hcju/isabelleps huggingface dataset:
terminatess_elim_cases_2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile>c # cs,css \<Down> t" obtains "\<Gamma>\<turnstile>c \<down> t" and "\<forall>ta. \<Gamma>\<turnstile> \<langle>c,t\<rangle> \<Rightarrow> ta \<longrightarrow> \<Gamma>\<turnstile>cs,css \<Down> ta"
span_span: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "module scale" shows "module.span scale (module.span scale A) = module.span scale A"
precompact_locally_finite_open_coverE: obtains W :: "nat \<Rightarrow> 'c set" where "manifold_eucl.dest.carrier = \<Union> (range W)" and "\<And>i. open (W i)" and "\<And>i. compact (closure (W i))" and "\<And>i. closure (W i) \<subseteq> manifold_eucl.dest.carrier" and "locally_finite_on manifold_eucl.dest.carrier UNIV W"
localRelS_def: shows "localRelS = (\<lambda>(u, v) (u', v') x y. u = 0 \<and> u' = (1::'a) \<and> v = v')"
dualization_rew_2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<not> \<diamond>F \<equiv> \<box>\<not> F"
q_Ind_def: fixes k :: "hf" shows "q_Ind k \<equiv> \<langle>htuple 6, k\<rangle>"
p_strict_mono: fixes x :: "real"   and y :: "real" assumes "x < y"   and "y \<le> t_stop" shows "p x < p y"
ivl_of_aform_err: fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e X" shows "x \<in>\<^sub>r ivl_of_aform_err p X"
AE_cong: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. x \<in> space M \<Longrightarrow> P x = Q x" shows "almost_everywhere M P = almost_everywhere M Q"
map_ext: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
assms: shows "A' \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>"
subdensity_finite_entropy: fixes T :: "'b \<Rightarrow> 'f"   and P :: "'b measure"   and Q :: "'f measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real"   and Y :: "'a \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> real" assumes "T \<in> P \<rightarrow>\<^sub>M Q"   and "finite_entropy P X f"   and "finite_entropy Q Y g"   and "Y = T \<circ> X" shows "AE x in P. g (T x) = 0 \<longrightarrow> f x = 0"
eval_G: fixes s :: "complex" shows "eval_fps G (s - 1) = g s"
assms_3: shows "k \<star> v \<cong> h \<star> u"
degree_map_poly_hom: fixes p :: "'a mod_ring poly" shows "degree (to_int_poly p) = degree p"
gbm_nodes_list_it_def: fixes g :: "'m1" shows "gbm_nodes_list_it g \<equiv> map_iterator_dom (m1.iteratei g)"
eNorm_cmult: fixes N :: "'b quasinorm"   and c :: "real"   and x :: "'b" shows "eNorm N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * eNorm N x"
abs_divide: fixes a :: "'a"   and b :: "'a" shows "\<bar>a / b\<bar> = \<bar>a\<bar> / \<bar>b\<bar>"
forward_terminating_iff2: fixes x :: "'a" assumes "path x" shows "forward_terminating x = (x\<^sup>T \<le> x\<^sup>\<star> ; - (x ; 1))"
K: fixes x :: "'d" shows "norm (g x) \<le> norm x * K"
prems_1: shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal r"
simps_21: fixes v1 :: "val"   and v2 :: "val" shows "eval_binop CondAnd v1 v2 = Bool (the_Bool v1 \<and> the_Bool v2)"
isomorphic_implies_hpar_4: fixes f :: "'a"   and f' :: "'a" assumes "B.isomorphic f f'" shows "trg\<^sub>B f = trg\<^sub>B f'"
simps_1: fixes x1 :: "'a"   and x2 :: "'c"   and y1 :: "'a"   and y2 :: "'c" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
f_2: shows "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"
prime_power_leD: fixes p :: "nat"   and i :: "nat" assumes "real p ^ i \<le> x"   and "prime p"   and "0 < i" shows "real p \<le> x"
bij_betw_inv_into: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "bij_betw (inv_into A f) B A"
min_less_iff_conj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and z :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
c3: assumes "ip \<in> kD rt"   and "sqn rt ip = \<pi>\<^sub>2 r"   and "\<pi>\<^sub>5 r < the (dhops rt ip)" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
one_chain_j_integrals: shows "one_chain_line_integral F {j} one_chain_typeI = one_chain_line_integral F {j} one_chain_typeII \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeI. line_integral_exists F {j} \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>one_chain_typeII. line_integral_exists F {j} \<gamma>)"
distinct_1: fixes x1 :: "\<omega>"   and x2 :: "\<Pi>\<^sub>1 set" shows "\<omega>\<nu> x1 \<noteq> \<alpha>\<nu> x2"
perm_length: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs <~~> ys" shows "length xs = length ys"
DynCom: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "\<Gamma>\<turnstile> (DynCom c # cs, css, Normal s) \<rightarrow> (c s # cs, css, Normal s)"
add_mono_thms_linordered_semiring_4: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i = j \<and> k = l" shows "i + k = j + l"
assms_2: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "B \<subseteq>\<^sub>\<circ> x"
projection_split_first_with_suffix: fixes \<tau> :: "'b list"   and X :: "'b set"   and x :: "'b"   and xs :: "'b list" assumes "\<tau> \<upharpoonleft> X = x # xs" shows "\<exists>\<alpha> \<beta>. \<tau> = \<alpha> @ [x] @ \<beta> \<and> \<alpha> \<upharpoonleft> X = [] \<and> \<beta> \<upharpoonleft> X = xs"
only_new_introduces_lids: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and r :: "'a"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "substitution.revision_step subst r s s'"   and "\<And>\<sigma> \<tau> \<E> v. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Ref (VE v)])" shows "LID\<^sub>G s' \<subseteq> LID\<^sub>G s"
INF_id_eq: fixes Inf :: "'c set \<Rightarrow> 'c"   and A :: "'c set" shows "Inf (id ` A) = Inf A"
updates_1: fixes k :: "nat" assumes "k < m" shows "gs2.gso k = (if k = i - 1 then gs1.gso i + gs1.\<mu> i (i - 1) \<cdot>\<^sub>v gs1.gso (i - 1) else if k = i then gs1.gso (i - 1) - map of_int_hom.vec_hom fs ! (i - 1) \<bullet> gs2.gso (i - 1) / \<parallel>gs2.gso (i - 1)\<parallel>\<^sup>2 \<cdot>\<^sub>v gs2.gso (i - 1) else gs1.gso k)"
Ord_mem_iff_lt: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" shows "(k \<^bold>\<in> l) = (k < l)"
borel_measurable_count_space: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" shows "f \<in> borel_measurable (count_space S)"
ANR_def: fixes S :: "'b set" shows "ANR S \<equiv> \<forall>U S'. S homeomorphic S' \<and> closedin (top_of_set U) S' \<longrightarrow> (\<exists>T. openin (top_of_set U) T \<and> S' retract_of T)"
NOT_IN_PRE_EFF_NOT_IN_PRE_EFF_PROJ: fixes v :: "'d"   and a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" shows "(v \<notin> fmdom' (fst a) \<longrightarrow> v \<notin> fmdom' (fst (action_proj a vs))) \<and> (v \<notin> fmdom' (snd a) \<longrightarrow> v \<notin> fmdom' (snd (action_proj a vs)))"
expr_typing_opE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and oper :: "pdf_operator"   and e :: "expr"   and t :: "pdf_type" assumes "\<Gamma> \<turnstile> oper $$ e : t" obtains ta :: "pdf_type" where "\<Gamma> \<turnstile> e : ta" and "op_type oper ta = Some t"
one_le_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq one a" shows "less_eq one (power.power one times a n)"
product_aforms_refine: shows "(product_aforms, product_listset) \<in> aforms_rel \<rightarrow> aforms_rel \<rightarrow> aforms_rel"
fw_int_aux_b: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int M n"   and "a \<le> n"   and "b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n 0 b c i j \<noteq> \<infinity>" shows "get_const (fw M n 0 b c i j) \<in> \<int>"
monom_subst_def: fixes \<sigma> :: "'f \<Rightarrow> ('d monom \<times> 'e) list"   and m :: "'f monom" shows "monom_subst \<sigma> m = monom_list_subst \<sigma> (monom_list m)"
scalar_prod_uminus_left: fixes v :: "'b vec"   and w :: "'b vec" assumes "dim_vec v = dim_vec w" shows "- v \<bullet> w = - (v \<bullet> w)"
set_between_id: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "set (between xs x x) = set xs - {x}"
below_sctn_coll_def: fixes XS :: "'c set"   and sctn :: "'c sctn" shows "below_sctn_coll XS sctn = (sets_of_coll XS ::: \<langle>\<langle>appr_rel\<rangle>list_wset_rel\<rangle>nres_rel) \<bind> (\<lambda>XS. FORWEAK XS (RETURN True) (\<lambda>X. below_sctn X sctn) (\<lambda>a b. RETURN (a \<and> b)))"
StdOMap_intro: fixes ops :: "('b, 'c, 'a, 'd) omap_ops_scheme" assumes "map_empty (map_op_\<alpha> ops) (map_op_invar ops) (map_op_empty ops)"   and "map_lookup (map_op_\<alpha> ops) (map_op_invar ops) (map_op_lookup ops)"   and "map_update (map_op_\<alpha> ops) (map_op_invar ops) (map_op_update ops)"   and "map_update_dj (map_op_\<alpha> ops) (map_op_invar ops) (map_op_update_dj ops)"   and "map_delete (map_op_\<alpha> ops) (map_op_invar ops) (map_op_delete ops)"   and "poly_map_iteratei (map_op_\<alpha> ops) (map_op_invar ops) (map_op_list_it ops)"   and "map_sng (map_op_\<alpha> ops) (map_op_invar ops) (map_op_sng ops)"   and "map_restrict (map_op_\<alpha> ops) (map_op_invar ops) (map_op_\<alpha> ops) (map_op_invar ops) (map_op_restrict ops)"   and "MapSpec.map_add (map_op_\<alpha> ops) (map_op_invar ops) (map_op_add ops)"   and "map_add_dj (map_op_\<alpha> ops) (map_op_invar ops) (map_op_add_dj ops)"   and "map_isEmpty (map_op_\<alpha> ops) (map_op_invar ops) (map_op_isEmpty ops)"   and "map_isSng (map_op_\<alpha> ops) (map_op_invar ops) (map_op_isSng ops)"   and "map_ball (map_op_\<alpha> ops) (map_op_invar ops) (map_op_ball ops)"   and "map_bex (map_op_\<alpha> ops) (map_op_invar ops) (map_op_bex ops)"   and "map_size (map_op_\<alpha> ops) (map_op_invar ops) (map_op_size ops)"   and "map_size_abort (map_op_\<alpha> ops) (map_op_invar ops) (map_op_size_abort ops)"   and "map_sel' (map_op_\<alpha> ops) (map_op_invar ops) (map_op_sel ops)"   and "map_to_list (map_op_\<alpha> ops) (map_op_invar ops) (map_op_to_list ops)"   and "list_to_map (map_op_\<alpha> ops) (map_op_invar ops) (map_op_to_map ops)"   and "poly_map_iterateoi (map_op_\<alpha> ops) (map_op_invar ops) (map_op_ordered_list_it ops)"   and "poly_map_rev_iterateoi (map_op_\<alpha> ops) (map_op_invar ops) (map_op_rev_list_it ops)"   and "map_min (map_op_\<alpha> ops) (map_op_invar ops) (map_op_min ops)"   and "map_max (map_op_\<alpha> ops) (map_op_invar ops) (map_op_max ops)"   and "MapSpec.map_to_sorted_list (map_op_\<alpha> ops) (map_op_invar ops) (map_op_to_sorted_list ops)"   and "map_to_rev_list (map_op_\<alpha> ops) (map_op_invar ops) (map_op_to_rev_list ops)" shows "StdOMap ops"
ctxt: shows "\<Theta>' = (\<lambda>\<sigma> p. \<Theta> \<union> (\<Union>q\<in>Procs. \<Union>Z. {(P q Z \<inter> {s. ((s, q), \<sigma>, p) \<in> r}, q, Q q Z, A q Z)}))"
\<sigma>_fresh_pub_img: shows "subterms\<^sub>s\<^sub>e\<^sub>t (subst_range \<sigma>) \<subseteq> {t. {} \<turnstile>\<^sub>c t} - subterms\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>s\<^sub>t S)"
mono_Un: fixes f :: "'b set \<Rightarrow> 'c set"   and A :: "'b set"   and B :: "'b set" assumes "order_class.mono f" shows "f A \<union> f B \<subseteq> f (A \<union> B)"
last_conv_nth: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "last xs = xs ! (length xs - 1)"
cEnum_set_def: shows "CENUM('a set) = (case ID CENUM('a) of None \<Rightarrow> None | Some (enum_a, enum_all_a, enum_ex_a) \<Rightarrow> Some (map set (subseqs enum_a), \<lambda>P. list_all P (map set (subseqs enum_a)), \<lambda>P. list_ex P (map set (subseqs enum_a))))"
x_times_c_geq_y_times_b_split_dotP: fixes X :: "(nat, rat) mapping"   and c :: "rat vec"   and b :: "rat vec"   and x :: "rat vec"   and y :: "rat vec" assumes "\<langle>X\<rangle> \<Turnstile>\<^sub>c x_times_c_geq_y_times_b c b"   and "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X" shows "b \<bullet> y \<le> c \<bullet> x"
stable_no_blocking_onI: fixes X :: "X3 set"   and ds :: "D2 set" assumes "\<And>h X''. \<lbrakk>X'' = CX3h h (X \<union> X''); X'' \<noteq> CX3h h X; X'' \<subseteq> Theorem_9.CD_on ds (X \<union> X'')\<rbrakk> \<Longrightarrow> False" shows "Theorem_9.stable_no_blocking_on ds X"
map_simp: fixes f :: "'j" assumes "\<chi>.J.arr f" shows "\<chi>.A.map f = a"
E: shows "\<forall>m. Sec_PriK m \<noteq> Rev_PriK"
l_3: shows "(v \<circ> a) \<longlonglongrightarrow> limsup (v \<circ> r \<circ> s)"
fold_set_fold: fixes y :: "'a"   and xs :: "'a list" shows "Finite_Set.fold (\<^bold>*) y (set xs) = fold (\<^bold>*) xs y"
LeftDerivation_skip_prefixword_ex: fixes u :: "symbol list"   and v :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and w :: "symbol list" assumes "LeftDerivation (u @ v) D w"   and "is_word u" shows "\<exists>w'. w = u @ w' \<and> LeftDerivation v (derivation_shift D (length u) 0) w'"
pset_pnorm_alt_Times: fixes s :: "'a rexp"   and r :: "'a rexp"   and t :: "'a rexp" assumes "s \<in> pset r" shows "pnTimes (pnorm_alt s) t = Times (pnorm_alt s) t"
p0: shows "\<guillemotleft>p0 : local.dom p0 \<rightarrow> a0\<guillemotright>"
simps: fixes u :: "'a Vector"   and v :: "'a Vector" shows "u \<bottom> v = ((u dot v) = (0::'a))"
conv_no_start_points_path: fixes x :: "'a" shows "no_start_points_path x = no_end_points_path (x\<^sup>T)"
diag_block_one_mat: fixes As :: "'b mat list" shows "diag_block_mat (map (\<lambda>A. 1\<^sub>m (dim_row A)) As) = 1\<^sub>m (sum_list (map dim_row As))"
intro: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" assumes "\<And>\<phi> \<psi> \<nu>. eq \<phi> \<psi> \<Longrightarrow> eq (af_letter \<phi> \<nu>) (af_letter \<psi> \<nu>)" shows "af_congruent_axioms eq"
card_ge_dim_independent: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "B \<subseteq> V"   and "vec.independent B"   and "vec.dim V \<le> card B" shows "V \<subseteq> vec.span B"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a \<sqinter> b \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
Disjoint_Eats_I: fixes H :: "fm set"   and x :: "tm"   and z :: "tm"   and y :: "tm" assumes "H \<turnstile> Disjoint x z"   and "insert (y IN z) H \<turnstile> Fls" shows "H \<turnstile> Disjoint (Eats x y) z"
rel_witness_spmf2: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf" assumes "rel_spmf A p q" shows "rel_spmf (\<lambda>(a, b') b. b = b' \<and> A a b') (rel_witness_spmf A (p, q)) q"
SubstForm_unique: fixes v :: "hf"   and y :: "hf"   and x :: "hf"   and x' :: "hf"   and e :: "(name, hf) finfun" assumes "is_Var v"   and "Term y"   and "Form x" shows "SubstForm v y x x' = (\<exists>t. y = \<lbrakk>\<guillemotleft>t\<guillemotright>\<rbrakk>e \<and> (\<exists>A. x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> x' = \<lbrakk>\<guillemotleft>A(decode_Var v::=t)\<guillemotright>\<rbrakk>e))"
af_letter_prop_congruent_2: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln"   and \<nu> :: "'a set" assumes "\<phi> \<sim>\<^sub>P \<psi>" shows "af_letter \<phi> \<nu> \<sim>\<^sub>P af_letter \<psi> \<nu>"
timpls_transformable_to_inv_Var_3: fixes TI :: "('e set \<times> 'e set) list"   and x :: "'h"   and t :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "timpls_transformable_to' TI (Var x) t" shows "t = Var x"
nsg_sg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> H" shows "G \<guillemotright> H "
prod_set_simps_2: fixes x :: "'d"   and y :: "'c" shows "snds (x, y) = {y}"
star_n_zero_num: shows "0 = star_n (\<lambda>n. 0::'b)"
alwaysI: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'g" assumes "\<And>i. P (\<sigma> |\<^sub>s i)" shows "(\<box>P) \<sigma>"
cring_simprules_14: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"
simps_4: fixes z :: "quat" shows "Im3 (Quaternions.cnj z) = - Im3 z"
hyperb_ineq_4_points': fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and deltaG :: "'a itself \<Rightarrow> real"   and e :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" assumes "class.Gromov_hyperbolic_space dist uniformity open deltaG" shows "Min {Gromov_product_at e x y, Gromov_product_at e y z, Gromov_product_at e z t} \<le> Gromov_product_at e x t + 2 * deltaG TYPE('a)"
tarski_absolute_space_axioms: shows "tarski_absolute_space C B"
case_eq_if: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and event :: "('b, 'c, 'd) event" shows "(case event of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = (if isTrans event then f1 (occurs_on event) (un_Trans2 event) (un_Trans3 event) else if isSend event then f2 (getId event) (occurs_on event) (partner event) (un_Send4 event) (un_Send5 event) (getMsg event) else if isRecv event then f3 (getId event) (occurs_on event) (partner event) (un_Recv4 event) (un_Recv5 event) (getMsg event) else if isSnapshot event then f4 (occurs_on event) else f5 (getId event) (occurs_on event) (partner event))"
tarjan_fin_def: fixes v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_fin v s = (let ll = if stack s = [] then lowlink s else let u = hd (stack s) in lowlink s(u \<mapsto> min (the_lowlink s u) (the_lowlink s v)); s' = s\<lparr>lowlink := ll\<rparr> in ASSERT (v \<in> set (tj_stack s)) \<bind> (\<lambda>_. ASSERT (distinct (tj_stack s)) \<bind> (\<lambda>_. if the_lowlink s v = the_discovered s v then ASSERT (scc_root s v (scc_of E v)) \<bind> (\<lambda>_. tj_stack_pop (tj_stack s) v \<bind> (\<lambda>(tjs, scc). NOOP (s'\<lparr>tj_stack := tjs, sccs := insert scc (sccs s)\<rparr>))) else ASSERT (\<not> scc_root s v (scc_of E v)) \<bind> (\<lambda>_. NOOP s'))))"
acyclic_5f_def: fixes x :: "'a" shows "acyclic_5f x \<equiv> \<forall>y z. y \<squnion> z \<le> x \<and> y \<sqinter> z = bot \<longrightarrow> y\<^sup>\<star> \<sqinter> z\<^sup>\<star> = (1::'a)"
user_accessible_mod_cache: fixes new_cache :: "cpu_cache"   and s :: "'b sparc_state"   and pa :: "word36" shows "user_accessible (s\<lparr>cache := new_cache\<rparr>) pa = user_accessible s pa"
319: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - - (- x \<sqinter> y) = - x"
ltake_ltl: fixes n :: "enat"   and xs :: "'b llist" shows "ltake n (ltl xs) = ltl (ltake (eSuc n) xs)"
h_def: shows "h \<equiv> \<lambda>x. f x / g x"
x2_notmem: shows "flow0 x t2 \<notin> {a--flow0 x t1}"
assert_opt_Some: fixes x :: "'c" shows "assert_opt (Some x) = return x"
SD_efficient_support_subset: fixes p :: "'alt pmf"   and p' :: "'alt pmf" assumes "SD_efficient R p"   and "set_pmf p' \<subseteq> set_pmf p"   and "p \<in> lotteries_on alts" shows "SD_efficient R p'"
mergesort_4: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "distinct xs" shows "distinct (mergesort f xs)"
hyps_3: shows "\<alpha>ah (map \<alpha> w1) [*] \<alpha>ah (map \<alpha> w2')"
cases: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and a :: "'b stream" assumes "ev \<phi> a" obtains xs :: "'b stream" where "a = xs__" and "\<phi> xs__"   | xs :: "'b stream" where "a = xs" and "ev \<phi> (stl xs)"
lincomb_elim_if: fixes S :: "'c set"   and P :: "'c \<Rightarrow> bool"   and c :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "S \<subseteq> carrier V"   and "\<And>v. v \<in> S \<Longrightarrow> \<not> P v"   and "c \<in> S \<rightarrow> carrier K" shows "lincomb (\<lambda>w. if P w then b w else c w) S = lincomb c S"
test_bit_eq_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.semiring_bit_syntax plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_bit_syntax.test_bit bit = bit"
UnionE: fixes A :: "'f"   and C :: "'f set set" assumes "A \<in> \<Union> C" obtains X :: "'f set" where "A \<in> X" and "X \<in> C"
borel_measurable_iff_less: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
vert_f21: shows "vertices f21 = u # snd (splitAt u (verticesFrom f v')) @ fst (splitAt v (verticesFrom f v')) @ v # ws"
simps_10: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Cod \<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>] = (Cod t \<^bold>\<otimes> Cod u) \<^bold>\<otimes> Cod v"
TR_submodule: fixes M :: "('e, 'a, 'f) Module_scheme"   and N :: "('g, 'a, 'f) Module_scheme"   and P :: "('e \<times> 'g, 'a, 'f) Module_scheme" assumes "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "submodule R P (TR\<^bsub>R\<^esub> M N P)"
h_def: shows "h = (\<lambda>x. g x - f x)"
add_pos_pos: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "(0::'a) < a + b"
yinv: shows "range (flow0 y) \<subseteq> \<omega>_limit_set x"
disjunctive_diff: fixes b :: "'a"   and a :: "'a" assumes "\<And>n. bit b n \<Longrightarrow> bit a n" shows "a - b = a AND NOT b"
tarjan_spec_def: fixes G :: "('c, 'd) graph_rec_scheme" shows "tarjan_spec G \<equiv> ASSERT (fb_graph G) \<bind> (\<lambda>_. SPEC (\<lambda>sccs. (\<forall>scc\<in>sccs. is_scc (g_E G) scc) \<and> \<Union> sccs = (g_E G)\<^sup>* `` g_V0 G))"
term_of_pair_injective: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and p :: "'d \<times> 'e"   and q :: "'d \<times> 'e" assumes "term_powerprod pair_of_term term_of_pair"   and "term_of_pair p = term_of_pair q" shows "p = q"
invertible_left_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> M" shows "inverse u \<cdot> u = \<one>"
rbl_pred: fixes n :: "nat"   and bin :: "int" shows "rbl_pred (rev (bin_to_bl n bin)) = rev (bin_to_bl n (bin - 1))"