hcju/isabelleps huggingface dataset:
t1: shows "items_le k (\<J> k (Suc u)) = items_le k (\<pi> k (\<T> k (Suc u)) (\<J> k u))"
D_def: fixes y :: "'s"   and n :: "nat" shows "D y n = \<bar>p x y n - pmf N y\<bar>"
d2_ex: fixes y :: "'a"   and t :: "real" assumes "(y, t) \<in> cball (x, 0) d2"   and "0 < t" shows "t \<in> existence_ivl0 y"
norm_pre_pure_iff_sng: fixes b :: "bool"   and f :: "'d Heap"   and Q :: "'d \<Rightarrow> assn" shows "<\<up> b> f <Q> = (b \<longrightarrow> <emp> f <Q>)"
cimage_cinsert: fixes f :: "'g \<Rightarrow> 'f"   and a :: "'g"   and B :: "'g cset" shows "cimage f (cinsert a B) = cinsert (f a) (cimage f B)"
rowvector_def: fixes v :: "('b, 'c) vec" shows "rowvector v = (\<chi>i j. v $ j)"
tendsto_ennrealI: fixes f :: "'b \<Rightarrow> real"   and x :: "real"   and F :: "'b filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"
scale_left_commute: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "a * (b * x) = b * (a * x)"
\<eta>_map_simp: fixes y :: "'c" assumes "C.ide y" shows "transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) C.map F'G'.\<eta>o y = \<phi>' y (F' y)"
bijI': fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. (f x = f y) = (x = y)"   and "\<And>y. \<exists>x. y = f x" shows "bij f"
lset_subset: fixes xs :: "'c llist"   and A :: "'c set" assumes "\<not> lset xs \<subseteq> A" shows "\<exists>n. enat n < llength xs \<and> xs $ n \<notin> A"
sup_float_def: fixes a :: "float"   and b :: "float" shows "sup a b = max a b"
set_replicate_Suc: fixes n :: "nat"   and x :: "'g" shows "set (replicate (Suc n) x) = {x}"
univ_poly_is_principal: fixes K :: "('c multiset \<Rightarrow> 'a) set" assumes "subfield K L" shows "principal_domain (K [X]\<^bsub>L\<^esub>)"
simps_4: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval (Times a\<^sub>1 a\<^sub>2) s = paval a\<^sub>1 s * paval a\<^sub>2 s"
ivl_rep_def: fixes X :: "'c set" shows "ivl_rep X = SPEC (\<lambda>(l, u). X = {l..u})"
coeff_smult: fixes a :: "'b"   and p :: "'b poly"   and n :: "nat" shows "coeff (Polynomial.smult a p) n = a * coeff p n"
Input: fixes a :: "name"   and x :: "name"   and P :: "pi" shows "a<x>.P \<longmapsto> a<x> \<prec> P"
polyn_c_max: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "polyn_expr R X (fst c) c = polyn_expr R X (c_max S c) c"
common_reparam_exists_def: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_reparam_exists one_chain1 one_chain2 \<equiv> \<exists>subdiv ps1 ps2. chain_reparam_chain' (one_chain1 - ps1) subdiv \<and> chain_reparam_chain' (one_chain2 - ps2) subdiv \<and> (\<forall>(k, \<gamma>)\<in>subdiv. \<gamma> C1_differentiable_on {0..1}) \<and> boundary_chain subdiv \<and> (\<forall>(k, \<gamma>)\<in>ps1. point_path \<gamma>) \<and> (\<forall>(k, \<gamma>)\<in>ps2. point_path \<gamma>)"
init_wit_blue_early_def: fixes s :: "'a"   and t :: "'a" shows "init_wit_blue_early s t \<equiv> if s = t then CIRC [] [s] else REACH t [s]"
7: shows "nth_least (liset A w) i < Suc (nth_least (liset A w) k')"
T_BIT_absch_le: fixes n :: "nat" assumes "n \<le> length qs" shows "T_BIT n \<le> 7 / 4 * real_of_int (T_A n) - 3 / 4 * real n"
d: fixes ii :: "nat" assumes "ii \<le> m" shows "d fs' ii = d fs ii"
measure_map_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf"   and A :: "'a set" shows "Sigma_Algebra.measure (measure_spmf (map_spmf f p)) A = Sigma_Algebra.measure (measure_spmf p) (f -` A)"
rels: shows "rel_interior S = {x \<in> S. \<forall>h\<in>F. a h \<bullet> x < b h}"
fail_env_def: shows "local.fail_env = EnvT (\<lambda>r. fail)"
lex_ord_pair_EqI: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> order option" assumes "tc.oalist_inv_raw xs"   and "tc.oalist_inv_raw ys"   and "\<And>k. k \<in> fst ` set xs \<union> fst ` set ys \<Longrightarrow> f k (lookup_pair_tc xs k) (lookup_pair_tc ys k) = Some Eq" shows "lex_ord_pair_tc f xs ys = Some Eq"
winding_number_subpath_combine: fixes g :: "real \<Rightarrow> complex"   and z :: "complex"   and u :: "real"   and v :: "real"   and w :: "real" assumes "path g"   and "z \<notin> path_image g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}" shows "winding_number (subpath u v g) z + winding_number (subpath v w g) z = winding_number (subpath u w g) z"
not_all: fixes P :: "'b \<Rightarrow> bool" shows "(\<not> (\<forall>x. P x)) = (\<exists>x. \<not> P x)"
blocks_def: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and R :: "'b set"   and p :: "'b" assumes "personal_quorums quorum_of" shows "personal_quorums.blocks quorum_of R p \<equiv> \<forall>Q. quorum_of p Q \<longrightarrow> Q \<inter> R \<noteq> {}"
eventually_const: fixes F :: "'b filter"   and P :: "bool" assumes "F \<noteq> bot" shows "(\<forall>\<^sub>F x in F. P) = P"
group_isomorphism_axioms: shows "group_isomorphism image.induced kernel.Partition ([\<cdot>]) (kernel.Class \<one>) (\<eta> ` G) (\<cdot>') \<one>'"
u_K4_imp_K2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace>\<rbrace>) \<in> set evs \<and> servKlife + Ts \<le> authKlife + Ta"
st_proj2_inv_continuous: shows "continuous_on UNIV st_proj2_inv"
explode_Empty: fixes x :: "ZF" shows "(explode x = {}) = (x = Empty)"
ground_subst_range_empty_fv: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a" assumes "ground (subst_range \<theta>)"   and "x \<in> subst_domain \<theta>" shows "fv (\<theta> x) = {}"
grid_dim_remove_outer: fixes b :: "(nat \<times> int) list"   and ds :: "nat set" shows "grid b ds = grid b {d \<in> ds. d < length b}"
evalOP_AbsV: fixes e :: "db" assumes "val (DBAbsV e)" shows "DBAbsV e \<Down> DBAbsV e"
lens_plus_swap: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'c" assumes "X \<bowtie> Y" shows "swap\<^sub>L ;\<^sub>L (X +\<^sub>L Y) = Y +\<^sub>L X"
assms: fixes kv :: "'a \<times> 'b" shows "fst (f kv) = fst kv"
arcs_G: shows "arcs (with_proj I') = arcs (with_proj H) \<union> {uv, swap_in (parcs I') uv} - {uw, swap_in (parcs H) uw, vw, swap_in (parcs H) vw}"
a2: shows "Re (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Re (f x))"
whileDecr: shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. enat (nat (s ''x'')) + 1} WHILE Less (N 0) (V ''x'') DO (SKIP;; SKIP;; ''x'' ::= Plus (V ''x'') (N (- 1))) {\<lambda>s. enat 0}"
blocking_on_imp_not_stable: fixes ds :: "'d set"   and X :: "'x set"   and h :: "'h"   and X'' :: "'x set" assumes "blocking_on ds X h X''" shows "\<not> stable_on ds X"
2: shows "(t - punit.lt (rep_list (bs ! i))) \<oplus> lt (bs ! i) \<prec>\<^sub>t lt p"
slim_simps_1: shows "pverts H.slim = H.slim_verts"
of_nat_0_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(zero = semiring_1.of_nat one plus zero n) = (0 = n)"
comp_null_2: fixes f :: "'a" shows "f \<cdot> null = null"
distinct_in_set_butlastD: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (butlast xs)"   and "distinct xs" shows "x \<noteq> last xs"
ideal_points_unique: fixes H :: "circline" assumes "is_poincare_line H" shows "ideal_points H = calc_ideal_points H"
simps_3: fixes v :: "'a" shows "cycle_of_list [v] = id"
assms_1: shows "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"
prob0_fst_qubits_def: fixes n :: "nat"   and v :: "complex mat" shows "prob0_fst_qubits n v \<equiv> \<Sum>j\<in>{k |k. k < 2 ^ (n + 1) \<and> (\<forall>i\<in>{0..<n}. \<not> select_index (n + 1) i k)}. (cmod (v $$ (j, 0)))\<^sup>2"
less_LiminfD: fixes y :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "y < Liminf F f" shows "\<forall>\<^sub>F x in F. y < f x"
analz_Un_SignSet: fixes K :: "msg set"   and G :: "msg set"   and H :: "msg set" assumes "K \<subseteq> range LtK"   and "Enc_keys_clean (G \<union> H)" shows "analz (SignSet G K \<union> H) \<subseteq> SignSet G K \<union> analz (G \<union> H)"
fmpred_add: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap" assumes "fmpred P m"   and "fmpred P n" shows "fmpred P (m ++\<^sub>f n)"
CAS_\<tau>red1r_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and e3 :: "'addr expr1" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h (Val v\<bullet>compareAndSwap(D\<bullet>F, e, e3), xs) (Val v\<bullet>compareAndSwap(D\<bullet>F, e', e3), xs')"
bl_shiftl1: fixes w :: "'b word" shows "to_bl (shiftl1 w) = tl (to_bl w) @ [False]"
cong_diff: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "Cong A B C D" shows "C \<noteq> D"
simps_2: fixes a :: "(int \<times> int \<times> protocol) set set"   and b :: "(int \<times> int \<times> protocol) set set"   and p :: "protocol \<times> int"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "wellformed_policy3Pr (AllowPortFromTo a b p # xs) = ((\<forall>r. r \<in> set xs \<longrightarrow> dom (Cp r) \<inter> dom (Cp (AllowPortFromTo a b p)) = {}) \<and> wellformed_policy3Pr xs)"
r_def: shows "r = run\<^sub>t (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) w"
undirected_def: fixes G :: "'b list_graph" shows "FiniteListGraph.undirected G \<equiv> \<lparr>nodesL = nodesL G, edgesL = remdups (edgesL G @ backlinks (edgesL G))\<rparr>"
has_bochner_integral_eq: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "has_bochner_integral M f x"   and "has_bochner_integral M f y" shows "x = y"
injectivity_5: fixes P :: "nat set set"   and j :: "nat"   and P' :: "nat set set"   and j' :: "nat"   and B' :: "nat set"   and k :: "nat"   and B'' :: "nat set"   and k' :: "nat"   and Q :: "nat set set"   and Q' :: "nat set set"   and f :: "nat \<Rightarrow> nat set"   and g :: "nat \<Rightarrow> nat set"   and P'' :: "nat set set"   and P''' :: "nat set set" assumes "(partition_on C P \<and> card P = j) \<and> partition_on C P' \<and> card P' = j'"   and "(B' \<subseteq> B \<and> card B' = k) \<and> B'' \<subseteq> B \<and> card B'' = k'"   and "partition_on B' Q \<and> partition_on B'' Q'"   and "f \<in> B - B' \<rightarrow>\<^sub>E P \<and> g \<in> B - B'' \<rightarrow>\<^sub>E P'"   and "P'' = (\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P \<and> P''' = (\<lambda>X. X \<union> {x \<in> B - B''. g x = X}) ` P'"   and "P'' \<union> Q = P''' \<union> Q'" shows "j = j'"
morphism_ringed_spaces_axioms_def: fixes X :: "'c set"   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_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 \<equiv> 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 (im_sheaf.im_sheaf X \<O>\<^sub>X f) (im_sheaf.im_sheaf_morphisms X \<rho>\<^sub>X f) b (im_sheaf.add_im_sheaf X add_str\<^sub>X f) (im_sheaf.mult_im_sheaf X mult_str\<^sub>X f) (im_sheaf.zero_im_sheaf X zero_str\<^sub>X f) (im_sheaf.one_im_sheaf X one_str\<^sub>X f) \<phi>\<^sub>f"
strict_order_nest: fixes r :: "('a \<times> 'a) set"   and R :: "((accent \<times> 'a) list \<times> (accent \<times> 'a) list) set" assumes "strict_order r"   and "strict_order R" shows "strict_order (nest r R)"
s_prod_test_aux2: fixes a :: "'a"   and A :: "'a set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" assumes "(a, A) \<in> R"   and "\<forall>a\<in>A. (a, {a}) \<in> S" shows "(a, A) \<in> R \<cdot> S"
ran_is_image: fixes M :: "'b \<Rightarrow> 'a option" shows "ran M = (the \<circ> M) ` dom M"
add_nonpos_eq_0_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<le> (0::'a)"   and "y \<le> (0::'a)" shows "(x + y = (0::'a)) = (x = (0::'a) \<and> y = (0::'a))"
fv\<^sub>s\<^sub>s\<^sub>t_subst_obtain_var: fixes x :: "'c"   and S :: "('d, 'c) stateful_strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "x \<in> fv\<^sub>s\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>y\<in>fv\<^sub>s\<^sub>s\<^sub>t S. x \<in> fv (\<delta> y)"
eq_app_right_class_eq: shows "UNIV // eq_app_right MN.language = (\<lambda>q. eq_app_right MN.language `` {MN.path_to q}) ` MN.accessible"
insert_iff: fixes a :: "'x"   and b :: "'x"   and A :: "'x set" shows "(a \<in> insert b A) = (a = b \<or> a \<in> A)"
homotopy_eqv_homotopic_triviality: fixes S :: "'a set"   and T :: "'b set"   and U :: "'c set" assumes "S homotopy_eqv T" shows "(\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> S \<and> continuous_on U g \<and> g ` U \<subseteq> S \<longrightarrow> homotopic_with_canon (\<lambda>x. True) U S f g) = (\<forall>f g. continuous_on U f \<and> f ` U \<subseteq> T \<and> continuous_on U g \<and> g ` U \<subseteq> T \<longrightarrow> homotopic_with_canon (\<lambda>x. True) U T f g)"
induct: fixes P :: "'a list \<Rightarrow> (real, 'a) vec list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "(real, 'a) vec list" assumes "\<And>uu_. P [] uu_"   and "\<And>k ps. P [k] ps"   and "\<And>k v va ps. P (v # va) ps \<Longrightarrow> P (k # v # va) ps" shows "P a0 a1"
w_def: shows "w = (\<lambda>(A, j). MPoly_Type.monom (monom_of_set A + sng j (k - card A)) ((- (1::'a)) ^ card A))"
FG_advice_\<mu>LTL_2: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "\<phi> \<in> \<mu>LTL" shows "\<phi>[X]\<^sub>\<mu> = \<phi>"
not_Some_eq2: fixes v :: "('d \<times> 'e) option" shows "(\<forall>x y. v \<noteq> Some (x, y)) = (v = None)"
induct: fixes x1 :: "Mode \<Rightarrow> 'Var set"   and x2 :: "('Var, 'AExp, 'BExp) Stmt"   and x3 :: "Mode \<Rightarrow> 'Var set"   and P :: "(Mode \<Rightarrow> 'Var set) \<Rightarrow> ('Var, 'AExp, 'BExp) Stmt \<Rightarrow> (Mode \<Rightarrow> 'Var set) \<Rightarrow> bool" assumes "\<turnstile> x1 { x2 } x3"   and "\<And>mds annos. P mds (Skip \<otimes> annos) (mds \<oplus> annos)"   and "\<And>x mds e annos. \<lbrakk>x \<notin> mds GuarNoWrite; aexp_vars e \<inter> mds GuarNoRead = {}\<rbrakk> \<Longrightarrow> P mds ((x \<leftarrow> e) \<otimes> annos) (mds \<oplus> annos)"   and "\<And>mds annos c\<^sub>1 mds'' c\<^sub>2 e. \<lbrakk>\<turnstile> mds \<oplus> annos { c\<^sub>1 } mds''; P (mds \<oplus> annos) c\<^sub>1 mds''; \<turnstile> mds \<oplus> annos { c\<^sub>2 } mds''; P (mds \<oplus> annos) c\<^sub>2 mds''; bexp_vars e \<inter> mds GuarNoRead = {}\<rbrakk> \<Longrightarrow> P mds (Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos) mds''"   and "\<And>mds' mds annos c e. \<lbrakk>mds' = mds \<oplus> annos; \<turnstile> mds' { c } mds'; P mds' c mds'; bexp_vars e \<inter> mds' GuarNoRead = {}\<rbrakk> \<Longrightarrow> P mds (While e c \<otimes> annos) mds'"   and "\<And>mds c\<^sub>1 mds' c\<^sub>2 mds''. \<lbrakk>\<turnstile> mds { c\<^sub>1 } mds'; P mds c\<^sub>1 mds'; \<turnstile> mds' { c\<^sub>2 } mds''; P mds' c\<^sub>2 mds''\<rbrakk> \<Longrightarrow> P mds (c\<^sub>1 ;; c\<^sub>2) mds''"   and "\<And>mds\<^sub>2 c mds\<^sub>2' mds\<^sub>1 mds\<^sub>1'. \<lbrakk>\<turnstile> mds\<^sub>2 { c } mds\<^sub>2'; P mds\<^sub>2 c mds\<^sub>2'; mds\<^sub>1 \<sqsubseteq> mds\<^sub>2; mds\<^sub>2' \<sqsubseteq> mds\<^sub>1'\<rbrakk> \<Longrightarrow> P mds\<^sub>1 c mds\<^sub>1'" shows "P x1 x2 x3"
slp_def: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" shows "n -as\<rightarrow>\<^bsub>sl\<^esub>* n' \<equiv> n -as\<rightarrow>* n' \<and> same_level_path as"
pow_len: fixes u :: "'b list"   and k :: "nat" shows "\<^bold>|u \<^sup>@ k\<^bold>| = k * \<^bold>|u\<^bold>|"
PDataSpace_DefaultPData: fixes D :: "'b dataspace" shows "PDataSpace (DefaultPData D) = D"
funpow_swap1: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
xor_compl_right: fixes x :: "'b"   and y :: "'b" shows "x XOR NOT y = NOT (x XOR y)"
return_bind: fixes x :: "'a \<times> 's"   and f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (return x) f = f x"
isContD: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "isCont f x" shows "f \<midarrow>x\<rightarrow> f x"
diag: shows "{} \<turnstile> \<delta> IFF Neg (PfP \<guillemotleft>\<delta>\<guillemotright>)"
inf_mono: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and a :: "'b::type"   and c :: "'b::type"   and b :: "'b::type"   and d :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (c::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (b::'b::type) (d::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type)) (inf (c::'b::type) (d::'b::type))"
singleton_None_Sup: shows "\<Squnion> {None} = None"
ht_loaded_locs_def: shows "ht_loaded_locs = prefixed ''hs_'' - {hs_load_pending, local.hs_pending, hs_mfence, hs_load_ht}"
d_OUT_plus_web: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<Gamma> :: "('k, 'l) web_scheme"   and g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" shows "d_OUT (f \<frown>\<^bsub>\<Gamma>\<^esub> g) x = d_OUT f x + d_OUT (g \<upharpoonleft> \<Gamma> / f) x"
S2: shows "nat_to_sch n = Pair_op (nat_to_sch (c_fst (c_snd n))) (nat_to_sch (c_snd (c_snd n)))"
ltakeWhile_mono: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "mono_llist (\<lambda>ltakeWhile. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> if P x then LCons x (ltakeWhile xs) else LNil)"
ide_def: fixes a :: "('obj, 'obj) concrete_category.arr" shows "ide a \<equiv> a \<cdot> a \<noteq> null \<and> (\<forall>f. (f \<cdot> a \<noteq> null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> null \<longrightarrow> a \<cdot> f = f))"
Sup_atLeast: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (ord.atLeast less_eq x) = top"
integral_nonneg: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
case_sum_expand_Inr_pointfree: fixes f :: "'a + 'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<circ> Inl = g" shows "case_sum g (f \<circ> Inr) = f"
weakly_precedes_e: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>"
\<gamma>_def: shows "\<gamma> \<equiv> circlepath 0 r"
snd_proj_prop: fixes P :: "'c \<Rightarrow> bool"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "\<forall>x\<in>snd_proj {(A, B) |A B. P B \<and> R A B}. P x"
\<sigma>: fixes l :: "nat"   and i :: "nat"   and j :: "nat" assumes "l \<le> length fs" shows "\<sigma> l i j = d l * (\<Sum>k<l. \<mu> i k * \<mu> j k * \<beta> fs k)"
farkas_coefficients_unsat: fixes cs :: "constraint set"   and C :: "(rat \<times> constraint) list" assumes "farkas_coefficients cs C" shows "\<nexists>v. v \<Turnstile>\<^sub>c\<^sub>s cs"
S2: shows "c_assoc_have_key (pr_gr (loc_upb (c_fst (c_snd n)) y)) (c_pair (c_fst (c_snd n)) y) = 0"
named_loop_def: fixes name :: "'f" shows "named_loop name = UNIV"
relcompI: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e" assumes "r a b"   and "s b c" shows "(r OO s) a c"
head_head: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "head (head p) = head p"
Subspace_dim_le: fixes U :: "'b set" assumes "Subspace U" shows "dim_R U \<le> dim_R V"
adjuster_triv: fixes n :: "nat"   and w :: "rat vec"   and us :: "rat vec list" shows "adjuster_triv n w (map (\<lambda>x. (x, \<parallel>x\<parallel>\<^sup>2)) us) = adjuster n w us"
intros_1: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" shows "t \<in> timpl_closure t TI"
coplanar_perm_13: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet C A D B"
is_ok_def: fixes result :: "('k, 'l) result" shows "is_ok result = (\<exists>x2. result = Ok x2)"