hcju/isabelleps huggingface dataset:
regular_event_implies_same_channel_snapshot_Recv_Recv: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isRecv ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
simps_2: fixes x :: "floatarith"   and a :: "floatarith"   and f :: "form"   and vs :: "real list" shows "interpret_form (Assign x a f) vs = (interpret_floatarith x vs = interpret_floatarith a vs \<longrightarrow> interpret_form f vs)"
P2_b: fixes x :: "'a"   and y :: "'a" assumes "x r\<rightarrow> y = (1::'a)"   and "y r\<rightarrow> x = (1::'a)" shows "x = y"
mset_subset_eq_add_mset_cancel: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(add_mset a A \<subseteq># add_mset a B) = (A \<subseteq># B)"
length_compl_rev_induct: fixes P :: "'a list \<Rightarrow> bool"   and l :: "'a list" assumes "P []"   and "\<And>l e. (\<And>ll. length ll \<le> length l \<Longrightarrow> P ll) \<Longrightarrow> P (l @ [e])" shows "P l"
lcliff_def: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and b :: "'a"   and xs :: "(accent \<times> 'a) list" shows "lcliff r a b xs = ((\<exists>as b' cs a' bs. snd ` set as \<subseteq> under r a \<and> b' = [(Macron, b)] \<and> snd ` set cs \<subseteq> under r a \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r a \<inter> under r b \<and> xs = as @ b' @ cs @ a' @ bs) \<or> (\<exists>cs a' bs. snd ` set cs \<subseteq> under r a \<union> under r b \<and> a' \<in> {[(Acute, a)], []} \<and> snd ` set bs \<subseteq> under r b \<and> xs = cs @ a' @ bs))"
Let_eq_iff: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and \<Gamma>' :: "(Vars.var \<times> exp) list"   and e' :: "exp" shows "(Terms.Let \<Gamma> e = Terms.Let \<Gamma>' e') = ([map (\<lambda>x. atom (fst x)) \<Gamma>]lst. (e, \<Gamma>) = [map (\<lambda>x. atom (fst x)) \<Gamma>']lst. (e', \<Gamma>'))"
fps_deriv_of_nat: fixes n :: "nat" shows "fps_deriv (of_nat n) = 0"
pp_arc_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.pp_arc f = (\<exists>e. f e \<noteq> bot \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot))"
Mul_AppendInv_def: shows "Mul_AppendInv \<equiv> \<lambda>s ind. \<forall>i\<ge>ind. i < length (M s) \<longrightarrow> i \<in> Reach (E s) \<longrightarrow> M s ! i = Black"
js_is_a_permutation: fixes js :: "nat list"   and k :: "nat" assumes "\<And>f h. associative f \<Longrightarrow> foldl1 f (map h js) = foldl1 f (map h [0..<k + 1])"   and "js \<noteq> []" shows "js <~~> [0..<k + 1]"
priv_is_asm_priv_ignores_old_vars: shows "ignores_old_vars priv_is_asm_priv"
UP_def: fixes \<mu> :: "'a" shows "UP \<mu> \<equiv> if B.arr \<mu> then MkArr \<^bold>\<langle>B.dom \<mu>\<^bold>\<rangle> \<^bold>\<langle>B.cod \<mu>\<^bold>\<rangle> \<mu> else null"
norm_triangle_ineq2: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm a - norm b \<le> norm (minus a b)"
Exit_not_weak_control_dependent: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g"   and n :: "'g" assumes "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "StrongPostdomination.weak_control_dependence sourcenode targetnode valid_edge Exit n Exit" shows "False"
fold_insert_idem2: fixes A :: "'a set"   and z :: "'b"   and x :: "'a" assumes "finite A" shows "Finite_Set.fold f z (insert x A) = Finite_Set.fold f (f x z) A"
gbm_nodes_list_it_impl: shows "graph_nodes_it gbm_\<alpha> gbm_invar gbm_nodes_list_it"
prv_cnj_imp_mono: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   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 num :: "'b set"   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 \<phi>1 :: "'c"   and \<phi>2 :: "'c"   and \<chi>1 :: "'c"   and \<chi>2 :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla" shows "prv (imp (imp \<phi>1 \<chi>1) (imp (imp \<phi>2 \<chi>2) (imp (cnj \<phi>1 \<phi>2) (cnj \<chi>1 \<chi>2))))"
assms: fixes i :: "nat" assumes "i < max_Var_floatariths f" shows "xs ! i = ys ! i"
simps: fixes e1 :: "'c"   and a1 :: "'d"   and r1 :: "nat"   and ts1 :: "('c, 'd) SkewBinomialTree list"   and e2 :: "'c"   and a2 :: "'d"   and r2 :: "nat"   and ts2 :: "('c, 'd) SkewBinomialTree list" shows "link (Node e1 a1 r1 ts1) (Node e2 a2 r2 ts2) = (if a1 \<le> a2 then Node e1 a1 (Suc r1) (Node e2 a2 r2 ts2 # ts1) else Node e2 a2 (Suc r2) (Node e1 a1 r1 ts1 # ts2))"
t_prod_def: fixes X :: "('a \<times> ('b \<times> 'a) list) set"   and Y :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod X Y = {t_fusion u v |u v. u \<in> X \<and> v \<in> Y \<and> Dioid_Models_New.last u = first v}"
ts: shows "deg_fun t < deg_fun s \<or> deg_fun t = deg_fun s \<and> ord t s"
rtrancl_path_distinct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "rtrancl_path r x xs y" obtains xs' :: "'b list" where "rtrancl_path r x xs' y" and "distinct (x # xs')" and "set xs' \<subseteq> set xs"
elims_3: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'd list \<Rightarrow> 'd"   and intP :: "'b \<Rightarrow> 'd list \<Rightarrow> bool"   and x :: "Preliminaries.var \<Rightarrow> 'd"   and xa :: "('a, 'b) atm" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP"   and "\<not> M.Struct.satA intF intP x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and T1 :: "'a trm" and T2 :: "'a trm" where "x = \<xi>__" and "xa = Eq T1 T2" and "M.Struct.int intF \<xi>__ T1 \<noteq> M.Struct.int intF \<xi>__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'd" and p :: "'b" and Tl :: "'a trm list" where "x = \<xi>" and "xa = Pr p Tl" and "\<not> intP p (map (M.Struct.int intF \<xi>) Tl)"
continuous_imp_tendsto: fixes x0 :: "'a"   and f :: "'a \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'a" assumes "isCont f x0"   and "x \<longlonglongrightarrow> x0" shows "(f \<circ> x) \<longlonglongrightarrow> f x0"
hyps: fixes x2a :: "('f, 'v) Term.term" assumes "x2a \<in> set ss" shows "x2a \<succeq> x2a"
in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VoV.in_hom f a b" obtains "VoV.arr f" and "VoV.dom f = a" and "VoV.cod f = b"
pow_generator_mod_int: fixes G :: "('a, 'b) cyclic_group_scheme"   and k :: "int" assumes "Cyclic_Group.cyclic_group G" shows "\<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> (k mod int (order G)) = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> k"
scale_eq_0_iff: fixes a :: "'a"   and x :: "'c" shows "(a *b x = (0::'c)) = (a = (0::'a) \<or> x = (0::'c))"
connected_space_def: fixes X :: "'c topology" shows "connected_space X \<equiv> \<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> topspace X \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {}"
merge_list_correct: fixes ls :: "'a list list"   and as :: "'a list list" assumes "\<And>l. l \<in> set ls \<Longrightarrow> sorted l"   and "\<And>l. l \<in> set as \<Longrightarrow> sorted l" shows "sorted (merge_list as ls) \<and> mset (merge_list as ls) = mset (concat (as @ ls)) \<and> set (merge_list as ls) = set (concat (as @ ls))"
fconst_on_def: fixes A :: "V"   and c :: "V" shows "fconst_on A c = set {[a, c]\<^sub>\<circ> |a. a \<in>\<^sub>\<circ> A}"
sfisp_sfis_eq: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" shows "sfisp f M = (\<lambda>x. x \<in> sfis f M)"
i_Exec_Stream_append_last: fixes xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'd"   and c :: "'e" assumes "xs \<noteq> []" shows "i_Exec_Comp_Stream trans_fun (xs \<frown> input) c = f_Exec_Comp_Stream trans_fun xs c \<frown> i_Exec_Comp_Stream trans_fun input (last (f_Exec_Comp_Stream trans_fun xs c))"
payload_inductive_cases_5: fixes X :: "msg"   and Y :: "msg" assumes "Enc X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
avl_delete: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "avl (delete x t)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> otway"   and "P []"   and "\<And>evsf X B. \<lbrakk>evsf \<in> otway; P evsf; X \<in> synth (analz (knows Spy evsf))\<rbrakk> \<Longrightarrow> P (Says Spy B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> otway; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> otway; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B \<lbrace>Nonce NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>\<rbrace> # evs1)"   and "\<And>evs2 NB B NA A X. \<lbrakk>evs2 \<in> otway; P evs2; Nonce NB \<notin> used evs2; Gets B \<lbrace>Nonce NA, Agent A, Agent B, X\<rbrace> \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B Server \<lbrace>Nonce NA, Agent A, Agent B, X, Crypt (shrK B) \<lbrace>Nonce NA, Nonce NB, Agent A, Agent B\<rbrace>\<rbrace> # evs2)"   and "\<And>evs3 KAB NA A B NB. \<lbrakk>evs3 \<in> otway; P evs3; Key KAB \<notin> used evs3; Gets Server \<lbrace>Nonce NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce NA, Agent A, Agent B\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NA, Nonce NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says Server B \<lbrace>Nonce NA, Crypt (shrK A) \<lbrace>Nonce NA, Key KAB\<rbrace>, Crypt (shrK B) \<lbrace>Nonce NB, Key KAB\<rbrace>\<rbrace> # evs3)"   and "\<And>evs4 B NA A X' NB X K. \<lbrakk>evs4 \<in> otway; P evs4; B \<noteq> Server; Says B Server \<lbrace>Nonce NA, Agent A, Agent B, X', Crypt (shrK B) \<lbrace>Nonce NA, Nonce NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs4; Gets B \<lbrace>Nonce NA, X, Crypt (shrK B) \<lbrace>Nonce NB, Key K\<rbrace>\<rbrace> \<in> set evs4\<rbrakk> \<Longrightarrow> P (Says B A \<lbrace>Nonce NA, X\<rbrace> # evs4)"   and "\<And>evso B NA X NB K. \<lbrakk>evso \<in> otway; P evso; Says Server B \<lbrace>Nonce NA, X, Crypt (shrK B) \<lbrace>Nonce NB, Key K\<rbrace>\<rbrace> \<in> set evso\<rbrakk> \<Longrightarrow> P (Notes Spy \<lbrace>Nonce NA, Nonce NB, Key K\<rbrace> # evso)" shows "P x"
map_2: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b list" shows "map f (x21 # x22) = f x21 # map f x22"
constant_transformation_def: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" shows "constant_transformation A B g \<equiv> (category A \<and> category B) \<and> constant_functor A B (partial_magma.dom B g) \<and> constant_functor A B (partial_magma.cod B g) \<and> constant_transformation_axioms B g"
f_2: shows "of_int_poly f = Polynomial.smult c G"
negligible_Union_nat: fixes S :: "nat \<Rightarrow> 'a set" assumes "\<And>n. negligible (S n)" shows "negligible (\<Union> (range S))"
s\<^sub>0_def: shows "s\<^sub>0 = (ts, Map.empty, start)"
seqr_assoc: fixes P :: "('d, 'g) urel"   and Q :: "('g, 'f) urel"   and R :: "('f, 'e) urel" shows "(P ;; Q) ;; R = P ;; Q ;; R"
stable: fixes X :: "X4 set" shows "StableNoDecomp.stable_on UNIV X = (X = {Xd1, Xd2} \<or> X = {Xd1', Xd2'})"
embed_bool_nneg: fixes P :: "'b \<Rightarrow> bool" shows "nneg \<guillemotleft> P \<guillemotright>"
finsert_absorb: fixes a :: "'b"   and A :: "'b fset" assumes "a |\<in>| A" shows "finsert a A = A"
sim_rel_def: shows "sim_rel \<equiv> sim_rel1 OO sim_rel2"
bits_of_int_simps_3: fixes k :: "int" assumes "k \<noteq> 0"   and "k \<noteq> - 1" shows "bits_of k = odd k # bits_of (k div 2)"
neg_divide_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less c zero" shows "less (divide b c) a = less (times a c) b"
reachable_refl_iff: fixes u :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> u) = (u \<in> verts (del_vert u))"
ensure_pre_insertion_validity_pure: fixes node :: "(_) node_ptr"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr option"   and h :: "(_) heap" shows "pure (local.a_ensure_pre_insertion_validity node ptr child) h"
r_shift_def: fixes f :: "recf" shows "r_shift f \<equiv> Cn (Suc (arity f)) f (map (\<lambda>i. recf.Id (Suc (arity f)) (Suc i)) [0..<arity f])"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> w"
T_filter_eq: fixes T_p :: "'a \<Rightarrow> nat"   and xs :: "'a list" shows "T_filter T_p xs = sum_list (map T_p xs) + length xs + 1"
assms_3: fixes x :: "nat" assumes "x\<^sub>1 \<le> x" shows "x\<^sub>0 \<le> t x"
anonymise_prefs_from_table: fixes agents :: "'a set"   and alts :: "'b set"   and xs :: "('a \<times> 'b set list) list" assumes "prefs_from_table_wf agents alts xs" shows "anonymous_profile (prefs_from_table xs) = mset (map snd xs)"
compatible_rtrancl_split: fixes NS :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "NS O S \<subseteq> S"   and "(x, y) \<in> (NS \<union> S)\<^sup>*" shows "(x, y) \<in> S O S\<^sup>* O NS\<^sup>* \<union> NS\<^sup>*"
Diff_eq: fixes A :: "'a set"   and B :: "'a set" shows "A - B = A \<inter> - B"
arr_def: fixes f :: "('o, 'a) arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
comm_ring_hom_axioms: shows "comm_ring_hom (map_poly hom)"
prems_4: shows "path_conforms_with_strategy p P' \<sigma>"
heron_step_real: fixes t :: "real"   and n :: "real" assumes "0 < t"   and "0 \<le> n" shows "sqrt n \<le> (t + n / t) / 2"
simps_1: fixes f :: "face"   and fs :: "face list"   and F :: "face list list" shows "replacefacesAt2 [] f fs F = F"
subset_eq: fixes A :: "'h set"   and B :: "'h set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
simple_rotations_surj: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "surj r"
simps_3: fixes \<Gamma> :: "name list"   and t :: "nterm"   and u :: "nterm" shows "nterm_to_term \<Gamma> (t $\<^sub>n u) = nterm_to_term \<Gamma> t $ nterm_to_term \<Gamma> u"
simps_2: fixes f :: "'c \<Rightarrow> ('c, 'd) Nested_Datatype.term"   and b :: "'d"   and ts :: "('c, 'd) Nested_Datatype.term list" shows "subst_term f (App b ts) = App b (subst_term_list f ts)"
assms_2: shows "g \<turnstile> ns\<Down>\<^sub>s s'"
f': fixes x :: "'a" assumes "x \<in> S" shows "inj (f' x)"
distinct_elim: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v1 :: "'a"   and w :: "'b"   and v2 :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G v ((v1, w, v2) # ps) v'" shows "((v1, w, v2) \<in> edges (rem_unPath ps G)) = ((v1, w, v2) \<in> edges G)"
strict_order_iff: fixes a :: "'k"   and b :: "'k" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
simps: fixes t1 :: "'a option list"   and t2 :: "'a option list" shows "merge t1 t2 = map merge_option (zip t1 t2)"
glbH_is_fixp: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and H :: "'b set" assumes "CLF cl f"   and "H = {x \<in> pset cl. (f x, x) \<in> order cl}" shows "glb H cl \<in> fix f (pset cl)"
infsetsum'_approx_sum: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and \<epsilon> :: "real" assumes "infsetsum'_converges f A"   and "0 < \<epsilon>" shows "\<exists>F. finite F \<and> F \<subseteq> A \<and> dist (sum f F) (infsetsum' f A) \<le> \<epsilon>"
invertible_interchange_rows: fixes a :: "'c"   and b :: "'c" shows "invertible (interchange_rows (Finite_Cartesian_Product.mat (1::'b)) a b)"
seq_implies_weakly_seq: fixes P :: "'c option process" assumes "sequential P" shows "weakly_sequential P"
relpow_impl_main: fixes new :: "'a list"   and "have" :: "'b"   and n :: "nat" shows "set_of (relpow_i new have n) = {uu_. \<exists>a b m. uu_ = b \<and> a \<in> set new \<and> m \<le> n \<and> (a, b) \<in> (rel \<inter> {(a, b). b \<notin> set_of have}) ^^ m} \<union> set_of have"
delayedR_correct_1: fixes u :: "'c \<Rightarrow> real"   and R :: "('c \<Rightarrow> real) set"   and I :: "'c \<Rightarrow> Regions.intv"   and r :: "('c \<times> 'c) set" assumes "u \<in> R"   and "R \<in> \<R>"   and "Regions.valid_region \<X> k I r"   and "\<forall>c\<in>\<X>. \<not> Regions.isConst (I c)"   and "Regions.region \<X> I r \<in> Succ \<R> R" shows "delayedR (Regions.region \<X> I r) u \<in> Regions.region \<X> I r"
fst_def: fixes prod :: "'f \<times> 'g" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
for_rec3_for_rec3': fixes f :: "'b \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "for_rec3 f a n k i j = for_rec3' f a n k i j"
simps_14: fixes M :: "char list"   and n :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" shows "succs (Invoke M n) \<tau> pc = (if fst \<tau> ! n = NT then [] else [pc + 1])"
partitions_bounds: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p partitions n" shows "p i \<le> n"
small_iff: fixes X :: "V set" shows "small X = (\<exists>x. X = elts x)"
wfP_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "wfP r = wf {(x, y). r x y}"
simps: fixes rects :: "'a rectangle list"   and i :: "nat" shows "ins_op rects i = INS (low (yint (rects ! i))) i (xint (rects ! i))"
DomainPI: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "r a b" shows "Domainp r a"
to_fun_eval: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and x :: "'c" assumes "UP_cring R"   and "f \<in> carrier (UP R)"   and "x \<in> carrier R" shows "UP_cring.to_fun R f x = eval R R (\<lambda>x. x) x f"
mem_fset_from_list: fixes x :: "'b"   and l :: "'b list" shows "(x |\<in>| fset_from_list l) = (x \<in> set l)"
x1: fixes x :: "'a" assumes "x \<in> space_as_set A" shows "x \<in> space_as_set B +\<^sub>M space_as_set C"
coeffs_eqI: fixes p :: "'b poly"   and xs :: "'b list" assumes "\<And>n. coeff p n = nth_default (0::'b) xs n"   and "no_trailing ((=) (0::'b)) xs" shows "coeffs p = xs"
F1set2_Lev1: fixes xf :: "(bd_type_F + bd_type_F) list"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and n :: "nat"   and xe :: "'d"   and xc :: "'d"   and xa :: "'d" assumes "xf \<in> Lev1 s1 s2 n xe"   and "rv1 s1 s2 xf xe = Inl xc"   and "xa \<in> F1set2 (s1 xc)" shows "xf @ [Inl (tobd_F12 s1 xc xa)] \<in> Lev1 s1 s2 (Suc n) xe"
assms_3: shows "G \<subseteq> Pow \<Omega>"
le_length: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs \<le>\<^sub>v ys" shows "length xs = length ys"
closure_sequential: fixes l :: "'d"   and S :: "'d set" shows "(l \<in> closure S) = (\<exists>x. (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> l)"
fps_shift_times_fps_X_power': fixes n :: "nat"   and f :: "'a fps" shows "fps_shift n (f * fps_X ^ n) = f"
rcos_equation: fixes H :: "'a set"   and ha :: "'a"   and a :: "'a"   and h :: "'a"   and b :: "'a"   and hb :: "'a" assumes "subgroup H (add_monoid G)"   and "ha \<oplus> a = h \<oplus> b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "h \<in> H"   and "ha \<in> H"   and "hb \<in> H" shows "hb \<oplus> a \<in> (\<Union>h\<in>H. {h \<oplus> b})"
gen_sqrt_lower_scaled: fixes x :: "real"   and u :: "real"   and n :: "nat" assumes "0 \<le> x"   and "0 < u" shows "sqrtl (x * u\<^sup>2) n / u \<le> sqrt x"
lang_rhs_union_distrib: fixes A :: "'b trm set"   and B :: "'b trm set" shows "lang_rhs A \<union> lang_rhs B = lang_rhs (A \<union> B)"
cod_char: fixes f :: "'a \<times> 'a" shows "H.VV.cod f = (if H.VV.arr f then H.VxV.cod f else H.VxV.null)"
lset_lSup: fixes Y :: "'c llist set" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) Y" shows "lset (lSup Y) = \<Union> (lset ` Y)"
eventually_frequently_const_simps_6: fixes C :: "bool"   and P :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<forall>\<^sub>F x in F. C \<longrightarrow> P x) = (C \<longrightarrow> eventually P F)"
fps_lr_inverse_0_iff_2: fixes f :: "'a fps"   and x :: "'a" shows "(fps_nth (fps_right_inverse f x) 0 = (0::'a)) = (x = (0::'a))"
auto_ll_on_open_axioms: shows "auto_ll_on_open f X"
ann_sem_def: fixes c :: "'b ann_com" shows "ann_sem c \<equiv> \<lambda>s. {t. (Some c, s) -*\<rightarrow> (None, t)}"
preserves_iso: fixes f :: "'a" assumes "A.iso f" shows "B.iso (F f)"
rev_ImageI: fixes a :: "'e"   and A :: "'e set"   and b :: "'f"   and r :: "('e \<times> 'f) set" assumes "a \<in> A"   and "(a, b) \<in> r" shows "b \<in> r `` A"
x_2: shows "fds_abs_converges f (complex_of_real x)"
sym_congruent_rel: fixes b :: "'a" shows "sym (congruent_rel b)"
equivp_reflp2: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivp R" shows "reflp R"
bicategorical_language_axioms: shows "bicategorical_language (\<cdot>) src trg"
Abs_coreflexive_inverse: fixes y :: "'b" assumes "y \<in> \<down>(1::'b)" shows "Rep_coreflexive (Abs_coreflexive y) = y"
basis_in_subset_carrier: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> X" shows "X \<subseteq> \<E>"
less_pos: fixes x :: "'a"   and y :: "'a" shows "(x < y) = (pos x < pos y)"
rep_list_mult_scalar: fixes c :: "'a \<Rightarrow>\<^sub>0 'b"   and r :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (c \<odot> r) = c * rep_list r"
comm_semigroup_axioms: shows "comm_semigroup f\<^sub>a"
inject: fixes x1 :: "'b"   and y1 :: "'b" shows "(Fin x1 = Fin y1) = (x1 = y1)"
integrable_real_mult_indicator: fixes A :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f" shows "integrable M (\<lambda>x. f x * indicat_real A x)"
val_Zp_p: shows "val_Zp ([p] \<cdot> \<one>) = 1"