hcju/isabelleps huggingface dataset:
P_L: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c"   and \<phi> :: "'c" assumes "Loeb_Assumptions var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (imp (HBL1.PP var subst P (enc (Loeb_Form.\<phi>L var fmla Fvars subst cnj imp bprv enc P \<phi>))) (HBL1.PP var subst P (enc \<phi>)))"
mhead_resTy_simp: fixes m :: "'a mhead_scheme" shows "resTy m = resT m"
ExistsIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E ExistsIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<tau>)" and "\<tau> \<le> Boolean[?]"
simps_3: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and ds :: "dir list" shows "position_of (App t s) (dir.Left # ds) = position_of t ds"
init: shows "dfa.init Power_dfa \<in> dfa.states Power_dfa"
power_int_1_left: fixes n :: "int" shows "(1::'a) powi n = (1::'a)"
not_add_less1: fixes i :: "nat"   and j :: "nat" shows "\<not> i + j < i"
orthogonal_transformation_linear: fixes f :: "'a \<Rightarrow> 'b" assumes "orthogonal_transformation f" shows "linear f"
sat_lit_def: fixes \<sigma> :: "nat \<Rightarrow> bool"   and l :: "lit" shows "sat_lit \<sigma> l = (\<sigma> (var l) = pos l)"
rep_number_non_zero_system_point: fixes x :: "'a" assumes "0 < \<B> rep x" shows "x \<in> \<V>"
gf: fixes x :: "'a" assumes "norm x = 1"   and "b \<bullet> x \<noteq> 1" shows "g (f x) = x"
happens_before_refl: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and P :: "'c prog" assumes "a \<in> actions E" shows "P,E \<turnstile> a \<le>hb a"
step: fixes \<phi> :: "'h stream \<Rightarrow> bool"   and xs :: "'h stream" assumes "ev \<phi> (stl xs)" shows "ev \<phi> xs"
distinct_map_fstD: fixes xs :: "('a \<times> 'b) list"   and x :: "'a"   and y :: "'b"   and z :: "'b" assumes "distinct (map fst xs)"   and "(x, y) \<in> set xs"   and "(x, z) \<in> set xs" shows "y = z"
leg0_in_hom_1: shows "\<guillemotleft>f : src\<^sub>C f \<rightarrow>\<^sub>C src\<^sub>C r\<guillemotright>"
hn_refine_cons_post: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and m :: "'b nres"   and Q' :: "assn" assumes "hn_refine P c Q R m"   and "Q \<Longrightarrow>\<^sub>t Q'" shows "hn_refine P c Q' R m"
leq_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(x, y) \<in> leq"   and "(y, z) \<in> leq" shows "(x, z) \<in> leq"
analz_image_pan_lemma: fixes P :: "nat"   and nE :: "nat set"   and H :: "msg set" assumes "Pan P \<in> analz (Key ` nE \<union> H) \<longrightarrow> Pan P \<in> analz H" shows "(Pan P \<in> analz (Key ` nE \<union> H)) = (Pan P \<in> analz H)"
max_Var_floatariths_map_const: fixes fa :: "floatarith"   and xs :: "'e list" shows "max_Var_floatariths (map (\<lambda>i. fa) xs) = (if xs = [] then 0 else max_Var_floatarith fa)"
L2_set_mono2: fixes L :: "'c set"   and K :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "finite L"   and "K \<subseteq> L" shows "L2_set f K \<le> L2_set f L"
elim_trivial_restriction: fixes l :: "'a"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option" assumes "l \<notin> LID\<^sub>S \<tau>" shows "\<tau>(l := None) = \<tau>"
nv2T: fixes x :: "Preliminaries.var" assumes "\<not> infTp (tpOfV x)"   and "x \<in> nv2T T" shows "\<exists>l. wtL l \<and> \<not> I.satL \<xi> l \<and> isGuard x l"
map_ident: fixes t :: "'d list" shows "map (\<lambda>x. x) t = t"
cases: fixes a1 :: "pat"   and a2 :: "POPLmarkRecord.type"   and a3 :: "binding list" assumes "\<turnstile> a1 : a2 \<Rightarrow> a3" obtains T :: "POPLmarkRecord.type" where "a1 = PVar T" and "a2 = T" and "a3 = [VarB T]"   | fps :: "(char list \<times> pat) list" and fTs :: "(char list \<times> POPLmarkRecord.type) list" and \<Delta> :: "binding list" where "a1 = PRcd fps" and "a2 = RcdT fTs" and "a3 = \<Delta>" and "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>"
ample_set_def: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and src :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b"   and A :: "'a set" assumes "ample_base ex en int ind src" shows "ample_base.ample_set ex en int ind src q A \<equiv> A \<subseteq> {a. en a q} \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<noteq> {}) \<and> (\<forall>a. A \<subset> {a. en a q} \<longrightarrow> a \<in> A \<longrightarrow> src (ex a q) q) \<and> (A \<subset> {a. en a q} \<longrightarrow> A \<subseteq> - transition_system_interpreted.visible ex en int) \<and> (\<forall>w. A \<subset> {a. en a q} \<longrightarrow> transition_system.path ex en w q \<longrightarrow> A \<inter> set w = {} \<longrightarrow> (\<forall>a\<in>A. \<forall>b\<in>set w. ind a b))"
mult_nonpos_nonneg: fixes a :: "'a"   and b :: "'a" assumes "a \<le> (0::'a)"   and "(0::'a) \<le> b" shows "a * b \<le> (0::'a)"
order_smult: fixes c :: "'a"   and x :: "'a"   and p :: "'a poly" assumes "c \<noteq> (0::'a)" shows "order x (smult c p) = order x p"
Domainp_iff: fixes T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a" shows "Domainp T x = (\<exists>y. T x y)"
fields_def: fixes P :: "'a prog"   and C :: "char list" shows "fields P C \<equiv> THE FDTs. P \<turnstile> C has_fields FDTs"
dropWhile_append3: fixes P :: "'f \<Rightarrow> bool"   and y :: "'f"   and xs :: "'f list"   and ys :: "'f list" assumes "\<not> P y" shows "dropWhile P (xs @ y # ys) = dropWhile P xs @ y # ys"
conc_def: fixes A :: "'a list set"   and B :: "'a list set" shows "A \<cdot> B = {xs @ ys |xs ys. xs \<in> A \<and> ys \<in> B}"
simps_4: fixes h :: "'heap"   and va :: "bool" shows "typeof\<^bsub>h\<^esub> (Bool va) = typeof (Bool va)"
trcl_concat: fixes c :: "'c"   and w1 :: "'d list"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set"   and w2 :: "'d list"   and c'' :: "'c" assumes "(c, w1, c') \<in> trcl t"   and "(c', w2, c'') \<in> trcl t" shows "(c, w1 @ w2, c'') \<in> trcl t"
BSI2: shows "BSI \<V>2 Tr\<^bsub>ES2\<^esub>"
s_init_def: fixes U0 :: "'e set" shows "s_init U0 \<equiv> RETURN (None, U0, \<lambda>u. if u \<in> U0 then Some [] else None)"
stupid_lemma: fixes cs :: "'a list" assumes "length cs = 3" shows "cs = [cs ! 0, cs ! 1, cs ! 2]"
Mem_non_refl: fixes H :: "fm set"   and x :: "tm"   and A :: "fm" assumes "H \<turnstile> x IN x" shows "H \<turnstile> A"
diff_rat_def: fixes q :: "rat"   and r :: "rat" shows "q - r = q + - r"
dg_is_arrD_4: fixes \<alpha> :: "V"   and \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "\<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
DF_def: shows "DF \<equiv> \<lambda>k. rec_nat ((enum N \<circ> Suc) ` {..<Suc k}) (\<lambda>r D. enum (nxt N (enum (nxt N (Max D)) (\<Sqinter> D - Suc 0))) ` {..<Suc k})"
simps_2: fixes x :: "real"   and y :: "real" shows "ms_exp_gt x (Some y) = (y < x)"
step_thread_Some_NoneD: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and t :: "'t"   and \<sigma>' :: "'a" assumes "\<alpha>.step_thread update_state s t' = \<lfloor>(t, None, \<sigma>')\<rfloor>" shows "\<exists>x ln n. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> 0 < ln $ n \<and> \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln \<and> \<sigma>' = update_state (K$ [], [], [], [], [], convert_RA ln)"
proj2_no_3_Col_def: fixes S :: "proj2 set" shows "proj2_no_3_Col S \<equiv> card S = 4 \<and> (\<forall>p\<in>S. \<not> proj2_set_Col (S - {p}))"
outs_plus_\<I>: fixes \<I>l :: "('a, 'c) \<I>"   and \<I>r :: "('b, 'd) \<I>" shows "outs_\<I> (\<I>l \<oplus>\<^sub>\<I> \<I>r) = outs_\<I> \<I>l <+> outs_\<I> \<I>r"
minus_splus_sminus_cancel: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s adds t"   and "t adds\<^sub>p v" shows "(t - s) \<oplus> (v \<ominus> t) = v \<ominus> s"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "\<tau>mredT a1 a2" obtains s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and tl :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action" where "a1 = s" and "a2 = s'" and "redT s tl s'" and "m\<tau>move s tl s'"
stut_fun2: fixes X :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b"   and Y :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "STUTINV X"   and "STUTINV Y" shows "STUTINV f<X, Y>"
iprange_single_def: fixes ip :: "'a word" shows "iprange_single ip \<equiv> WordInterval ip ip"
sorted_smult_spvec: fixes a :: "(nat \<times> 'a) list"   and y :: "'a" assumes "sorted_spvec a" shows "sorted_spvec (smult_spvec y a)"
ab_group_add_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" shows "ab_group_add_ow U plus zero minus uminus \<equiv> (minus_ow U minus \<and> uminus_ow U uminus) \<and> comm_monoid_add_ow U plus zero \<and> ab_group_add_ow_axioms U plus zero minus uminus"
overlapping_def: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.overlapping subst s t = (\<exists>u. term.matches subst s u \<and> term.matches subst t u)"
DBM_reset_int_preservation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and d :: "'c"   and k :: "nat" assumes "dbm_int M n"   and "d \<in> \<int>"   and "0 < k"   and "k \<le> n" shows "dbm_int (reset M n k d) n"
set_upt: fixes i :: "nat"   and j :: "nat" shows "ran\<^sub>l [i..<j] = {i..<j}"
b_def: shows "b = insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var (Suc 0))"
same_set_semiJoin: fixes x :: "nat set \<times> 'a option list set"   and other :: "nat set \<times> 'a option list" shows "fst (semiJoin x other) = fst x"
valid_path_ltl': fixes v :: "'a"   and Ps :: "'a llist" assumes "valid_path (LCons v Ps)" shows "valid_path Ps"
pair_def: fixes d :: "('fun, 'c) Term.term \<times> ('fun, 'c) Term.term" shows "pair d \<equiv> case d of (t, t') \<Rightarrow> Fun Pair [t, t']"
hyps: fixes p :: "(nat \<times> int) list"   and ds :: "nat set" shows "set (gridgen p ds l) = lgrid p ds (level p + l)"
A11': fixes X :: "(real, 'a) vec set"   and Y :: "(real, 'a) vec set"   and a :: "(real, 'a) vec" assumes "\<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B\<^sub>\<real> a x y" shows "\<exists>b. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B\<^sub>\<real> x b y"
insert: fixes a :: "'a"   and A :: "'a set" shows "F (insert a A) = a \<^bold>* F A"
d: shows "complex_roots_of_int_poly3 p \<equiv> if degree p = length rr then rrts else if degree p - length rr = 2 then rrts @ croots2 (map_poly (\<lambda>r. Complex r 0) (real_of_int_poly p div (\<Prod>x\<leftarrow>rr. [:- x, 1:]))) else rrts @ concat (map (\<lambda>x. [x, cnj x]) (map pair_to_complex (filter_list_length (map_prod tighten_bounds_3 tighten_bounds_3) (\<lambda>(r, i). 0 \<in>\<^sub>c ipoly_complex_interval p (Complex_Interval (get_itvl_3 r) (get_itvl_3 i))) ((degree p - length rr) div 2) (concat (map (\<lambda>rx. map (Pair rx) (pos_imaginary_parts_3 p)) (real_parts_3 p))))))"
listI: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set" assumes "length xs = n"   and "set xs \<subseteq> A" shows "xs \<in> list n A"
mult_oner: fixes x :: "'a" shows "x \<cdot> (1::'a) = x"
fold_Cons: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" shows "fold f (x # xs) = (fold f xs \<circ>\<circ> f) x"
assms_2: shows "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IBool True"
M_def: shows "M \<equiv> vector [vector [1, 0, 0], vector [0, 1, 0], vector [0, 0, - 1]]"
word_plus_mcs_4': fixes x :: "'b word"   and v :: "'b word"   and w :: "'b word" assumes "x + v \<le> x + w"   and "x \<le> x + v" shows "v \<le> w"
erf_def: fixes z :: "'a" shows "erf z = (\<Sum>n. of_real (erf_coeffs n) * z ^ n)"
right_option_def: fixes g :: "ZF"   and opt :: "ZF" shows "right_option g opt \<equiv> Elem opt (Snd g)"
elims_2: fixes x :: "'a mat" assumes "square_mat x" obtains A :: "'a mat" where "x = A" and "dim_col A = dim_row A"
path_strategies_strategy: fixes P :: "'a llist"   and n :: "nat" assumes "lset P \<subseteq> winning_region p"   and "enat n < llength P" shows "strategy p (path_strategies P $ n)"
\<gamma>_3: shows "pathfinish \<gamma> = pathfinish g"
1: shows "v \<in> {v \<in> fmdom' (f ++ g). v \<in> fmdom' (fst a) \<and> fmlookup (fst a) v = fmlookup (f ++ g) v}"
take_bit_int_less_self_iff: fixes n :: "nat"   and k :: "int" shows "(take_bit n k < k) = (2 ^ n \<le> k)"
R4_1: shows "pmf (sds R4) a = pmf (sds R21) a"
High_not_in_\<C>: fixes mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and x :: "'Var" assumes "dma mem\<^sub>1 x = High" shows "x \<notin> \<C>"
is_aligned_triv: fixes n :: "nat" shows "is_aligned (2 ^ n) n"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<leadsto> a2" obtains x :: "name" and t\<^sub>1 :: "trm" and t\<^sub>2 :: "trm" where "a1 = App (Lam [x].t\<^sub>1__) t\<^sub>2__" and "a2 = t\<^sub>1__[x::=t\<^sub>2__]"   | t\<^sub>1 :: "trm" and t\<^sub>1' :: "trm" and t\<^sub>2 :: "trm" where "a1 = App t\<^sub>1 t\<^sub>2" and "a2 = App t\<^sub>1' t\<^sub>2" and "t\<^sub>1 \<leadsto> t\<^sub>1'"
assms_1: shows "raw_match_term t u subs = Some subs'"
count_roots_correct: fixes p :: "real poly" shows "count_roots p = card {x. poly p x = 0}"
exhaust: fixes y :: "'a msstream" obtains x1 :: "'a" and x2 :: "'a msstream" where "y = MSSCons x1 x2"
lookup_Abs_poly_mapping_when_finite: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite S" shows "lookup (Abs_poly_mapping (\<lambda>x. f x when x \<in> S)) = (\<lambda>x. f x when x \<in> S)"
lunit_simp: fixes a :: "'d" assumes "D.ide a" shows "D'.lunit a = \<r>\<^sub>D[a]"
arbitrary_def: fixes \<U> :: "'d set set" shows "arbitrary \<U> \<equiv> True"
assms_1: fixes S1 :: "'a"   and S2 :: "'a"   and x :: "char list"   and e :: "aexp" assumes "S1 \<le> S2" shows "f x e S1 \<le> f x e S2"
unwrap_def: shows "unwrap \<equiv> \<Lambda> r e. SK\<cdot>(r\<cdot>e)"
UP_simps_2: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr V\<^sub>B \<mu>" shows "strictified_bicategory.src V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (strictified_bicategory.UP V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B \<mu>) = strictified_bicategory.UP\<^sub>0 V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (src\<^sub>B \<mu>)"
product_prob_space_def: fixes M :: "'d \<Rightarrow> 'e measure" shows "product_prob_space M \<equiv> product_sigma_finite M \<and> product_prob_space_axioms M"
Ballot_nzero: shows "\<forall>p. 0 \<notin> Ballot p"
intros: fixes X :: "'d set"   and u :: "'d \<Rightarrow> real"   and I :: "'d \<Rightarrow> intv"   and X\<^sub>0 :: "'d set"   and r :: "('d \<times> 'd) set" assumes "\<forall>x\<in>X. 0 \<le> u x"   and "\<forall>x\<in>X. intv_elem x u (I x)"   and "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}"   and "\<forall>x\<in>X\<^sub>0. \<forall>y\<in>X\<^sub>0. ((x, y) \<in> r) = (frac (u x) \<le> frac (u y))" shows "u \<in> region X I r"
union_disjoint: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "sum g (A \<union> B) = sum g A + sum g B"
notin: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T"
le_in_le_max: fixes M :: "'a multiset"   and N :: "'a multiset"   and a :: "'a" assumes "M \<le> N"   and "a \<in># M" shows "a \<le> Max_mset N"
mtf2_moves_to_frontm1: fixes xs :: "'a list"   and q :: "'a" assumes "distinct xs"   and "q \<in> set xs" shows "index (mtf2 (length xs - 1) q xs) q = 0"
induct: fixes P :: "'a list \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "nat" assumes "\<And>uu_. P [] uu_"   and "\<And>x xs. P (x # xs) 0"   and "\<And>x xs n. P xs n \<Longrightarrow> P (x # xs) (Suc n)" shows "P a0 a1"
foldr_append: fixes f :: "'c \<rightarrow> 'b \<rightarrow> 'b"   and a :: "'b"   and xs :: "['c]"   and ys :: "['c]" shows "foldr\<cdot>f\<cdot>a\<cdot>(xs ++ ys) = foldr\<cdot>f\<cdot>(foldr\<cdot>f\<cdot>a\<cdot>ys)\<cdot>xs"
antisym: fixes b :: "'c \<times> 'd"   and a :: "'c \<times> 'd" assumes "less_eq_prod b a"   and "less_eq_prod a b" shows "a = b"
uncurry_preserves_functors: fixes F :: "'a1 \<Rightarrow> ('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>]) F" shows "functor (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (uncurry F)"
homeomorphic_arc_image_interval: fixes g :: "real \<Rightarrow> 'c"   and a :: "real"   and b :: "real" assumes "arc g"   and "a < b" shows "path_image g homeomorphic {a..b}"
iMOD_iprev': fixes t :: "nat"   and r :: "nat"   and m :: "nat" assumes "t \<in> [ r, mod m ]" shows "iprev (t + m) [ r, mod m ] = t"
v_horiz_disj: shows "\<V> \<inter> two_chain_horizontal_boundary two_chain = {}"
satL_e: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and l :: "('a, 'b) lit" shows "U.satL \<xi> l = F.satL \<xi> l"
assms_3: fixes i :: "'c" assumes "i \<in> I" shows "f i \<in> hom_deg_set z A"
assms_1: shows "vs1 \<subseteq> fmdom' fma"
simps_3: fixes P :: "('a, 'b, 'c) psi" shows "pTau\<cdot>P = \<tau>.P"
A: shows "sp_impl_subj_subj s1 (Step.partition tid) (Step.partition partner) = sp_impl_subj_subj s2 (Step.partition tid) (Step.partition partner)"
advantage_single_def: fixes \<A> :: "unit \<Rightarrow> ('x \<times> 'hash, 'x, 'hash) gpv" shows "advantage_single \<A> = spmf (game_single \<A>) True"
eq_resource_on_trans: fixes A :: "'i set"   and res :: "('i, 'j) resource"   and res' :: "('i, 'j) resource"   and res'' :: "('i, 'j) resource" assumes "A \<turnstile>\<^sub>R res \<sim> res'"   and "A \<turnstile>\<^sub>R res' \<sim> res''" shows "A \<turnstile>\<^sub>R res \<sim> res''"
exec_plan_Append: fixes s :: "('a, 'b) fmap"   and as_a :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as_b :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" shows "exec_plan s (as_a @ as_b) = exec_plan (exec_plan s as_a) as_b"
4: shows "times_top_class.total (top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top))"
right_minus_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(minus a b = zero) = (a = b)"
holomorphic_perzeta': fixes q' :: "real"   and A :: "complex set" assumes "q' \<notin> \<int>" shows "perzeta q' holomorphic_on A"
o_eq_dest: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'e \<Rightarrow> 'c"   and d :: "'b \<Rightarrow> 'e"   and v :: "'b" assumes "a \<circ> b = c \<circ> d" shows "a (b v) = c (d v)"
approx_5: fixes i :: "nat" shows "a < ereal (l i)"
distributed_borel_measurable: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> ennreal" assumes "distributed M N X f" shows "f \<in> borel_measurable N"
eq_fin_rev: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* (rev w\<^sub>1) (rev w\<^sub>2)"
size_hp: fixes h :: "'b heap" assumes "is_root' h" shows "size_hp h = size (hom h)"
set_zmset_eq_empty_iff: fixes M :: "'b zmultiset" shows "(set_zmset M = {}) = (M = {#}\<^sub>z)"
Main: fixes n :: "nat" assumes "N \<le> n" shows "B (2 * n) \<le> 2 * C0 / a * B n"
run_asset_iii_rel: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "s \<Turnstile> fst (run_asset_iii n s)"
simps_3: shows "consequent impI = [Fun imp [X, Y]]"
fmap_def: fixes f :: "'c \<Rightarrow> 'd" shows "fmap f = hom (\<lambda>i. var (f i))"
bal_tree\<^sub>d_del: fixes t :: "'b tree234"   and x :: "'b" assumes "bal t" shows "bal (tree\<^sub>d (del x t))"
two_power_sum: fixes m :: "nat" shows "sum ((^) 2) {i. i < Suc m} = 2 ^ Suc m - 1"
True_in_eps_star: fixes p :: "bool list"   and q :: "bool list"   and A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" shows "((True # p, q) \<in> eps (RegExp2NAe.star A)) = ((\<exists>r. q = True # r \<and> (p, r) \<in> eps A) \<or> fin A p \<and> q = True # start A)"
completion_into_space: fixes M :: "'b measure" shows "{uu_. \<exists>S N N'. uu_ = S \<union> N \<and> S \<in> sets M \<and> N' \<in> null_sets M \<and> N \<subseteq> N'} \<subseteq> Pow (space M)"
assms_2: shows "e \<le> v * - v\<^sup>T"