hcju/isabelleps huggingface dataset:
Rep_lifted_boolean_algebra_cases: fixes y :: "('c, 'd) phi \<Rightarrow> 'c" assumes "y \<in> {xf. True}" obtains x :: "('c, 'd) lifted_boolean_algebra" where "y = Rep_lifted_boolean_algebra x"
div_left_ineq: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x < y * z" shows "x div z < y"
Trivial: fixes a :: "ennreal" assumes "a \<noteq> \<top>" shows "a = (1 + a) * (a / (1 + a))"
swapEnv_defs_2: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "FixVars TYPE('a) TYPE('b)" shows "f \<circ> g = (\<lambda>x. f (g x))"
invertible_times_non_zero: fixes M :: "((real, 'a) vec, 'a) vec"   and v :: "(real, 'a) vec" assumes "invertible M"   and "v \<noteq> 0" shows "M *v v \<noteq> 0"
GMVT: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "a < b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"   and "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable at x"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"   and "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable at x" shows "\<exists>g'c f'c c. (g has_real_derivative g'c) (at c) \<and> (f has_real_derivative f'c) (at c) \<and> a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
f_repr: shows "list_repr 0 ([], fs_init) (map ((!) fs_init) [0..<m])"
diamond_bot_right_edge_def: shows "diamond_bot_right_edge = (1, \<lambda>x. (diamond_x (1 / 2 * x + 1 / 2), - (- diamond_x (1 / 2 * x + 1 / 2) + d / 2)))"
fst_gb_schema_dummy_components: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set" assumes "struct_spec sel ap ab compl"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)" shows "component_of_term ` Keys (fst ` set (fst (gb_schema_dummy data D bs ps))) = component_of_term ` Keys (args_to_set (gs, bs, ps))"
rank_def: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and v :: "nat \<times> 'b" shows "rank A w v \<equiv> GREATEST k. v \<in> graph A w k"
assms_2: fixes x :: "'c" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x" shows "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
per_rootD: fixes u :: "'b list"   and r :: "'b list" assumes "u \<le>p r\<^sup>\<omega>" shows "u \<le>p r \<cdot> u"
dfs_invariant: fixes S :: "'b"   and xs :: "'a list"   and I :: "'b \<Rightarrow> bool" assumes "invariant S"   and "list_all is_node xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> memb x S; is_node x; invariant S; I S\<rbrakk> \<Longrightarrow> I (ins x S)" shows "I (dfs S xs)"
Basic: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and f :: "'f \<Rightarrow> 'f"   and Q :: "'f set"   and A :: "'f set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. f s \<in> Q} Basic f Q,A"
wtl_pres: fixes "is" :: "'b list"   and c :: "'a list"   and s :: "'a"   and pc :: "nat" assumes "pres_type step (length is) A"   and "cert_ok c (length is) \<top> \<bottom> A"   and "s \<in> A"   and "wtl is c 0 s \<noteq> \<top>"   and "pc < length is" shows "wtl (take pc is) c 0 s \<in> A"
empty_iff: fixes c :: "'l" shows "(c \<in> {}) = False"
rep_eq: fixes x :: "('a, 'b) fmap" shows "lookup (Pm_fmap x) = lookup0 x"
measure_UNION: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "disjoint_family A"   and "emeasure M (\<Union> (range A)) \<noteq> \<infinity>" shows "(\<lambda>i. Sigma_Algebra.measure M (A i)) sums Sigma_Algebra.measure M (\<Union> (range A))"
ChamberComplexAutomorphism_axioms_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ChamberComplexAutomorphism_axioms X f \<equiv> \<forall>v. v \<notin> \<Union> X \<longrightarrow> f v = v"
prm_frame_def: fixes pi :: "('e \<times> 'e) list"   and x :: "'d frame" shows "pi \<bullet> x \<equiv> Abs_frame (pi \<bullet> Rep_frame x)"
lm046: fixes X :: "'a set"   and Y :: "'a set"   and R :: "('a \<times> 'b) set" assumes "X \<inter> Y = {}" shows "R `` X = Outside' Y R `` X"
LeftDerivationIntro_implies_LeftDerivation: fixes \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationIntro \<alpha> i r ix D j \<gamma>" shows "LeftDerivation \<alpha> ((i, r) # D) \<gamma>"
make_set_nil: fixes p :: "nat \<Rightarrow> 'a \<Rightarrow> bool" shows "make_set 0 p = {}"
mismatchNil: fixes a :: "name"   and P :: "pi" shows "[a\<noteq>a]P \<sim> \<zero>"
lottery_prob_carrier: fixes p :: "'a pmf"   and A :: "'a set" assumes "p \<in> lotteries_on A" shows "measure_pmf.prob p A = 1"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> cs : [] _> [T_i32]"
coprime_hom: fixes x :: "'a poly poly"   and y' :: "'a poly poly" shows "comm_monoid_mult_class.coprime (poly_y_x x) y' = comm_monoid_mult_class.coprime x (inv poly_y_x y')"
mat_monoid_def: fixes nr :: "nat"   and nc :: "nat"   and b :: "'d" shows "mat_monoid nr nc b \<equiv> \<lparr>carrier = Collect (mat nr nc), mult = mat_plus, one = mat0 nr nc, \<dots> = b\<rparr>"
inserted_ids_last_del: fixes xs :: "('c \<times> ('c, 'd) list_op) list"   and oid :: "'c"   and ref :: "'c" shows "inserted_ids (xs @ [(oid, Delete ref)]) = inserted_ids xs"
inf_sup_aci_1: fixes inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and x :: "'h"   and y :: "'h" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
prepend_queue_rep: fixes a :: "'a"   and q :: "'a queue" shows "linearize (prepend_queue a q) = a # linearize q"
assms: fixes i :: "nat" shows "fresh_floatariths ea i \<or> i < length ys \<and> i < length zs \<and> ys ! i = zs ! i"
scaleQ_scaleQ: fixes a :: "'a"   and b :: "'a"   and x :: "'a['b, 'c]" shows "a *\<^sub>Q b *\<^sub>Q x = a \<cdot> b *\<^sub>Q x"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs \<down> Suc i = xs \<down> i @ [xs ! i]"
take_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list"   and y :: "'b" assumes "n \<le> m" shows "take n (xs[m := y]) = take n xs"
Eprim: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (EPrim f e1 e2) \<rho> = {v. \<exists>n1 n2. VNat n1 \<in> E e1 \<rho> \<and> VNat n2 \<in> E e2 \<rho> \<and> v = VNat (f n1 n2)}"
empty_correct: fixes q :: "('c, 'd) BinomialHeap" shows "(to_mset q = {#}) = (q = local.empty)"
add_mset_add_single: fixes a :: "'a"   and A :: "'a multiset" shows "add_mset a A = A + {#a#}"
le_mb: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<le> mb"
init_imp_inv_init_imp_prop_safe: fixes c :: "('b, 't, 'loc) Combined.configuration" assumes "InitConfig c" shows "inv_init_imp_prop_safe c"
dim_a: shows "dim_row A = dim_row (two_block_non_interfering A B) - n"
gkernTr2: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e"   and y :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> gker\<^bsub>F,G\<^esub> f"   and "y \<in> gker\<^bsub>F,G\<^esub> f" shows "x \<cdot>\<^bsub>F\<^esub> y \<in> gker\<^bsub>F,G\<^esub> f"
weak_complete_lattice_axioms: shows "weak_complete_lattice (fpl \<P> \<H>)"
invariant_factor_decomposition_idirprod: obtains gs :: "'a list" where "set gs \<subseteq> carrier G" and "cycle gs" and "is_idirprod (carrier G) (\<lambda>g. generate G {g}) (set gs)" and "successively (dvd) (map ord gs)" and "card (set gs) \<le> card gen" and "\<one> \<notin> set gs"
Alls_def: shows "Alls = {(FAll Pos A # Gamma, {instanceF x A # Gamma}) |A x Gamma. x \<notin> freeVarsFL (FAll Pos A # Gamma)}"
hyps_1: fixes i :: "nat" assumes "n = j - (i + 1)"   and "trace init t final"   and "i < j"   and "j \<le> length t"   and "last (msgs (S t i) cid) = Marker"   and "Marker \<in> set (msgs (S t i) cid)"   and "Marker \<in> set (msgs (S t j) cid)"   and "\<forall>k. i \<le> k \<and> k < j \<longrightarrow> (\<nexists>u u' m. t ! k = Send cid p q u u' m)"   and "channel cid = Some (p, q)" shows "last (msgs (S t j) cid) = Marker"
t_list_of_B_log_bound: fixes t :: "'b tree" assumes "braun t" shows "t_list_of_B t \<le> 3 * (nat \<lceil>log 2 (real (size t + 1))\<rceil> + 1) * size t"
table_of_fieldsD: fixes C :: "qtname"   and c :: "'a cbody_scheme"   and efn :: "fspec"   and f :: "Decl.field" assumes "table_of (map (\<lambda>(fn, ft). ((fn, C), ft)) (cfields c)) efn = Some f" shows "declclassf efn = C \<and> table_of (cfields c) (fname efn) = Some f"
mod_eq_dvd_iff: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "(modulo a c = modulo b c) = dvd.dvd times c (minus a b)"
PiM_eqI: fixes M :: "'c \<Rightarrow> 'd measure"   and I :: "'c set"   and P :: "('c \<Rightarrow> 'd) measure" assumes "product_sigma_finite M"   and "finite I"   and "sets P = sets (Pi\<^sub>M I M)"   and "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" shows "P = Pi\<^sub>M I M"
moebius_commute: fixes x :: "'a"   and n :: "nat" shows "x * moebius_mu n = moebius_mu n * x"
induced_pow_fun_subset_ordering_iso: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A" shows "OrderingSetIso dual_order.greater_eq dual_order.greater dual_order.greater_eq dual_order.greater (Pow A) ((`) f)"
trans: shows "\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. R x y \<longrightarrow> R y z \<longrightarrow> R x z"
subtop_lemma: fixes A :: "'b set set"   and B :: "'b set set" assumes "A = (\<Union>t\<in>B. {t \<inter> \<Union> A})" shows "subtopology (topo A) (topo B)"
normalize_keep_corthogonal: fixes vs :: "complex vec list"   and n :: "nat" assumes "corthogonal vs"   and "set vs \<subseteq> carrier_vec n" shows "corthogonal (map vec_normalize vs)"
IH: fixes y :: "V" assumes "y \<in> elts x" shows "Transset (TC y)"
real_linear: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "less a b \<or> a = b \<or> less b a"
case_prod_beta': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
subst_bv2_preserves_beta: fixes r :: "Core.term"   and s :: "Core.term"   and k :: "nat"   and u :: "Core.term" assumes "r \<rightarrow>\<^sub>\<beta> s" shows "subst_bv2 r k u \<rightarrow>\<^sub>\<beta> subst_bv2 s k u"
list_ex_length: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "list_ex P xs = (\<exists>n<length xs. P (xs ! n))"
acc_inv_image: fixes r :: "('b \<times> 'b) set"   and f :: "'c \<Rightarrow> 'b" assumes "acc r" shows "acc (inv_image r f)"
is_extensional: fixes f :: "'a" assumes "\<not> A.arr f" shows "G f = B.null"
semiring_normalization_rules_15: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lx :: "'a"   and ly :: "'a"   and rx :: "'a"   and ry :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times (times lx ly) (times rx ry) = times rx (times (times lx ly) ry)"
simps_109: fixes x9 :: "Term.var"   and x41 :: "expr"   and x42 :: "ref_ty" shows "Acc x9 \<noteq> x41 InstOf x42"
Cond: fixes P :: "'d set"   and b :: "'d set"   and P\<^sub>1 :: "'d set"   and P\<^sub>2 :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "P \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Cond b c\<^sub>1 c\<^sub>2 Q,A"
has_vector_derivative_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real"   and x :: "real"   and X :: "real set" shows "(part_circlepath z r s t has_vector_derivative \<i> * complex_of_real r * (complex_of_real t - complex_of_real s) * exp (\<i> * complex_of_real (linepath s t x))) (at x within X)"
get_child_nodes_is_l_get_child_nodes: shows "l_get_child_nodes DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs"
assms_1: shows "f1 \<in> \<Theta>[F](g1)"
Inter_UNIV_conv_1: fixes A :: "'b set set" shows "(\<Inter> A = UNIV) = (\<forall>x\<in>A. x = UNIV)"
a_obs'': shows "action_obs (justifying_exec (J (Suc n))) a'' = NormalAction (ReadMem ad al v'')"
assms_3: shows "rep_list (sig_trd bs (poly_of_pair p)) \<noteq> 0"
tt_monom_mult: fixes c :: "'c"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "p \<noteq> 0" shows "punit.tt (local.punit.monom_mult c t p) = local.punit.splus t (punit.tt p)"
simps_3: fixes z :: "octo" shows "Im2 (Octonions.cnj z) = - Im2 z"
valid_node_cases: fixes prog :: "cmd"   and n :: "w_node" assumes "While_CFG.valid_node prog n" obtains "n = (_Entry_)"   | "n = (_Exit_)"   | "While_CFGExit.inner_node prog n"
assms_3: shows "\<forall>x\<in>U\<^sub>3. g x \<in> U"
intros_1: fixes G :: "'b list set" shows "\<epsilon> \<in> \<langle>G\<rangle>"
R_idem2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R (R P) = R P"
range_vars_subst_compose_subset: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "range_vars (\<sigma> \<circ>\<^sub>s \<tau>) \<subseteq> range_vars \<sigma> - subst_domain \<tau> \<union> range_vars \<tau>"
Residues_Mult_def: fixes p :: "'a" shows "Residues_Mult p = \<lparr>carrier = {x \<in> {1::'a..p}. coprime x p}, monoid.mult = \<lambda>x y. x * y mod p, one = 1::'a\<rparr>"
der_g: fixes x :: "(real, 'm) vec" assumes "x \<in> S" shows "(g has_derivative g' x) (at x within S)"
singular_chain_def: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_chain p X c \<equiv> Poly_Mapping.keys c \<subseteq> singular_simplex_set p X"
snd_echelon_form_of_column_k_det_eq: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and n :: "'b"   and A :: "(('b, 'c) vec, 'd) vec"   and i :: "nat"   and k :: "nat" shows "snd (echelon_form_of_column_k_det bezout (n, A, i) k) = echelon_form_of_column_k bezout (A, i) k"
intFF_def: fixes f :: "'fsym"   and al :: "univ list" shows "intFF f al \<equiv> toFull (resOf f) (intF f (map2 fromFull (arOf f) al))"
n_zeros_typeof: fixes ts :: "t list"   and vs :: "v list" assumes "n_zeros ts = vs" shows "ts = map typeof vs"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>D f = n \<and> f \<cdot>\<^sub>D n = n"
is_morphism_of_sheaves: shows "morphism_sheaves_of_rings Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf \<phi>\<^sub>f"
fps_deriv_fps_integral: fixes a :: "'a fps"   and a0 :: "'a" shows "fps_deriv (fps_integral a a0) = a"
homomorphism: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "return_spmf f \<diamondop> return_spmf x = return_spmf (f x)"
map_pre_coGcoF\<^sub>M_def: shows "map_pre_coGcoF\<^sub>M \<equiv> \<lambda>f1 f2 f3. BNF_Composition.id_bnf \<circ> map_coF0 f2 f3 \<circ> BNF_Composition.id_bnf"
hd: shows "hd \<HH> = {\<one>\<^bsub>G\<^esub>}"
mult_cancel_left: fixes f1 :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "f1 \<in> \<Theta>[F](g1)"   and "\<forall>\<^sub>F x in F. g1 x \<noteq> (0::'b)" shows "((\<lambda>x. f1 x * f2 x) \<in> O[F](\<lambda>x. g1 x * g2 x)) = (f2 \<in> O[F](g2))"
apath_append_iff: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list"   and v :: "'a" shows "G.apath u (p @ q) v = (G.apath u p (last (pawalk_verts u p)) \<and> G.apath (last (pawalk_verts u p)) q v \<and> set (pawalk_verts u p) \<inter> set (tl (pawalk_verts (last (pawalk_verts u p)) q)) = {})"
sym_Un_converse: fixes r :: "('b \<times> 'b) set" shows "sym (r\<^sup>\<leftrightarrow>)"
vmc_path_axioms: shows "vmc_path G P v0 p (\<sigma>(v := w))"
sup1CI: fixes B :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a \<Rightarrow> bool" assumes "\<not> B x \<Longrightarrow> A x" shows "sup A B x"
subterm_trm_ord_eq: fixes t :: "'a trm"   and p :: "indices list"   and s :: "'a trm" assumes "subterm t p s" shows "s = t \<or> (s, t) \<in> trm_ord"
measurable_translation_subtract: fixes S :: "'a set"   and a :: "'a" assumes "S \<in> lmeasurable" shows "(\<lambda>x. x - a) ` S \<in> lmeasurable"
nat_ivl_Suc': fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "m \<le> Suc n" shows "sum g {m..Suc n} = g (Suc n) + sum g {m..n}"
elims_2: fixes x :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xa :: "'e option"   and xb :: "'e option" assumes "option_le x xa xb" obtains P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and y :: "'e option" where "x = P__" and "xa = None" and "xb = y__"   | P :: "'e \<Rightarrow> 'e \<Rightarrow> bool" and xc :: "'e" and y :: "'e" where "x = P" and "xa = Some xc" and "xb = Some y" and "P xc y"
assms_1: shows "OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) less_eq'' less'' Q g"
zero_i_suc_k: shows "is_zero_row_upt_k i (Suc k) B"
add: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "f \<in> R"   and "g \<in> R" shows "(\<lambda>x. f x + g x) \<in> R"
smaps_over_singleton: fixes k :: "'a"   and A :: "'a set"   and vs :: "'b set"   and B :: "'b set" assumes "k \<in> A"   and "vs \<subseteq> B" shows "{ k := vs}. \<in> smaps_over A B"
cases: fixes a1 :: "'node SDG_node"   and a2 :: "'node SDG_node list"   and a3 :: "'node SDG_node" assumes "a1 is-a2\<rightarrow>\<^sub>d* a3" obtains n :: "'node SDG_node" where "a1 = na__" and "a2 = []" and "a3 = na__" and "valid_SDG_node na__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and n' :: "'node SDG_node" where "a1 = nb__" and "a2 = nsa__ @ [n''a__]" and "a3 = n'a__" and "nb__ is-nsa__\<rightarrow>\<^sub>d* n''a__" and "n''a__ s\<longrightarrow>\<^bsub>cd\<^esub> n'a__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and V :: "'var" and n' :: "'node SDG_node" where "a1 = nc__" and "a2 = nsb__ @ [n''b__]" and "a3 = n'b__" and "nc__ is-nsb__\<rightarrow>\<^sub>d* n''b__" and "n''b__ s-V\<rightarrow>\<^sub>d\<^sub>d n'b__" and "n''b__ \<noteq> n'b__"   | n :: "'node SDG_node" and ns :: "'node SDG_node list" and n'' :: "'node SDG_node" and p :: "'pname" and n' :: "'node SDG_node" where "a1 = n" and "a2 = ns @ [n'']" and "a3 = n'" and "n is-ns\<rightarrow>\<^sub>d* n''" and "n'' s-p\<rightarrow>\<^bsub>sum\<^esub> n'"
lt_in_keys: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0" shows "lt p \<in> keys p"
f_Exec_Stream_take: fixes n :: "nat"   and trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp_Stream trans_fun xs c \<down> n = f_Exec_Comp_Stream trans_fun (xs \<down> n) c"
ask_state_def: fixes ask :: "('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and f :: "'f \<Rightarrow> ('d, 'e) stateT" shows "ask_state ask f = StateT (\<lambda>s. ask (\<lambda>r. run_state (f r) s))"
hv_cmult: fixes a :: "real"   and x :: "(real, 3) vec" shows "Hv (a *\<^sub>R x) = quat_of_real a * Hv x"
pop_hd: fixes a :: "'a"   and u :: "'a list" shows "g (a # u) = g [a] \<cdot> g u"
flip_Cons_pos: fixes n :: "nat"   and x :: "bool"   and xs :: "bool list" assumes "0 < n" shows "flip (x # xs) n = x # flip xs (n - 1)"
set_mrbst_push_down: fixes t' :: "'a tree"   and t1 :: "'a tree"   and x :: "'a"   and t2 :: "'a tree" assumes "t' \<in> set_pmf (mrbst_push_down t1 x t2)"   and "bst t1"   and "bst t2"   and "\<And>y. y \<in> set_tree t1 \<Longrightarrow> y < x"   and "\<And>y. y \<in> set_tree t2 \<Longrightarrow> x < y" shows "set_tree t' = {x} \<union> set_tree t1 \<union> set_tree t2"
lt_max: fixes p :: "('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x option \<Rightarrow>\<^sub>0 nat" assumes "lookup p v \<noteq> (0::'b)" shows "pp_pm.extended_ord v (pp_pm.extended_ord.lpp p)"
Gets_imp_knows_Spy_parts: fixes B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Gets B X \<in> set evs"   and "evs \<in> ns_public" shows "X \<in> parts (knows Spy evs)"
poly_list_rel_def: fixes R :: "('a \<times> char list multiset) set" shows "poly_list_rel R = {(xs, ys). (xs, ys) \<in> \<langle>R \<times>\<^sub>r int_rel\<rangle>list_rel O list_mset_rel \<and> 0 \<notin># snd `# ys}"
fold_Nil: fixes f :: "'h \<Rightarrow> 'g \<Rightarrow> 'g" shows "fold f [] = id"
all_edge_cases: shows "\<gamma> = (\<lambda>x. twoC (x, 0)) \<or> \<gamma> = (\<lambda>y. twoC (1, y)) \<or> \<gamma> = (\<lambda>x. twoC (x, 1)) \<or> \<gamma> = (\<lambda>y. twoC (0, y))"
ThrowRule: fixes P :: "'d set"   and A :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and F :: "'f set"   and Q :: "'d set" assumes "P \<subseteq> A" shows "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnExpr P) Throw Q, A"
eval_on_carrier: fixes p :: "nat \<Rightarrow> 'a"   and S :: "('c, 'd) ring_scheme"   and phi :: "'a \<Rightarrow> 'c"   and s :: "'c" assumes "p \<in> carrier P" shows "eval R S phi s p = (\<Oplus>\<^bsub>S\<^esub>i\<in>{..deg R p}. phi (coeff P p i) \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> i)"
asymptotics: fixes x :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> x"   and "b \<in> set bs" shows "akra_bazzi_asymptotics b hb e p x"
mult_zero_left: fixes x :: "'a" assumes "x \<in> E" shows "0 \<cdot> x = (0::'a)"
ms_aux_hd_exp_MSLNil: shows "ms_aux_hd_exp MSLNil = None"
trace_sets_def: fixes s :: "char list"   and X :: "'c set" shows "trace_sets s X = sets_of_coll (X ::: clw_rel appr_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN ()) (\<lambda>X. RETURN (trace_set s (Some X))) (\<lambda>_ _. RETURN ()))"
comp_def: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'd" shows "f \<circ> g = (\<lambda>x. f (g x))"