hcju/isabelleps huggingface dataset:
seq_compact_eq_compact: fixes U :: "'b set" shows "seq_compact U = compact U"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b stream" shows "i.R (smap f v) = f ` i.R v"
eval_fps_expansion: fixes f :: "complex \<Rightarrow> complex"   and z0 :: "complex"   and r :: "ereal"   and z :: "complex" assumes "f holomorphic_on eball z0 r"   and "z \<in> eball z0 r" shows "eval_fps (fps_expansion f z0) (z - z0) = f z"
simple_syntactic_or_def: fixes P :: "'a \<Rightarrow> bool" shows "simple_syntactic_or P \<equiv> \<forall>t u. P (app t u) = (P t \<or> P u)"
YMapInvImage: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and x :: "ZF" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "x |\<in>| (F @@ X)" shows "NT YMapInv C X F x : YFtor C @@ X \<Longrightarrow> F"
matrix_change_of_basis_def: fixes X :: "(('a, 'b) vec, 'b) vec"   and Y :: "(('a, 'b) vec, 'b) vec" shows "matrix_change_of_basis X Y = (\<chi>i j. coord Y (X $ j) $ i)"
subset_image_iff: fixes B :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
split_paired_Bex_Sigma: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and P :: "'a \<times> 'b \<Rightarrow> bool" shows "(\<exists>z\<in>Sigma A B. P z) = (\<exists>x\<in>A. \<exists>y\<in>B x. P (x, y))"
card_idx_state: shows "|UNIV| <o |UNIV|"
stable_on_CD_on: fixes ds :: "'b set"   and X :: "'a set" assumes "Singleton_for_d.stable_on ds X" shows "Singleton_for_d.CD_on ds X = X"
prems_2: shows "output_iface (routing_table_semantics (r # rs) k) = output_port"
disj_left_commute: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj b (disj a c) = disj a (disj b c)"
map_graph_fn_comp: fixes G :: "('b, 'd) labeled_graph"   and f :: "'e \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'e" shows "map_graph_fn G (f \<circ> g) = map_graph_fn (map_graph_fn G g) f"
cases: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and a1 :: "('a, 'b, 'c, 'd) converter"   and a2 :: "('a, 'b, 'c, 'd) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C a1 \<sim> a2" obtains conv :: "('a, 'b, 'c, 'd) converter" and conv' :: "('a, 'b, 'c, 'd) converter" where "a1 = conv" and "a2 = conv'" and "\<And>q. q \<in> outs_\<I> \<I> \<Longrightarrow> eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r. r \<in> responses_\<I> \<I> q)) (eq_\<I>_converter \<I> \<I>')) \<I>' (run_converter conv q) (run_converter conv' q)"
simps_2: fixes x :: "char list" shows "FV (Fvar x) = {x}"
abs_eq: shows "1\<^sub>h\<^sub>c = Abs_complex_homo_coords 1\<^sub>v"
idem_normalize: fixes a :: "'b" shows "gcd a a = normalize a"
vrat_of_rat_transfer: fixes a :: "rat" shows "cr_vrat (a\<^sub>\<rat>) a"
face_cycle_sets_def: shows "face_cycle_sets = face_cycle_set ` arcs (with_proj (list_digraph G_list))"
aNeg_aEval: fixes a :: "atom"   and L :: "real list" shows "aEval a L = (\<not> aEval (aNeg a) L)"
cycle_free_up_to_loop_dest: fixes i :: "nat"   and n :: "nat"   and xs :: "nat list"   and k :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a" assumes "i \<le> n"   and "set xs \<subseteq> {0..k}"   and "cycle_free_up_to m k n" shows "(0::'a) \<le> len m i i xs"
coeff_monom_mult: fixes a :: "'b"   and d :: "nat"   and p :: "'b poly"   and i :: "nat" shows "coeff (monom a d * p) i = (if d \<le> i then a * coeff p (i - d) else (0::'b))"
AddSA_CompFun_the3: fixes S' :: "'a"   and SA :: "('a, 'b, 'c) seqauto"   and S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S' \<notin> States SA"   and "S \<noteq> S'"   and "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S') = the (CompFun A S')"
inv: shows "mopup_inv (Suc 0, l, r) lm n ires"
Y_def: shows "(Y::'a::refinement_lattice set) = {(F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) y |y::'a::refinement_lattice. F y \<sqsubseteq> (x::'a::refinement_lattice)}"
subspace_eq_real2: shows "module_on.subspace (*\<^sub>R) = subspace"
keys_monomial_pmdl: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'t" assumes "is_monomial_set F"   and "p \<in> pmdl F"   and "t \<in> keys p" obtains f :: "'t \<Rightarrow>\<^sub>0 'c" where "f \<in> F" and "f \<noteq> 0" and "lt f adds\<^sub>t t"
PROJ_DOM_PRE_EFF_SUBSET_DOM: fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" shows "fmdom' (fst (action_proj a vs)) \<subseteq> fmdom' (fst a) \<and> fmdom' (snd (action_proj a vs)) \<subseteq> fmdom' (snd a)"
get_document_ptr_simp4: fixes document_ptr :: "(_) document_ptr"   and element_ptr :: "(_) element_ptr"   and f :: "('a, 'b, 'c, 'e, 'h option) RElement_scheme"   and h :: "(_) heap" shows "get document_ptr (put element_ptr f h) = get document_ptr h"
results_gpv_sub_gvps: assumes "gpv' \<in> sub_gpvs \<I> gpv" shows "results_gpv \<I> gpv' \<subseteq> results_gpv \<I> gpv"
convex_halfspace_lt: fixes a :: "'c"   and b :: "real" shows "convex {x. a \<bullet> x < b}"
section_imp_embedding_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "section_map X Y f" shows "embedding_map X Y f"
web_of_network_sel_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and e :: "'a \<times> 'a"   and e' :: "'a \<times> 'a" shows "edge (web_of_network \<Delta>) e e' = (e \<in> \<^bold>E\<^bsub>\<Delta>\<^esub> \<and> e' \<in> \<^bold>E\<^bsub>\<Delta>\<^esub> \<and> snd e = fst e')"
assms: shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"
row_transpose: fixes i :: "'f"   and A :: "(('d, 'f) vec, 'e) vec" shows "row i (Finite_Cartesian_Product.transpose A) = column i A"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list" shows "take n (take m xs) = take (min n m) xs"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelEQ Enc"
FOREACHci_refine_genR: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "(('c set \<times> 'e) \<times> 'd set \<times> 'f) set"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres"   and R' :: "('e \<times> 'f) set" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "((S, \<sigma>0), \<alpha> ` S, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi> it \<sigma>; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; x' = \<alpha> x; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; x \<in> it; x' \<in> it'; c \<sigma>; c' \<sigma>'\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"   and "\<And>\<sigma> \<sigma>'. \<lbrakk>\<Phi> {} \<sigma>; \<Phi>' {} \<sigma>'; (({}, \<sigma>), {}, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; it \<noteq> {}; it' \<noteq> {}; \<not> c \<sigma>; \<not> c' \<sigma>'\<rbrakk> \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'" shows "FOREACH\<^sub>C\<^bsup>\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R' (FOREACH\<^sub>C\<^bsup>\<Phi>'\<^esup> S' c' f' \<sigma>0')"
b_e_type_unop_testop_1: fixes \<C> :: "t_context"   and e :: "b_e"   and ts :: "t list"   and ts' :: "t list"   and t :: "t"   and iop :: "unop_i"   and fop :: "unop_f"   and testop :: "testop" assumes "\<C> \<turnstile> [e] : ts _> ts'"   and "e = Unop_i t iop \<or> e = Unop_f t fop \<or> e = Testop t testop" shows "\<exists>ts''. ts = ts'' @ [t] \<and> ts' = ts'' @ [arity_1_result e]"
"case": fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "(case OptionT x of OptionT x \<Rightarrow> f x) = f x"
assms_3: fixes xs :: "'a option list"   and x :: "'a" assumes "P xs" shows "P (Some x # xs)"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "atom fm"   and y :: "atom fm" assumes "liftFm x xa xb = y" obtains d :: "nat" and amount :: "nat" and f :: "atom fm" where "x = d" and "xa = amount" and "xb = f" and "y = map_fm_binders (\<lambda>a x. liftAtom (d + x) amount a) f 0"
filterlim_at_bot_lt: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" shows "filterlim f at_bot F = (\<forall>Z<c. \<forall>\<^sub>F x in F. f x \<le> Z)"
inf_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = x \<sqinter> y \<sqinter> z"
intros_3: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> hmultiset"   and coef_sym :: "'b \<Rightarrow> nat \<Rightarrow> hmultiset"   and t :: "('b, 'a) tm"   and s :: "('b, 'a) tm" assumes "tkbo_coefs ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym coef_sym"   and "tkbo_coefs.ge_tpoly ground_heads_var \<delta> arity_sym wt_sym (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym t) (tkbo_coefs.wt ground_heads_var \<delta> arity_sym wt_sym coef_sym s)"   and "ground_heads.gt_hd gt_sym ground_heads_var (head t) (head s)" shows "tkbo_coefs.gt ground_heads_var gt_sym \<delta> extf arity_sym wt_sym coef_sym t s"
assms: shows "simplex [x_times_c_geq_y_times_b c b] = Inr x"
f_join_nth2: fixes n :: "nat"   and I :: "nat set"   and xs :: "'b list" assumes "n < card (I \<down>< length xs)" shows "xs \<Join>\<^sub> I ! n = xs ! (I \<rightarrow> n)"
RETURN_leof_RES_iff: fixes x :: "'c"   and Y :: "'c set" shows "(RETURN x \<le>\<^sub>n RES Y) = (x \<in> Y)"
pmf_of_alist_support_aux_1: fixes \<mu> :: "('c \<times> real) list"   and x :: "'c" assumes "\<forall>(uu_, p)\<in>set \<mu>. 0 \<le> p" shows "0 \<le> (case map_of \<mu> x of None \<Rightarrow> 0 | Some p \<Rightarrow> p)"
disc_13: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "isRecv (Recv x31 x32 x33 x34 x35 x36)"
backward_finite_path_root_acyclic_terminating_iff: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "is_acyclic x = (x ; r = 0)"
AllowedActs_Join: fixes F :: "'c program"   and G :: "'c program" shows "AllowedActs (F \<squnion> G) = AllowedActs F \<inter> AllowedActs G"
concE: fixes w :: "'a list"   and A :: "'a list set"   and B :: "'a list set" assumes "w \<in> A @@ B" obtains u :: "'a list" and v :: "'a list" where "u \<in> A" and "v \<in> B" and "w = u @ v"
Abs_process_inverse: fixes y :: "('c event list \<times> 'c event set) set \<times> 'c event list set" assumes "y \<in> {p. is_process p}" shows "Rep_process (Abs_process y) = y"
lub_prod: fixes S :: "nat \<Rightarrow> 'b \<times> 'c" assumes "chain S" shows "(\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
dep: shows "\<forall>a\<in>set as. depends\<^sub>0 a"
lookup_pair_map2_val_pair: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and g :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'd) list"   and h :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd"   and k0 :: "'a" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "tc.map2_val_compat g"   and "tc.map2_val_compat h"   and "\<And>zs. tc.oalist_inv_raw zs \<Longrightarrow> g zs = map_val_pair (\<lambda>k v. f k v (0::'c)) zs"   and "\<And>zs. tc.oalist_inv_raw zs \<Longrightarrow> h zs = map_val_pair (\<lambda>k. f k (0::'b)) zs"   and "\<And>k. f k (0::'b) (0::'c) = (0::'d)" shows "lookup_pair_tc (map2_val_pair_tc f g h xs ys) k0 = f k0 (lookup_pair_tc xs k0) (lookup_pair_tc ys k0)"
simps_1: fixes A :: "('c, 'd) ioa"   and s :: "'c"   and p :: "'d \<times> 'c"   and p' :: "'d \<times> 'c"   and ps :: "('d \<times> 'c) list" shows "is_exec_frag_of A (s, Cons p (Cons p' ps)) = (snd p' \<midarrow>fst p\<midarrow>A\<longrightarrow> snd p \<and> is_exec_frag_of A (s, Cons p' ps))"
iterate_op_plus: fixes m :: "nat"   and k :: "nat" shows "((+) k ^^ m) 0 = k * m"
octo_trad: fixes x :: "real"   and y :: "real"   and z :: "real"   and w :: "real"   and u :: "real"   and v :: "real"   and q :: "real"   and g :: "real" shows "Octo x y z w u v q g = x *\<^sub>R e0 + y *\<^sub>R e1 + z *\<^sub>R e2 + w *\<^sub>R e3 + u *\<^sub>R e4 + v *\<^sub>R e5 + q *\<^sub>R e6 + g *\<^sub>R e7"
induct: fixes P :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c) \<Rightarrow> 'b match_expr \<Rightarrow> bool"   and a0 :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and a1 :: "'b match_expr" assumes "\<And>uu_. P uu_ MatchAny"   and "\<And>disc sel a. P (disc, sel) (Match a)"   and "\<And>disc sel a. P (disc, sel) (MatchNot (Match a))"   and "\<And>C ms1 ms2. \<lbrakk>P C ms1; \<And>x xa y. \<lbrakk>x = primitive_extractor C ms1; (xa, y) = x\<rbrakk> \<Longrightarrow> P C ms2\<rbrakk> \<Longrightarrow> P C (MatchAnd ms1 ms2)"   and "\<And>uv_ va. P uv_ (MatchNot (MatchNot va))"   and "\<And>uv_ va vb. P uv_ (MatchNot (MatchAnd va vb))"   and "\<And>uv_. P uv_ (MatchNot MatchAny)" shows "P a0 a1"
simps_2: fixes a :: "'a"   and n :: "nat"   and p :: "'a point" shows "point_mult a (Suc n) p = local.add a p (point_mult a n p)"
fun_upd_apply: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and y :: "'a"   and z :: "'c" shows "(f(x := y)) z = (if z = x then y else f z)"
span_is_subset2: fixes S :: "'a vec set" assumes "S \<subseteq> carrier_vec n" shows "local.span S \<subseteq> carrier_vec n"
IH: fixes F :: "'fmla set" assumes "F \<subseteq> fmla"   and "finite F"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "prv (eqv (rawpsubst (sdsj F) txs) (sdsj ((\<lambda>\<phi>. rawpsubst \<phi> txs) ` F)))"
finfun_All_All: fixes P :: "'d \<Rightarrow>f bool" shows "finfun_All P = All (($) P)"
poincare_map_def: fixes P :: "'a set"   and x :: "'a" shows "poincare_map P x = flow0 x (return_time P x)"
inv_safe_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.inv_safe summary results_in c = (\<forall>loc1 loc2 t s. 0 < zcount (c_pts c loc1) t \<and> s \<in>\<^sub>A graph.path_weight summary loc1 loc2 \<and> dataflow_topology.worklists_vacant_to summary results_in c (results_in t s) \<longrightarrow> (\<exists>t'\<le>results_in t s. t' \<in>\<^sub>A frontier (c_imp c loc2)))"
unit_factor_is_unit: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "a \<noteq> zero" shows "dvd.dvd times (unit_factor a) one"
asubg_inc_zero: fixes A :: "('c, 'd) aGroup_scheme"   and H :: "'c set" assumes "aGroup A"   and "A +> H" shows "\<zero>\<^bsub>A\<^esub> \<in> H"
gen_iapath_triv_arc: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "pair_sym_digraph G"   and "pre_digraph.gen_iapath (with_proj G) (pverts G) u p v" shows "(u, v) \<in> parcs G"
tauLaw4StatImpRight: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel" shows "\<Psi> \<rhd> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q \<lessapprox><Rel> (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q"
refTE: fixes a :: "ty" assumes "is_refT a" obtains "a = NT"   | C :: "String.literal" where "a = Class C"   | A :: "ty" where "a = A\<lfloor>\<rceil>"
subset_UnE: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "C \<subseteq> A \<union> B" obtains A' :: "'c set" and B' :: "'c set" where "A' \<subseteq> A" and "B' \<subseteq> B" and "C = A' \<union> B'"
deltafun0: fixes a :: "'a" shows "(0::'b) \<delta> a = 0"
factorsI: fixes fs :: "'a list"   and a :: "'a" assumes "\<forall>f\<in>set fs. irreducible G f"   and "foldr (\<otimes>) fs \<one> = a" shows "factors G fs a"
assms_1: shows "\<forall>i s t. (t \<in> \<K> M i s) = (s \<in> \<K> M i t)"
HRB_slice_def: fixes S :: "'node SDG_node set" shows "HRB_slice S \<equiv> {n'. \<exists>n\<in>S. n' \<in> combine_SDG_slices (sum_SDG_slice1 n)}"
constructive_security2_axioms: shows "constructive_security2 real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w"
equiv_iff_qle: fixes x :: "'b"   and y :: "'b" assumes "I x"   and "I y" shows "subsumed x y = x \<preceq> y"
G2: shows "G2 = \<lparr>nodes = G2_nodes, edges = {(n, e, n'). (n, e, n') \<in> edges (del_unEdge v w v' G) \<and> n \<in> G2_nodes \<and> n' \<in> G2_nodes}\<rparr>"
abs_eq: fixes xa :: "int"   and x :: "int" shows "(Word.Word xa < Word.Word x) = (take_bit LENGTH('k) xa < take_bit LENGTH('k) x)"
div_le_mono2: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m"   and "m \<le> n" shows "k div n \<le> k div m"
example_fault_domain: shows "asc_fault_domain M\<^sub>S M\<^sub>I 2"
graph_union_def: fixes G\<^sub>1 :: "('e, 'f) labeled_graph"   and G\<^sub>2 :: "('e, 'f) labeled_graph" shows "graph_union G\<^sub>1 G\<^sub>2 = LG (edges G\<^sub>1 \<union> edges G\<^sub>2) (vertices G\<^sub>1 \<union> vertices G\<^sub>2)"
frag_extend_cmul: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and c :: "int"   and x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_extend f (frag_cmul c x) = frag_cmul c (frag_extend f x)"
nat_sum_has_integral_floor: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "m < n" shows "((\<lambda>x. f (nat \<lfloor>x\<rfloor>)) has_integral sum f {m..<n}) {real m..real n}"
L1_I_1: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f" shows "f \<in> space\<^sub>N (\<LL> 1 M)"
tell_option_def: fixes tell :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and w :: "'f"   and m :: "('d, 'e) optionT" shows "tell_option tell w m = OptionT (tell w (run_option m))"
distinct_185: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x131 :: "expr"   and x132 :: "expr"   and x133 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> if (x131) x132 else x133"
rm_vars_dom_eq': fixes V :: "'a set"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "subst_domain (rm_vars (UNIV - V) s) = subst_domain s \<inter> V"
sel_2: fixes x1 :: "'g"   and x2 :: "'f" shows "snd (x1, x2) = x2"
map_add_upd: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d" shows "f ++ g(x \<mapsto> y) = (f ++ g)(x \<mapsto> y)"
intros_17: fixes v\<^sub>1 :: "Syntax.term"   and v\<^sub>2 :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and va :: "mode" assumes "value v\<^sub>1"   and "value v\<^sub>2" shows "\<lless>\<pi>, Prj2 (Syntax.Pair v\<^sub>1 v\<^sub>2)\<ggreater> va\<rightarrow> \<lless>\<pi>, v\<^sub>2\<ggreater>"
wf_subtypeE: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and U :: "POPLmark.type" assumes "\<Gamma> \<turnstile> T <: U" obtains "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U"
hom_induced_empty_hom: fixes p :: "int"   and X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "hom_induced p X {} X' {} f \<in> hom (homology_group p X) (homology_group p X')"
hp: shows "h\<^sub>2 a = \<lfloor>(C, S)\<rfloor>"
HPhase0Read_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "p \<noteq> q"   and "HInv4b s q" shows "HInv4b s' q"
Times_in_interior_subtopology: fixes x :: "'c"   and y :: "'d"   and U :: "('c \<times> 'd) set"   and S :: "'c set"   and T :: "'d set" assumes "(x, y) \<in> U"   and "openin (top_of_set (S \<times> T)) U" obtains V :: "'c set" and W :: "'d set" where "openin (top_of_set S) V" and "x \<in> V" and "openin (top_of_set T) W" and "y \<in> W" and "V \<times> W \<subseteq> U"
supp_Rep_formula: fixes x :: "('a, 'b, 'c, 'd) formula" shows "supp (Rep_formula x) = supp x"
otau_completeTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> ocnet_sos S" obtains R :: "nat set" and m :: "'f" where "((\<sigma>, s), R:*cast(m), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"   | "((\<sigma>, s), \<tau>\<^sub>n, \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
P_def': fixes x :: "'a" shows "P x = \<langle>t', x\<rangle> *\<^sub>C t'"
valid_K_cfg: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and cfg' :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "cfg \<in> MDP.valid_cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "cfg' \<in> MDP.valid_cfg"
power_minor_var: fixes X :: "'a list set"   and k :: "nat"   and n :: "nat" assumes "\<forall>w\<in>X. k \<le> |w|" shows "\<forall>w\<in>X\<^bsup>Suc n\<^esup>. n \<cdot> k \<le> |w|"
induct: fixes P :: "'a tree \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "'a tree"   and a1 :: "'a" assumes "\<And>w\<^sub>b b a. P (Leaf w\<^sub>b b) a"   and "\<And>w w\<^sub>b b w\<^sub>c c a. P (Node w (Leaf w\<^sub>b b) (Leaf w\<^sub>c c)) a"   and "\<And>w v va vb t\<^sub>2 a. \<lbrakk>P (Node v va vb) a; P t\<^sub>2 a\<rbrakk> \<Longrightarrow> P (Node w (Node v va vb) t\<^sub>2) a"   and "\<And>w t\<^sub>1 v va vb a. \<lbrakk>P t\<^sub>1 a; P (Node v va vb) a\<rbrakk> \<Longrightarrow> P (Node w t\<^sub>1 (Node v va vb)) a" shows "P a0 a1"
ex_forward: fixes P' :: "'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x. P' x"   and "\<And>x. P' x \<Longrightarrow> P x" shows "\<exists>x. P x"
N_2: fixes k :: "nat" shows "N k < N (Suc k)"
Suc: fixes bin :: "int" shows "rbl_pred (rev (bin_to_bl_aux n bin [])) = rev (bin_to_bl_aux n (bin - 1) [])"