hcju/isabelleps huggingface dataset:
typeof_opubx_correct: fixes opubx :: "opubx"   and xs :: "(dynamic, nat, bool) unboxed list"   and codomain :: "Unboxed.type option" assumes "typeof_opubx opubx = (map typeof xs, codomain)" shows "\<exists>y. eval_ubx opubx xs = Some y \<and> typeof y = codomain"
RS_simps_4: shows "Maps.Dom rs.R\<^sub>0 = src (tab\<^sub>0 r)"
assms: shows "algebraic_semidom_class.coprime (h - [:c1:]) (h - [:c2:])"
l_zero: fixes x :: "'a list set" assumes "x \<in> carrier (Rupt K p)" shows "\<zero>\<^bsub>Rupt K p\<^esub> \<oplus>\<^bsub>Rupt K p\<^esub> x = x"
decT_decT: fixes n :: "nat"   and k :: "nat"   and n' :: "nat"   and T :: "POPLmarkRecord.type" shows "\<down>\<^sub>\<tau> n k (\<down>\<^sub>\<tau> n' (k + n) T) = \<down>\<^sub>\<tau> (n + n') k T"
tail_app_iso: fixes hom :: "('g, 'h, 'f, 'e) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "tail (app_iso hom G) = iso_tail hom"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "three" assumes "h2 x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if i = j then One else if i + 1 = j then Two else Zero)"
2: fixes s :: "'state"   and vl :: "'a"   and s1 :: "'state"   and vl1 :: "'value list" assumes "\<Delta> s vl s1 vl1" shows "\<Delta>' s vl s1 vl1"
evaluate_list_singleton_errD: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and err :: "v error_result" assumes "evaluate_list ck env s [e] (s', Rerr err)" shows "evaluate ck env s e (s', Rerr err)"
wellformed_induce_subgraph: fixes vs :: "'a set" shows "wf_digraph ((G \<restriction> S) \<restriction> vs)"
simps_2: fixes m :: "nat"   and ms :: "nat list" shows "normal (m # ms) = (0 < m)"
le_iff_le_implies: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<le> x \<leadsto> y)"
deg_monom_le: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "deg R (monom (UP R) a n) \<le> n"
product_concat_map: fixes xs :: "'c list"   and ys :: "'d list" shows "List.product xs ys = concat (map (\<lambda>x. map (Pair x) ys) xs)"
cobounded_outside: fixes S :: "'a set" assumes "bounded S" shows "bounded (- outside S)"
y_def: shows "y = rec_nat (0::'a) (\<lambda>n x. \<xi> (T n) x (inverse (real 3 ^ n)))"
less_Suc_unat_less_bound: fixes n :: "nat"   and x :: "'b word" assumes "n < Suc (unat x)" shows "n < 2 ^ LENGTH('b)"
\<epsilon>\<^sub>h_gt_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym" shows "0 < of_nat \<epsilon>"
set_integral_divide_zero: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and a :: "'a" shows "\<integral>t\<in>A. (f t / a)\<partial>M = set_lebesgue_integral M A f / a"
intros: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c" assumes "r a b" shows "r\<inverse>\<inverse> b a"
simps_3: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "remdups_adj (x # y # xs) = (if x = y then remdups_adj (x # xs) else x # remdups_adj (y # xs))"
scaleC_1_right: fixes x :: "complex" shows "x *\<^sub>C (1::'a) = of_complex x"
target: fixes T1 :: "'a"   and T2 :: "'a"   and TRel :: "('a \<times> 'a) set"   and Enc :: "'b \<Rightarrow> 'a" assumes "(T1, T2) \<in> TRel" shows "(TargetTerm T1, TargetTerm T2) \<in> encoding.indRelRT Enc TRel"
assms: shows "natural_isomorphism (\<cdot>) (\<cdot>) F A.map \<phi>"
a_Inj2_2: fixes b :: "Syntax.term list"   and ba :: "Syntax.term list"   and bb :: "Syntax.term" assumes "\<lless>b, eV\<ggreater> V\<rightarrow> \<lless>ba, bb\<ggreater>" shows "\<exists>e' \<pi> eP'. \<lless>[], e\<ggreater> I\<rightarrow> \<lless>[], e'\<ggreater> \<and> (\<forall>\<pi>\<^sub>P. \<lless>\<pi>\<^sub>P, eP\<ggreater> P\<rightarrow> \<lless>\<pi>\<^sub>P @ \<pi>, eP'\<ggreater>) \<and> ({$$} \<turnstile> e', eP', bb : \<tau>\<^sub>2 \<and> b = \<pi> @ ba \<or> (\<exists>s s'. closed s \<and> closed s' \<and> \<pi> = [s] \<and> b = [s'] @ ba \<and> s \<noteq> s' \<and> hash s = hash s'))"
exhaust: fixes y :: "('a, 'b) efsym" obtains x1 :: "'a" where "y = Oldf x1"   | x2 :: "'b" where "y = Wit x2"
Dg: shows "(g has_field_derivative g') (at (f x))"
basic_gt_and_gt_le_gt: shows "(\<lambda>t s. t >\<^sub>t s \<and> local.kbo_std.gt t s) \<le> local.kbo_std.gt"
dim_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "dim_vec (Matrix.vec n f) = n"
intros_1: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set" shows "qReachable cfg1 Q cfg1"
Exit_not_control_dependent: fixes n :: "'node"   and n' :: "'node" assumes "n controls n'" shows "n' \<noteq> (_Exit_)"
cong_ex_bigtheta: fixes f1 :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f2 :: "'b \<Rightarrow> 'c"   and g1 :: "'b \<Rightarrow> 'c"   and g2 :: "'b \<Rightarrow> 'c" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> O[F](g1)) = (f2 \<in> O[F](g2))"
encode_0_iff: fixes e :: "nat"   and n :: "nat" shows "(local.encode e n = 0) = (n = 0)"
map: fixes f :: "'f \<Rightarrow> 'e"   and x :: "unit \<Rightarrow> 'f" shows "map_future f (future.fork x) = future.fork (f \<circ> x)"
set_less_eq_aux_refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "ord.set_less_eq_aux less_eq A A = (A \<in> infinite_complement_partition)"
runiq_wrt_eval_rel: fixes R :: "('c \<times> 'd) set" shows "runiq R = (\<forall>x. R `` {x} \<subseteq> {R ,, x})"
simplex_sing: fixes n :: "int"   and a :: "'b" shows "(n simplex {a}) = (n = 0)"
is_empty_def: fixes A :: "'b multiset" shows "Multiset.is_empty A = (A = {#})"
induct: fixes P :: "'a array \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "'a array"   and a1 :: "nat" assumes "\<And>a n. (\<not> array_length a \<le> n \<Longrightarrow> P a (n + 1)) \<Longrightarrow> P a n" shows "P a0 a1"
H_eq: fixes P :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set" assumes "P \<subseteq> Id"   and "Q \<subseteq> Id" shows "rel_kat.H P X Q = rel_antidomain_kleene_algebra.H P X Q"
face_cycle_succ_no_arc: fixes a :: "'b" assumes "a \<notin> arcs H" shows "H.face_cycle_succ a = a"
BNTr10: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "S_inductive_set D"   and "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x" shows "Algebra2.Sup D (\<Union> (WWa D f a)) \<in> \<Union> (WWa D f a)"
stern_brocot_def: shows "stern_brocot = unfold_tree (\<lambda>(lb, ub). mediant (lb, ub)) (\<lambda>(lb, ub). (lb, mediant (lb, ub))) (\<lambda>(lb, ub). (mediant (lb, ub), ub)) ((0, 1), 1, 0)"
init_fin_\<tau>divergeD: fixes final :: "'c \<Rightarrow> bool"   and r :: "('e, 'f, 'c, 'd, 'g, 'h) semantics"   and t :: "'f"   and \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and status :: "status"   and x :: "'c"   and m :: "'d" assumes "\<tau>trsys.\<tau>diverge (multithreaded_base.init_fin final r t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move) ((status, x), m)" shows "\<tau>trsys.\<tau>diverge (r t) \<tau>move (x, m) \<and> status = Running"
lincomb_Nil: fixes rs :: "'a list"   and ms :: "'b list"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "rs = [] \<or> ms = []" shows "scalar_mult.lincomb smult rs ms = (0::'b)"
set_cases_1: fixes e :: "'d"   and a :: "('d, 'e) optionT"   and thesis :: "bool" assumes "e \<in> phantom_optionT a" shows "thesis"
LIM_def: fixes f :: "'c \<Rightarrow> 'd"   and L :: "'d"   and a :: "'c" shows "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)"
FSbImorph_def: fixes h :: "'h \<Rightarrow> 'i"   and hA :: "'j \<Rightarrow> 'k"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'h, 'j, 'm) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'l, 'i, 'k, 'n) model_scheme" shows "FSbImorph h hA MOD MOD' \<equiv> ipresIGWlsAll h hA MOD MOD' \<and> ipresIGCons h hA MOD MOD' \<and> ipresIGFreshAll h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "src g = src f"
t0_le: fixes i :: "nat" assumes "t i \<le> t1" shows "t 0 \<le> t1"
simps_2: fixes r1 :: "'b rexp"   and r2 :: "'b rexp" shows "mkeps (Times r1 r2) = Seq (mkeps r1) (mkeps r2)"
dyn_def: shows "\<forall>c x. dy c x = 0"
Trivial_Aheap_simp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "Trivial_Aheap \<Gamma> e\<cdot>a = (\<lambda>x. up\<cdot>0) f|` domA \<Gamma>"
perm_cont2cont: fixes f :: "'c \<Rightarrow> 'd"   and p :: "perm" assumes "cont f" shows "cont (\<lambda>x. p \<bullet> f x)"
g_def: shows "g = (\<lambda>y. fact (Suc k) * (x * z) ^ Suc k / (1 - (1 - x * y) * z) ^ (k + 2))"
path_take_to: fixes l1 :: "'vtx"   and l2 :: "'vtx"   and xs :: "('vtx \<times> 'lbl \<times> 'vtx) list"   and m :: "nat"   and l2' :: "'vtx" assumes "path l1 l2 xs"   and "m < length xs"   and "(case xs ! m of (s, l, t) \<Rightarrow> t) = l2'" shows "path l1 l2' (take (m + 1) xs)"
set_eq_fold: fixes x :: "'a"   and xs :: "'a list" shows "Min (set (x # xs)) = fold min xs x"
connex_axioms: shows "connex A (\<sqsubseteq>)"
tw_cmpser_is_w_cmpser: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "tw_cmpser G n f" shows "w_cmpser G n f"
REC_le_rule: fixes body :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'c"   and x' :: "'e"   and R :: "('c \<times> 'e) set"   and M :: "'e \<Rightarrow> 'd" assumes "trimono body"   and "(x, x') \<in> R"   and "\<And>f x x'. \<lbrakk>\<And>x x'. (x, x') \<in> R \<Longrightarrow> f x \<le> M x'; (x, x') \<in> R\<rbrakk> \<Longrightarrow> body f x \<le> M x'" shows "REC body x \<le> M x'"
simps_1: fixes e :: "'b"   and a :: "'c"   and ts :: "('b, 'c) SkewBinomialTree list" shows "tree_invar (Node e a 0 ts) = (ts = [])"
normalized_sort_code: fixes cs :: "(String.literal \<times> String.literal) list"   and s :: "String.literal set" shows "normalized_sort (set cs) s = exenormalized_sort cs s"
wlsFSb_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "wlsFSb MOD \<equiv> gWlsAllDisj MOD \<and> gWlsAbsIsInBar MOD \<and> gConsPresGWls MOD \<and> gSubstAllPresGWlsAll MOD \<and> gFreshCls MOD \<and> gSubstCls MOD \<and> gAbsRen MOD"
sum_over_product_right: fixes y :: "i"   and z :: "i"   and x :: "i" assumes "O y z" shows "P ((x \<oplus> y) \<otimes> (x \<oplus> z)) (x \<oplus> y \<otimes> z)"
basis_reduction_add_rows_def: fixes n :: "nat"   and upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "LLL_Impl.basis_reduction_add_rows n upw i state = (if upw then LLL_Impl.basis_reduction_add_rows_loop n state i i (small_fs_state state) else state)"
simps_41: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x31@[x32] \<noteq> Stop"
poly_mult_monom_assoc: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a"   and n :: "nat" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "a \<in> carrier R" shows "poly_mult (poly_mult (local.monom a n) p) q = poly_mult (local.monom a n) (poly_mult p q)"
metric_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real" shows "metric dist \<equiv> (\<forall>x y. (dist x y = 0) = (x = y)) \<and> (\<forall>x z y. dist x z \<le> dist y x + dist y z)"
simps_3: fixes u :: "'a local.term"   and v :: "'a local.term"   and w :: "'a local.term"   and x :: "'a local.term" shows "(u \<^bold>\<otimes> v) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (w \<^bold>\<otimes> x) = u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w \<^bold>\<otimes> v \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> x"
CollectIteratorTE: 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 CollectIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" and \<upsilon> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "to_nonunique_collection \<tau>' \<upsilon>" and "update_element_type \<upsilon> (to_single_type \<rho>) \<tau>"
assms_2: fixes l :: "'a llist"   and a :: "'a" assumes "l \<in> A\<^sup>\<omega>"   and "a \<in> A"   and "s = a ## l" shows "R"
arr_map: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah"   and S :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and \<iota> :: "('ag \<Rightarrow> 'ah) \<Rightarrow> 'ai"   and f :: "'ah" assumes "cones_functor J C D S \<iota>"   and "partial_magma.arr C f" shows "partial_magma.arr S (cones_functor.map J C D S \<iota> f)"
eq: shows "map (\<lambda>_. 0::'b) [0..<length ts] = map (\<lambda>_. 0::'b) ts"
tree_ordering_right: fixes X :: "(nat \<times> nat) \<times> nat \<times> nat"   and Y :: "(nat \<times> nat) \<times> nat \<times> nat" assumes "Det X = 1"   and "Det Y = 1"   and "0 < snd (mediant X)" shows "rat_of (mediant X) < rat_of (mediant (X \<otimes> UR \<otimes> Y))"
get_parent_def: shows "get_parent = (\<lambda>node_ptr. Heap_Error_Monad.bind (check_in_heap (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)) (\<lambda>_. Heap_Error_Monad.bind (Heap_Error_Monad.bind object_ptr_kinds_M (filter_M (\<lambda>ptr. Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. return (node_ptr \<in> set children))))) (\<lambda>parent_ptrs. if parent_ptrs = [] then return None else return (Some (hd parent_ptrs)))))"
assms_2: assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> r" shows "P"
vec_of_poly_rev_shifted_dim: fixes p :: "'b poly"   and n :: "nat"   and s :: "nat"   and j :: "nat" shows "dim_vec (Missing_Lemmas.vec_of_poly_rev_shifted p n s j) = n"
normalized_tableau_unique_eq_for_lvar: fixes t :: "(nat \<times> linear_poly) list" assumes "\<triangle> t" shows "\<forall>x\<in>lvars t. \<exists>!p. (x, p) \<in> set t"
simps_53: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "int \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "int \<Rightarrow> 'a"   and f5 :: "CTypeId \<Rightarrow> nat \<Rightarrow> 'a"   and f6 :: "Arraytype \<Rightarrow> nat \<Rightarrow> 'a"   and f7 :: "'a"   and x51 :: "CTypeId"   and x52 :: "nat" shows "(case objV x51 x52 of boolV x \<Rightarrow> f1 x | intgV x \<Rightarrow> f2 x | shortV x \<Rightarrow> f3 x | byteV x \<Rightarrow> f4 x | objV x xa \<Rightarrow> f5 x xa | arrV x xa \<Rightarrow> f6 x xa | nullV \<Rightarrow> f7) = f5 x51 x52"
simps: fixes a1 :: "'e"   and a2 :: "'f"   and r :: "('e \<times> 'g) set"   and s :: "('g \<times> 'f) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
igAbsCongUSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igAbsCongUSTR MOD \<equiv> \<forall>xs x x' y X X'. igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSubst MOD xs (igVar MOD xs y) x X = igSubst MOD xs (igVar MOD xs y) x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
measurable_on_componentwise_UNIV: fixes f :: "'a \<Rightarrow> 'b" shows "(f measurable_on UNIV) = (\<forall>i\<in>Basis. (\<lambda>x. (f x \<bullet> i) *\<^sub>R i) measurable_on UNIV)"
CSP2_Inf: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Union> {{p. P p} |P. P \<in> action.relation_of ` A} is CSP2 healthy"
n'_stl: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "distr (n' N M y) (stream_space (count_space UNIV)) stl = n N M y"
length_replicate: fixes n :: "nat"   and x :: "'f" shows "length (replicate n x) = n"
omega_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<omega> \<le> y\<^sup>\<omega>"
linear_scale: fixes f :: "'b \<Rightarrow> 'c"   and r :: "real"   and b :: "'b" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
matrix_to_iarray_transpose: fixes A :: "(('c, 'e) vec, 'd) vec" shows "matrix_to_iarray (Finite_Cartesian_Product.transpose A) = transpose_iarray (matrix_to_iarray A)"
rep_ltln\<^sub>L_def: shows "rep_ltln\<^sub>L \<equiv> quot_type.rep Rep_ltln\<^sub>L"
empty_set_less_aux: fixes A :: "'a set" shows "({} \<sqsubset>' A) = (A \<noteq> {} \<and> finite A)"
simps_1: shows "T_del_min \<langle>\<rangle> = 1"
equalityD1: fixes A :: "'d set"   and B :: "'d set" assumes "A = B" shows "A \<subseteq> B"
a0: shows "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then (- 1) ^ f (i div 2) / sqrt 2 ^ (n + 1) else (- 1) ^ (f (i div 2) + 1) / sqrt 2 ^ (n + 1))))"
coinduction_helper: fixes v0a :: "'a" assumes "v0a \<in> V"   and "\<nexists>v. greedy_conforming_path p \<sigma> \<sigma>' v0a = LCons v LNil" shows "\<exists>P'. greedy_conforming_path p \<sigma> \<sigma>' v0a = LCons v0a P' \<and> \<not> lnull P' \<and> v0a \<rightarrow> lhd P' \<and> lhd P' \<in> V \<and> (\<exists>v. P' = greedy_conforming_path p \<sigma> \<sigma>' v \<and> v \<in> V)"
subset_Un_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (A \<union> B = B)"
lnorm_triangle: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and M :: "'a measure"   and g :: "'a \<Rightarrow> real" assumes "f \<in> lspace M (ennreal p)"   and "g \<in> lspace M (ennreal p)"   and "1 \<le> p" shows "lnorm M p (\<lambda>x. f x + g x) \<le> lnorm M p f + lnorm M p g"
mk_finite_char_def: fixes C :: "'a set set" shows "mk_finite_char C = {S. \<forall>S'\<subseteq>S. finite S' \<longrightarrow> S' \<in> C}"
assms_3: shows "(i, Add i e1) \<in> set (node_deliver_messages xs)"
B: fixes a :: "'a" assumes "a \<in> A" shows "\<exists>b\<in>B. a \<le> b"
next_events_def: fixes P :: "'e process"   and xs :: "'e list" shows "next_events P xs \<equiv> {x. xs @ [x] \<in> traces P}"
card_list_def: fixes xs :: "'b list" shows "card_list xs = card (set xs)"
surj_signed_lconjaction: fixes s :: "'a" shows "surj (signed_lconjaction s)"
cases: fixes a1 :: "('fun, 'var) strand_step list set \<times> ('fun, 'var) strand_step list"   and a2 :: "('fun, 'var) strand_step list set \<times> ('fun, 'var) strand_step list" assumes "a1 \<Rightarrow>\<^sup>\<bullet> a2" obtains \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) strand_step list" where "a1 = (\<S>''__, \<A>''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>''__ [], \<A>''__)" and "[] \<in> \<S>''__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) strand_step list" where "a1 = (\<S>'''__, \<A>'''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>'''__ (send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__), \<A>'''__ @ [receive\<langle>t__\<rangle>\<^sub>s\<^sub>t])" and "send\<langle>t__\<rangle>\<^sub>s\<^sub>t # S__ \<in> \<S>'''__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) strand_step list" where "a1 = (\<S>''''__, \<A>''''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>''''__ (receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__), \<A>''''__ @ [send\<langle>ta__\<rangle>\<^sub>s\<^sub>t])" and "receive\<langle>ta__\<rangle>\<^sub>s\<^sub>t # Sa__ \<in> \<S>''''__"   | a :: "poscheckvariant" and t :: "('fun, 'var) Term.term" and t' :: "('fun, 'var) Term.term" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) strand_step list" where "a1 = (\<S>'''''__, \<A>'''''__)" and "a2 = (update\<^sub>s\<^sub>t \<S>'''''__ (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__), \<A>'''''__ @ [\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])" and "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # Sb__ \<in> \<S>'''''__"   | X :: "'var list" and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" and S :: "('fun, 'var) strand_step list" and \<S> :: "('fun, 'var) strand_step list set" and \<A> :: "('fun, 'var) strand_step list" where "a1 = (\<S>, \<A>)" and "a2 = (update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])" and "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
almost_full_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "almost_full_on P A = (\<forall>f\<in>SEQ A. good P f)"
run_exn: fixes f :: "'b Heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b" assumes "run f \<sigma> \<sigma>' r"   and "is_exn \<sigma>" obtains "\<sigma>' = \<sigma>"
assms: fixes a :: "'a"   and b :: "'a" assumes "a \<in> range f"   and "b \<in> range g" shows "m a b = m' a b"
VInterE3: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A" obtains "\<And>y. y \<in>\<^sub>\<circ> A \<Longrightarrow> a \<in>\<^sub>\<circ> y"
insert_clause'_def: shows "insert_clause' \<equiv> Abs_comp_fun_idem insert_clause"
chkchop_compat_arg: fixes s'' :: "('s, 'v) tm" assumes "hsize s'' \<le> hsize s'"   and "chkchop (>\<^sub>t) s'' s" shows "chkchop (>\<^sub>t) (App s'' t') (App s t)"
conforms_absorb: fixes a :: "abopt"   and b :: "st"   and G :: "prog"   and L :: "lenv"   and j :: "jump" assumes "(a, b)\<Colon>\<preceq>(G, L)" shows "(absorb j a, b)\<Colon>\<preceq>(G, L)"
Lxy_empty: fixes S :: "'a set" shows "Lxy [] S = []"
IH: fixes y :: "nat"   and \<M> :: "'a set set" assumes "y < n"   and "finite \<M>"   and "\<And>C. C \<in> \<M> \<Longrightarrow> polytope C"   and "\<And>C. C \<in> \<M> \<Longrightarrow> aff_dim C \<le> int y"   and "\<And>C F. \<lbrakk>C \<in> \<M>; F face_of C\<rbrakk> \<Longrightarrow> F \<in> \<M>"   and "\<And>C1 C2. \<lbrakk>C1 \<in> \<M>; C2 \<in> \<M>\<rbrakk> \<Longrightarrow> C1 \<inter> C2 face_of C1" shows "\<exists>\<T>. simplicial_complex \<T> \<and> (\<forall>K\<in>\<T>. aff_dim K \<le> int y) \<and> \<Union> \<T> = \<Union> \<M> \<and> (\<forall>C\<in>\<M>. \<exists>F. finite F \<and> F \<subseteq> \<T> \<and> C = \<Union> F) \<and> (\<forall>K\<in>\<T>. \<exists>C. C \<in> \<M> \<and> K \<subseteq> C)"
almost_full_onD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "almost_full_on P A"   and "\<And>i. f i \<in> A" obtains i :: "nat" and j :: "nat" where "i < j" and "P (f i) (f j)"
assms_2: shows "\<forall>\<^sub>F x in F. - exp (- 1) \<le> f x"
ereal_Limsup_uminus: fixes net :: "'b filter"   and f :: "'b \<Rightarrow> ereal" shows "Limsup net (\<lambda>x. - f x) = - Liminf net f"
borel_measurable_if: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) \<in> borel_measurable lebesgue) = (f \<in> borel_measurable (lebesgue_on S))"
Minr_range: fixes Rr :: "nat list \<Rightarrow> bool"   and xs :: "nat list"   and w :: "nat" shows "Minr Rr xs w \<le> w \<or> Minr Rr xs w = Suc w"
IH: shows "h_minus_aux j (source \<Delta>, y) = 0"