n"
size_image_mset: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset" shows "size (image_mset f M) = size M"
galois_connection_axioms_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "galois_connection_axioms G \<equiv> \<forall>x y. x \<in> carrier \<X>\<^bsub>G\<^esub> \<longrightarrow> y \<in> carrier \<Y>\<^bsub>G\<^esub> \<longrightarrow> (\<pi>\<^sup>*\<^bsub>G\<^esub> x \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> \<pi>\<^sub>*\<^bsub>G\<^esub> y)"
smult_zero: fixes r :: "'r" assumes "r \<in> R" shows "r \<cdot> (0::'m) = (0::'m)"
Neg_Imp_I: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A"   and "insert B H \<turnstile> Fls" shows "H \<turnstile> Neg (A IMP B)"
lift_def: fixes h :: "'h \<Rightarrow> 'g"   and inp :: "'f \<Rightarrow> 'h option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
assms_2: shows "\<phi> i y\<^sub>0 \<down>"
assms_2: fixes x :: "'a" assumes "x \<in> set xs" shows "x \<le> a"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'b"   and xa :: "('c, 'a) uexpr"   and xb :: "('d, 'a) uexpr"   and xc :: "('e, 'a) uexpr"   and xd :: "('f, 'a) uexpr" shows "\<lbrakk>qtop x xa xb xc xd\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b) (\<lbrakk>xc\<rbrakk>\<^sub>e b) (\<lbrakk>xd\<rbrakk>\<^sub>e b))"
norm_generalize_instr: fixes instr :: "('dyn, 'd, 'e, 'f, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr" shows "norm_instr (Subx.generalize_instr instr) = norm_instr instr"
ff_def: shows "ff \<equiv> \<lambda>x. if x \<in> ball a r \<inter> T then f x else x"
adjuster_already_in_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "corthogonal us"   and "w \<in> local.span (set us)" shows "adjuster n w us + w = 0\<^sub>v n"
gen_succ_def: shows "Digraph_Map_Impl.gen_succ \<equiv> \<lambda>as xs. concat (map (\<lambda>(a, b). if a \<in> set xs then [b] else []) as)"
satisfies_consistent: fixes s :: "('a, 'b) state" shows "\<Turnstile> s \<longrightarrow> \<diamond> s"
path_connected_Un: fixes S :: "'a set"   and T :: "'a set" assumes "path_connected S"   and "path_connected T"   and "S \<inter> T \<noteq> {}" shows "path_connected (S \<union> T)"
group_chain_group: fixes p :: "nat"   and X :: "'b topology" shows "Group.group (chain_group p X)"
inter_add_right1: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<notin># N" shows "N \<inter># add_mset x M = N \<inter># M"
at_calc_2: fixes x :: "'a" assumes "at TYPE('a)" shows "[] \<bullet> x = x"
mid_two_sides: fixes M :: "'p"   and A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "M Midpoint A B"   and "\<not> Col A B X"   and "M Midpoint X Y" shows "A B TS X Y"
XD_XH_lfp_F_2: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set" assumes "stable_pair_on ds XD_XH" shows "snd XD_XH \<subseteq> snd (lfp_F ds)"
vardict_inv_empty: fixes ss :: "(integer \<times> edge list) iarray"   and proc :: "nat \<times> edgeIndex \<times> procArg list \<times> varDecl list" shows "vardict_inv ss proc (lm.empty ())"
vector_matrix_row: fixes x :: "('a, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" shows "x v* A = (\<Sum>i\<in>UNIV. x $ i *s A $ i)"
vdisjnt_singleton1: fixes a :: "V"   and A :: "V" shows "vdisjnt (set {a}) A = (a \<notin>\<^sub>\<circ> A)"
R_def: shows "R = {(V, V'). V = V' \<and> (V = [] \<or> V = [x:=\<^bsub>\<iota>\<^esub> e])}"
pmf_map_inj: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a pmf"   and x :: "'a" assumes "inj_on f (set_pmf M)"   and "x \<in> set_pmf M" shows "pmf (map_pmf f M) (f x) = pmf M x"
oexp_empty: assumes "Field r = {}"   and "Field s \<noteq> {}" shows "oexp = {}"