hcju/isabelleps huggingface dataset:
that: fixes i :: "nat" assumes "Q (\<sigma> |\<^sub>s i)"   and "\<forall>k<i. P (\<sigma> |\<^sub>s k)" shows "thesis"
closure_of_openin_Int_closure_of: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" assumes "openin X S" shows "X closure_of (S \<inter> X closure_of T) = X closure_of (S \<inter> T)"
inf2I: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "A x y"   and "B x y" shows "(A \<sqinter> B) x y"
set_12: fixes x4 :: "('b, 'a, 'c) expr" shows "LID\<^sub>E (Ref x4) = LID\<^sub>E x4"
div_free_CHAOS: fixes A :: "'a set" shows "D (CHAOS A) = {}"
rtrancl_path_simps_2: fixes R :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and a :: "'k"   and bs :: "'k list"   and y :: "'k" shows "rtrancl_path R x (a # bs) y = (R x a \<and> rtrancl_path R a bs y)"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_on_left_moving_B m rs (l, r) = (\<exists>ml mr nl nr rn. l = Bk \<up> nl @ Oc # Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Bk \<up> nr @ Bk # Bk # Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc rs \<and> 0 < mr)"
bnd_g_1: fixes s :: "'s"   and f :: "ennreal" shows "g s f \<le> b"
absorb1: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" shows "max a b = a"
merge_append: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'e) list" shows "map_of (xs @ ys) = map_of (AList.merge ys xs)"
simps_3: fixes c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "t_find_closest_pair (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va) = 1 + (let p\<^sub>1 = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va) in t_find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va) + (if dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then t_find_closest_pair (c\<^sub>0, c\<^sub>1) (v # va) else t_find_closest_pair (p\<^sub>0, p\<^sub>1) (v # va)))"
LeftDerivationIntro_implies_nonterminal: fixes \<alpha> :: "symbol list"   and i :: "nat"   and e :: "'a \<times> symbol \<times> symbol list"   and ix :: "nat"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro \<alpha> i (snd e) ix E j \<gamma>" shows "is_nonterminal (\<alpha> ! i)"
count_remdups_mset_eq_1: fixes a :: "'b"   and A :: "'b multiset" shows "(a \<in># remdups_mset A) = (count (remdups_mset A) a = 1)"
A: shows "[(y mod p)\<^sup>2 = x] (mod p)"
length_stake: fixes n :: "nat"   and s :: "'c stream" shows "length (stake n s) = n"
sel_14: fixes x51 :: "'b mrexp2"   and x52 :: "'b mrexp2"   and x53 :: "bool"   and x54 :: "bool" shows "nul (Times2 x51 x52 x53 x54) = x54"
wset_find_path_impl_def: fixes node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat"   and node_def_hash_size :: "nat"   and node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and succi :: "'b \<Rightarrow> 'b list"   and U0' :: "'b list"   and Pimpl :: "'b \<Rightarrow> bool" shows "wset_find_path_impl node_hash_impl node_def_hash_size node_eq_impl succi U0' Pimpl \<equiv> LIST_FOREACH' (RETURN (id U0')) (\<lambda>x. True) (\<lambda>x (xa, xb). RETURN (map2set_insert (Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl) x xa, list_map_update node_eq_impl x [] xb)) (Impl_Array_Hash_Map.ahm_empty node_def_hash_size, []) \<bind> (\<lambda>(x, xa). RETURN (None, x, xa)) \<bind> (\<lambda>x. WHILE\<^sub>T (\<lambda>(xb, xc, xd). is_None xb \<and> \<not> list_map_isEmpty xd) (\<lambda>(xb, xc, xd). RETURN (list_map_pick_remove xd) \<bind> (\<lambda>((xe, xf), xg). if Pimpl xe then RETURN (Some (rev xf, xe), xc, xg) else LIST_FOREACH' (RETURN (id (rev (succi xe)))) (\<lambda>xh. True) (\<lambda>xh (xi, xj). if map2set_memb (Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl) xh xi then RETURN (xi, xj) else RETURN (map2set_insert (Impl_Array_Hash_Map.ahm_update node_eq_impl node_hash_impl) xh xi, (xh, xe # xf) # xj)) (xc, xg) \<bind> (\<lambda>(xh, xi). RETURN (None, xh, xi)))) x \<bind> (\<lambda>(xa, xb, xc). RETURN xa))"
suffix_nth: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "suffix k x n = x (k + n)"
continuous_on_coordinatewise_then_product: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "\<And>i. continuous_on S (\<lambda>x. f x i)" shows "continuous_on S f"
bind_le_nofailI: fixes m :: "'c nres"   and f :: "'c \<Rightarrow> 'd nres"   and m' :: "'d nres" assumes "nofail m"   and "\<And>x. RETURN x \<le> m \<Longrightarrow> f x \<le> m'" shows "m \<bind> f \<le> m'"
Simulation_def: fixes A :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and B :: "('c, 'e) action" shows "A \<preceq>S B \<equiv> \<forall>a b. action.relation_of B (a, b) \<longrightarrow> action.relation_of A (Simul S a, Simul S b)"
equiintegrable_reflect: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a" assumes "F equiintegrable_on cbox a b" shows "(\<lambda>f. f \<circ> uminus) ` F equiintegrable_on cbox (- b) (- a)"
add_mset_remove_trivial: fixes x :: "'b"   and M :: "'b multiset" shows "remove1_mset x (add_mset x M) = M"
case_1: fixes f1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and x11 :: "'d"   and x12 :: "'d" shows "(case Branch x11 x12 of Branch x xa \<Rightarrow> f1 x xa | Leaf x \<Rightarrow> f2 x) = f1 x11 x12"
flip_subst_fmla: fixes y :: "name"   and \<phi> :: "fmla"   and x :: "name" assumes "atom y \<sharp> \<phi>" shows "(x \<leftrightarrow> y) \<bullet> \<phi> = \<phi>(x::=Var y)"
inv_swap_id: fixes a :: "'a"   and b :: "'a" shows "inv (a \<rightleftharpoons>\<^sub>F b) = a \<rightleftharpoons>\<^sub>F b"
total_preorder_on_axioms: shows "total_preorder_on alts' Ri'"
not_equiv_trans1: fixes eqv :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "class.equiv eqv"   and "\<not> eqv x y"   and "eqv y z" shows "\<not> eqv x z"
complete_def: fixes n :: "nat"   and seed :: "natural \<times> natural" shows "local.complete n seed = (let (A, seed') = local.build n seed; (hits, seed'') = local.lookup n (A, seed') in Let (local.test A (\<lambda>x. x \<le> of_nat n)) (Pair hits))"
ir_valid_Seq1: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c1 :: "com"   and c2 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P (c1;; c2) c' Q" shows "ir_valid P c1 c' (\<lambda>t t'. \<exists>s s'. P s s' \<and> (c1, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t' \<and> (\<exists>u. (c2, t) \<Rightarrow> u \<and> Q u t'))"
set_lbounds: fixes as :: "atom list" shows "set (lbounds as) = {uu_. \<exists>i k ks. uu_ = (i, ks) \<and> Le i (k # ks) \<in> set as \<and> 0 < k}"
skl1_inv4_init: shows "init skl1 \<subseteq> skl1_inv4"
currentD_finite: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and e :: "'v \<times> 'v" assumes "current \<Gamma> f" shows "f e \<noteq> \<top>"
embe_def: fixes P :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "embe P = (\<lambda>s. Inf {enat n |n. P (part s, n)})"
Flow_axioms_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "Flow_axioms c s t f \<equiv> \<forall>v\<in>Graph.V c - {s, t}. sum f (Graph.incoming c v) \<le> sum f (Graph.outgoing c v)"
proj2_line_through_def: fixes p :: "proj2"   and q :: "proj2" shows "proj2_line_through p q \<equiv> SOME l. proj2_incident p l \<and> proj2_incident q l"
ExFalso: fixes H :: "fm set"   and A :: "fm" assumes "H \<turnstile> Fls" shows "H \<turnstile> A"
vint_ls_def: shows "vint_ls = ZFC_in_HOL.set {[a, b]\<^sub>\<circ> |a b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat> \<and> int_of_vint a < int_of_vint b}"
nterm_to_term_frees: fixes \<Gamma> :: "name list"   and t :: "nterm" shows "frees (nterm_to_term \<Gamma> t) = frees t |-| fset_of_list \<Gamma>"
inclusion_L: fixes A :: "'a set" shows "A \<subseteq> Cn A"
sat_MatchP_rec: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and I :: "\<I>"   and r :: "formula Regex.regex" shows "sat \<sigma> V v i (MatchP I r) = (mem 0 I \<and> Regex.eps (sat \<sigma> V v) i r \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> (\<exists>s\<in>Regex.rpd (sat \<sigma> V v) i r. sat \<sigma> V v (i - 1) (MatchP (subtract (\<Delta> \<sigma> i) I) s)))"
b: shows "R1 \<subseteq> upRules \<and> R = Ax \<union> R1"
vector_add_ldistrib: fixes c :: "'a"   and x :: "('a, 'b) vec"   and y :: "('a, 'b) vec" shows "c *s (x + y) = c *s x + c *s y"
ideal_subset1: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "I \<subseteq> carrier R"
inf_eq_cases: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<sqinter> - y = - x \<sqinter> - z"   and "- - x \<sqinter> - y = - - x \<sqinter> - z" shows "- y = - z"
g': shows "g' = subdivFace' g f v 0 (tl is)"
good_strategies: fixes \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "\<sigma> \<in> good v" shows "strategy p \<sigma>"
list_of_array_id: fixes l :: "'a list" shows "list_of_array (array_of_list l) = l"
gen_ideal_subset: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "x \<in> R" shows "comm_ring.gen_ideal R multiplication x \<subseteq> R"
unique_connecting_path_properties_1: fixes G :: "('c, 'd) Graph_scheme"   and v :: "'c"   and w :: "'c" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "Graph.walk G (v \<leadsto>\<^bsub>G\<^esub> w) \<and> distinct (v \<leadsto>\<^bsub>G\<^esub> w)"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> TargetTerm TQ"
Res: fixes R :: "pi"   and S :: "pi"   and y :: "name" assumes "(R, S) \<in> Rel" shows "(<\<nu>y>R, <\<nu>y>S) \<in> Rel"
trace_decrease_mul_adj: fixes \<rho> :: "complex mat"   and dims :: "nat list"   and x :: "complex mat" assumes "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "x \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"   and "adjoint x * x \<le>\<^sub>L 1\<^sub>m (state_sig.d dims)" shows "trace (x * \<rho> * adjoint x) \<le> trace \<rho>"
nth_list_of_fset_mem: fixes i :: "nat"   and s :: "'b fset" assumes "i < size s" shows "list_of_fset s ! i |\<in>| s"
l11_62_existence_bis: 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 P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C P" shows "\<exists>X. Tarski_neutral_dimensionless.OrthAt Bet Cong X A B C X P"
dg_dghm_id_is_dghm: shows "dghm_id \<BB> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
\<Delta>''_sel_4: shows "sink \<Delta>'' = Vertex (sink \<Delta>)"
at_least_at_most_lower: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> \<lbrace>a..b\<rbrace>\<^bsub>L\<^esub>" shows "a \<sqsubseteq>\<^bsub>L\<^esub> x"
fold_rev: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f y \<circ> f x = f x \<circ> f y" shows "fold f (rev xs) = fold f xs"
IH: fixes ts :: "'a wordinterval list" assumes "ts \<noteq> []" shows "partitioningIps ss ts \<noteq> []"
in_spanE: fixes v :: "complex vec"   and W :: "complex vec set" assumes "v \<in> local.span W" shows "\<exists>a A. v = lincomb a A \<and> finite A \<and> A \<subseteq> W"
cmod_bound: fixes u :: "'a" assumes "u \<in> closed_segment w z" shows "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n"
inject_1: fixes x11 :: "'d"   and x12 :: "'d"   and x13 :: "('d, 'e, 'f) psi"   and y11 :: "'d"   and y12 :: "'d"   and y13 :: "('d, 'e, 'f) psi" shows "(RIn_Rep x11 x12 x13 = RIn_Rep y11 y12 y13) = (x11 = y11 \<and> x12 = y12 \<and> x13 = y13)"
tendsto_scaleR: fixes f :: "'c \<Rightarrow> real"   and a :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x *\<^sub>R g x) \<longlongrightarrow> a *\<^sub>R b) F"
eval_in_hom_2: fixes t :: "'c local.term" assumes "Arr t" shows "\<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Dom t\<rbrace> \<Rightarrow> \<lbrace>Cod t\<rbrace>\<guillemotright>"
strict_inv_image: fixes r :: "('c \<times> 'c) set"   and f :: "'b \<Rightarrow> 'c" shows "strict (inv_image r f) = inv_image (strict r) f"
assms_2: shows "row x A \<in> {row i A |i. row i A \<noteq> 0}"
mk_vec_def: fixes f :: "'e \<Rightarrow> 'd" shows "HMA_Connect.mk_vec f = (\<chi>i. f i)"
has_as_product_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'j"   and a :: "'j" shows "D.J.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J J__ D a \<pi>"
card_g1: shows "(k, k1) \<in> cards s g1"
ldistinct_llist_of: fixes xs :: "'c list" shows "ldistinct (llist_of xs) = distinct xs"
ex_perpendicular: fixes H :: "circline"   and z :: "complex_homo" assumes "is_poincare_line H"   and "z \<in> unit_disc" shows "\<exists>H'. is_poincare_line H' \<and> perpendicular H H' \<and> z \<in> circline_set H'"
list_encode_inverse: fixes x :: "nat list" shows "list_decode (list_encode x) = x"
simps_3: fixes x1 :: "'a"   and x2 :: "'a" shows "Pos x1 \<noteq> Neg x2"
trans_genPrefix: fixes r :: "('b \<times> 'b) set" assumes "trans r" shows "trans (genPrefix r)"
T_splay_simps_4: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and A :: "'a tree"   and B :: "'a tree"   and CD :: "'a tree" assumes "x < a"   and "x < b" shows "T_splay x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle> = (if A = \<langle>\<rangle> then 1 else T_splay x A + 1)"
implem_LtK2: fixes implem :: "chan \<Rightarrow> msg"   and x :: "chan" assumes "semivalid_implem implem" shows "implem x \<notin> range LtK"
valof_valof2: fixes e' :: "exp"   and s :: "loc \<Rightarrow> nat"   and v :: "nat"   and s' :: "loc \<Rightarrow> nat"   and c1 :: "com"   and c2 :: "com"   and e :: "exp" assumes "(e', s) -|-> (v, s')"   and "e' = VALOF c1;;c2 RESULTIS e" shows "(VALOF c1 RESULTIS VALOF c2 RESULTIS e, s) -|-> (v, s')"
w_simps_4: shows "trg w = src s\<^sub>0"
rel_fun_eq_rel: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool" shows "rel_fun (=) R = (\<lambda>f g. \<forall>x. R (f x) (g x))"
maxsimp_is_singleton: fixes S :: "'b set"   and x :: "'b set set" assumes "CoxeterComplex S"   and "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) x" shows "\<exists>w\<in>\<langle>S\<rangle>. dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w} = x"
sorted_descending_append: fixes xs :: "'b list"   and ys :: "'b list" shows "sorted_descending (xs @ ys) = (sorted_descending xs \<and> sorted_descending ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. y \<le> x))"
is_none_filter_option: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "Option.is_none (filter_option P x) = (Option.is_none x \<or> \<not> P (the x))"
many_strongly_connected_iff_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.relation_algebra_rtc minus uminus inf less_eq less sup bot top composition converse unit star" shows "(star x = star (converse x)) = less_eq (converse x) (star x)"
conjugate_zero: fixes conjugate :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "conjugate zero = zero"
euclidean_size_times_unit: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" shows "euclidean_size (a * b) = euclidean_size b"
act_act_inv: fixes G :: "('d, 'e) monoid_scheme"   and act :: "'f \<Rightarrow> 'd \<Rightarrow> 'f"   and g :: "'d"   and b :: "'f" assumes "action G act"   and "g \<in> carrier G" shows "act (act b g) (inv\<^bsub>G\<^esub> g) = b"
ls_nds_rel_sv: fixes R :: "('a \<times> ('b, 'c) node_scheme) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>ls_nds_rel)"
uniform_limitI: fixes S :: "'b set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and l :: "'b \<Rightarrow> 'c"   and F :: "'a filter" assumes "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e" shows "uniform_limit S f l F"
nat_power_complex_def: fixes n :: "nat"   and z :: "complex" shows "nat_power n z = of_nat n powr z"
poly_IVT_neg: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "0 < poly p a"   and "poly p b < 0" shows "\<exists>x>a. x < b \<and> poly p x = 0"
ennreal_less_one_iff: fixes x :: "real" shows "(ennreal x < 1) = (x < 1)"
comp_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y \<squnion> x * z = x * (y \<squnion> z)"
nff_\<alpha>_append_default: fixes dflt :: "'a"   and l :: "'a list"   and n :: "nat" shows "nff_\<alpha> dflt (l @ replicate n dflt) = nff_\<alpha> dflt l"
assms_2: shows "\<Union> p \<subseteq> cbox a b"
eqI: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<sigma> :: "'b \<Rightarrow> 'c"   and \<sigma>' :: "'b \<Rightarrow> 'c" assumes "natural_transformation A B F G \<sigma>"   and "natural_transformation A B F G \<sigma>'"   and "\<And>a. partial_magma.ide A a \<Longrightarrow> \<sigma> a = \<sigma>' a" shows "\<sigma> = \<sigma>'"
vars\<^sub>s\<^sub>s\<^sub>t_is_fv\<^sub>s\<^sub>s\<^sub>t_bvars\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('e, 'd) stateful_strand_step list" shows "vars\<^sub>s\<^sub>s\<^sub>t S = fv\<^sub>s\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>s\<^sub>t S"
check_s_ok: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) set"   and s :: "'s"   and s' :: "'s" assumes "invariant_s t as s"   and "check_s s = Inr s'" shows "checked_s t as s'"
g_def: fixes x :: "real" shows "??.Stirling_Formula.g x = ln_Gamma x - (x - 1 / 2) * ln x + x"
fmlookup_restrict_fset: fixes A :: "'d fset"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup (fmrestrict_fset A m) x = (if x |\<in>| A then fmlookup m x else None)"
fps_catalan_fps_binomial: shows "fps_catalan = 1 / 2 * (1 - (fps_binomial ((1::'a) / (2::'a)) oo - 4 * fps_X)) / fps_X"
mid_err_def: fixes ivl :: "float interval" shows "mid_err ivl = (real_of_float (lower ivl + upper ivl) / 2, real_of_float (upper ivl - lower ivl) / 2)"
realfun_mon_conv_add: fixes x :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> real"   and a :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and b :: "'b \<Rightarrow> real" assumes "x\<up>y"   and "a\<up>b" shows "(\<lambda>m w. x m w + a m w)\<up>(\<lambda>w. y w + b w)"
assms: shows "low_equal s1 s2 \<and> \<not> snd (execute_instr_sub1 instr s1) \<and> \<not> snd (execute_instr_sub1 instr s2) \<and> t1 = snd (fst (execute_instr_sub1 instr s1)) \<and> t2 = snd (fst (execute_instr_sub1 instr s2))"
a0i_defs_8: shows "a0i_trans \<equiv> \<Union>h d ds. (a0i_running h d \<union> a0i_commit h d \<union> a0i_corrupt ds)\<^sup>="
NSDERIV_NSLIM_iff2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" shows "NSDERIV f x :> D = (\<lambda>z. (f z - f x) / (z - x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S D"
older_seniors_stable: fixes x :: "nat"   and n :: "nat"   and m :: "nat"   and m' :: "nat" assumes "x \<le> n"   and "older_seniors x n = older_seniors x (n + m + m')" shows "older_seniors x n = older_seniors x (n + m)"
subset: shows "{(a, b). \<exists>c. g a b c \<noteq> \<^bold>1} \<times> {c. \<exists>a b. g a b c \<noteq> \<^bold>1} \<subseteq> D"
gcd_factorial_def: fixes a :: "'a"   and b :: "'a" shows "gcd_factorial a b = (if a = (0::'a) then normalize b else if b = (0::'a) then normalize a else normalize (\<Prod>\<^sub># (prime_factorization a \<inter># prime_factorization b)))"
Healthy_if: fixes P :: "'c upred"   and H :: "'c upred \<Rightarrow> 'c upred" assumes "P is H" shows "H P = P"
strengthen_pre: fixes P' :: "state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool"   and c :: "com"   and Q :: "state \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile> {P} c {Q}" shows "\<turnstile> {P'} c {Q}"
homotopic_with_def: fixes P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "homotopic_with P X Y f g \<equiv> \<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x. h (0, x) = f x) \<and> (\<forall>x. h (1, x) = g x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x)))"
LTL_GF_infinitely_many_suffixes: fixes w :: "nat \<Rightarrow> 'c set"   and \<phi> :: "'c ltl" shows "w \<Turnstile> G (F \<phi>) = (\<exists>\<^sub>\<infinity>i. suffix i w \<Turnstile> \<phi>)"
I_mono: fixes it :: "'d set"   and \<sigma> :: "'e"   and it' :: "'d set" assumes "I it \<sigma>"   and "it' \<subseteq> it"   and "it \<subseteq> S" shows "I it' \<sigma>"
simps_1: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "successively P [] = True"
hyps_1: fixes q :: "(symbol \<times> character list) list" assumes "q \<in> \<P> k u" shows "\<forall>i<length q. \<exists>u. q ! i \<in> \<Z> (charslength (take i q)) u"
0: fixes V :: "((('a, 'b, 'c) prot_fun, 'b prot_atom) Term.term \<times> nat) set"   and S :: "(('a, 'b, 'c) prot_fun, (('a, 'b, 'c) prot_fun, 'b prot_atom) Term.term \<times> nat) stateful_strand_step list"   and S' :: "(('a, 'b, 'c) prot_fun, (('a, 'b, 'c) prot_fun, 'b prot_atom) Term.term \<times> nat) stateful_strand_step list" assumes "wf'\<^sub>s\<^sub>s\<^sub>t V S"   and "f S' \<subseteq> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S \<union> V" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (S @ S')"
f_2: shows "f ` sphere a r \<subseteq> sphere b s"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>" shows "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))"
bangSC: fixes P :: "pi" shows "!P \<sim> P \<parallel> !P"