hcju/isabelleps huggingface dataset:
slim_paths_in_G_E: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "pair_sym_digraph G"   and "(u, p, v) \<in> pair_pre_digraph.slim_paths G" obtains "pre_digraph.gen_iapath (with_proj G) (verts3 (with_proj G)) u p v" and "u \<noteq> v"
Vfrom_sup: fixes a :: "V"   and i :: "V"   and j :: "V" shows "Vfrom a (i \<union>\<^sub>\<circ> j) = Vfrom a i \<union>\<^sub>\<circ> Vfrom a j"
wls_psubstAbs_substAbs_compose: fixes us :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and ys :: "'varSort"   and Y :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and y :: "'var" assumes "wlsAbs (us, s) A"   and "wls (asSort ys) Y"   and "wlsEnv rho" shows "A $[Y / y]_ys $[rho] = A $[rho [y \<leftarrow> Y #[rho]]_ys]"
gen_subdiv_3: shows "\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>"
expectation_gpv_map_gpv': fixes fail :: "ennreal"   and \<I> :: "('a, 'b) \<I>"   and f :: "'c \<Rightarrow> ennreal"   and g :: "'d \<Rightarrow> 'c"   and h :: "'e \<Rightarrow> 'a"   and k :: "'b \<Rightarrow> 'f"   and gpv :: "('d, 'e, 'f) gpv" shows "expectation_gpv fail \<I> f (map_gpv' g h k gpv) = expectation_gpv fail (map_\<I> h k \<I>) (f \<circ> g) gpv"
mult_mv_nonzero: fixes v :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" assumes "v \<noteq> vec_zero"   and "mat_det A \<noteq> 0" shows "A *\<^sub>m\<^sub>v v \<noteq> vec_zero"
cases: 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 t :: "'c"   and ta :: "('b, 'c, 'f, 'd, 'e, 'g) thread_action" assumes "final_thread.actions_ok final s t ta" obtains "lock_ok_las (locks s) t \<lbrace>ta\<rbrace>\<^bsub>l\<^esub>" and "thread_oks (thr s) \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" and "final_thread.cond_action_oks final s t \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>" and "wset_actions_ok (wset s) t \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>" and "interrupt_actions_ok (interrupts s) \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>"
round_robin_invar_correct: fixes \<sigma> :: "'queue \<times> nat"   and T :: "'t set" assumes "local.round_robin_invar \<sigma> T" shows "Round_Robin.round_robin_invar (round_robin_\<alpha> \<sigma>) T"
neg_mod_bound: fixes l :: "int"   and k :: "int" assumes "l < 0" shows "l < k mod l"
wf_mbufn_add: fixes i :: "nat"   and js :: "nat list"   and Ps :: "(nat \<Rightarrow> event_data option list set \<Rightarrow> bool) list"   and buf :: "event_data option list set list list"   and js' :: "nat list"   and xss :: "event_data option list set list list" assumes "wf_mbufn i js Ps buf"   and "list_all3 list_all2 Ps (map2 upt js js') xss"   and "list_all2 (\<le>) js js'" shows "wf_mbufn i js' Ps (mbufn_add xss buf)"
b_2: fixes i :: "nat" shows "(LEAST a. a \<in> A \<and> card (A \<inter> {..<a}) = i) \<in> A"
bits_div_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide zero a = zero"
natural_transformation_axioms: shows "natural_transformation C D Adj.GFG.map G (G \<circ> \<epsilon>)"
set_zip_map: fixes f :: "'d \<Rightarrow> 'b"   and xs :: "'d list"   and g :: "'d \<Rightarrow> 'c" shows "set (zip (map f xs) (map g xs)) = (\<lambda>x. (f x, g x)) ` set xs"
indep_sets_Dynkin: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set" assumes "indep_sets F I" shows "indep_sets (\<lambda>i. Dynkin (space M) (F i)) I"
assms: shows "typ_of1 Ts (t $ u) = Some ty"
vec_plusI_def: fixes pl :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and v :: "'b list"   and w :: "'b list" shows "vec_plusI pl v w = map (\<lambda>xy. pl (fst xy) (snd xy)) (zip v w)"
liftAll_mono: fixes phi :: "'a \<Rightarrow> bool"   and chi :: "'a \<Rightarrow> bool"   and inp :: "'b \<Rightarrow> 'a option" assumes "\<And>v. phi v \<Longrightarrow> chi v"   and "liftAll phi inp" shows "liftAll chi inp"
relH_def: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap" shows "relH as h h' = (in_range (h, as) \<and> in_range (h', as) \<and> (\<forall>t. \<forall>a\<in>as. refs h t a = refs h' t a \<and> arrays h t a = arrays h' t a))"
fs_int_indpt_axioms: shows "fs_int_indpt n fs"
pd_nodes_1: fixes y :: "'n"   and x :: "'n" assumes "y pd\<rightarrow> x" shows "y \<in> local.nodes"
\<phi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<phi> (b, a) \<in> C.hom b a \<rightarrow> Hom.set (b, a)"
axioms_4: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b" assumes "inverse_functors A B F G" shows "functor A B G"
exists_vec_append: fixes n :: "nat"   and m :: "nat"   and P :: "'b vec \<Rightarrow> bool" shows "(\<exists>x\<in>carrier_vec (n + m). P x) = (\<exists>x1\<in>carrier_vec n. \<exists>x2\<in>carrier_vec m. P (x1 @\<^sub>v x2))"
cauchy_ine_in: fixes u :: "nat \<Rightarrow> 'c"   and N :: "'c quasinorm" assumes "\<And>n. u n \<in> space\<^sub>N N" shows "cauchy_ine\<^sub>N N u = cauchy_in\<^sub>N N u"
"1_1": shows "w \<le>\<^sub>F\<^sub>I v"
partition_onE: fixes A :: "'c set"   and P :: "'c set set" assumes "partition_on A P" obtains "\<And>p. p \<in> P \<Longrightarrow> p \<noteq> {}" and "\<Union> P = A" and "\<And>p p'. \<lbrakk>p \<in> P; p' \<in> P; p \<noteq> p'\<rbrakk> \<Longrightarrow> p \<inter> p' = {}"
trace_0: shows "trace (mat (0::'d)) = (0::'d)"
subordinate_partition_of_unity: fixes S :: "'c set"   and \<C> :: "'c set set" assumes "S \<subseteq> \<Union> \<C>"   and "\<And>T. T \<in> \<C> \<Longrightarrow> open T"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. U \<inter> V \<noteq> {}}" obtains F :: "'c set \<Rightarrow> 'c \<Rightarrow> real" where "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x\<in>S. 0 \<le> F U x)" and "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0" and "\<And>x. x \<in> S \<Longrightarrow> supp_sum (\<lambda>W. F W x) \<C> = 1" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}"
msubset_of_eq: shows "msubset_of A F = image_mset f (mset_set A)"
contour_integral_linepath_Reals_eq: fixes a :: "complex"   and b :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "a \<in> \<real>"   and "b \<in> \<real>"   and "Re a < Re b" shows "contour_integral (linepath a b) f = integral {Re a..Re b} (\<lambda>x. f (complex_of_real x))"
Ball_Collect: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
ex1_cross_ratio: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and c :: "complex_homo" assumes "u \<noteq> v"   and "u \<noteq> w"   and "v \<noteq> w" shows "\<exists>!z. cross_ratio z u v w = c"
trace_add_linear: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n" shows "trace (A + B) = trace A + trace B"
ord_Zp_p_pow: fixes n :: "nat" shows "ord_Zp ([p] \<cdot> \<one> [^] n) = int n"
inverse_aform_err: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real"   and Y :: "(real \<times> real pdevs) \<times> real"   and x :: "real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "inverse_aform_err p X = Some Y"   and "x \<in> aform_err e X" shows "inverse x \<in> aform_err e Y"
proj0_iff: fixes x :: "'a" shows "(proj x = (0::'a)) = (x = (0::'a))"
sim2: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and P' :: "pi" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "P \<longmapsto> \<tau> \<prec> P'"   and "(P', Q) \<in> Rel" shows "P \<leadsto><Rel> \<tau>.Q"
i_take_ilist_slice_eq_list_slice: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "0 < k" shows "ilist_slice f k \<Down> n = list_slice (f \<Down> (n * k)) k"
least_method: shows "P \<turnstile> last Cs has least M = (Ts', T', pns', body') via Ds"
mdc_leq_ldc_aux: 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 C' :: "char list"   and Cs' :: "char list list" assumes "is_subobj P (C, C' # rev Cs')" shows "P \<turnstile> C \<preceq>\<^sup>* last (C' # rev Cs')"
dH_indistinguishable_def: fixes E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and DA :: "'b \<Rightarrow> 'e"   and d :: "'e"   and H :: "('e \<times> 'a) set"   and e1 :: "'a"   and e2 :: "'a" shows "WHATWHERE_Secure_Programs.dH_indistinguishable E DA d H e1 e2 \<equiv> \<forall>m m'. WHATWHERE.dH_equal E DA d H m m' \<longrightarrow> E e1 m = E e2 m'"
simps_6: fixes f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd \<Rightarrow> 'd" shows "rec_nat f1 f2 0 = f1"
le_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
ZFfunCompAppDomCod: fixes f :: "ZF"   and g :: "ZF" assumes "isZFfun f"   and "isZFfun g"   and "|dom|g = |cod|f" shows "|dom|f |o| g = |dom|f \<and> |cod|f |o| g = |cod|g"
prems_7: shows "\<forall>k. i \<le> k \<and> k < j \<and> regular_event (t ! k) \<longrightarrow> occurs_on (t ! k) \<noteq> p"
card_le_dim_spanning: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and Basis :: "'c set"   and B :: "'c set"   and V :: "'c set" assumes "finite_dimensional_vector_space scale Basis"   and "B \<subseteq> V"   and "V \<subseteq> module.span scale B"   and "finite B"   and "card B \<le> vector_space.dim scale V" shows "\<not> module.dependent scale B"
jb_def: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "local.jb A i j \<equiv> (Suc i = j \<longrightarrow> A $$ (i, j) \<in> {0::'a, 1::'a}) \<and> (i \<noteq> j \<longrightarrow> Suc i \<noteq> j \<or> A $$ (i, i) \<noteq> A $$ (j, j) \<longrightarrow> A $$ (i, j) = (0::'a))"
alpha_qSwap_preserves_qGood2: fixes zs :: "'varSort"   and y :: "'var"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs' :: "'varSort"   and y' :: "'var"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"   and "qGood X'" shows "qGood X"
icd_imp_cd: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "n icd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "n cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
1: shows "fmdom' (snd a) \<subseteq> vs"
sisoL_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list" assumes "PL_Indis cval indis" shows "PL_Indis.sisoL aval tval cval indis cl \<equiv> \<forall>c\<in>set cl. PL_Indis.siso aval tval cval indis c"
insert_no_failure: fixes i :: "'c option"   and xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool" assumes "i = None \<or> (\<exists>i'. i = Some i' \<and> i' \<in> fst ` set xs)" shows "\<exists>xs'. insert xs e i = Some xs'"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "Ik.satC \<xi> \<equiv> list_ex (Ik.satL \<xi>)"
compet_BIT4: fixes init :: "'c list"   and qs :: "'c list" assumes "init \<noteq> []"   and "distinct init" shows "T_on_rand' BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs \<le> 7 / 4 * real (T_opt init qs)"
approx_floatariths_Elem: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and ra :: "floatarith list"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and X :: "((real \<times> real pdevs) \<times> real) list"   and vs :: "real list" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "approx_floatariths_aformerr p ra VS = Some X"   and "vs \<in> aforms_err e VS" shows "interpret_floatariths ra vs \<in> aforms_err e X"
pderiv_eq_0_iff: fixes p :: "'a poly" shows "(pderiv p = 0) = (Polynomial.degree p = 0)"
kE_satPB: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"   and "Struct.wtE intT (InfModel.kE intT \<xi>)" shows "Struct.satPB (InfModel.intFF arOf resOf intT intF) (InfModel.intPF parOf intT intP) \<xi> \<Phi>"
list_all_simps_2: fixes P :: "'d \<Rightarrow> bool" shows "list_all P [] = True"
rep_eq: fixes x :: "nat"   and xa :: "uint64" shows "Rep_uint64 (drop_bit x xa) = drop_bit x (Rep_uint64 xa)"
shows_list_gen_map: fixes f :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and sep :: "char list"   and r :: "char list"   and g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "shows_list_gen f e l sep r (map g xs) = shows_list_gen (f \<circ> g) e l sep r xs"
lbint_lim: shows "(\<lambda>i. interval_lebesgue_integral lborel (ereal (l i)) (ereal (u i)) f) \<longlonglongrightarrow> C"
arithmetic_simps_50: fixes a :: "'a" shows "a + (0::'a) = a"
list_update_overwrite: fixes xs :: "'c list"   and i :: "nat"   and x :: "'c"   and y :: "'c" shows "xs[i := x, i := y] = xs[i := y]"
NotifyWatchesLoopConflictFlagEffect: fixes state :: "State"   and Wl :: "nat list"   and literal :: "Literal"   and newWl :: "nat list" assumes "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> 0 \<le> c \<and> c < length (getF state)"   and "InvariantConsistent (getM state)"   and "\<forall>c. c \<in> set Wl \<longrightarrow> Some literal = getWatch1 state c \<or> Some literal = getWatch2 state c"   and "literalFalse literal (elements (getM state))"   and "uniq Wl" shows "let state' = notifyWatches_loop literal Wl newWl state in getConflictFlag state' = (getConflictFlag state \<or> (\<exists>clause. clause \<in> set Wl \<and> clauseFalse (getF state ! clause) (elements (getM state))))"
subst3: fixes subst :: "'c \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'c"   and xvec :: "name list"   and Tvec :: "'d list"   and T :: "'c"   and x :: "name" assumes "substType subst"   and "length xvec = length Tvec"   and "distinct xvec"   and "set xvec \<subseteq> supp T"   and "x \<sharp> subst T xvec Tvec" shows "x \<sharp> Tvec"
case_prodI: fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'f"   and b :: "'g" assumes "f a b" shows "case (a, b) of (c, d) \<Rightarrow> f c d"
mulex_on_union': fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and M :: "'b multiset"   and N :: "'b multiset"   and K :: "'b multiset" assumes "mulex_on P A M N"   and "K \<in> multisets A" shows "mulex_on P A (M + K) (N + K)"
sum_eq: shows "(\<Sum>j\<le>k - 1. ws ! j * unity_root_sum k (m - int j)) = (\<Sum>j\<in>{nat m}. ws ! j * unity_root_sum k (m - int j))"
Rep_downsets_inject: fixes x :: "'b downsets"   and y :: "'b downsets" shows "(Rep_downsets x = Rep_downsets y) = (x = y)"
Ionescu_Tulcea_def: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> 'b measure"   and M :: "nat \<Rightarrow> 'b measure" shows "Ionescu_Tulcea P M \<equiv> (\<forall>i. P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)) \<and> (\<forall>i x. x \<in> space (Pi\<^sub>M {0..<i} M) \<longrightarrow> prob_space (P i x))"
Cauchy_integral_formula_global_weak: fixes U :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and \<gamma> :: "real \<Rightarrow> complex" assumes "open U"   and "f holomorphic_on U"   and "z \<in> U"   and "polynomial_function \<gamma>"   and "path_image \<gamma> \<subseteq> U - {z}"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "\<And>w. w \<notin> U \<Longrightarrow> winding_number \<gamma> w = 0" shows "((\<lambda>w. f w / (w - z)) has_contour_integral complex_of_real (2 * pi) * \<i> * winding_number \<gamma> z * f z) \<gamma>"
T7: shows "P InAngle A B C \<and> D E F CongA A B P"
open_contains_ball: fixes S :: "'a set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
map_single: fixes c :: "'a"   and f :: "'a \<Rightarrow> 'b"   and x :: "'c" assumes "c = (0::'a) \<Longrightarrow> f (0::'a) = (0::'b)" shows "Poly_Mapping.map f (Poly_Mapping.single x c) = Poly_Mapping.single x (f c)"
concurrent_ops_commute_def: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and interp :: "'c \<Rightarrow> 'd \<Rightarrow> 'd option"   and xs :: "'c list" assumes "happens_before hb_weak hb" shows "happens_before.concurrent_ops_commute hb interp xs \<equiv> \<forall>x y. {x, y} \<subseteq> set xs \<longrightarrow> happens_before.concurrent hb x y \<longrightarrow> interp x \<rhd> interp y = interp y \<rhd> interp x"
prems: fixes i :: "nat" assumes "i < Suc n" shows "\<exists>j. suffix (index i + j) w \<Turnstile>\<^sub>n af (formula i) (w [index i \<rightarrow> index i + j])[X]\<^sub>\<nu>"
S_def: shows "S = {x \<in> space M. \<forall>N. real (card {n \<in> {..<N}. \<exists>l\<in>{1..n}. u n ((Tinv ^^ n) x) - u (n - l) ((Tinv ^^ (n - l)) x) \<le> - delta l * real l}) \<le> e * real N}"
coding_tm_hf: fixes t :: "tm"   and e :: "(name, hf) finfun" assumes "coding_tm t" shows "coding_hf \<lbrakk>t\<rbrakk>e"
A: shows "of_int_poly a dvd of_int_poly (prod_list xs)"
equivalent_iff_not_distinguished: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and Q :: "'b" assumes "indexed_nominal_ts TYPE('a) satisfies transition" shows "indexed_nominal_ts.logically_equivalent TYPE('a) satisfies transition P Q = (\<nexists>x. indexed_nominal_ts.is_distinguishing_formula satisfies transition x P Q)"
mimic_trans: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>1' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 \<approx> s\<^sub>1'"   and "s\<^sub>1 \<leadsto>\<^sup>* s\<^sub>2" shows "\<exists>s\<^sub>2'. s\<^sub>2 \<approx> s\<^sub>2' \<and> s\<^sub>1' \<leadsto>\<^sup>* s\<^sub>2'"
appliedLearn_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedLearn stateA stateB \<equiv> \<exists>c. formulaEntailsClause (getF stateA) c \<and> vars c \<subseteq> vars (getF stateA) \<union> vars (elements (getM stateA)) \<and> getF stateB = getF stateA @ [c] \<and> getM stateB = getM stateA"
sums_def_le: fixes f :: "nat \<Rightarrow> 'b"   and s :: "'b" shows "f sums s = (\<lambda>n. sum f {..n}) \<longlonglongrightarrow> s"
Units_inv_Units: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> Units Zp" shows "inv x \<in> Units Zp"
poly_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> '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 p :: "'a list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.poly plus zero times p = semiring_0.poly plus zero times []) = (\<forall>c\<in>set p. c = zero)"
IH: assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> {0..1}" shows "1 - sum f A \<le> (\<Prod>x\<in>A. 1 - f x)"
g_2: shows "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"
frequently_disj_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<exists>\<^sub>F x in F. P x \<or> Q x) = (frequently P F \<or> frequently Q F)"
es_def_2: shows "Lfilled i l (vs @ [$Return]) LK"
zero_closed_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "zero_closed f \<equiv> (\<forall>x. f x (0::'c) = (0::'d)) \<and> (\<forall>y. f (0::'b) y = (0::'d))"
inorder_adjust: fixes t :: "('c \<times> nat) tree" assumes "t \<noteq> \<langle>\<rangle>"   and "pre_adjust t" shows "Tree2.inorder (adjust t) = Tree2.inorder t"
CH_def: fixes X :: "X4 set" shows "StableNoDecomp.CH X = (\<Union>h. CX4h h X)"
bex_Un: fixes A :: "'d set"   and B :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "(\<exists>x\<in>A \<union> B. P x) = ((\<exists>x\<in>A. P x) \<or> (\<exists>x\<in>B. P x))"
disc_20: fixes x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "\<not> isSnapshot (RecvMarker x51 x52 x53)"
Qw_cmpser_def: fixes G :: "('e, 'f) Group_scheme"   and f :: "nat \<Rightarrow> 'e set"   and l :: "nat" shows "Qw_cmpser G f l = \<natural>\<^bsub>G\<^esub>f l / f (Suc l)"
SPN_in_hom: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "Span.in_hom (SPN \<mu>) (Span.mkIde \<lbrakk>\<lbrakk>tab\<^sub>0 (local.dom \<mu>)\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 (local.dom \<mu>)\<rbrakk>\<rbrakk>) (Span.mkIde \<lbrakk>\<lbrakk>tab\<^sub>0 (cod \<mu>)\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 (cod \<mu>)\<rbrakk>\<rbrakk>)"
eq: shows "SyntaxN.Ex k' ((k' \<leftrightarrow> k) \<bullet> A) = SyntaxN.Ex k A"
sel: fixes x2 :: "'a" shows "the (pure x2) = x2"
env_restr_cong: fixes m :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and S' :: "'b set" assumes "\<And>x. edom m \<subseteq> S \<inter> S' \<union> - S \<inter> - S'" shows "m f|` S = m f|` S'"
smcf_ObjMap_vrange: shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> smc_0\<lparr>Obj\<rparr>"
induct: fixes P :: "('fun, 'var) Term.term set \<Rightarrow> ('fun, 'var) strand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) Term.term set"   and a1 :: "('fun, 'var) strand_step list" assumes "\<And>M. P M []"   and "\<And>M t S. (\<And>x. P M S) \<Longrightarrow> P M (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M t S. (\<And>x. P (insert (t \<cdot> x) M) S) \<Longrightarrow> P M (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M uu_ t t' S. (\<And>x. P M S) \<Longrightarrow> P M (\<langle>uu_: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>M X F S. (\<And>x. P M S) \<Longrightarrow> P M (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0 a1"
keys_plus_ninv_comm_monoid_add: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "keys (s + t) = keys s \<union> keys t"
simps_3: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and e1 :: "('c, 'd) expr"   and e2 :: "('c, 'd) expr" shows "compile (Binop f e1 e2) = compile e2 @ compile e1 @ [Apply f]"
le_iff_inf_bot: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<sqinter> - y = \<bottom>)"
simps_3: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "nat" assumes "project \<Sigma> project" shows "project.lang \<Sigma> project lookup n One = {[]}"
simps_3: fixes \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and p :: "'d"   and m :: "'c" shows "map_match_tac \<beta> p (Match m) = TernaryValue (\<beta> m p)"