hcju/isabelleps huggingface dataset:
lm057: fixes bids :: "'b \<times> 'c set \<Rightarrow> 'a"   and N :: "'b set"   and G :: "'c set"   and a :: "('b \<times> 'c set) set" shows "sum (summedBidVector bids N G) (a \<inter> Domain (summedBidVectorRel bids N G)) = sum snd (summedBidVectorRel bids N G || a)"
degree_inner_scaleR_pdevs_le: fixes l :: "'b" shows "degree (inner_scaleR_pdevs l One_pdevs) \<le> degree One_pdevs"
fempty_fsubsetI: fixes x :: "'b fset" shows "{||} |\<subseteq>| x"
conjuncts_in_maximal: fixes C :: "'b fm set set"   and V :: "'b fm set"   and p :: "'b fm"   and q :: "'b fm" assumes "consistency C"   and "V \<in> C"   and "maximal V C" shows "((p \<^bold>\<and> q) \<in> V) = (p \<in> V \<and> q \<in> V)"
simps_2: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" shows "P \<turnstile> typeof\<^bsub>h\<^esub> Null = \<lfloor>NT\<rfloor>"
yun_2: fixes a :: "rat poly"   and i :: "nat" assumes "(a, i) \<in> set pis" shows "square_free a \<and> degree a \<noteq> 0"
iMODb_card: fixes m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < m" shows "card [ r, mod m, c ] = Suc c"
S_def: shows "S = {(t, u). 0 < u \<and> coprime (int u) t \<and> \<bar>real_of_int t - real u * sqrt (real D)\<bar> < 1 / real u}"
drop_append: fixes n :: "nat"   and xs :: "'g list"   and ys :: "'g list" shows "drop n (xs @ ys) = drop n xs @ drop (n - length xs) ys"
euler_consistent_traj_set: fixes h :: "real"   and t :: "real"   and u :: "real"   and x :: "real \<Rightarrow> 'a"   and R :: "'a set" assumes "0 \<le> h"   and "t + h \<le> u"   and "{t..u} \<subseteq> {0..ha__}"   and "\<And>s. s \<in> {t..u} \<Longrightarrow> (x has_vector_derivative ode (x s)) (at s within {t..u})"   and "\<And>s. s \<in> {t..u} \<Longrightarrow> x s \<in> CX"   and "\<And>s. s \<in> {t..u} \<Longrightarrow> discrete_evolution (euler_increment (\<lambda>_. ode)) (t + h) t (x t) + (h\<^sup>2 / 2) *\<^sub>R blinfun_apply (case (s, x s) of (t, x) \<Rightarrow> ode_d1 x o\<^sub>L snd_blinfun) (1, ode (x s)) \<in> R"   and "bounded R"   and "closed R"   and "convex R" shows "x (t + h) \<in> R"
ground_aux_mono: fixes A :: "atom set"   and B :: "atom set"   and t :: "tm" assumes "A \<subseteq> B"   and "ground_aux t A" shows "ground_aux t B"
multisetUnionLeMono2: fixes r :: "('b \<times> 'b) set"   and b1 :: "'b multiset"   and b2 :: "'b multiset"   and a :: "'b multiset" assumes "trans r"   and "multiset_le b1 b2 r" shows "multiset_le (a + b1) (a + b2) r"
3: shows "3 = Suc (Suc (Suc 0))"
elims: fixes x :: "'b tree"   and y :: "'b tree" assumes "hchild x = y" obtains l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = (if height l \<le> height r then r else l)"   | "x = \<langle>\<rangle>" and "y = undefined"
map_upds_in_expansion_map_swap: fixes tab :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and vn :: "'b"   and z :: "'a"   and tab' :: "'b \<Rightarrow> 'a option" assumes "(tab(xs [\<mapsto>] ys)) vn = Some z"   and "tab vn \<noteq> Some z" shows "(tab'(xs [\<mapsto>] ys)) vn = Some z"
subset_msetE: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "A \<subset># B" obtains "A \<subseteq># B" and "\<not> B \<subseteq># A"
map_of_lookup: fixes T :: "(('a \<times> 'b) \<times> 'c) tree" assumes "sorted1 (Tree2.inorder T)" shows "Map.map_of (Tree2.inorder T) = lookup T"
euclidean_representation_sum': fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and b :: "'a"   and f :: "'a \<Rightarrow> real" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(b = comm_monoid_add.sum plus zero (\<lambda>i. scaleR (f i) i) Basis) = (\<forall>i\<in>Basis. f i = inner b i)"
Lyndon_fac_longest_dec: fixes w :: "'a list" shows "concat (LynFac w) = w"
intro: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" assumes "myss x y"   and "myss' a"   and "myss2_axioms x y a z" shows "myss2 x y a z"
closed_def: fixes P :: "instr list" shows "closed P = (exits P \<subseteq> {size P})"
prod_cases4: fixes y :: "'e \<times> 'f \<times> 'g \<times> 'h" obtains a :: "'e" and b :: "'f" and c :: "'g" and d :: "'h" where "y = (a, b, c, d)"
abs_conv_abscissa_higher_deriv: fixes n :: "nat"   and f :: "'a fds" shows "abs_conv_abscissa ((fds_deriv ^^ n) f) = abs_conv_abscissa f"
skl1_inv2_def: shows "skl1_inv2 \<equiv> {s. \<forall>gnx A B Rb Ni. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<longrightarrow> in_progressS (progress s Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<longrightarrow> sklvl1.guessed_frame Rb xgnx = Some gnx \<longrightarrow> sklvl1.guessed_frame Rb xni = Some Ni \<longrightarrow> can_signal s A B \<longrightarrow> 0 < signalsInit s (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)}"
assms_1: shows "con \<phi> \<psi> \<pi>l"
image_comp: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'g \<Rightarrow> 'f"   and r :: "'g set" shows "f ` g ` r = (f \<circ> g) ` r"
eval_throw_nonVal: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap" assumes "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>throw a,s'\<rangle>" shows "val_of e = None"
unstream_simps_3: fixes g :: "('e, 'f) generator"   and s :: "'f"   and x :: "'e"   and s' :: "'f" assumes "generator g s = Yield x s'" shows "unstream g s = x # unstream g s'"
map_replicate_const: fixes k :: "'a"   and lst :: "'b list" shows "map (\<lambda>x. k) lst = replicate (List.length lst) k"
mult_right_sub_dist_sup_left: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<sqinter> z \<le> (x \<squnion> y) \<sqinter> z"
subst_not_in_dom: fixes x :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'b option"   and mem :: "'a \<Rightarrow> 'b" assumes "x \<notin> dom \<sigma>" shows "subst \<sigma> mem x = mem x"
subst_fv_img_alt_def: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "range_vars s = \<Union> (fv ` {t. \<exists>v. s v = t \<and> t \<noteq> Var v})"
axis_eq_0_iff: fixes m :: "'b"   and x :: "'a" shows "(axis m x = 0) = (x = (0::'a))"
image_constant: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "\<And>i. i \<in> I \<Longrightarrow> f i = y" shows "f ` I = (if I = {} then {} else {y})"
all_positions_suffix_take: fixes i :: "nat"   and w :: "'a list"   and x :: "'a list" assumes "i \<le> length w"   and "i \<le> length x"   and "\<forall>j<i. x ! j = w ! (length w + j - i)" shows "suffix (take i x) w"
monic_degree_m: fixes f :: "int poly" assumes "monic f" shows "pl.degree_m f = degree f"
list_of_oalist_reorder: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and list_of_oalist :: "'c \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and oalist_of_list :: "('b \<times> 'd) list \<times> 'a \<Rightarrow> 'c"   and ko :: "'a"   and xs :: "'c" assumes "oalist_abstract rep_key_order list_of_oalist oalist_of_list" shows "list_of_oalist (oalist_abstract.reorder rep_key_order list_of_oalist oalist_of_list ko xs) = (oalist_raw.sort_oalist_aux rep_key_order ko (list_of_oalist xs), ko)"
assms_3: shows "tfr\<^sub>s\<^sub>e\<^sub>t N"
True': shows "a \<le> b \<and> poly p a = 0 \<and> p \<noteq> 0 \<or> a = b \<and> p = 0"
assms_2: fixes s :: "'a"   and x :: "'b"   and s' :: "'a" assumes "run_state m s = (x, s')" shows "strict_mono_state (f x)"
nb: shows "numbound0 (snd (rsplit0 (Add a b)))"
appliedConflict_def: fixes stateA :: "State"   and stateB :: "State" shows "appliedConflict stateA stateB \<equiv> \<exists>clause. getConflictFlag stateA = False \<and> formulaEntailsClause (getF stateA) clause \<and> clauseFalse clause (elements (getM stateA)) \<and> getF stateB = getF stateA \<and> getM stateB = getM stateA \<and> getConflictFlag stateB = True \<and> getC stateB = clause"
Loop: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and b :: "val"   and s1 :: "state"   and c :: "stmt"   and s2 :: "state"   and l :: "label"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>b\<rightarrow> s1"   and "if the_Bool b then G\<turnstile>s1 \<midarrow>c\<rightarrow> s2 \<and> G\<turnstile>abupd (absorb (Cont l)) s2 \<midarrow>l\<bullet> While(e) c\<rightarrow> s3 else s3 = s1" shows "G\<turnstile>Norm s0 \<midarrow>l\<bullet> While(e) c\<rightarrow> s3"
f2_def: fixes uu :: "nat llist llist list" shows "f2 uu = ()"
continuous_at_imp_continuous_within: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "isCont f x" shows "continuous (at x within s) f"
subgroup_indicator_trivial: fixes a :: "'a" assumes "a \<in> carrier G" shows "subgroup_indicator G {\<one>} a = ord a"
P920_example_gfp_F_value: shows "P920_example_gfp_F UNIV = ({(D1, H1), (D1, H2), (D2, H2)}, {(D1, H1), (D2, H1), (D2, H2)})"
rel_refl: fixes Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a stream" assumes "\<And>x. Ra x x" shows "stream_all2 Ra x x"
cases: fixes a1 :: "nat"   and a2 :: "s"   and a3 :: "v list"   and a4 :: "e list"   and a5 :: "t list" assumes "\<turnstile>_ a1 a2;a3;a4 : a5" obtains s :: "s" and \<S> :: "s_context" and i :: "nat" and vs :: "v list" and es :: "e list" and ts :: "t list" where "a1 = i" and "a2 = s" and "a3 = vs" and "a4 = es" and "a5 = ts" and "store_typing s \<S>" and "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts"
id_eq_prop_prop_6_b: fixes v :: "i"   and F :: "\<Pi>\<^sub>3"   and G :: "\<Pi>\<^sub>3"   and H :: "\<Pi>\<^sub>3" shows "[F \<^bold>= G \<^bold>& G \<^bold>= H \<^bold>\<rightarrow> F \<^bold>= H in v]"
completeness_game_def: fixes h :: "'a \<times> 'a"   and w :: "nat"   and e :: "nat" shows "chaum_ped_sigma.completeness_game h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (check h a e z)))"
irreducible_monic_factor: fixes p :: "'c poly" assumes "0 < degree p" shows "\<exists>q r. irreducible q \<and> p = q * r \<and> monic q"
cPar2B: fixes Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi"   and P :: "pi"   and C :: "'a" assumes "Q \<longmapsto> a<\<nu>x> \<prec> Q'"   and "x \<sharp> P"   and "\<And>C. F C Q a x Q'" shows "F C (P \<parallel> Q) a x (P \<parallel> Q')"
multiplicity_dvd': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "nat"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "n \<le> factorial_semiring.multiplicity times one p x" shows "dvd.dvd times (power.power one times p n) x"
Reds_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 E :: "char list \<Rightarrow> ty option" shows "Reds P E = {((es, s), es', s'). P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>}"
i_th_suc: fixes i :: "nat discr Box"   and x :: "'a"   and xs :: "'a Stream" assumes "i \<noteq> \<bottom>" shows "(x && xs) !! (i + 1) = xs !! i"
prems_5: shows "snd (cs (S t i) cid) = NotStarted"
const_FINFUNC: assumes "Field r \<noteq> {}" shows "const \<in> FINFUNC"
simps_11: fixes v :: "tm" shows "qfree (Lt v) = True"
simps_5: fixes t :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr" shows "valid_sops_expr t (Binop f e\<^sub>1 e\<^sub>2) = (valid_sops_expr t e\<^sub>1 \<and> valid_sops_expr t e\<^sub>2)"
update_correct'_2: fixes m :: "(uint32, ('a, 'b) assoc_list) RBT.rbt"   and k :: "'a"   and v :: "'b" assumes "HashMap_Impl.invar m" shows "HashMap_Impl.invar (HashMap_Impl.update k v m)"
insert_Diff1: fixes x :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "x \<in> B" shows "(x \<triangleright> A) - B = A - B"
winding_number_circlepath: fixes w :: "complex"   and z :: "complex"   and r :: "real" assumes "cmod (w - z) < r" shows "winding_number (circlepath z r) w = 1"
HRB_slice_valid_node: fixes n :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n \<in> HRB_slice S" shows "valid_SDG_node n"
mul_eq_def: fixes r :: "('b \<times> 'b) set" shows "mul_eq r = {(M, N). \<exists>I J K. M = I + K \<and> N = I + J \<and> set_mset K \<subseteq> r \<down>m J}"
construct_lhs_matches_solve_for_lhs: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list"   and signs :: "rat list list" assumes "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)" shows "w_vec p qs signs = solve_for_lhs p qs subsets (M_mat signs subsets)"
class_add_subcls_rev: fixes D :: "char list"   and D' :: "char list"   and C :: "char list"   and cdec :: "jvm_method class"   and P :: "jvm_prog" assumes "class_add P (C, cdec) \<turnstile> D \<preceq>\<^sup>* D'"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> D \<preceq>\<^sup>* D'"
rep_eq: fixes x :: "integer" shows "Code_Target_Nat.Nat x = nat (int_of_integer x)"
\<beta>\<^sub>\<tau>: shows "\<guillemotleft>\<a>[t\<^sub>1, t\<^sub>0u\<^sub>1.p\<^sub>1, chine] \<cdot> the_\<nu> \<cdot> (local.inv \<omega>.the_\<nu> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1) \<cdot> \<a>\<^sup>-\<^sup>1[t\<^sub>1, \<omega>.chine, r\<^sub>0s\<^sub>1.p\<^sub>1] : t\<^sub>1 \<star> \<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>1 \<star> chine\<guillemotright>"
assms_3: shows "((s, vs), t, ws) \<in> step M"
res_fold_A_\<C>_A_im_fold_A_\<C>_A_im: shows "restrict1 fold_A (\<Union> A) \<turnstile> ChamberComplex.chamber_system A = fold_A \<turnstile> ChamberComplex.chamber_system A"
filterlim_at_bot_dense: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" shows "filterlim f at_bot F = (\<forall>Z. \<forall>\<^sub>F x in F. f x < Z)"
main_prem_of_def: fixes inference :: "'b inference" shows "main_prem_of inference = (case inference of Infer x1 x2 x3 \<Rightarrow> x2)"
strategyproofness: shows "(pmf (sds R35) c < pmf (sds R21) c \<or> pmf (sds R35) c + (pmf (sds R35) b + pmf (sds R35) a) < pmf (sds R21) c + (pmf (sds R21) b + pmf (sds R21) a)) \<or> pmf (sds R35) c = pmf (sds R21) c \<and> pmf (sds R35) c + (pmf (sds R35) b + pmf (sds R35) a) = pmf (sds R21) c + (pmf (sds R21) b + pmf (sds R21) a)"
g_pair_is_pr: shows "g_pair \<in> PrimRec2"
Abs_lgenerator'_inverse: fixes y :: "'c \<Rightarrow> ('d, 'c) step" assumes "y \<in> {g. productive g}" shows "lgenerator (Abs_lgenerator' y) = y"
sfoldl_sappend: fixes f :: "'b \<rightarrow> 'a \<rightarrow> 'b"   and z :: "'b"   and xs :: "[:'a:]"   and ys :: "[:'a:]" assumes "f\<cdot>\<bottom> = \<bottom>" shows "sfoldl\<cdot>f\<cdot>z\<cdot>(xs :@ ys) = sfoldl\<cdot>f\<cdot>(sfoldl\<cdot>f\<cdot>z\<cdot>xs)\<cdot>ys"
iT_Div_card_ge_aux: fixes d :: "nat"   and I :: "nat set"   and n :: "nat" assumes "0 < d"   and "finite I"   and "Max I div d \<le> n" shows "card I div d + (if card I mod d = 0 then 0 else Suc 0) \<le> card (I \<oslash> d)"
valid_cfgD: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg" shows "cfg \<in> R_G.cfg_on (state cfg)"
of_nat_0_le_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
one_and_eq: fixes a :: "'a" shows "(1::'a) AND a = a mod (2::'a)"
that: fixes u :: "'a \<Rightarrow> real"   and e :: "real" assumes "s (flow0 x (u x)) = 0"   and "u x = return_time {x \<in> S. s x = 0} x"   and "\<And>y. y \<in> cball x e \<Longrightarrow> s (flow0 y (u y)) = 0"   and "continuous_on (cball x e) u"   and "(\<lambda>t. (t, u t)) ` cball x e \<subseteq> Sigma X existence_ivl0"   and "0 < e"   and "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)))) o\<^sub>L (Ds (poincare_map {x \<in> S. s x = 0} x) o\<^sub>L flowderiv x (return_time {x \<in> S. s x = 0} x)) o\<^sub>L embed1_blinfun)) (at x)" shows "thesis"
1: shows "\<guillemotleft>unit a : map\<^sub>0 a \<Rightarrow>\<^sub>D F a\<guillemotright> \<and> D.iso (unit a) \<and> unit a \<cdot>\<^sub>D \<i>\<^sub>D[map\<^sub>0 a] = (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a)) \<cdot>\<^sub>D (unit a \<star>\<^sub>D unit a)"
can_exec_snocD: fixes root :: "(att \<times> char list, att, nat) env"   and xs :: "operation list"   and y :: "operation" assumes "can_exec root (xs @ [y])" shows "\<exists>root' root''. root \<Midarrow>xs\<Rightarrow> root' \<and> root' \<midarrow>y\<rightarrow> root''"
NotAllD: fixes v :: "i"   and \<phi> :: "'d \<Rightarrow> \<o>" assumes "\<not> [\<^bold>\<forall>x. \<phi> x in v]" shows "\<exists>x. \<not> [\<phi> x in v]"
subst_eqD: shows "Fsubst (foldr (||) \<Delta> FF) \<sigma> = foldr (||) (map (\<lambda>\<phi>. Fsubst \<phi> \<sigma>) \<Delta>) FF"
pCons_1: shows "a \<noteq> (0::'c) \<or> p \<noteq> 0"
vimage_is_vempty: fixes r :: "V"   and A :: "V" shows "(r `\<^sub>\<circ> A = 0) = vdisjnt (\<D>\<^sub>\<circ> r) A"
l7_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B" shows "Tarski_neutral_dimensionless.Midpoint Bet Cong M B A"
finite_rvars: fixes c :: "com" shows "finite (rvars c)"
external_invariant_preservation: fixes Sa :: "('f, 'e) spec"   and Q :: "'e set"   and pi :: "'g \<Rightarrow> 'e"   and Sc :: "('h, 'g) spec" assumes "oreach Sa \<subseteq> Q"   and "implements pi Sa Sc" shows "pi ` oreach Sc \<subseteq> Q"
map_G_rel_cong: fixes L1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and L2 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra2 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and xf :: "('n, 'p, 'r, 't, 'v, 'x, 'z) G"   and yf :: "('o, 'q, 's, 'u, 'w, 'y, 'z) G"   and L1' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and x :: "'n \<Rightarrow> 'aa"   and y :: "'o \<Rightarrow> 'ab"   and L2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and xa :: "'p \<Rightarrow> 'ac"   and ya :: "'q \<Rightarrow> 'ad"   and Co1' :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and xb :: "'r \<Rightarrow> 'ae"   and yb :: "'s \<Rightarrow> 'af"   and Co2' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and xc :: "'t \<Rightarrow> 'ag"   and yc :: "'u \<Rightarrow> 'ah"   and Contra1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and xd :: "'ai \<Rightarrow> 'v"   and yd :: "'aj \<Rightarrow> 'w"   and Contra2' :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and xe :: "'ak \<Rightarrow> 'x"   and ye :: "'al \<Rightarrow> 'y" assumes "rel_G L1 L2 Co1 Co2 Contra1 Contra2 xf yf"   and "\<And>x y. L1 x y \<Longrightarrow> L1' (x x) (y y)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xb x) (yb y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xc x) (yc y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xd x) (yd y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xe x) (ye y)" shows "rel_G L1' L2' Co1' Co2' Contra1' Contra2' (map_G x xa xb xc xd xe xf) (map_G y ya yb yc yd ye yf)"
vector_mult_component: fixes x :: "('b, 'c) vec"   and y :: "('b, 'c) vec"   and i :: "'c" shows "(x * y) $ i = x $ i * y $ i"
IH2: assumes "Vagree \<nu> \<nu>' (FVDiff t2)"   and "Iagree I J {Inl x |x. x \<in> SIGT t2}" shows "frechet I t2 (fst \<nu>) (snd \<nu>) = frechet J t2 (fst \<nu>') (snd \<nu>')"
monad_state_prob_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_state_prob return bind get put sample \<equiv> monad_state return bind get put \<and> monad_prob return bind sample \<and> monad_state_prob_axioms get sample"
induct: fixes P :: "MIR.num \<Rightarrow> bool"   and a0 :: "MIR.num" assumes "\<And>j. P (C j)"   and "\<And>n c t. (\<And>x. P t) \<Longrightarrow> P (CN n c t)"   and "\<And>c t s. (\<And>x. P s) \<Longrightarrow> P (CF c t s)"   and "\<And>c t. (\<And>x. P t) \<Longrightarrow> P (Mul c t)"   and "\<And>v. P (Bound v)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v. P (Floor v)" shows "P a0"
meet_connection: fixes x :: "nat"   and y :: "nat" shows "(x \<le> y) = (nat.meet x y = x)"
rb_aux_is_sig_GB_in: fixes z :: "nat"   and u :: "'t" shows "is_sig_GB_in dgrad (set (fst (fst (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z))))) u"
simps: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and a1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "\<R>p \<Gamma> a1 a2 = ((\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> (case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<R>\<^sub>1p \<Gamma> x xa xaa xaaa xaaaa xaaaaa) xa) lc') \<or> (\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> (case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<R>\<^sub>2p \<Gamma> x xa xaa xaaa xaaaa xaaaaa) xa) lc') \<or> (\<exists>lc lc'. a1 = lc \<and> a2 = lc' \<and> lc \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> lc'))"
prefs_from_table_wfI: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list" assumes "agents \<noteq> {}"   and "alts \<noteq> {}"   and "distinct (map fst xss)"   and "set (map fst xss) = agents"   and "\<And>xs. xs \<in> set (map snd xss) \<Longrightarrow> \<Union> (set xs) = alts"   and "\<And>xs. xs \<in> set (map snd xss) \<Longrightarrow> is_finite_weak_ranking xs" shows "prefs_from_table_wf agents alts xss"
wf_word_append: fixes \<Sigma> :: "nat \<Rightarrow> 'b set"   and n :: "nat"   and ws :: "'b list"   and vs :: "'b list" shows "alphabet.wf_word \<Sigma> n (ws @ vs) = (alphabet.wf_word \<Sigma> n ws \<and> alphabet.wf_word \<Sigma> n vs)"
norm_column_divide_by_norm: fixes a :: "'c"   and A :: "(('a, 'c) vec, 'b) vec" assumes "column a A \<noteq> 0" shows "norm (column a (divide_by_norm A)) = 1"
tm_abs_decomp: shows "tm_abs prec I a t = TaylorModel (mid abs_bound)\<^sub>p (centered abs_bound)"
arc_balancedI_trail: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "trail u p v" shows "arc_balanced u p v"
image_atLeastZeroLessThan_int: fixes u :: "int" assumes "0 \<le> u" shows "{0..<u} = int ` {..<nat u}"