hcju/isabelleps huggingface dataset:
base: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<I> :: "('e, 'f) \<I>" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out" shows "c input \<in> sub_gpvs \<I> gpv"
know_composition_neg1: fixes P :: "specID"   and m :: "KS"   and Q :: "specID"   and PQ :: "specID" assumes "\<not> know P m"   and "\<not> know Q m"   and "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "correctCompositionIn PQ" shows "\<not> know PQ m"
inprod_cvec_bilinear1: fixes z' :: "complex \<times> complex"   and k :: "complex"   and z :: "complex \<times> complex"   and w :: "complex \<times> complex" assumes "z' = k *\<^sub>s\<^sub>v z" shows "\<langle>z',w\<rangle> = cnj k * \<langle>z,w\<rangle>"
step2: shows "mat (row_length M1 * row_length M2) (length M1 * length M2) (M1 \<otimes> M2)"
graph_union_iff: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" shows "(graph_union G\<^sub>1 G\<^sub>2 = G\<^sub>2) = (edges G\<^sub>1 \<subseteq> edges G\<^sub>2 \<and> vertices G\<^sub>1 \<subseteq> vertices G\<^sub>2)"
nom_at_in_bridge: fixes a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and i :: "'c"   and xs :: "(nat \<times> nat) set"   and k :: "'c"   and j :: "'c" assumes "Nom i at a in' set branch" shows "Nom i at a in' set (mapi_branch (bridge k j xs) branch)"
mod_pure: fixes b :: "bool"   and h :: "heap \<times> nat set" shows "(h \<Turnstile> \<up> b) = (snd h = {} \<and> b)"
zero_cap_simp: fixes u :: "nat"   and v :: "nat"   and c :: "nat \<times> nat \<Rightarrow> 'a" assumes "(u, v) \<notin> Graph.E c" shows "c (u, v) = (0::'a)"
inj_on_imp_bij_betw: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "bij_betw f A (f ` A)"
redp_red_eq: fixes P :: "(char list list \<times> char list exp) prog" shows "redp P = red' P"
isShortestPath_alt: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "isShortestPath u p v = (isSimplePath u p v \<and> length p = min_dist u v)"
quotient_imp_continuous_map: fixes X :: "'b topology"   and X' :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "quotient_map X X' f" shows "continuous_map X X' f"
simps_2: fixes r :: "'a rexp" shows "seq [r] = r"
Acc_property: shows "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w = accepting_pair\<^sub>R \<MM>.\<delta>\<^sub>\<R> \<MM>.q\<^sub>\<R> (\<MM>.fail\<^sub>R \<union> \<MM>.merge\<^sub>R (the (\<pi> (G \<psi>))), \<MM>.succeed\<^sub>R (the (\<pi> (G \<psi>)))) w"
lifelock_free_def: fixes P :: "'d process" shows "lifelock_free P \<equiv> CHAOS UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
C: shows "C \<in> carrier_mat (dim_row A) (dim_col A)"
gen_ramanujan_sum_distrib_right: fixes a :: "nat"   and b :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> complex"   and g :: "nat \<Rightarrow> complex" assumes "0 < a"   and "0 < b"   and "0 < m"   and "coprime b m"   and "multiplicative_function f"   and "multiplicative_function g" shows "s f g m (a * b) = s f g m a"
multi_self_add_other_not_self: fixes M :: "'b multiset"   and x :: "'b" shows "(M = x, M) = False"
intro: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>j. \<not> partial_magma.arr J j" shows "empty_diagram_axioms J"
delete_conv': fixes k :: "'c"   and al :: "('c \<times> 'd) list" shows "map_of (AList.delete k al) = (map_of al)(k := None)"
intro: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'e spmf"   and S2 :: "'b \<Rightarrow> 'f \<Rightarrow> 'e spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf" assumes "\<And>m1 m2. lossless_spmf (R1 m1 m2)"   and "\<And>m1 out1. lossless_spmf (S1 m1 out1)"   and "\<And>m1 m2. lossless_spmf (R2 m1 m2)"   and "\<And>m2 out2. lossless_spmf (S2 m2 out2)"   and "\<And>m1 m2. lossless_spmf (funct m1 m2)" shows "sim_det_def R1 S1 R2 S2 funct"
assoc'_simps_3: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "trg\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h] = trg\<^sub>D f"
split_vals_e_const_list: fixes vs :: "v list" shows "split_vals_e ($$* vs) = (vs, [])"
prfx: fixes p' :: "'a list" assumes "prefix (p @ [m]) p'" shows "augment_list (joint_strategy \<sigma>\<^sub>e \<sigma>) p' = augment_list (joint_strategy (\<sigma>\<^sub>e(p := m)) \<sigma>) p'"
pasz: shows "path_image \<gamma> \<subseteq> U - {z}"
that: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "punit.is_Groebner_basis G"   and "ideal G = ideal F"   and "G \<subseteq> P[X]"   and "\<And>g. g \<in> G \<Longrightarrow> \<exists>q. g = (\<Sum>f\<in>F. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b)" shows "thesis"
compl_fam_def: fixes S :: "'e \<Rightarrow> 'a set"   and I :: "'e set" shows "compl_fam S I = (\<forall>i\<in>I. complementary (S i) (IDirProds G S (I - {i})))"
extender_fin_sat: fixes S :: "'b formula set"   and n :: "nat" assumes "fin_sat S" shows "fin_sat (extender n S)"
embed_dir_inj: fixes x :: "'b \<times> sort_dir"   and y :: "'b \<times> sort_dir" shows "(embed_dir x = embed_dir y) = (x = y)"
snoc0: fixes ys :: "'a list" shows "match (name $$ xs) (name $$ ys) = matchs xs ys"
Process_eq_spec_optimized: fixes P :: "'c process"   and Q :: "'c process" shows "(P = Q) = (D P = D Q \<and> (D P = D Q \<longrightarrow> F P = F Q))"
FirstOccurrenceExists: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and start :: "'a \<Rightarrow> 'b"   and exec :: "('a, 'c, 'b) configuration list"   and trace :: "('a, 'c) message list"   and msg :: "('a, 'c) message" assumes "execution trans sends start exec trace"   and "enabled (last exec) msg"   and "\<exists>p. isReceiverOf p msg" shows "\<exists>n. execution.firstOccurrence exec trace msg n"
nn_integral_density: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and g :: "'d \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (density M f) g = \<integral>\<^sup>+ x. f x * g x \<partial>M"
inj_sur_semi_swap_short: fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_sur z"   and "is_inj x"   and "z \<le> y\<^sup>T ; x" shows "x \<le> y ; z"
Necessary_expl: shows "\<lfloor>\<lambda>w. \<forall>x. Necessary x w = (\<^bold>\<box>Abstract x \<^bold>\<or> Godlike x) w\<rfloor>"
midpoint_in_closed_segment: fixes a :: "'a"   and b :: "'a" shows "midpoint a b \<in> closed_segment a b"
sumlist_carrier: fixes xs :: "'a list" assumes "set xs \<subseteq> carrier R" shows "R.sumlist xs \<in> carrier R"
nc_ccomp_closed: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<parallel> (y \<sqinter> nc) \<le> nc"
pure_conj: fixes P :: "bool"   and Q :: "bool" shows "\<up> (P \<and> Q) = \<up> P * \<up> Q"
j_less: shows "mod_type_class.from_nat j < mod_type_class.from_nat (j + 1)"
below_antisym: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.po below"   and "below x y"   and "below y x" shows "x = y"
elims: fixes x :: "exetheory"   and y :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "translate_theory x = y" obtains \<Sigma> :: "exesignature" and ax :: "Core.term list" where "x = ExeTheory \<Sigma> ax" and "y = (if exe_sig_conds \<Sigma> then (translate_signature \<Sigma>, set ax) else ((Map.empty, Map.empty, {}, [STR ''A'' \<mapsto> [STR ''A'' \<mapsto> [insert STR ''A'' full_sort]]]), {}))"
that: fixes f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec" assumes "orthogonal_transformation f"   and "2 \<le> CARD('n) \<Longrightarrow> det (matrix f) = 1"   and "f (norm a *\<^sub>R axis k 1) = a" shows "thesis"
p'_zero: shows "ego2.p' 0 = v\<^sub>e"
cr_complex_homo_def: shows "cr_complex_homo \<equiv> \<lambda>x. (=) (abs_complex_homo x)"
ex_un_null: shows "\<exists>!n. \<forall>f. Right.comp n f = n \<and> Right.comp f n = n"
zero_eq_add_iff_both_eq_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(zero = plus x y) = (x = zero \<and> y = zero)"
emeasure_cball: shows "emeasure lborel (cball c r) = ennreal (unit_ball_vol (real DIM('a)) * r ^ DIM('a))"
int: assumes "(\<lambda>x. \<Sum>i. f i x) \<in> borel_measurable M"   and "\<And>i. (\<lambda>x. \<Sum>i<i. f i x) \<in> borel_measurable M" shows "(\<lambda>i. LINT x|M. (\<Sum>i<i. f i x)) \<longlonglongrightarrow> LINT x|M. (\<Sum>i. f i x)"
rep_eq: fixes x :: "'vtx"   and xa :: "'vtx" shows "set_antichain (path_weight x xa) = minimal_antichain {xb. path_weightp x xa xb}"
simps_1: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "diagonal_to_Smith_aux_PQ [] bezout (P, A, Q) = (P, A, Q)"
fact1: shows "spmf (connect \<A> (RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv_f) (Void, None, Map.empty))) True = spmf (connect \<A> (RES (lazy_channel_insec' \<oplus>\<^sub>O lazy_channel_send' \<oplus>\<^sub>O lazy_channel_recv_f') (False, Void, None, Map.empty))) True"
is_proper_prefix_def: fixes u :: "'a list"   and v :: "'a list" shows "is_proper_prefix u v = (\<exists>w. w \<noteq> [] \<and> u @ w = v)"
comb: fixes args :: "term list"   and name :: "name" assumes "linears args"   and "t = name $$ args" shows "thesis"
map_sel_2: fixes a :: "('g, 'h) tllist"   and f1 :: "'g \<Rightarrow> 'i"   and f2 :: "'h \<Rightarrow> 'j" assumes "\<not> is_TNil a" shows "thd (tmap f1 f2 a) = f1 (thd a)"
L_transform_formula_Act: fixes f :: "'d"   and \<alpha> :: "'c"   and x :: "('a, 'b, 'c, 'd) FL_Formula.formula" shows "L_transform_formula (FL_Formula.Act f \<alpha> x) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Act\<^sub>\<alpha> (L_action.Act \<alpha>) (L_transform_formula x))"
primeideal_iff_prime: fixes p :: "'a list" assumes "p \<in> carrier (K [X]) - {\<zero>\<^bsub>K [X]\<^esub>}" shows "primeideal (PIdl\<^bsub>K [X]\<^esub> p) (K [X]) = pprime K p"
mono_Mprefix0: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq> Q x" shows "Mprefix A P \<sqsubseteq> Mprefix A Q"
I_def: shows "I = Sigma {..n} (atLeastAtMost 1)"
n_val_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "n_val K v = (\<lambda>x\<in>carrier K. THE l. l * Lv K v = v x)"
mod_hom_axioms_def: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and N :: "('e, 'i, 'j) module_scheme"   and f :: "'g \<Rightarrow> 'i" shows "mod_hom_axioms R M N f \<equiv> f \<in> module_hom R M N"
inject: fixes x2 :: "loc"   and y2 :: "loc" shows "(Addr x2 = Addr y2) = (x2 = y2)"
extra_wt_App: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "extra_wt \<rho> (App s t) = extra_wt \<rho> s + extra_wt \<rho> t"
reflexive_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "reflexive A less_eq \<equiv> \<forall>x. x \<in> A \<longrightarrow> less_eq x x"
cases: fixes a1 :: "nat \<times> nat \<times> thread"   and a2 :: "nat \<times> nat \<times> thread" assumes "astep_sng a1 a2" obtains c :: "nat" and n :: "nat" where "a1 = (c__, n__, INIT)" and "a2 = (c__, n__ + 1, WAIT n__)"   | c :: "nat" and k :: "nat" and n :: "nat" where "a1 = (ca__, na__, WAIT k__)" and "a2 = (ca__, na__, WAIT k__)" and "ca__ \<noteq> k__"   | c :: "nat" and n :: "nat" where "a1 = (cb__, nb__, WAIT cb__)" and "a2 = (cb__, nb__, HOLD cb__)"   | c :: "nat" and n :: "nat" and k :: "nat" where "a1 = (cc__, nc__, HOLD ka__)" and "a2 = (cc__, nc__, REL ka__)"   | c :: "nat" and n :: "nat" and k :: "nat" where "a1 = (c, n, REL k)" and "a2 = (k + 1, n, INIT)"
subcls1p_subcls1_eq: fixes P :: "'a prog" shows "subcls1p P = subcls1' P"
dist_ExcessTab: fixes g :: "graph" shows "distinct (map fst (ExcessTable g (vertices g)))"
Bessel_inequality: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and I :: "nat set" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S"   and "finite I" shows "(\<Sum>i\<in>I. (orthonormal_coeff S w f i)\<^sup>2) \<le> (l2norm S f)\<^sup>2"
add_del_arc_collapse: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" shows "pre_digraph.add_arc (pre_digraph.del_arc G a) a = pre_digraph.add_arc G a"
select_convs_1: fixes bar1 :: "'a"   and bar2 :: "'b"   and bar3 :: "'c"   and bar21 :: "'b \<times> 'a"   and bar32 :: "'c \<times> 'b"   and bar31 :: "'c \<times> 'a"   and more :: "'d" shows "bar1 \<lparr>bar1 = bar1, bar2 = bar2, bar3 = bar3, bar21 = bar21, bar32 = bar32, bar31 = bar31, \<dots> = more\<rparr> = bar1"
nuc_Sup_closed_var: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "nucleus f" shows "f (\<Squnion> (f ` X)) = f (\<Squnion> X)"
square_integrable_add: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "(\<lambda>x. f x + g x) square_integrable S"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "bij_betw \<alpha> S T" shows "bijective \<alpha> S T"
invar_init: fixes src :: "nat"   and dst :: "nat" assumes "src \<noteq> dst"   and "src \<in> V"   and "finite V" shows "nf_invar c src dst [src \<mapsto> src] {src} {} 0"
domain_eq: shows "Domainp (pcr_segment (=)) = (\<lambda>x. fst x \<le> snd x)"
comp_def: fixes g :: "'a"   and f :: "'a" shows "R.comp g f = (if right f f \<and> right f g \<and> cod f = local.dom g then g \<cdot> f else null)"
sum_length_filter_compl: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (filter P xs) + length (filter (\<lambda>x. \<not> P x) xs) = length xs"
SimplicialComplex_\<Sigma>: shows "SimplicialComplex \<Sigma>"
eq_case: shows "{Var y EQ k, Var z IN Var y, OrdP k} \<turnstile> Var z SUBS Var y"
closure_eq: fixes S :: "'d set" shows "(closure S = S) = closed S"
rtranclp_converseI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "r\<^sup>*\<^sup>* y x" shows "r\<inverse>\<inverse>\<^sup>*\<^sup>* x y"
pr_confirm_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Crypt (sesK K) (Number 0) \<in> parts (A \<union> spies evs)"   and "Key (sesK K) \<notin> A" shows "\<exists>n run X. Says n run 5 X (Card n) (Crypt (sesK K) (Passwd n)) \<in> set evs \<and> Says n run 5 (Card n) X (Crypt (sesK K) (Number 0)) \<in> set evs"
af_F_prefix_prop: fixes \<phi> :: "'b ltln"   and w :: "'b set list"   and w' :: "'b set list" shows "af (F\<^sub>n \<phi>) w \<longrightarrow>\<^sub>P af (F\<^sub>n \<phi>) (w' @ w)"
3: shows "Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y"
split_int_less_real: fixes a :: "int"   and b :: "real" shows "(real_of_int a < b) = (a < \<lfloor>b\<rfloor> \<or> a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b)"
set_entails_clause_def: fixes S :: "'b literal set set"   and C :: "'b literal set" shows "set_entails_clause S C = (\<forall>I. fo_interpretation I \<longrightarrow> validate_clause_set I S \<longrightarrow> validate_clause I C)"
subpath_def: fixes g :: "('a, 'b) rgraph_scheme"   and v1 :: "'a"   and es :: "'a edge list"   and v2 :: "'a" shows "subpath g v1 es v2 \<equiv> ces v1 es v2 \<and> v1 \<in> vertices g \<and> set es \<subseteq> edges g"
simps: fixes db :: "bool discr"   and v\<^sub>1 :: "CValue"   and v\<^sub>2 :: "CValue" shows "CB_project\<cdot>(CB\<cdot>db)\<cdot>v\<^sub>1\<cdot>v\<^sub>2 = (if undiscr db then v\<^sub>1 else v\<^sub>2)"
IH: assumes "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (G \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<delta>) \<subseteq> subst_domain \<sigma>" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G \<subseteq> subst_domain \<sigma> \<union> subst_domain \<delta>"
trace_rest_eq_complete: fixes rest1 :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) rest_scheme"   and rest2 :: "('h, 'b, 'c, 'd, 'e, 'f, 'i) rest_scheme"   and IA :: "'c set"   and IU :: "'d set"   and p :: "'a spmf"   and q :: "'h spmf" assumes "trace_rest_eq rest1 rest2 IA IU p q" obtains S :: "'a spmf \<Rightarrow> 'h spmf \<Rightarrow> bool" where "S p q" and "\<And>p q ia. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (rfunc_adv rest1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_adv rest2 s2 ia))" and "\<And>p q ia oa. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_adv rest1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_adv rest2 s2 ia)) oa)" and "\<And>p q iu. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (rfunc_usr rest1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (rfunc_usr rest2 s2 iu))" and "\<And>p q iu ou. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. rfunc_usr rest1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. rfunc_usr rest2 s2 iu)) ou)"
span_breakdown_eq: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and a :: "'b"   and S :: "'b set" assumes "module scale" shows "(x \<in> module.span scale (insert a S)) = (\<exists>k. x - scale k a \<in> module.span scale S)"
ordLess_or_ordIso: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Well_order r"   and "Well_order r'" shows "r <o r' \<or> r' <o r \<or> r =o r'"
mset_le_irrefl: fixes M :: "'b multiset"   and R :: "bool" assumes "M < M" shows "R"
Q_def: shows "Q = (\<Prod>i<n. [:- x i, 1:])"
epred_conv_minus: fixes n :: "enat" shows "epred n = n - 1"
Par_Wbis_discrL1: fixes c1 :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>w d"   and "discr c2" shows "c1 | c2 \<approx>w d"
a_coset_add_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M +> g +> h = M +> g \<oplus> h"
simps_1: fixes xa :: "'o"   and x :: "'a" shows "lookup_pair xa [] x = (0::'c)"
finitely_generated_ideal_def: fixes I :: "'b set" shows "finitely_generated_ideal I = (ideal I \<and> (\<exists>S. finite S \<and> ideal_generated S = I))"
bot_notin_R: fixes c :: "'a filter set" assumes "c \<in> Chains {(b, a). a \<noteq> bot \<and> a \<le> b \<and> b \<le> F}" shows "bot \<notin> c"
that: fixes a :: "V"   and A' :: "V" assumes "A = vinsert a A'"   and "a \<notin>\<^sub>\<circ> A'" shows "thesis"
simps_2: fixes b :: "bexp"   and c :: "com"   and n :: "nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" shows "wpw b c (Suc n) Q l s = (bval b s \<and> (\<exists>s'. (c, s) \<Rightarrow> s' \<and> wpw b c n Q l s'))"
FaceDivsionGraph_one_final_but: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list"   and f\<^sub>1 :: "face"   and f\<^sub>2 :: "face"   and g' :: "graph"   and r :: "nat" assumes "minGraphProps g"   and "pre_splitFace g u v f vs"   and "(f\<^sub>1, f\<^sub>2, g') = splitFace g u v f vs"   and "r \<noteq> v"   and "before (verticesFrom f r) u v"   and "r \<in> \<V> f"   and "one_final_but g (if r = u then {} else Edges (r # between (vertices f) r u @ [u]))" shows "one_final_but g' (Edges (r # between (vertices f\<^sub>2) r v @ [v]))"
Empty_is_rbt: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.is_rbt less rbt.Empty"
elements_mat_h_def': fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "elements_mat_h A = {A $h i $h j |i j. True}"
le_opt_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b option"   and y :: "'b option"   and z :: "'b option" assumes "order r"   and "x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>Opt.le r\<^esub> z" shows "x \<sqsubseteq>\<^bsub>Opt.le r\<^esub> z"
sign_changes_def: fixes xs :: "'a list" shows "sign_changes xs = length (remdups_adj (filter (\<lambda>x. x \<noteq> (0::'a)) (map sgn xs))) - 1"
skl1_inv1_init: shows "init skl1 \<subseteq> skl1_inv1"
isin_set: fixes t :: "'b tree"   and x :: "'b" assumes "Sorted_Less.sorted (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
simps_2: fixes v :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and va :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.meldUniq (v # va) [] = v # va"
span_span: fixes A :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "local.punit.pmdl (local.punit.pmdl A) = local.punit.pmdl A"
card_le_pick_le: fixes n :: "nat"   and S :: "nat set"   and i :: "nat" assumes "n < card S"   and "i \<le> pick S n" shows "card {a \<in> S. a < i} \<le> n"
IH: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat list"   and xc :: "nat"   and xd :: "bool" assumes "((x, xa, xb), n, i, is) \<in> measures [\<lambda>(n, i, is). n - i, \<lambda>(n, i, is). if is = [] then 1 else 0]"   and "2 \<le> xa"   and "2 \<le> x"   and "\<And>xb. \<lbrakk>2 \<le> xb; xb < xa\<rbrakk> \<Longrightarrow> \<not> xb dvd x"   and "\<And>x. \<lbrakk>xa \<le> x; x < xc; prime x\<rbrakk> \<Longrightarrow> x \<in> set xb"   and "xa \<le> xc"   and "sorted xb"   and "distinct xb"   and "candidate_invariant xc"   and "set xb \<subseteq> {xa..<xc}"   and "xd = prime_nat_main x xc xb" shows "xd = prime x"
return_time_least: fixes P :: "'a set"   and x :: "'a"   and s :: "real" assumes "returns_to P x"   and "closed P"   and "0 < s"   and "s < return_time P x" shows "flow0 x s \<notin> P"
sufficiency: assumes "\<forall>A. admits_triangular_reduction A"   and "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})" shows "\<forall>A. admits_diagonal_reduction A"
T1_neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y"   and "x \<in> carrier"   and "y \<in> carrier" obtains u :: "'a set" and v :: "'a set" where "u \<in> nhds x" and "v \<in> nhds y" and "y \<notin> u" and "x \<notin> v"
assms_2: shows "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"
execn_final_notin_to_exec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and T :: "('e, 'f) xstate set" assumes "\<forall>n. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>T" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T"
head_subst: fixes \<rho> :: "'b \<Rightarrow> ('a, 'b) tm"   and s :: "('a, 'b) tm" shows "head (subst \<rho> s) = head (subst \<rho> (Hd (head s)))"
maxsimp_vertices: fixes x :: "'a set" assumes "maxsimp x" shows "order.greater_eq (\<Union> X) x"
csubspace_is_convex: fixes M :: "'a set" assumes "csubspace M" shows "convex M"