hcju/isabelleps huggingface dataset:
scalar_product_lower_bound: fixes b :: "nat list"   and a :: "nat"   and "is" :: "int list" assumes "\<forall>x\<in>set b. len x \<le> a" shows "(2 ^ a - 1) * sum_list (concat (map (\<lambda>i. if i < 0 then [i] else []) is)) \<le> scalar_product b is"
plus_adds_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus (plus a b) c" shows "plus.adds plus a c"
pause_option_def: fixes pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and out :: "'c"   and c :: "'d \<Rightarrow> ('a, 'b) optionT" shows "pause_option pause out c = OptionT (pause out (\<lambda>i. run_option (c i)))"
sorted_list_of_set_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
assms_4: fixes x :: "nat" assumes "x < length qs" shows "finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))"
freshImEnvAt_def: fixes xs :: "'varSort"   and x :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var" shows "freshImEnvAt xs x rho ys y \<equiv> rho ys y = None \<and> (ys \<noteq> xs \<or> y \<noteq> x) \<or> (\<exists>Y. rho ys y = Some Y \<and> fresh xs x Y)"
proved_imp_Pf: fixes H :: "fm set"   and \<alpha> :: "fm"   and e :: "(name, hf) finfun" assumes "H \<turnstile> \<alpha>"   and "H = {}" shows "Pf \<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e"
vseq_implies_hpar_2: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.seq \<nu> \<mu>" shows "Span.trg \<nu> = Span.trg \<mu>"
path_in_V: fixes xs :: "'a list" assumes "path xs" shows "set xs \<subseteq> V"
HC_intros_7: fixes F :: "'b formula"   and H :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H ((F::'b formula) \<^bold>\<rightarrow> (H::'b formula)) \<^bold>\<rightarrow> ((G::'b formula) \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<or> G) \<^bold>\<rightarrow> H"
fun_upd_def: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'d"   and b :: "'e" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
zero_less_one: fixes one :: "'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
list_all_iff: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a list" shows "list_all P x = Ball (set x) P"
\<epsilon>': shows "\<guillemotleft>UP.unit (trg f) \<cdot>\<^sub>S \<xi>' \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (f, g)) : S.UP (f \<star> g) \<Rightarrow>\<^sub>S S.UP (trg f)\<guillemotright>"
deps_insert: fixes M :: "('c, 'd) message set"   and m :: "('c, 'd) message" assumes "\<Union> (deps ` M) \<subseteq> I ` insert_messages M"   and "deps m \<subseteq> I ` insert_messages M" shows "\<Union> (deps ` (M \<union> {m})) \<subseteq> I ` insert_messages (M \<union> {m})"
add_new_coeff_r: fixes p :: "(char list multiset \<times> int) multiset"   and x :: "char list multiset \<times> int"   and q :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" shows "add_poly_p (p, add_mset x q, r) (p, q, add_mset x r)"
LAssReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>V:=e',s'\<rangle>"
SkipRuleTC: fixes p :: "'a set"   and q :: "'a set" assumes "p \<subseteq> q" shows "[p]   SKIP   [q]"
ct: fixes x :: "nat" shows "c (x, t) = (if n < x \<and> x \<le> 2 * n + 1 then g (x - n - 1) else 0)"
par_strict_left_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B ParStrict C D" shows "B A ParStrict C D"
execution_simulation_trace_simulation: fixes f :: "nat \<Rightarrow> nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list \<times> value list) list" assumes "execution_simulation f e1 s1 r1 e2 s2 r2 (map (\<lambda>(l, i, o). (l, i)) t)" shows "trace_simulation f e1 s1 r1 e2 s2 r2 t"
Hermite_Lindemann_aux2: fixes X :: "complex set"   and \<beta> :: "complex \<Rightarrow> int" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> \<beta> x \<noteq> 0"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic x"   and "(\<Sum>x\<in>X. of_int (\<beta> x) * exp x) = 0" shows "X = {}"
dghm_up_ObjMap_app_component: fixes a :: "V"   and \<CC> :: "V"   and i :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and \<phi> :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "dghm_up I \<AA> \<CC> \<phi>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<lparr>i\<rparr> = \<phi> i\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>"
adhCE: fixes T :: "'b set set"   and x :: "'b"   and A :: "'b set"   and U :: "'b set" assumes "x adh\<^bsub>T\<^esub> A" obtains "U \<notin> nhds\<^bsub>T\<^esub> x"   | "U \<inter> A \<noteq> {}"
env_restr_eqD: fixes S :: "'c set"   and m\<^sub>1 :: "'c \<Rightarrow> 'd"   and m\<^sub>2 :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "m\<^sub>1 f|` S = m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x = m\<^sub>2 x"
wset_thread_ok_NotifyAllI: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and w :: "'d \<Rightarrow> 'e wait_set_status"   and w' :: "'d \<Rightarrow> 'e wait_set_status" assumes "wset_thread_ok ws ts" shows "wset_thread_ok (\<lambda>t. if ws t = \<lfloor>w t\<rfloor> then \<lfloor>w' t\<rfloor> else ws t) ts"
ncols_move_matrix_le: fixes A :: "'b matrix"   and j :: "int"   and i :: "int" shows "ncols (move_matrix A j i) \<le> nat (int (ncols A) + i)"
lex_two_order_pair: fixes s1 :: "('b \<times> 'b) set"   and ns1 :: "('b \<times> 'b) set"   and s2 :: "('c \<times> 'c) set"   and ns2 :: "('c \<times> 'c) set" assumes "order_pair s1 ns1"   and "order_pair s2 ns2" shows "order_pair (lex_two s1 ns1 s2) (lex_two s1 ns1 ns2)"
IH_3: assumes "finite B"   and "\<And>a b. \<lbrakk>a \<in> A; b \<in> B\<rbrakk> \<Longrightarrow> g a b \<in> carrier G" shows "(\<Otimes>a\<in>A. finprod G (g a) B) = (\<Otimes>a\<in>A \<times> B. case a of (a, b) \<Rightarrow> g a b)"
simps: fixes checkp :: "'b \<Rightarrow> 'a"   and cs :: "'c"   and s :: "'b" shows "checkpoint_cs checkp (cs, s) = checkp s"
gt_a0_ge_1: fixes x :: "ant" assumes "0 < x" shows "1 \<le> x"
long: shows "\<^bold>|w\<^bold>| < \<^bold>|x\<^bold>| + \<^bold>|x\<^bold>|"
card_Un_Int: fixes A :: "'d set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
rs: shows "listrelp (\<lambda>t u. t \<rightarrow>\<^sub>s u \<and> (\<forall>r. u \<rightarrow>\<^sub>\<beta> r \<longrightarrow> t \<rightarrow>\<^sub>s r)) rs rs'"
ENFsr_offending_set_P_representation: fixes F :: "('v \<times> 'v) set" assumes "F \<in> set_offending_flows G nP" shows "F = {(s, r). (s, r) \<in> edges G \<and> \<not> P (nP s) s (nP r) r}"
weak_ranking_index_unique: fixes xs :: "'c set list"   and i :: "nat"   and j :: "nat"   and x :: "'c" assumes "is_weak_ranking xs"   and "i < length xs"   and "j < length xs"   and "x \<in> xs ! i"   and "x \<in> xs ! j" shows "i = j"
integral_discrete_difference: fixes X :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M"   and "\<And>x. \<lbrakk>x \<in> space M; x \<notin> X\<rbrakk> \<Longrightarrow> f x = g x" shows "integral\<^sup>L M f = integral\<^sup>L M g"
restrict_top_ifex_ordered_invar: fixes b :: "'a ifex"   and var :: "'a"   and val :: "bool" assumes "ifex_ordered b" shows "ifex_ordered (restrict_top b var val)"
degree_mod_less: fixes y :: "'a poly"   and x :: "'a poly" assumes "y \<noteq> 0" shows "x mod y = 0 \<or> degree (x mod y) < degree y"
comp_arr_ide: fixes a :: "'s"   and f :: "'s" assumes "S.ide a"   and "S.seq f a" shows "f \<cdot> a = f"
simps: fixes xs :: "'varSort"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" shows "qGoodAbs (qAbs xs x X) = qGood X"
of_nat_gt_0: fixes k :: "nat" assumes "of_nat k \<noteq> (0::'a)" shows "0 < k"
in_fset_mdef: fixes F :: "'b fset"   and x :: "'b" shows "(x |\<in>| F) = (x |\<notin>| F - {|x|} \<and> F = insert_fset x (F - {|x|}))"
card_of_refl: fixes A :: "'c set" shows "|A| =o |A|"
Cauchy_Schwarz_ineq2: fixes x :: "'c"   and y :: "'c" shows "cmod \<langle>x, y\<rangle> \<le> norm x * norm y"
inf_le1: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (inf_prod x y) x"
TensorDiag_preserves_Diag_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Diag C t"   and "monoidal_language.Diag C u" shows "monoidal_language.Dom C (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = monoidal_language.Dom C t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> monoidal_language.Dom C u"
show_l_zero: fixes x :: "'a vec"   and a :: "'a vec" assumes "x \<in> carrier_vec n"   and "a \<in> carrier_vec n" shows "(a + x = x) = (a = 0\<^sub>v n)"
pure_eventual_until_simp: fixes \<mu> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "pure_eventual \<mu>" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<mu> = w \<Turnstile>\<^sub>n \<mu>"
hypsubst: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
ipurge_tr_aux_single_event: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and x :: "'a" shows "ipurge_tr_aux I D U [x] = (if \<exists>v\<in>U. (v, D x) \<in> I then [] else [x])"
fps_to_fls_eq_imp_fps_eq: fixes f :: "'a fps"   and g :: "'a fps" assumes "fps_to_fls f = fps_to_fls g" shows "f = g"
induct: fixes P :: "trieP \<Rightarrow> bool"   and a0 :: "trieP" assumes "P LfP"   and "\<And>ps b l r. \<lbrakk>P l; P r\<rbrakk> \<Longrightarrow> P (NdP ps b (l, r))" shows "P a0"
hd_in_set: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "lead_coeff xs \<in> set xs"
append_simpl3: fixes y :: "'d list"   and i :: "nat"   and z :: "'e"   and zs :: "'e list" assumes "length y < i" shows "i < length (z # zs) * length y \<longrightarrow> i - length y < length zs * length y"
acquire_locks_conv: fixes L :: "('b \<times> nat) option"   and t :: "'b"   and n :: "nat" shows "acquire_locks L t n = (case L of None \<Rightarrow> case n of 0 \<Rightarrow> None | Suc m \<Rightarrow> \<lfloor>(t, m)\<rfloor> | \<lfloor>(t', m)\<rfloor> \<Rightarrow> \<lfloor>(t', n + m)\<rfloor>)"
periodic_arg_plus_distrib: fixes x :: "'a"   and m :: "int"   and n :: "int" shows "g x (of_int (m + n)) = g (g x (of_int n)) (of_int m)"
closed_map_imp_subset_topspace: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "closed_map X1 X2 f" shows "f ` topspace X1 \<subseteq> topspace X2"
subst_list1_eq_map_substs: fixes l :: "exp list" shows "\<forall>\<sigma>. subst_list1 \<sigma> l = map (substs \<sigma>) l"
assms_3: shows "X\<^sub>2 \<approx>\<^sub>L Y\<^sub>2"
in_set_conv_iff: fixes x :: "'c"   and n :: "nat"   and xs :: "'c list" shows "(x \<in> set (take n xs)) = (\<exists>i<n. i < length xs \<and> xs ! i = x)"
distinct_19: fixes x111 :: "'b ltln"   and x112 :: "'b ltln" shows "true\<^sub>n \<noteq> x111 M\<^sub>n x112"
LeftDerivation_skip_prefix: fixes u :: "symbol list"   and v :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and w :: "symbol list" assumes "LeftDerivation (u @ v) D w"   and "derivation_ge D (length u)" shows "LeftDerivation v (derivation_shift D (length u) 0) (drop (length u) w)"
braun: shows "braun \<langle>l, x, r\<rangle>"
sub_composition_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> I"   and "b \<in> I" shows "addA a b \<in> I"
maxsimp_vertices: fixes S :: "'b set"   and w :: "'b"   and s :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "s \<in> S" shows "w +o \<langle>S - {s}\<rangle> \<in> dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w}"
summable: shows "summable (\<lambda>n. norm (fps_nth f (Suc n)) * \<delta> ^ Suc n)"
column_rank_def: fixes A :: "((real, 'a) vec, 'b) vec" shows "rank A = dim (columns A)"
IH_2: fixes x :: "nat"   and xa :: "nat"   and xaa :: "'a"   and x21 :: "nat"   and x22 :: "nat list" assumes "x = dim_row A"   and "xa = dim_col A"   and "i < x \<and> j < xa"   and "xaa = A $$ (i, j)"   and "xaa = (0::'a)"   and "concat (map (\<lambda>i'. if A $$ (i', j) \<noteq> (0::'a) then [i'] else []) [Suc i..<x]) = x21 # x22"   and "swaprows i x21 A \<in> carrier_mat nr nc"   and "swaprows i x21 B \<in> carrier_mat nr nc2" shows "gauss_jordan_main (mat\<^sub>h (swaprows i x21 A)) (mat\<^sub>h (swaprows i x21 B)) i j = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan_main (swaprows i x21 A) (swaprows i x21 B) i j)"
FactGroup_int_pow: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and k :: "int" assumes "H \<lhd> G"   and "a \<in> carrier G" shows "(H #>\<^bsub>G\<^esub> a) [^]\<^bsub>G Mod H\<^esub> k = H #>\<^bsub>G\<^esub> a [^]\<^bsub>G\<^esub> k"
set_union_code: fixes xs :: "'a list"   and ys :: "'a list" shows "set xs \<union> set ys = set (xs @ ys)"
assms_3: shows "g \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
MultiSplit: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, k, ll, t) \<in> MStep"   and "1 \<le> k" shows "\<exists>n m r l1. (M, l, s, n, l1, r) \<in> Step \<and> (M, l1, r, m, ll, t) \<in> MStep \<and> Suc m + n = k"
characteristic_weak_formula_is_characteristic: fixes \<tau> :: "'b"   and satisfies :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and transition :: "'c \<Rightarrow> ('b, 'c) residual \<Rightarrow> bool"   and Q :: "'c"   and P :: "'c" assumes "indexed_weak_nominal_ts TYPE('a) \<tau> satisfies transition" shows "nominal_ts.valid satisfies transition Q (indexed_weak_nominal_ts.characteristic_weak_formula \<tau> satisfies transition P) = weak_nominal_ts.weakly_bisimilar satisfies transition \<tau> P Q"
exhaust: fixes y :: "'a atom" obtains x11 :: "predicate" and x12 :: "'a list" where "y = predAtm x11 x12"   | x21 :: "'a" and x22 :: "'a" where "y = Eq x21 x22"
return_bind: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'm" shows "bind (return x) f = f x"
power_commutes: fixes a :: "'b \<Rightarrow> 'b"   and n :: "nat" shows "fpower a n \<circ> a = a \<circ> fpower a n"
absolutely_integrable_componentwise_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "(f absolutely_integrable_on A) = (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A)"
XI': fixes xvec :: "name list"   and \<Psi>' :: "'b"   and Ra :: "('a, 'b, 'c) psi"   and Pa :: "('a, 'b, 'c) psi"   and Qa :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "xvec \<sharp>* \<Psi>'"   and "\<And>A\<^sub>R \<Psi>\<^sub>R. \<lbrakk>extractFrame Ra = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>; A\<^sub>R \<sharp>* \<Psi>'; A\<^sub>R \<sharp>* Pa; A\<^sub>R \<sharp>* Qa; A\<^sub>R \<sharp>* C\<rbrakk> \<Longrightarrow> \<Psi>' \<otimes> \<Psi>\<^sub>R \<rhd> Pa \<sim> Qa" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>Pa \<parallel> Ra, \<lparr>\<nu>*xvec\<rparr>Qa \<parallel> Ra) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<sim> Q)}"
eval_fps_minus: fixes z :: "'a"   and f :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f" shows "eval_fps (- f) z = - eval_fps f z"
WTrt_elim_cases_8: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and T :: "ty" assumes "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sF{D} : T" obtains "P \<turnstile> C has F,Static:T in D"
ENF_notevalmodel_imp_offending_not_empty: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P"   and "\<not> sinvar G nP" shows "set_offending_flows G nP \<noteq> {}"
path_image_circlepath_nonneg: fixes r :: "real"   and z :: "complex" assumes "0 \<le> r" shows "path_image (circlepath z r) = sphere z r"
dbl_inc_simps_5: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and k :: "Num.num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit1 k)"
arr_char: fixes f :: "'a \<times> 'a" shows "Cop.VV.arr f = (arr (fst f) \<and> arr (snd f) \<and> Cop.src (fst f) = Cop.trg (snd f))"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and xa :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "cblinfun_apply (x + xa) = (\<lambda>xb. cblinfun_apply x xb + cblinfun_apply xa xb)"
prod_fact: fixes k :: "nat" shows "real (\<Prod>{1..<1 + k}) = fact k"
cases: fixes r :: "'a stateful_policy" obtains hosts :: "'a set" and flows_fix :: "('a \<times> 'a) set" and flows_state :: "('a \<times> 'a) set" where "r = \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state\<rparr>"
disjoint: shows "vars1 \<inter> (vars2 \<union> vars3) = {}"
arr_char: fixes f :: "(bool, unit) arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> UNIV \<and> Cod f \<in> UNIV \<and> Map f \<in> (if Dom f = Cod f then {()} else {}))"
older_seniors_card_bounded: fixes x :: "nat"   and n :: "nat" assumes "\<not> sink (token_run x n)"   and "x \<le> n" shows "card (older_seniors x n) < card (reach \<Sigma> \<delta> q\<^sub>0 - Collect sink)"
assms: shows "abc_steps_l (0, lm) B n = (sa, lm')"
iterate_is_sng_def: fixes it :: "(nat \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" shows "iterate_is_sng it = (iterate_size_abort it 2 = 1)"
mono_Ndet1: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "D (Q |-| S) \<subseteq> D (P |-| S)"
all_final_except_def: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and Ts :: "'c set" shows "final_thread.all_final_except final s Ts \<equiv> \<forall>t. final_thread.not_final_thread final s t \<longrightarrow> t \<in> Ts"
nat_pow_0: fixes x :: "'b" shows "x ^ 0 = (1::'b)"
superset_sound: fixes C :: "fterm literal set"   and C' :: "fterm literal set"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "C \<subseteq> C'"   and "eval\<^sub>c F G C" shows "eval\<^sub>c F G C'"
typing_alt_simps_16: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and k :: "call_kind"   and op :: "typeop"   and \<sigma> :: "'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E TypeOperationCall a k op \<sigma> : \<tau>) = (\<exists>\<tau>'. \<Gamma> \<turnstile>\<^sub>E a : \<tau>' \<and> typeop_type k op \<tau>' \<sigma> \<tau>)"
join_least: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and x :: "'b"   and z :: "'b"   and y :: "'b" assumes "class.SL_top le join Top"   and "le x z"   and "le y z" shows "le (join x y) z"
succ_is_dep: fixes m :: "('c, 'd) insert_message"   and i :: "'c" shows "S m = \<lbrakk>i\<rbrakk> \<longrightarrow> i \<in> deps (Insert m)"
simps_19: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "'a exp"   and A :: "'a set option" shows "\<D> (INIT C (Cs,b) \<leftarrow> e) A = \<D> e A"
init_def: shows "init \<equiv> on_init param \<bind> (\<lambda>e. RETURN (empty_state e))"
simple_integral_indicator_only: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets M" shows "integral\<^sup>S M (indicator A) = emeasure M A"
b: shows "0 \<le> limsup (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x))"
transcendental_def: fixes K :: "'a set"   and x :: "'a" shows "transcendental K x = inj_on (\<lambda>p. local.eval p x) (carrier (K [X]))"
Compl_disjoint: fixes A :: "'d set" shows "A \<inter> - A = {}"
pnet_sos_def: shows "pnet_sos \<equiv> \<lambda>S T. {(xb, xa, x). pnet_sosp (\<lambda>x xa xb. (x, xa, xb) \<in> S) (\<lambda>x xa xb. (x, xa, xb) \<in> T) xb xa x}"
comp_eqI': fixes f :: "'a"   and g :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "seq f g"   and "f = f'"   and "g = g'" shows "f \<cdot>\<^sub>B g = f' \<cdot> g'"
assms_3: shows "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
sv_hom_meet: fixes x :: "'a"   and y :: "'a" shows "(d x \<sqinter> d y) \<cdot> U = d x \<cdot> U \<sqinter> d y \<cdot> U"
exec_mark: shows "\<Gamma>\<turnstile> \<langle>mark_guards f (Catch c1 c2),s\<rangle> =n\<Rightarrow> t"
l1_refines_a0i: shows "refines (R01ia \<inter> reach a0i \<times> (l1_inv1 \<inter> l1_inv2)) med01ia a0i l1"
semilattice_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semilattice f \<equiv> abel_semigroup f \<and> semilattice_axioms f"
n_s_narrow_rep: fixes X :: "'a set"   and S1 :: "'a \<Rightarrow> 'av"   and S2 :: "'a \<Rightarrow> 'av"   and x :: "'a" assumes "finite X"   and "S1 = S2 on - X"   and "\<forall>x. S2 x \<le> S1 x"   and "\<forall>x. S1 x \<triangle> S2 x \<le> S1 x"   and "S1 x \<noteq> S1 x \<triangle> S2 x" shows "(\<Sum>x\<in>X. n (S1 x \<triangle> S2 x)) < (\<Sum>x\<in>X. n (S1 x))"
ltl_to_rabin_correct_exposed': shows "\<PP>\<^sub>\<infinity> \<psi> \<G> w = accept\<^sub>R'"
in_keys_lookup_in_range: fixes k :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'd" assumes "k \<in> keys f" shows "lookup f k \<in> Poly_Mapping.range f"
galois_aux4: fixes x :: "'a"   and y :: "'a" shows "(- x + y = 1) = (x \<le> y)"
mono: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "m \<le> n" shows "f n \<le> f m"