hcju/isabelleps huggingface dataset:
weight_spmf_parametric: fixes A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (=) weight_spmf weight_spmf"
generate_subset_change_eqI: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> carrier G"   and "B \<subseteq> carrier G"   and "C \<subseteq> carrier G"   and "generate G A = generate G B" shows "generate G (A \<union> C) = generate G (B \<union> C)"
assms_3: fixes s :: "'sort" shows "(\<exists>xs. s = asSort xs) \<or> witOpS s (f s) R"
not_subsumed: shows "\<nexists>D. D \<in> partial_evaluation S C \<and> subsumes D R"
embed_le_1_alt: fixes G :: "'a \<Rightarrow> bool"   and s :: "'a" shows "0 \<le> 1 - \<guillemotleft> G \<guillemotright> s"
rep_eq: fixes x :: "circline"   and xa :: "complex_homo" shows "on_circline x xa = on_circline_clmat_hcoords (rep_circline x) (rep_complex_homo xa)"
ts''_def_1: shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c] : ts _> ts''"
assms: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> (subst_domain \<gamma> \<union> range_vars \<gamma>) = {}"
commute: fixes a :: "'k"   and b :: "'k" shows "a + b = b + a"
morphism_ringed_spaces_def: fixes X :: "'c set"   and is_open\<^sub>X :: "'c set \<Rightarrow> bool"   and \<O>\<^sub>X :: "'c set \<Rightarrow> 'd set"   and \<rho>\<^sub>X :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str\<^sub>X :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and one_str\<^sub>X :: "'c set \<Rightarrow> 'd"   and Y :: "'e set"   and is_open\<^sub>Y :: "'e set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'e set \<Rightarrow> 'f set"   and \<rho>\<^sub>Y :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and d :: "'f"   and add_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str\<^sub>Y :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and one_str\<^sub>Y :: "'e set \<Rightarrow> 'f"   and f :: "'c \<Rightarrow> 'e"   and \<phi>\<^sub>f :: "'e set \<Rightarrow> 'f \<Rightarrow> 'd" shows "morphism_ringed_spaces X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X 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 f \<phi>\<^sub>f \<equiv> Comm_Ring.im_sheaf X is_open\<^sub>X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y f \<and> ringed_space 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 \<and> morphism_ringed_spaces_axioms X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X 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 f \<phi>\<^sub>f"
assms: fixes i :: "nat" shows "f i \<noteq> (0::'a)"
openin_path_component_locally_path_connected: fixes S :: "'a set"   and x :: "'a" assumes "locally path_connected S" shows "openin (top_of_set S) (path_component_set S x)"
event_can_go_back_if_no_sender: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> occurs_on ev'"   and "can_occur ev' c'"   and "\<not> isRecvMarker ev'"   and "\<not> isSend ev" shows "can_occur ev' c"
assms_1: shows "typ_of t = Some (uty \<rightarrow> tty)"
infinite_regularCard_exists: fixes R :: "('c \<times> 'c) set set" assumes "\<forall>r\<in>R. Card_order r" shows "\<exists>A. infinite A \<and> regularCard |A| \<and> (\<forall>r\<in>R. r <o |A|)"
simps_2: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and t1 :: "'a MFOTL.trm"   and t2 :: "'a MFOTL.trm"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Eq t1 t2) j = j"
getFresh: fixes P :: "pvar set" assumes "finite P" shows "getFresh P \<notin> P"
valid_path_compose_holomorphic: fixes g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "valid_path g"   and "f holomorphic_on S"   and "open S"   and "path_image g \<subseteq> S" shows "valid_path (f \<circ> g)"
hyps_1: shows "P \<turnstile> \<langle>e0,s0,b0\<rangle> \<rightarrow> \<langle>e1,s1,b1\<rangle>"
list_of_code: fixes as :: "'b iarray" shows "IArray.list_of as = map ((!!) as) [0..<IArray.length as]"
strong_del_point_index_alt: fixes ps :: "'a set"   and p :: "'a" assumes "ps \<subseteq> del_point p" shows "str_del_point_blocks p index ps = \<B> index ps - filter_mset ((\<in>) p) \<B> index ps"
differentiable_imp_continuous_on: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set" assumes "f differentiable_on s" shows "continuous_on s f"
rk2_param_spec_def: shows "rk2_param_spec = SPEC (\<lambda>r. 0 < r \<and> r \<le> 1)"
subset_Zorn: fixes A :: "'b set set" assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
ab_semigroup_mult_hom_axioms: shows "ab_semigroup_mult_hom (map_poly complex_of_real)"
deriv_sat_preserving: fixes \<Gamma> :: "'b inference set"   and Rf :: "'b literal multiset set \<Rightarrow> 'b literal multiset set"   and Ri :: "'b literal multiset set \<Rightarrow> 'b inference set"   and Ns :: "'b literal multiset set llist" assumes "sat_preserving_redundancy_criterion \<Gamma> Rf Ri"   and "chain (redundancy_criterion.derive \<Gamma> Rf) Ns"   and "satisfiable (lhd Ns)" shows "satisfiable (Sup_llist Ns)"
seq_char: fixes g :: "(bool, 'arr list) concrete_category.arr"   and f :: "(bool, 'arr list) concrete_category.arr" shows "seq g f = (arr f \<and> arr g \<and> Dom g = Cod f)"
top_unique: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and a :: "'a" assumes "class.order_top less_eq less top" shows "less_eq top a = (a = top)"
single_gen_free_def: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e" shows "single_gen_free A a \<equiv> \<forall>n. n \<noteq> 0 \<longrightarrow> \<zero>\<^bsub>A\<^esub> \<noteq> n\<triangleright>a\<^bsub>A\<^esub>"
progress_LN: fixes j :: "nat"   and lholed :: "Lholed"   and k :: "nat"   and es :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and tvs :: "t list" assumes "Lfilled j lholed [$Br (j + k)] es"   and "\<S>\<bullet>\<C> \<turnstile> es : [] _> ts"   and "label \<C> ! k = tvs" shows "\<exists>lholed' vs \<C>'. Lfilled j lholed' (vs @ [$Br (j + k)]) es \<and> \<S>\<bullet>\<C>' \<turnstile> vs : [] _> tvs \<and> const_list vs"
leq_def: fixes x :: "'c"   and y :: "'c" shows "(x \<le>\<^sub>\<K> y) = (x + y = y)"
hom_boundary3: shows "\<exists>d. ((\<forall>p X S c. c \<notin> carrier (relative_homology_group p X S) \<longrightarrow> d p X S c = \<one>\<^bsub>homology_group (p - 1) (subtopology X S)\<^esub>) \<and> (\<forall>p X S. d p X S \<in> hom (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))) \<and> (\<forall>p X S c. singular_relcycle p X S c \<and> 1 \<le> p \<longrightarrow> d (int p) X S (homologous_rel_set p X S c) = homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c)) \<and> (\<forall>p X S. d p X S = d p X (topspace X \<inter> S))) \<and> (\<forall>p X S c. d p X S c \<in> carrier (homology_group (p - 1) (subtopology X S))) \<and> (\<forall>p\<le>0. d p = (\<lambda>q r t. undefined))"
dim_col_of_dagger: fixes M :: "complex Matrix.mat" shows "dim_col M\<^sup>\<dagger> = dim_row M"
sim_delta_wf_state: fixes s :: "'s" assumes "wf_state s" shows "\<forall>s'\<in>set (sim_delta s). wf_state s'"
commutativity2_lemma: fixes F :: "'b program"   and T :: "'b set"   and L :: "'b set set"   and B :: "'b set" assumes "\<And>act s t. \<lbrakk>act \<in> Acts F; s \<in> T; (s, t) \<in> relcl L\<rbrakk> \<Longrightarrow> s \<in> B \<or> t \<in> B \<or> (funof act s, funof act t) \<in> relcl L"   and "\<And>act. act \<in> Acts F \<Longrightarrow> single_valued act"   and "\<And>act. act \<in> Acts F \<Longrightarrow> Domain act = UNIV"   and "lattice L"   and "B \<in> L"   and "T \<in> L"   and "F \<in> stable T" shows "commutes F T B L"
nonzero_eq_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero" shows "(a = divide b c) = (times a c = b)"
size_gen: fixes xa :: "'a \<Rightarrow> nat"   and x :: "'a list" shows "size_iarray xa (IArray x) = size_list xa x + Suc 0"
intros: fixes R1 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b"   and b :: "'c"   and R2 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'d"   and d :: "'e" assumes "R1 a b"   and "R2 c d" shows "rel_prod R1 R2 (a, c) (b, d)"
r_parallel''_3: fixes j :: "nat"   and x :: "nat"   and i :: "nat" assumes "eval r_phi [j, x] \<down> \<and> eval r_phi [i, x] \<up>" shows "eval r_parallel [i, j, x] \<down>= prod_encode (1, the (eval r_phi [j, x]))"
lfp_omega_lowerbound: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c \<Rightarrow> 'b" assumes "mono f"   and "Omega_fun f g A \<le> A" shows "omega_fun f \<circ> g \<le> A"
intro: fixes A :: "'a set" assumes "(0::'a) \<in> A"   and "(1::'a) \<in> A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x + y \<in> A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x * y \<in> A"   and "\<And>x. x \<in> A \<Longrightarrow> - x \<in> A" shows "ring_closed A"
elims_2: fixes x :: "('a, 'b) rbt" assumes "cl_inv x" obtains "x = Leaf"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l__ R k__ v__ r__" and "cl_inv l__ \<and> cl_inv r__ \<and> cl l__ = B \<and> cl r__ = B"   | l :: "('a, 'b) rbt" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l B k v r" and "cl_inv l \<and> cl_inv r"
Diff_eq_empty_iff: fixes A :: "'e set"   and B :: "'e set" shows "(A - B = {}) = (A \<subseteq> B)"
ge_refl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.non_strict_order less_eq" shows "less_eq x x"
up_correct: fixes v :: "'e \<Rightarrow> nat"   and n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)" shows "[up M]\<^bsub>v,n\<^esub> = [M]\<^bsub>v,n\<^esub>\<^sup>\<up>"
sel: fixes x2 :: "'h" shows "the (Some x2) = x2"
assms_3: shows "\<forall>i. i < i + (1::'a) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'a)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'b)) < (LEAST n. A $ (i + (1::'a)) $ n \<noteq> (0::'b))"
measurable_completion: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> completion M \<rightarrow>\<^sub>M N"
content_mult: fixes p :: "'a poly"   and q :: "'a poly" shows "content (p * q) = content p * content q"
prems: shows "\<forall>x\<in>insert x xs. \<forall>a\<in>ys. x < a"
cod_dom: fixes f :: "'a" assumes "C.arr f" shows "C.cod (C.dom f) = C.dom f"
numeral_Integer_eq: fixes k :: "num" shows "numeral k = MkI\<cdot>(numeral k)"
addfunset0: fixes A :: "'a set" shows "addfunset A 0 = 0"
AI_wn_def: fixes c :: "com" shows "AI_wn_ivl c = pfp_wn (step_ivl \<top>) (bot c)"
eq_pow_exp: fixes u :: "'b list"   and k :: "nat"   and m :: "nat" assumes "u \<noteq> \<epsilon>" shows "(u \<^sup>@ k = u \<^sup>@ m) = (k = m)"
eexp_ineq_5: fixes x :: "ereal" shows "0 \<le> eexp x"
LCons: fixes x :: "'a"   and xs' :: "'a llist"   and y :: "'b"   and ys' :: "'b llist" assumes "xs = LCons x xs'"   and "ys = LCons y ys'"   and "P x y"   and "llist_all2 P xs' ys'" shows "thesis"
induct: fixes P :: "('fun, 'var) Term.term set \<Rightarrow> (('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set \<Rightarrow> ('fun, 'var) stateful_strand_step list \<Rightarrow> bool"   and a0 :: "('fun, 'var) Term.term set"   and a1 :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) set"   and a2 :: "('fun, 'var) stateful_strand_step list" assumes "\<And>M D. P M D []"   and "\<And>M D t S. (\<And>x. P M D S) \<Longrightarrow> P M D (send\<langle>t\<rangle> # S)"   and "\<And>M D t S. (\<And>x. P (insert (t \<cdot> x) M) D S) \<Longrightarrow> P M D (receive\<langle>t\<rangle> # S)"   and "\<And>M D uu_ t t' S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<langle>uu_: t \<doteq> t'\<rangle> # S)"   and "\<And>M D t s S. (\<And>x. P M (insert (t \<cdot> x, s \<cdot> x) D) S) \<Longrightarrow> P M D (insert\<langle>t,s\<rangle> # S)"   and "\<And>M D t s S. (\<And>x. P M (D - {(t \<cdot> x, s \<cdot> x)}) S) \<Longrightarrow> P M D (delete\<langle>t,s\<rangle> # S)"   and "\<And>M D uv_ t s S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<langle>uv_: t \<in> s\<rangle> # S)"   and "\<And>M D X F F' S. (\<And>x. P M D S) \<Longrightarrow> P M D (\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: F'\<rangle> # S)" shows "P a0 a1 a2"
has_bochner_integral_divide: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x / y) (x / y)"
equiintegrable_halfspace_restrictions_le: fixes F :: "('a \<Rightarrow> 'b) set"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "F equiintegrable_on cbox a b"   and "f \<in> F"   and "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm (f x)" shows "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then h x else (0::'b)}) equiintegrable_on cbox a b"
rel_frontier_convex_hull_explicit: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "rel_frontier (convex hull S) = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> (\<exists>x\<in>S. u x = 0) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
stacks_eq_iff: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "(stack s = []) = (tj_stack s = [])"
of_int_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus k = (if k < 0 then uminus (semiring_1.of_nat one plus zero (nat (- k))) else semiring_1.of_nat one plus zero (nat k))"
continuous_on_real_sqrt: fixes s :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "continuous_on s f" shows "continuous_on s (\<lambda>x. sqrt (f x))"
less_r_def: fixes a :: "'a"   and b :: "'a" shows "less_r a b = (lesseq_r a b \<and> \<not> lesseq_r b a)"
gfp_exp_lemma2: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)"   and "\<And>P Q. \<lbrakk>unitary P; unitary Q; P \<tturnstile> Q\<rbrakk> \<Longrightarrow> t P \<tturnstile> t Q" shows "gfp_exp t \<tturnstile> t (gfp_exp t)"
hd_Cons_tl: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
binomial_maximum: fixes n :: "nat"   and k :: "nat" shows "n choose k \<le> n choose n div 2"
fps_sum_nth: fixes f :: "'b \<Rightarrow> 'a fps"   and S :: "'b set"   and n :: "nat" shows "sum f S $ n = (\<Sum>k\<in>S. f k $ n)"
\<section>: fixes y :: "'b" assumes "norm (f a - y) < r"   and "y \<in> frontier (f ` S)" shows "False"
iagreement_implies_niagreement: shows "a0i_agreement \<subseteq> a0n_agreement"
dsum_zero_func: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_zero I A \<in> carr_dsumag I A"
trivial_FGModuleHom: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a" assumes "\<And>r. r \<in> FG \<Longrightarrow> smult' r (0::'a) = (0::'a)" shows "FGModuleHom G (\<cdot>) V smult' 0"
aI_def: fixes v :: "Value" shows "aI v = (case v of intgV i \<Rightarrow> i)"
a0i_defs_6: shows "a0i \<equiv> \<lparr>init = a0i_init, trans = a0i_trans, obs = id\<rparr>"
L_transition_AC_fresh: fixes \<alpha> :: "'act"   and F :: "'effect fs_set"   and f :: "'effect"   and P :: "'state"   and P\<^sub>L' :: "('state, 'effect) L_state" assumes "bn \<alpha> \<sharp>* (F, f, P)" shows "AC (f, F, P) \<rightarrow>\<^sub>L \<langle>L_action.Act \<alpha>,P\<^sub>L'\<rangle> = (\<exists>P'. P\<^sub>L' = EF (L (\<alpha>, F, f), P') \<and> P \<rightarrow> \<langle>\<alpha>,P'\<rangle>)"
step_Nil_iff: fixes xs :: "'a list"   and r :: "'a"   and act :: "nat \<times> nat list" shows "(step xs r act = []) = (xs = [])"
assms_1: shows "\<exists>x. Ord x \<and> P x"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a AND b) AND c = a AND b AND c"
SIod_def: fixes D :: "'c Order"   and T :: "'c set" shows "SIod D T = \<lparr>carrier = T, rel = {(a, b). (a, b) \<in> rel D \<and> a \<in> T \<and> b \<in> T}\<rparr>"
C: shows "con_comp_map D E p q x = Some (the (con_comp_map D E p q x))"
unadjoin_unique': fixes y :: "'a"   and h :: "nat"   and x :: "'a" assumes "y \<in> H"   and "h < subgroup_indicator G H a"   and "x = y \<otimes> a [^] h" shows "unadjoin x = (y, h)"
downset_set_prop_var2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and X :: "'b set"   and y :: "'b" assumes "class.preorder less_eq less"   and "x \<in> ord.downset_set less_eq X"   and "less_eq y x" shows "y \<in> ord.downset_set less_eq X"
wt_init: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and C :: "qtname" shows "(E,dt\<Turnstile>Init C\<Colon>\<surd>) = is_class (prg E) C"
chine_eq_apex: shows "\<nu>\<pi>.chine = \<nu>\<pi>.apex"
lookup_Abs_poly_mapping: fixes f :: "'a \<Rightarrow> 'b" assumes "finite {x. f x \<noteq> (0::'b)}" shows "lookup (Abs_poly_mapping f) = f"
tW_cmpser_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "tW_cmpser G n g = (if n = 0 then td_gchain G 0 g else tD_gchain G n g \<and> (\<forall>l\<le>n - 1. \<natural>\<^bsub>G\<^esub>g l \<triangleright> g (Suc l)))"
set_minus_plus: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" shows "(a - b \<in> C) = (a \<in> b +o C)"
twoCisTypeI_5: shows "g1 piecewise_C1_differentiable_on {a..b}"
vimage_eq_set_mult_kern_right: fixes A :: "'a set" assumes "A \<subseteq> carrier G" shows "{x \<in> carrier G. h x \<in> h ` A} = A <#> kernel G H h"
all_consts_def: shows "all_consts = heads_of rs |\<union>| C"
continuous_on_open: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on S f = (\<forall>T. openin (top_of_set (f ` S)) T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
deg_one: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "deg R \<one>\<^bsub>UP R\<^esub> = 0"
mult_ac_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
assms_2: fixes k' :: "int"   and \<gamma>' :: "real \<Rightarrow> real \<times> real" assumes "(k', \<gamma>') \<in> set ((1, \<gamma>) # l)" shows "valid_path \<gamma>'"
map_snd_zip_take: fixes xs :: "'h list"   and ys :: "'g list" shows "map snd (zip xs ys) = take (min (length xs) (length ys)) ys"
gt_imp_ge: fixes x :: "'a"   and y :: "'a" assumes "delta_gt \<delta> x y" shows "y \<le> x"
assms_4: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<And>\<xi>. \<xi> \<in> elts \<alpha> \<Longrightarrow> P \<xi>" shows "P (SUP \<xi>\<in>elts \<alpha>. \<xi>)"
conv_no_start_end_points: fixes x :: "'a" shows "no_start_end_points x = no_start_end_points (x\<^sup>T)"
incid_inters_right: fixes P :: "Points"   and l :: "Lines"   and m :: "Lines" assumes "P \<in> inters l m" shows "incid P m"
src_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.hseq \<mu> \<nu>" shows "src\<^sub>C (\<mu> \<star>\<^sub>C \<nu>) = src\<^sub>C \<nu>"
clock_def: fixes state :: "'a state" shows "clock state = (case state of make_state x1 x2 x3 x4 x5 \<Rightarrow> x1)"
measurable_bounded_by_integrable_imp_integrable: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable (lebesgue_on S)"   and "g integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "S \<in> sets lebesgue" shows "f integrable_on S"
eq: fixes n :: "nat" shows "outer_measure_of M (A \<inter> C n) = emeasure M (E n \<inter> C n)"
Top_def: fixes po :: "'b potype" shows "Top po \<equiv> greatest (\<lambda>x. True) po"
assms_2: shows "vsa \<subseteq> fmdom' fma"
list_order_def: fixes ops :: "('b \<times> 'b option) list"   and x :: "'b"   and y :: "'b" shows "list_order ops x y \<equiv> \<exists>xs ys zs. interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
lub_least: fixes A :: "'a set"   and z :: "'a" assumes "Complete_Partial_Order.chain leq A"   and "\<And>x. x \<in> A \<Longrightarrow> leq x z" shows "leq (lub A) z"
streamValue1: fixes D :: "nat"   and z :: "nat"   and str :: "nat \<Rightarrow> 'a"   and t :: "nat"   and x :: "'a"   and j :: "nat" assumes "\<forall>j\<le>D + z. str (t + j) = x"   and "j \<le> D" shows "str (t + j + z) = x"
Goedel_I: fixes e :: "(name, hf) finfun" obtains \<delta> :: "fm" where "{} \<turnstile> \<delta> IFF Neg (PfP \<guillemotleft>\<delta>\<guillemotright>)" and "\<not> {} \<turnstile> \<delta>" and "\<not> {} \<turnstile> Neg \<delta>" and "eval_fm e \<delta>" and "ground_fm \<delta>"
mapset_some: fixes t :: "(int \<times> 'b) Tree"   and i :: "int"   and a :: "'b" assumes "valid_tmap t" shows "(mapOf t i = Some a) = ((i, a) \<in> setOf t)"
MGF_MGFn_iff: fixes G :: "prog"   and A :: "state triple set"   and t :: "Term.term" shows "G,A\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>} = (\<forall>n. G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>})"
writer_abs_def: shows "writer_abs \<equiv> PRJ('e\<cdot>'d writer) oo EMB('d\<^sub>\<bottom> \<otimes> 'e\<^sub>\<bottom>)"
vdiff_vempty_right: fixes A :: "V" shows "A -\<^sub>\<circ> 0 = A"
sl_impl: shows "correct_subseqs_foldr_impl (\<lambda>x. map_prod (mul_const x) ((#) x)) sl_impl sli"
emeasure_extend_measure: fixes M :: "'b measure"   and \<Omega> :: "'b set"   and I :: "'c set"   and G :: "'c \<Rightarrow> 'b set"   and \<mu> :: "'c \<Rightarrow> ennreal"   and \<mu>' :: "'b set \<Rightarrow> ennreal"   and i :: "'c" assumes "M = extend_measure \<Omega> I G \<mu>"   and "\<And>i. i \<in> I \<Longrightarrow> \<mu>' (G i) = \<mu> i"   and "G ` I \<subseteq> Pow \<Omega>"   and "positive (sets M) \<mu>'"   and "countably_additive (sets M) \<mu>'"   and "i \<in> I" shows "emeasure M (G i) = \<mu> i"
closed_Int_compact: fixes s :: "'c set"   and t :: "'c set" assumes "closed s"   and "compact t" shows "compact (s \<inter> t)"
map_converter_parallel_converter2: fixes f :: "'a \<Rightarrow> 'i"   and f' :: "'b \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'c"   and g' :: "'l \<Rightarrow> 'd"   and h :: "'m \<Rightarrow> 'e"   and h' :: "'n \<Rightarrow> 'f"   and k :: "'g \<Rightarrow> 'o"   and k' :: "'h \<Rightarrow> 'p"   and conv1 :: "('i, 'k, 'm, 'o) converter"   and conv2 :: "('j, 'l, 'n, 'p) converter" shows "map_converter (map_sum f f') (map_sum g g') (map_sum h h') (map_sum k k') (parallel_converter2 conv1 conv2) = parallel_converter2 (map_converter f g h k conv1) (map_converter f' g' h' k' conv2)"
fmlookup_ran'_iff: fixes y :: "'a"   and m :: "('b, 'a) fmap" shows "(y \<in> fmran' m) = (\<exists>x. fmlookup m x = Some y)"