hcju/isabelleps huggingface dataset:
nn_integral_disjoint_pair_countspace: fixes B :: "'h set"   and C :: "'h set"   and f :: "'h \<Rightarrow> ennreal" assumes "B \<inter> C = {}" shows "set_nn_integral (count_space UNIV) (B \<union> C) f = set_nn_integral (count_space UNIV) B f + set_nn_integral (count_space UNIV) C f"
prems_2: shows "(x # xvec) \<sharp>* (y # yvec)"
Fract_same: fixes a :: "'c" shows "Fraction_Field.Fract a a = (1 when a \<noteq> (0::'c))"
distinct_zipI1: fixes xs :: "'k list"   and ys :: "'l list" assumes "distinct xs" shows "distinct (zip xs ys)"
arg_cis: fixes x :: "real" assumes "x \<in> {- pi<..pi}" shows "arg (cis x) = x"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> yahalom"   and "KK \<subseteq> - range shrK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
wr: fixes \<sigma> :: "'a"   and s :: "'b"   and \<sigma>' :: "'a"   and s' :: "'b"   and a :: "'c" assumes "P (\<sigma>, s)"   and "P (\<sigma>', s')"   and "Q ((\<sigma>, s), a, \<sigma>', s')"   and "S \<sigma> \<sigma>' a" shows "R ((\<sigma>, s), a, \<sigma>', s')"
simps: shows "(if x < l \<or> r < x then (l, r, sr) else case local.tighten_poly_bounds l r sr of (l', xa, xb) \<Rightarrow> local.tighten_poly_bounds_for_x l' xa xb) = (l', r', sr')"
obj_simps_5: fixes a :: "'a" assumes "B.obj a" shows "B.cod a = a"
pp_last_pp_fusion: fixes x :: "'a ppath"   and y :: "'a ppath" assumes "pp_last x = pp_first y" shows "pp_last (pp_fusion x y) = pp_last y"
matrix_works: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec"   and x :: "('b, 'c) vec" assumes "linear (*s) (*s) f" shows "matrix f *v x = f x"
map_def: fixes f :: "'a \<times> 'a" shows "\<pi>.map f = (if CC.arr f then CC.comp (\<pi> (CC.cod f)) (\<Delta>o\<Pi>.map f) else CC.null)"
degree_map_poly: fixes f :: "'b \<Rightarrow> 'c"   and p :: "'b poly" assumes "\<And>x. x \<noteq> (0::'b) \<Longrightarrow> f x \<noteq> (0::'c)" shows "degree (map_poly f p) = degree p"
Agent_univ: shows "UNIV = {Child0, Child1, Child2}"
absorb_iff1: fixes b :: "'a"   and a :: "'a" shows "(b \<le> a) = (a \<squnion> b = a)"
asIMOD_igWlsInp: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> (('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'b) option" assumes "sWlsNE SEM" shows "igWlsInp (asIMOD SEM) delta inp = ((\<forall>val. liftAll (\<lambda>X. sWlsVal SEM val \<or> X val = undefined) inp) \<and> (\<forall>val. sWlsVal SEM val \<longrightarrow> sWlsInp SEM delta (lift (\<lambda>X. X val) inp)))"
fch_equivalence: fixes h :: "'a" assumes "h \<sqinter> id h \<le> top \<and> h \<sqinter> top \<le> - top" shows "(top \<le> top \<sqinter> top \<and> top \<sqinter> top \<sqinter> (top \<sqinter> top) \<le> top \<sqinter> top) \<and> local.comp_inf.symmetric (top \<sqinter> top)"
prems_3: shows "ccw'.sortedP (fst x) (map snd (butlast (x # y # z # w # ws)))"
approx_floatariths_outer: fixes p :: "nat"   and ea :: "floatarith list"   and as :: "(real \<times> real pdevs) list"   and XS :: "(real \<times> real pdevs) list"   and vs :: "real list" assumes "approx_floatariths p ea as = Some XS"   and "vs \<in> Joints as" shows "interpret_floatariths ea vs @ vs \<in> Joints (XS @ as)"
find_Some: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "find P xs = Some x" shows "x \<in> set xs \<and> P x"
finite_dimensional_vector_space_pair_def: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and B1 :: "'d set"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and B2 :: "'e set" shows "finite_dimensional_vector_space_pair s1 B1 s2 B2 \<equiv> finite_dimensional_vector_space s1 B1 \<and> finite_dimensional_vector_space s2 B2"
mdc_hd_path: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "Subobjs P C Cs"   and "C \<in> set Cs"   and "wf_prog wf_md P" shows "C = hd Cs"
arr_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "arr (mkarr t)"
UP_subring_taylor: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "subring S R"   and "g \<in> carrier (UP R)"   and "\<And>n. g n \<in> S"   and "a \<in> S" shows "taylor_expansion R a g = taylor_expansion (R\<lparr>carrier := S\<rparr>) a g"
simps_5: fixes bs :: "real list"   and a :: "Ferrack.num"   and b :: "Ferrack.num" shows "Inum bs (Add a b) = Inum bs a + Inum bs b"
hyps_4: shows "TQ \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> TR"
Cinfinite_csum1: fixes r1 :: "('k \<times> 'k) set"   and r2 :: "('l \<times> 'l) set" assumes "Cinfinite r1" shows "Cinfinite (r1 +c r2)"
extremum_strict: fixes a :: "'b word" shows "\<not> - 1 < a"
enum_add: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d" assumes "index_less index key"   and "0 < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "foldl (+) 0 (enum xs index key n mi ma) = length xs"
equiv_allrel: fixes A :: "'b set" shows "equiv A (A \<times> A)"
cmp_fun_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f ` A = g ` f ` A"
function_ring_def: fixes S :: "'c set"   and R :: "('d, 'e) ring_scheme" shows "function_ring S R = \<lparr>carrier = S \<rightarrow>\<^sub>E carrier R, mult = function_mult S R, one = function_one S R, zero = function_zero S R, add = function_add S R, smult = function_scalar_mult S R\<rparr>"
f: shows "f \<in> m \<rightarrow>\<^sub>M N"
induct: fixes P :: "('f, 'b) Term.term \<Rightarrow> ('f, 'b) Term.term \<Rightarrow> bool"   and a0 :: "('f, 'b) Term.term"   and a1 :: "('f, 'b) Term.term" assumes "\<And>s t. (\<And>x21 x22 x21a x22a i. \<lbrakk>vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> weight t \<le> weight s; \<not> weight t < weight s; s = Fun x21 x22; t = Fun x21a x22a; \<not> pr_strict' (x21, length x22) (x21a, length x22a); pr_weak' (x21, length x22) (x21a, length x22a); i < length x22; i < length x22a\<rbrakk> \<Longrightarrow> P (x22 ! i) (x22a ! i)) \<Longrightarrow> P s t" shows "P a0 a1"
less_sets_alLeast: fixes A :: "'a set"   and a :: "'a" shows "(A \<lless> {a..}) = (\<forall>x\<in>A. x < a)"
mset_pos_as_neg: fixes M :: "'b zmultiset" shows "zmset_of (mset_pos M) = zmset_of (mset_neg M) + M"
map_id: fixes t :: "('d, 'e, 'f) gpv" shows "map_gpv id id t = t"
euler_product_fds_zeta: assumes "1 < s \<bullet> 1" shows "(\<lambda>n. \<Prod>p\<le>n. if prime p then inverse (1 - 1 / nat_power p s) else 1) \<longlonglongrightarrow> eval_fds fds_zeta s"
tendsto_unique': fixes "open" :: "'b set \<Rightarrow> bool"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" assumes "class.t2_space open"   and "F \<noteq> bot" shows "\<exists>\<^sub>\<le>\<^sub>1 l. filterlim f (topological_space.nhds open l) F"
sdiv_minus_dividend: fixes a :: "int"   and b :: "int" shows "- a sdiv b = - (a sdiv b)"
edge_det: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and a :: "'g"   and a' :: "'g" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a = targetnode a'" shows "a = a'"
rsub_3: shows "r ` {x. x (Suc n) = 0} \<subseteq> {x. x (Suc n) = 0}"
prems: shows "\<forall>a\<in>set (z # les). case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
TER_plus_web': fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current \<Gamma> g"   and "wave \<Gamma> g" shows "\<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f \<union> TER\<^bsub>\<Gamma>\<^esub> g) \<subseteq> TER\<^bsub>\<Gamma>\<^esub> (f \<frown>\<^bsub>\<Gamma>\<^esub> g)"
LtK_synth: fixes K :: "ltkey"   and H :: "msg set" assumes "LtK K \<in> synth H" obtains "LtK K \<in> H"
abs_summable_on_norm_iff: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "((\<lambda>x. norm (f x)) abs_summable_on A) = (f abs_summable_on A)"
isVar_castVar: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isVar s X" shows "asSort (fst (castVar s X)) = s \<and> Var (fst (castVar s X)) (snd (castVar s X)) = X"
root_free_def: fixes p :: "'a poly" shows "root_free p = (degree p = 1 \<or> (\<forall>x. poly p x \<noteq> (0::'a)))"
Plus_def: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme" shows "Plus fsm1 fsm2 = \<lparr>states = states fsm1 + states fsm2, init = init fsm1 + init fsm2, final = final fsm1 + final fsm2, next = \<lambda>st x st'. (\<exists>st1 st1'. st = HF.Inl st1 \<and> st' = HF.Inl st1' \<and> next fsm1 st1 x st1') \<or> (\<exists>st2 st2'. st = HF.Inr st2 \<and> st' = HF.Inr st2' \<and> next fsm2 st2 x st2')\<rparr>"
induct: fixes P :: "(complex \<times> nat) list \<Rightarrow> bool"   and a0 :: "(complex \<times> nat) list" assumes "P []"   and "\<And>x i xs. \<lbrakk>x \<in> \<real> \<Longrightarrow> P xs; \<And>xa xb xc. \<lbrakk>x \<notin> \<real>; xa = cnj x; xb = delete_cnj xa (Suc i) xs; xc = map_poly Re ([:- x, 1:] * [:- xa, 1:])\<rbrakk> \<Longrightarrow> P xb\<rbrakk> \<Longrightarrow> P ((x, i) # xs)" shows "P a0"
MethodDefs_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list" shows "MethodDefs P C M \<equiv> {(Cs, mthd). Subobjs P C Cs \<and> (\<exists>Bs fs ms. class P (last Cs) = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>mthd\<rfloor>)}"
dioph_dfs_def: fixes n :: "nat"   and ks :: "int list"   and l :: "int" shows "dioph_dfs n ks l = dioph_dfs.dfs n ks (dioph_empt ks l) [l]"
assoc'_naturality: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>\<^sup>-\<^sup>1[cod \<mu>, cod \<nu>, cod \<tau>] \<cdot> (\<mu> \<star> \<nu> \<star> \<tau>) = ((\<mu> \<star> \<nu>) \<star> \<tau>) \<cdot> \<a>\<^sup>-\<^sup>1[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
albega: shows "\<bar>k\<bar> = \<alpha>\<^sup>2 \<and> \<bar>l\<bar> = \<beta>\<^sup>2 \<and> \<bar>k\<^sup>2 + l\<^sup>2\<bar> = \<gamma>\<^sup>2"
in_the_default_empty_conv: fixes x :: "'a"   and m :: "'a list option" shows "(x \<in> set (the_default [] m)) = (\<exists>xs. m = Some xs \<and> x \<in> set xs)"
interval_not_empty: fixes a :: "'a"   and b :: "'a" assumes "interval r a b \<noteq> {}" shows "(a, b) \<in> r"
cfov: shows "f O ov \<subseteq> ov \<union> s \<union> d"
R1_imp_total1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<R>" shows "total1 f"
distinct_vertices: fixes a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex" assumes "distinct [a, b, c, d]" shows "\<forall>e. e \<in> {a, b, c, d}"
inner_vec_def: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec" shows "x \<bullet> y = (\<Sum>i\<in>UNIV. x $ i \<bullet> y $ i)"
set_disconnected_nodes_get_child_nodes_is_l_set_disconnected_nodes_get_child_nodes: shows "l_set_disconnected_nodes_get_child_nodes set_disconnected_nodes_locs get_child_nodes_locs"
map_of_update_with_aux: fixes v :: "'e"   and k :: "'d"   and f :: "'e \<Rightarrow> 'e"   and ps :: "('d \<times> 'e) list" shows "map_of (AList.update_with_aux v k f ps) = map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)"
coboundedI1: fixes a :: "'e"   and c :: "'e"   and b :: "'e" assumes "a \<le> c" shows "inf a b \<le> c"
not_greater_Greatest: fixes P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "(GREATEST x. P x) < k" shows "\<not> P k"
dn: shows "d = b - n * x \<and> 2 * \<bar>d\<bar> < x"
unitary_mat_mat: fixes M :: "complex mat" assumes "unitary M" shows "inverts_mat M\<^sup>\<dagger> M"
interchange_rows_j: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'f" shows "interchange_rows A i j $ j = A $ i"
config: shows "BIT_inv (Partial_Cost_Model.config'_rand BIT s qs) (last qs) [x0, y0]"
substitute''_Cons: fixes zs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and x :: "'b"   and xs' :: "'b list"   and xs :: "'b list"   and T :: "'b set"   and ys :: "'b list" assumes "zs \<in> paths (f x)"   and "xs' \<in> xs \<otimes> zs"   and "substitute'' (f_nxt f T x) T xs' ys" shows "substitute'' f T (x # xs) (x # ys)"
val_mkIF: fixes x :: "'d"   and t1 :: "'d ifex"   and t2 :: "'d ifex"   and s :: "'d \<Rightarrow> bool" shows "val_ifex (mkIF x t1 t2) s = val_ifex (IF x t1 t2) s"
stateful_strand_step_subst_inv_cases_4: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and S :: "('a, 'b) stateful_strand_step list"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "insert\<langle>t,s\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>t' s'. t = t' \<cdot> \<sigma> \<and> s = s' \<cdot> \<sigma> \<and> insert\<langle>t',s'\<rangle> \<in> set S"
D_invar'_def: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a"   and Q :: "'a \<Rightarrow> nat option"   and V :: "'a \<Rightarrow> nat option" shows "Dijkstra.D_invar' w s Q V \<equiv> \<exists>D S. D.coupling Q V D S \<and> Dijkstra_Invar w s D S"
is_option_of_imp_games: fixes opt :: "ZF"   and g :: "ZF" assumes "(opt, g) \<in> is_option_of" shows "opt \<in> games_lfp \<and> g \<in> games_lfp"
assms_3: fixes xs :: "'a llist"   and y :: "'a" assumes "\<not> lnull xs"   and "y \<in> lset (ltl xs)"   and "P y (ltl xs)" shows "P y xs"
cEnum_prod_def: shows "CENUM('a \<times> 'b) = (case ID CENUM('a) of None \<Rightarrow> None | Some (enum_a, enum_all_a, enum_ex_a) \<Rightarrow> case ID CENUM('b) of None \<Rightarrow> None | Some (enum_b, enum_all_b, enum_ex_b) \<Rightarrow> Some (List.product enum_a enum_b, \<lambda>P. enum_all_a (\<lambda>x. enum_all_b (\<lambda>y. P (x, y))), \<lambda>P. enum_ex_a (\<lambda>x. enum_ex_b (\<lambda>y. P (x, y)))))"
even_Suc_div_two: fixes n :: "nat" assumes "even n" shows "Suc n div 2 = n div 2"
discI_2: fixes sum :: "'a + 'b"   and x2 :: "'b" assumes "sum = Inr x2" shows "\<not> isl sum"
assms: shows "h \<turnstile> get_ancestors ptr \<rightarrow>\<^sub>r ancestors"
Minf_le: fixes y :: "'a extended" shows "-\<infinity> \<le> y"
abs_of_pos: fixes abs :: "'a \<Rightarrow> '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 a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less zero a" shows "abs a = a"
norm_triangle_half_l: fixes x :: "'b"   and y :: "'b"   and e :: "real"   and x' :: "'b" assumes "norm (x - y) < e / 2"   and "norm (x' - y) < e / 2" shows "norm (x - x') < e"
subgroup_generated_group_carrier: shows "subgroup_generated B (carrier B) = B"
not_None_eq: fixes x :: "'m option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
cipher: fixes \<eta> :: "nat" shows "\<I>_real \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res CIPHER.res \<eta> \<surd>"
event_can_go_back_if_no_sender: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "distributed_system.can_occur channel trans send recv ev' c'"   and "\<not> isRecvMarker ev'"   and "\<not> isSend ev" shows "distributed_system.can_occur channel trans send recv ev' c"
subspaceI: fixes S :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "0 \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x + y \<in> S"   and "\<And>c x. x \<in> S \<Longrightarrow> c \<cdot> x \<in> S" shows "phull.subspace S"
reachable_arc_trans: fixes u :: "'a"   and v :: "'a"   and e :: "'b"   and w :: "'a" assumes "u \<rightarrow>\<^sup>* v"   and "arc e (v, w)" shows "u \<rightarrow>\<^sup>* w"
weight_scale_spmf': fixes r :: "real"   and p :: "'c spmf" assumes "0 \<le> r"   and "r \<le> 1" shows "weight_spmf (scale_spmf r p) = r * weight_spmf p"
ground_resolution_with_selection_axioms: shows "ground_resolution_with_selection (\<lambda>_. {#})"
SpecAnno: fixes P :: "'d set"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and Q :: "'d set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com" assumes "P \<subseteq> {s. \<exists>Z. s \<in> P' Z \<and> Q' Z \<subseteq> Q \<and> A' Z \<subseteq> A}"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c Z (Q' Z),(A' Z)"   and "\<forall>Z. c Z = c undefined" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P specAnno P' c Q' A' Q,A"
simps_1: fixes t :: "'d" shows "dbm_entry_bound (Le t) = t"
div_free_CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P: shows "D (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) = {}"
case_cong: fixes xstate :: "('d, 'e) xstate"   and xstate' :: "('d, 'e) xstate"   and f1 :: "'d \<Rightarrow> 'f"   and g1 :: "'d \<Rightarrow> 'f"   and f2 :: "'d \<Rightarrow> 'f"   and g2 :: "'d \<Rightarrow> 'f"   and f3 :: "'e \<Rightarrow> 'f"   and g3 :: "'e \<Rightarrow> 'f"   and f4 :: "'f"   and g4 :: "'f" assumes "xstate = xstate'"   and "\<And>x1. xstate' = Normal x1 \<Longrightarrow> f1 x1 = g1 x1"   and "\<And>x2. xstate' = Abrupt x2 \<Longrightarrow> f2 x2 = g2 x2"   and "\<And>x3. xstate' = Fault x3 \<Longrightarrow> f3 x3 = g3 x3"   and "xstate' = Stuck \<Longrightarrow> f4 = g4" shows "(case xstate of Normal x1 \<Rightarrow> f1 x1 | Abrupt x2 \<Rightarrow> f2 x2 | Fault x3 \<Rightarrow> f3 x3 | Stuck \<Rightarrow> f4) = (case xstate' of Normal x1 \<Rightarrow> g1 x1 | Abrupt x2 \<Rightarrow> g2 x2 | Fault x3 \<Rightarrow> g3 x3 | Stuck \<Rightarrow> g4)"
poly_deg_zero: shows "poly_deg 0 = (0::'a)"
preserves_cod: fixes f :: "'c \<times> 'c" assumes "C.CC.arr f" shows "C.CC.cod (C.CC.map f) = C.CC.map (C.CC.cod f)"
is_set_to_list_def: shows "is_set_to_list \<equiv> is_set_to_sorted_list (\<lambda>_ _. True)"
assms_2: shows "\<forall>n\<ge>m. f n \<in> S"
summable_minus_ereal: fixes f :: "nat \<Rightarrow> ereal" assumes "\<not> {- \<infinity>, \<infinity>} \<subseteq> range f"   and "summable f" shows "summable (\<lambda>n. - f n)"
partCompat_def: fixes P :: "'c set set"   and theta :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "partCompat P theta f \<equiv> \<forall>I\<in>P. Resumption_Based.compat I theta f"
that: fixes B :: "real" assumes "1 \<le> B"   and "\<forall>\<^sub>F e in at_right 0. \<forall>x\<in>cball t0 e \<times> cball x0 e. norm (f x) \<le> B" shows "thesis"
elims_3: fixes x :: "'a"   and xa :: "'a list" assumes "\<not> exhaustive_above x xa" obtains xb :: "'a" where "x = xb__" and "xa = []" and "\<not> \<not> proper_interval (Some xb__) None"   | xb :: "'a" and y :: "'a" and ys :: "'a list" where "x = xb" and "xa = y # ys" and "\<not> (\<not> proper_interval (Some xb) (Some y) \<and> exhaustive_above y ys)"
map_default_eq_id: fixes d :: "'e"   and f :: "'d \<Rightarrow> 'e" shows "map_default d ((\<lambda>a. Some (f a)) |` {a. f a \<noteq> d}) = f"
Z_alt_def: fixes n :: "nat"   and m :: "nat" shows "Z n m = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i} \<times> {\<pi>. \<pi> permutes {0..<n}}"
step_sound: fixes R :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<times> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and S :: "('sf, 'sc, 'sz) step" assumes "step_ok R i S"   and "0 \<le> i"   and "i < length (fst R)"   and "sound R" shows "sound (step_result R (i, S))"
EphK_Sign: fixes X :: "ephkey"   and X' :: "msg"   and Y' :: "msg" shows "EphK X \<noteq> Sign X' Y'"
safe_def: fixes c :: "('loc, 't) Propagate.configuration" shows "safe c \<equiv> \<forall>loc1 loc2 t s. 0 < zcount (c_pts c loc1) t \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<longrightarrow> (\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c loc2))"
E4: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> \<diamond>F"
hcomp_in_hhom: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d"   and \<nu> :: "'d"   and c :: "'d" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>"
lists_def: shows "lists \<equiv> \<lambda>A. {x. listsp (\<lambda>x. x \<in> A) x}"
finite_induct_select: fixes S :: "'c set"   and P :: "'c set \<Rightarrow> bool" assumes "finite S"   and "P {}"   and "\<And>T. \<lbrakk>T \<subset> S; P T\<rbrakk> \<Longrightarrow> \<exists>s\<in>S - T. P (insert s T)" shows "P S"
calculus_axioms: shows "calculus Bot Inf (\<Turnstile>) Red_I Red_F"
is_map_from_is_homomorphism: fixes U :: "'c set"   and V :: "'c set" assumes "is_open\<^sub>Y U"   and "is_open\<^sub>Y V"   and "V \<subseteq> U" shows "Set_Theory.map (\<rho>\<^sub>Y U V) (\<O>\<^sub>Y U) (\<O>\<^sub>Y V)"
permute_multiset_1: fixes p :: "perm" shows "p \<bullet> {#} = {#}"
higher_differentiable_on_le: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" assumes "higher_differentiable_on S f m"   and "n \<le> m" shows "higher_differentiable_on S f n"
assms_4: shows "x \<le> - forest_components h \<sqinter> - - g"
elims_1: fixes x :: "('a, 'b) message"   and y :: "bool" assumes "is_delete x = y" obtains m :: "('a, 'b) insert_message" where "x = Insert m__" and "\<not> y"   | m :: "'a delete_message" where "x = Delete m" and "y"
psubstEnv_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodEnv rho"   and "goodEnv rho'" shows "goodEnv rho &[rho']"
in_inv_image: fixes x :: "'a"   and y :: "'a"   and r :: "('c \<times> 'c) set"   and f :: "'a \<Rightarrow> 'c" shows "((x, y) \<in> inv_image r f) = ((f x, f y) \<in> r)"