hcju/isabelleps huggingface dataset:
exhaust: fixes y :: "'e seq_action" obtains x1 :: "'e" where "y = broadcast x1"   | x21 :: "nat set" and x22 :: "'e" where "y = groupcast x21 x22"   | x31 :: "nat" and x32 :: "'e" where "y = unicast x31 x32"   | x4 :: "nat" where "y = \<not>unicast x4"   | x5 :: "'e" where "y = send x5"   | x6 :: "nat" where "y = deliver x6"   | x7 :: "'e" where "y = receive x7"   | "y = \<tau>\<^sub>s"
downset_iso_iff: fixes x :: "'a"   and y :: "'a" shows "(ord.downset (\<lambda>x y. y \<le> x) x \<subseteq> ord.downset (\<lambda>x y. y \<le> x) y) = (y \<le> x)"
ExecAssignNoVar: fixes v :: "char list"   and \<sigma> :: "char list \<Rightarrow> nat option"   and e :: "expr" assumes "v \<notin> dom \<sigma>" shows "Assign v e \<rhd> \<sigma> \<leadsto> None"
WhileStrongPostdomination_aux: fixes prog :: "cmd" shows "StrongPostdomination sourcenode targetnode kind (valid_edge prog) (_Entry_) (_Exit_)"
simps: fixes I :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a" shows "invar ({I} WHILE b DO c {P}) = I"
simps_1: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" shows "sharing_consistent \<S> \<O> [] = True"
start_prog_Start_soconf: fixes C :: "char list"   and M :: "char list"   and P :: "jvm_prog"   and h :: "heap" shows "class_add P (start_class C M),h,Start \<turnstile>\<^sub>s Map.empty \<surd>"
f_supp: shows "emeasure M {x \<in> space M. 0 < f x} < \<infinity>"
update_by_fun_gr_pair_eq_update_by_fun_pair: fixes xs :: "('a \<times> 'c) list"   and k :: "'a"   and f :: "'c \<Rightarrow> 'c" assumes "oalist_inv_raw xs" shows "update_by_fun_gr_pair k f xs = update_by_fun_pair k f xs"
Trg_Dom_Cod: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Trg trg (bicategorical_language.Dom V src trg t) = bicategorical_language.Trg trg t \<and> bicategorical_language.Trg trg (bicategorical_language.Cod V src trg t) = bicategorical_language.Trg trg t"
an_neq_minf: fixes n :: "nat" shows "an n \<noteq> - \<infinity>"
HEndPhase0_blocksOf_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "p \<noteq> q" shows "blocksOf s' q \<subseteq> blocksOf s q"
fls_E: fixes H :: "fmla set"   and A :: "fmla" shows "insert fls H \<turnstile> A"
ret_no_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" assumes "IFC prob"   and "\<pi> n = IFC_def.return prob" shows "\<not> IFC_def.is_cdi prob n \<pi> k"
max_atm_imp_Interp: fixes C :: "'b literal multiset"   and N :: "'b literal multiset set"   and A :: "'b" assumes "C \<in> N"   and "Pos A \<in># C"   and "A = Max (atms_of C)"   and "{#} = {#}" shows "Interp N C \<Turnstile> C"
positive_mult: fixes x :: "rat"   and y :: "rat" assumes "Rat.positive x"   and "Rat.positive y" shows "Rat.positive (x * y)"
assms_5: fixes u :: "'a"   and v :: "'a" shows "\<exists>w z. g ` cbox u v = cbox w z"
mat_of_row_carrier_1: fixes y :: "'b Matrix.vec"   and n :: "nat" assumes "y \<in> carrier_vec n" shows "mat_of_row y \<in> carrier_mat 1 n"
continuous_on_const_minus_ennreal: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and a :: "ennreal" assumes "continuous_on A f" shows "continuous_on A (\<lambda>x. a - f x)"
ldrop_inf: fixes xs :: "'c llist" shows "ldrop \<infinity> xs = LNil"
min_dist_split_2: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and d1 :: "nat"   and w :: "nat"   and d2 :: "nat"   and v :: "nat" assumes "Graph.dist c u d1 w"   and "Graph.dist c w d2 v"   and "Graph.min_dist c u v = d1 + d2" shows "Graph.min_dist c w v = d2"
"case": fixes f :: "'b list \<Rightarrow> 'a"   and x :: "'b list" shows "(case Array x of Array x \<Rightarrow> f x) = f x"
discI_2: fixes event :: "('a, 'b, 'c) event"   and x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" assumes "event = Send x21 x22 x23 x24 x25 x26" shows "isSend event"
poincare_distance_formula_def: fixes u :: "complex"   and v :: "complex" shows "poincare_distance_formula u v = arcosh (poincare_distance_formula' u v)"
list_dnf_to_dnf_list_dnf: fixes \<phi> :: "'b ltln" shows "list_dnf_to_dnf (list_dnf \<phi>) = dnf \<phi>"
numadd_nb: fixes t :: "MIR.num"   and s :: "MIR.num" assumes "numbound0 t"   and "numbound0 s" shows "numbound0 (numadd (t, s))"
prems: fixes x :: "'a" assumes "x \<in> insert x__ S" shows "set (coeffs (f x)) \<subseteq> \<real>"
hcomp_simps_2: fixes \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.hseq \<nu> \<mu>" shows "trg\<^sub>D (\<nu> \<star>\<^sub>D \<mu>) = trg\<^sub>D \<nu>"
nn_integral_pmf: fixes A :: "'b set"   and p :: "'b pmf" shows "\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space A = emeasure (measure_pmf p) A"
smooth_on_subset: fixes k :: "enat"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "k-smooth_on T f"   and "S \<subseteq> T" shows "k-smooth_on S f"
fast_dice_roll_alt_conv_while: fixes v :: "nat"   and c :: "nat" shows "local.fdr_alt v c = map_spmf snd (local.fdr_step v c \<bind> loop_spmf.while (\<lambda>(v, c). n \<le> c) (\<lambda>(v, c). local.fdr_step (v - n) (c - n)))"
bn_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and z :: "Vars.var" shows "domA \<Gamma>[y::h=z] = domA \<Gamma>"
hoare_triple_def_var: fixes p :: "'a"   and c :: "'b"   and q :: "'a" shows "(\<iota> p \<cdot> c \<le> c \<cdot> \<iota> q) = (\<iota> p \<cdot> c \<cdot> ! (\<iota> q) = (0::'b))"
eventually_gt_at_top: fixes c :: "'b" shows "eventually ((<) c) at_top"
mask_eq_take_bit_minus_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "mask n = take_bit n (uminus one)"
hbase_ext_def: shows "hbase_ext \<equiv> \<lambda>b. {x. hbase_extp b x}"
frontier_ball: fixes e :: "real"   and a :: "'a" assumes "0 < e" shows "frontier (ball a e) = sphere a e"
less_le_trans: fixes x :: "'t"   and y :: "'t"   and z :: "'t" assumes "x \<prec>\<^sub>t y"   and "y \<preceq>\<^sub>t z" shows "x \<prec>\<^sub>t z"
domo_narrow_subset: fixes S1 :: "'a Abs_State.st option"   and S2 :: "'a Abs_State.st option" shows "domo (S1 \<triangle> S2) \<subseteq> domo S1 \<union> domo S2"
linordered_field_no_ub: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less x y"
put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and setter :: "('v \<Rightarrow> 'v) \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'o, 'r option) RElement_scheme"   and v :: "'v" assumes "CharacterDataClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr setter v"
cw_len_length: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option"   and w :: "'d list" assumes "source_code M b fi X enc dec" shows "foldr (\<lambda>x. (+) (code_word_length enc x)) w 0 = length (enc w)"
measure_unat: fixes p :: "'b word" assumes "p \<noteq> 0" shows "unat (p - 1) < unat p"
iMODb_iFROM_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and n :: "nat" shows "([ r', mod m', c' ] \<subseteq> [n\<dots>]) = (n \<le> r')"
snd_Gauss_Jordan_upt_k_PA: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "snd (Gauss_Jordan_upt_k_PA A k) = Gauss_Jordan_upt_k A k"
A_eq: shows "A = A1 \<union> A2 \<union> A3"
cball_trivial: fixes x :: "'c" shows "cball x 0 = {x}"
matches_dest_2: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and m' :: "'b match_expr"   and a :: "action"   and p :: "'c" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m' a p" shows "False"
pre_subdivFace'_preFaceDiv: fixes g :: "graph"   and f :: "face"   and v' :: "nat"   and v :: "nat"   and n :: "nat"   and u :: "nat"   and vol :: "nat option list" assumes "pre_subdivFace' g f v' v n (Some u # vol)"   and "f \<in> \<F> g"   and "f \<bullet> v = u \<longrightarrow> n \<noteq> 0"   and "\<V> f \<subseteq> \<V> g" shows "pre_splitFace g v u f [countVertices g..<countVertices g + n]"
sds_lowering_axioms: shows "sds_lowering agents alts sds agents' alts'"
order_refl: fixes x :: "'a word" shows "x \<le>s x"
prems_3: shows "f \<in> insert x F \<rightarrow> UNIV"
subset: shows "\<E> (TER g) \<subseteq> \<E> (TER f)"
map_of_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "map_of (entries t) = rbt_lookup t"
redundant_scc_phis: fixes g :: "'g"   and P :: "'val set"   and scc :: "'val set"   and x :: "'val" assumes "redundant_set g P"   and "scc \<in> condensation_nodes g P"   and "x \<in> scc" shows "phi g x \<noteq> None"
box_eq_empty_1: fixes a :: "'b"   and b :: "'b" shows "(box a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i \<le> a \<bullet> i)"
isomorphic_implies_hpar_1: fixes f :: "'d"   and f' :: "'d" assumes "f \<cong>\<^sub>D f'" shows "D.ide f"
distinct_3: fixes x3 :: "char list list" shows "No_ics \<noteq> Called x3"
atLeastLessThan_empty: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and b :: "'g"   and a :: "'g" assumes "class.order less_eq less"   and "less_eq b a" shows "ord.atLeastLessThan less_eq less a b = {}"
iso_runit: fixes f :: "'a" assumes "B.ide f" shows "B.iso (B.runit f)"
simps_2: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and ifce :: "iface" shows "oiface_rewrite ipassmt (Match (OIface ifce)) = ipassmt_iface_replace_dstip_mexpr ipassmt ifce"
list_assn_aux_eqlen_simp_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l :: "'a list"   and l' :: "'b list"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> list_assn P l l'" shows "length l' = length l"
assms: shows "nop_bij A (1\<^sub>\<nat>) f"
that: fixes p :: "'a"   and q :: "'a" assumes "d = p * a + q * b" shows "thesis"
preserves_cod: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "functor A B F"   and "partial_magma.arr A f" shows "partial_magma.cod B (F f) = F (partial_magma.cod A f)"
HT_def: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" shows "HT \<pi> P c Q \<equiv> \<forall>s\<^sub>0. P s\<^sub>0 \<longrightarrow> wp \<pi> c (Q s\<^sub>0) s\<^sub>0"
bin_to_bl_aux_bintr: fixes n :: "nat"   and m :: "nat"   and bin :: "int"   and bl :: "bool list" shows "bin_to_bl_aux n (take_bit m bin) bl = replicate (n - m) False @ bin_to_bl_aux (min n m) bin bl"
cospan: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and h :: "'b"   and k :: "'b" assumes "cospan_of_maps_in_bicategory_of_spans V H \<a> \<i> src trg h k" shows "trg h = trg k"
scale_cancel_left: fixes x :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and y :: "('a \<Rightarrow> real) \<Rightarrow> real"   and a :: "real" assumes "x \<in> sub.tangent_space p"   and "y \<in> sub.tangent_space p" shows "(a *\<^sub>R x = a *\<^sub>R y) = (x = y \<or> a = 0)"
ru_3: shows "poly (real_of_int_poly (poly_real_alg_1 (p, l, r))) (real_of_1 (p, l, r)) = 0"
simplices_map: shows "order.greater_eq X (g \<turnstile> X)"
cone_smult: fixes a :: "'b"   and Vs :: "'b vec set"   and n :: "nat"   and x :: "'b vec" assumes "(0::'b) \<le> a"   and "Vs \<subseteq> carrier_vec n"   and "x \<in> gram_schmidt.cone n Vs" shows "a \<cdot>\<^sub>v x \<in> gram_schmidt.cone n Vs"
increasing_def: fixes f :: "'b \<Rightarrow> 'c" shows "increasing f \<equiv> \<Inter>z. stable {s. z \<le> f s}"
outstanding_refs_mono_pred: fixes P :: "'b memref \<Rightarrow> bool"   and P' :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" assumes "\<forall>r. P r \<longrightarrow> P' r" shows "outstanding_refs P sb \<subseteq> outstanding_refs P' sb"
simps_3: fixes x21 :: "'a tree"   and x22 :: "'a"   and x23 :: "'a tree" shows "\<langle>x21, x22, x23\<rangle> \<noteq> \<langle>\<rangle>"
extreme_points_of_translation_subtract: fixes a :: "'b"   and S :: "'b set" shows "{x. x extreme_point_of (\<lambda>x. x - a) ` S} = (\<lambda>x. x - a) ` {x. x extreme_point_of S}"
times_upcast_upper: fixes x :: "int"   and y :: "int" assumes "- 2147483648 \<le> x"   and "- 2147483648 \<le> y"   and "x \<le> 2147483648"   and "y \<le> 2147483648" shows "x * y \<le> 4611686018427387904"
assms_4: shows "\<nexists>sfs. sh\<^sub>1 D = \<lfloor>(sfs, Done)\<rfloor>"
gfp_exp_upperbound: fixes P :: "'a \<Rightarrow> real"   and t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "P \<tturnstile> t P"   and "unitary P" shows "P \<tturnstile> gfp_exp t"
strictly_preserves_tensor: fixes f :: "'c"   and g :: "'c" assumes "C.arr f"   and "C.arr g" shows "F (f \<otimes>\<^sub>C g) = F f \<otimes>\<^sub>D F g"
False: shows "memory_write asi address (st_byte_mask instr address) (st_data0 instr curr_win rd address s) s \<noteq> None"
inter_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "inter_list xs ys = filter (\<lambda>x. x \<in> set xs \<and> x \<in> set ys) xs"
oldDefs_def: fixes g :: "'a"   and n :: "'b" shows "oldDefs g n = var g ` defs g n"
distinct_42: fixes x61 :: "Arraytype"   and x62 :: "nat" shows "nullV \<noteq> arrV x61 x62"
functor_axioms: shows "functor B.VVV.comp (\<cdot>\<^sub>B) B.HoVH"
trivial_factors_imp_splitted: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "\<And>q. \<lbrakk>q \<in> carrier (poly_ring R); pirreducible (carrier R) q; q pdivides p\<rbrakk> \<Longrightarrow> degree q \<le> 1" shows "splitted p"
functor_axioms: shows "functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) H"
locally_reflects_ide: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A a\<guillemotright>"   and "B.ide (F f)" shows "A.ide f"
lsemimodule2: fixes next_behaviour :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and next_stimulus :: "'c \<times> 'd \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c"   and a :: "'d" assumes "c2ka next_behaviour next_stimulus" shows "next_behaviour (s \<oplus> t) a = next_behaviour s a + next_behaviour t a"
assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<odot> y \<odot> z = x \<odot> (y \<odot> z)"
lex_list_well_founded: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "well_founded order" shows "well_founded (lexp order)"
Star_unfoldL: fixes r :: "'a language" shows "Star r = Plus One (Times r (Star r))"
faithful_functor_axioms: shows "faithful_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F"
simps_2: fixes x :: "'a list"   and xs :: "'a list list"   and ys :: "'a list list" shows "x # xs \<otimes> ys = vec_mat_Tensor x ys @ (xs \<otimes> ys)"
S_intersections_defs_4: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection2_rep p q \<equiv> S_intersection_coeff2 p q *\<^sub>R p + q"
project_aHom: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and j :: "'d" assumes "\<forall>k\<in>I. aGroup (A k)"   and "j \<in> I" shows "\<pi>\<^bsub>I,A,j\<^esub> \<in> aHom (a\<Pi>\<^bsub>I\<^esub> A) (A j)"
biQuots_def: fixes xs :: "'b list"   and ys :: "'b list"   and A :: "'b list set" shows "biQuots xs ys A = {zs. xs @ zs @ rev ys \<in> A}"
map_cong: fixes x :: "('d, 'e) nondetT"   and ya :: "('d, 'e) nondetT"   and f1 :: "'d \<Rightarrow> 'f"   and g1 :: "'d \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'g"   and g2 :: "'e \<Rightarrow> 'g" assumes "x = ya"   and "\<And>z1. z1 \<in> phantom_nondetT ya \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> set_nondetT ya \<Longrightarrow> f2 z2 = g2 z2" shows "map_nondetT' f1 f2 x = map_nondetT' g1 g2 ya"
intros_1: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" shows "star r x x"
indep_vars_def2: fixes M :: "'c measure"   and M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> 'e"   and I :: "'d set" assumes "prob_space M" shows "prob_space.indep_vars M M' X I = ((\<forall>i\<in>I. X i \<in> M \<rightarrow>\<^sub>M M' i) \<and> prob_space.indep_sets M (\<lambda>i. {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
res_to_cong_simps_4: fixes m :: "int" assumes "residues m" shows "\<one>\<^bsub>residue_ring m\<^esub> = 1 mod m"
step'_2: fixes c :: "int" assumes "c = round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)"   and "\<mu>_small_row i fs (Suc j)" shows "\<mu>_small_row i (fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]) j"
degree_m_eq_monic: fixes f :: "int poly" assumes "monic f"   and "1 < m" shows "degree_m f = degree f"
is_iso_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
wf_rulesetD_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and r :: "'b rule"   and rs :: "'b rule list" assumes "wf_ruleset \<gamma> p (r # rs)" shows "wf_ruleset \<gamma> p [r]"
SUP_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd \<times> 'e"   and D :: "'c \<Rightarrow> 'd \<times> 'e" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup_prod (C ` A) = Sup_prod (D ` B)"
frequently_eventually_mono: fixes Q :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool" assumes "frequently Q F"   and "eventually P F"   and "\<And>x. \<lbrakk>P x; Q x\<rbrakk> \<Longrightarrow> R x" shows "frequently R F"
induct: fixes x1 :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and x2 :: "'d"   and x3 :: "('e \<Rightarrow> 'f) set"   and x4 :: "'d"   and x5 :: "('e \<Rightarrow> 'f) set"   and P :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint) \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) set \<Rightarrow> 'd \<Rightarrow> ('e \<Rightarrow> 'f) set \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2, x3\<rangle> \<leadsto>* \<langle>x4, x5\<rangle>"   and "\<And>A l Z. P A l Z l Z"   and "\<And>A l Z l' Z' l'' Z''. \<lbrakk>A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>; A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>; P A l' Z' l'' Z''\<rbrakk> \<Longrightarrow> P A l Z l'' Z''" shows "P x1 x2 x3 x4 x5"
fact1: shows "get_trap_set s = {} \<or> reset_trap_val s \<noteq> False \<or> get_ET (cpu_reg_val PSR s) \<noteq> 0"
run_shift_elim: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) list"   and s :: "(nat \<times> 'b) stream"   and p :: "nat \<times> 'b" assumes "grun A w V (r @- s) p" obtains "gpath A w V r p" and "grun A w V s (gtarget r p)"
P'_nosubset: shows "\<forall>X\<in>P'. \<not> X \<subseteq> B"
unr: shows "prv (imp (instInp Pinp \<langle>\<phi>'\<rangle>) (instInp Pinp \<langle>\<phi>\<rangle>))"
bl1: shows "bl \<in># filter_mset ((\<subseteq>) ps) B"
map_graph_def: fixes f :: "'a \<Rightarrow> 'b option" shows "map_graph f = {(x, y) |x y. (x, Some y) \<in> fun_graph f}"
dynmethd_rec: fixes G :: "prog"   and dynC :: "qtname"   and c :: "class"   and statC :: "qtname"   and sig :: "sig" assumes "class G dynC = Some c"   and "ws_prog G" shows "dynmethd G statC dynC sig = (if G\<turnstile>dynC\<preceq>\<^sub>C statC then case methd G statC sig of None \<Rightarrow> None | Some statM \<Rightarrow> case methd G dynC sig of None \<Rightarrow> dynmethd G statC (super c) sig | Some dynM \<Rightarrow> if G \<turnstile> qmdecl sig dynM overrides qmdecl sig statM \<or> dynM = statM then Some dynM else dynmethd G statC (super c) sig else None)"
SNa_elim: fixes M :: "trm"   and P :: "trm \<Rightarrow> bool" assumes "SNa M" shows "(\<forall>M. (\<forall>N. M \<longrightarrow>\<^sub>a N \<longrightarrow> P N) \<longrightarrow> P M) \<longrightarrow> P M"
bits_induct: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "\<And>a. divide a (numeral.numeral one plus (num.Bit0 num.One)) = a \<Longrightarrow> P a"   and "\<And>a b. \<lbrakk>P a; divide (plus (zero_neq_one.of_bool one zero b) (times (numeral.numeral one plus (num.Bit0 num.One)) a)) (numeral.numeral one plus (num.Bit0 num.One)) = a\<rbrakk> \<Longrightarrow> P (plus (zero_neq_one.of_bool one zero b) (times (numeral.numeral one plus (num.Bit0 num.One)) a))" shows "P a"
ev_blocksD: fixes n :: "nat"   and A :: "'b mat"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "??.Jordan_Normal_Form_Existence.ev_blocks n A"   and "i < j"   and "j < k"   and "k < n"   and "A $$ (k, k) = A $$ (i, i)" shows "A $$ (j, j) = A $$ (i, i)"