hcju/isabelleps huggingface dataset:
tree_map_of_invar: fixes T :: "(('a \<times> 'b) \<times> color) tree"   and pres :: "('a \<times> 'b) list" assumes "M.invar T" shows "M.invar (M.tree_map_of' T pres)"
atLeast1_atMost_eq: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {Suc 0..n} = (\<Sum>k<n. g (Suc k))"
assms_3: shows "Domain a \<inter> X \<noteq> {}"
S2: shows "Struct wtFsym wtPsym arOf resOf intT intF2 intP"
L5: fixes x :: "nat" shows "(p x \<in> V) = (x \<in> A)"
select_append': fixes ys :: "'b list"   and zs :: "'b list"   and k :: "nat" assumes "\<And>y z. \<lbrakk>y \<in> set ys; z \<in> set zs\<rbrakk> \<Longrightarrow> y \<le> z"   and "k < length ys + length zs" shows "select k (ys @ zs) = (if k < length ys then select k ys else select (k - length ys) zs)"
Iff_Ann1: fixes r :: "'a pfm"   and r' :: "'a pfm"   and p :: "'a pfm" assumes "\<turnstile>\<^sub>! (r \<^bold>\<longleftrightarrow>\<^sub>! r')"   and "static p" shows "\<turnstile>\<^sub>! ([r]\<^sub>! p \<^bold>\<longleftrightarrow>\<^sub>! [r']\<^sub>! p)"
set_valued_functor_def: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and S :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" shows "set_valued_functor C S F \<equiv> category C \<and> replete_set_category S \<and> functor C S F"
divide_nonpos_pos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less zero y" shows "less_eq (divide x y) zero"
indep_in_subI: fixes \<E> :: "'a set"   and X :: "'a set"   and \<E>' :: "'a set" assumes "\<E>.indep_in \<E> X" shows "\<E>.indep_in \<E>' (X \<inter> \<E>')"
D_2: shows "\<forall>x y. x < y \<longrightarrow> (\<exists>d\<in>D. x < d \<and> d \<le> y)"
subspace_imp_convex: fixes s :: "'c set" assumes "subspace s" shows "convex s"
hyps_1: fixes kdt :: "'a kdt" assumes "h = height kdt"   and "KD_Tree.size_kdt kdt = 2 ^ height kdt" shows "KD_Tree.complete kdt"
g_nconst: shows "\<exists>\<^sub>F w in at z. g w \<noteq> 0"
int_option_enumeration_inj: shows "inj int_option_enumeration"
measurable_stream_filtration1: fixes i :: "nat"   and n :: "enat"   and M :: "'b measure" assumes "enat i \<le> n" shows "(\<lambda>\<omega>. \<omega> !! i) \<in> stream_filtration M n \<rightarrow>\<^sub>M M"
less_eq_vname_def: fixes v1 :: "vname"   and v2 :: "vname" shows "(v1 \<le> v2) = (v1 < v2 \<or> v1 = v2)"
mat_extension_def: fixes dims :: "nat list"   and vars :: "nat set"   and m :: "'b mat" shows "mat_extension dims vars m = tensor_mat dims vars m (1\<^sub>m (partial_state.d2 dims vars))"
dim_unique: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set"   and n :: "nat" assumes "B \<subseteq> V"   and "V \<subseteq> vec.span B"   and "vec.independent B"   and "card B = n" shows "vec.dim V = n"
take_bit_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit 0 a = zero"
R32_d: shows "pmf (sds R32) d = pmf (sds R28) a"
ccard_inc: fixes p :: "nat \<Rightarrow> bool"   and n' :: "nat"   and n :: "nat" assumes "p (Suc n')"   and "n \<le> n'" shows "card {i. n < i \<and> i \<le> Suc n' \<and> p i} = Suc (card {i. n < i \<and> i \<le> n' \<and> p i})"
rank_subadditive: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n nc"   and "B \<in> carrier_mat n nc" shows "vec_space.rank n (A + B) \<le> vec_space.rank n A + vec_space.rank n B"
that_2: fixes y :: "'b" assumes "x = Some y"   and "run (s y) \<sigma> \<sigma>' r" shows "thesis"
uJ_AE: fixes n :: "nat"   and m :: "nat" shows "AE x in MJ. uJ (n + m) x \<le> uJ n x + uJ m ((TJ ^^ n) x)"
mat_mult_scalar: fixes A :: "(('e, 'h) Finite_Cartesian_Product.vec, 'g) Finite_Cartesian_Product.vec"   and B :: "(('e, 'f) Finite_Cartesian_Product.vec, 'h) Finite_Cartesian_Product.vec" shows "A ** B = local.mk_mat (\<lambda>i j. scalar_product (row i A) (column j B))"
assms_2: shows "\<tau>s \<turnstile>\<^sub>\<tau> t : \<tau>2"
eq1: shows "connect_obsf (absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C))) (obsf_resource (sim_inner |\<^sub>= 1\<^sub>C \<rhd> ideal)) = connect_obsf \<A> (obsf_resource (sim_outer \<odot> sim_inner |\<^sub>= 1\<^sub>C \<rhd> ideal))"
elims: fixes x :: "('d \<times> 'e) list"   and xa :: "('d, 'e, 'f) FSM"   and xb :: "'d list set"   and y :: "('d \<times> 'e) list set set" assumes "N x xa xb = y" obtains io :: "('d \<times> 'e) list" and M :: "('d, 'e, 'f) FSM" and V :: "'d list set" where "x = io" and "xa = M" and "xb = V" and "y = {V'' \<in> Perm V M. map fst (mcp' io V'') = mcp' (map fst io) V}"
next_not_nil_right: fixes x :: "'node" shows "(x, nil) \<notin> next"
lsl_comp_pres: fixes x :: "'b"   and y :: "'b" shows "\<nu> (x \<cdot> y) = \<nu> x \<cdot> \<nu> y"
ln_ln_primorial_asymp_equiv: shows "(\<lambda>x. ln (ln (real (primorial x)))) \<sim>[at_top] ln"
assoc_mult_mat: fixes A :: "'a mat"   and n\<^sub>1 :: "nat"   and n\<^sub>2 :: "nat"   and B :: "'a mat"   and n\<^sub>3 :: "nat"   and C :: "'a mat"   and n\<^sub>4 :: "nat" assumes "A \<in> carrier_mat n\<^sub>1 n\<^sub>2"   and "B \<in> carrier_mat n\<^sub>2 n\<^sub>3"   and "C \<in> carrier_mat n\<^sub>3 n\<^sub>4" shows "A * B * C = A * (B * C)"
wf_induct_rule: fixes r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "wf r"   and "\<And>x. (\<And>y. (y, x) \<in> r \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
flow_has_vector_derivative: fixes t :: "real"   and x0a :: "'c"   and t0 :: "real"   and x0 :: "'c \<Rightarrow>\<^sub>L 'c" assumes "t \<in> mvar.existence_ivl x0a t0 x0" shows "(mvar.flow x0a t0 x0 has_vector_derivative vareq x0a t o\<^sub>L mvar.flow x0a t0 x0 t) (at t)"
set_zip_rightD: fixes x :: "'a"   and y :: "'b"   and xs :: "'a list"   and ys :: "'b list" assumes "(x, y) \<in> set (zip xs ys)" shows "y \<in> set ys"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b \<le> b"
res: shows "res = prime_factorization_nat_main n jj is ps"
strict_coboundedI1: fixes c :: "'b"   and a :: "'b"   and b :: "'b" assumes "c < a" shows "c < max a b"
that: fixes T :: "'a set" assumes "countable T"   and "T \<subseteq> S"   and "S \<subseteq> closure T" shows "thesis"
case_y_in_range: shows "\<forall>x\<in>cbox a b. integral (cbox (g1 x) (g2 x)) (\<lambda>y. g (y, x)) = integral UNIV (\<lambda>y. f (y, x))"
wellorder_Least_lemma_2: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "class.wellorder less_eq less"   and "P k" shows "less_eq (ord.Least less_eq P) k"
simps_1: fixes xs :: "'a list"   and e :: "'a" shows "insert_rga xs (e, None) = insert_body xs e"
streams_sets: fixes X :: "'b set"   and M :: "'b measure" assumes "X \<in> sets M" shows "streams X \<in> sets (stream_space M)"
stern_brocot_iterate_aux_decompose: fixes m :: "(nat \<times> nat) \<times> nat \<times> nat"   and path :: "dir list" shows "\<exists>m''. m \<otimes> m'' = root (traverse_tree path (stern_brocot_iterate_aux m)) \<and> Det m'' = 1"
list_vec: fixes xs :: "'c list" shows "list_of_vec (vec_of_list xs) = xs"
IH_readable_1: shows "cnf F \<union> \<Union> (cnf ` set_mset \<Gamma>) \<turnstile> \<box>"
lenlex_append2: fixes R :: "('a \<times> 'a) set"   and us :: "'a list"   and xs :: "'a list"   and ys :: "'a list" assumes "irrefl R" shows "((us @ xs, us @ ys) \<in> lenlex R) = ((xs, ys) \<in> lenlex R)"
simps_2: fixes cs :: "'edge list"   and as :: "'edge list"   and a :: "'edge" shows "upd_rev_cs cs (as @ [a]) = (case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> case cs of [] \<Rightarrow> upd_rev_cs cs as | c' # cs' \<Rightarrow> upd_rev_cs cs' as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> upd_rev_cs (a # cs) as | _ \<Rightarrow> upd_rev_cs cs as)"
intros_4: fixes x :: "W.var"   and \<Gamma> :: "(W.var \<times> tyS) list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "ty"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "ty" assumes "x \<sharp> \<Gamma>"   and " \<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1 "   and " (x, close \<Gamma> T\<^sub>1) # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2 "   and "set (ftv T\<^sub>1 - ftv \<Gamma>) \<sharp>* T\<^sub>2" shows " \<Gamma> \<turnstile> Let x be t\<^sub>1 in t\<^sub>2 : T\<^sub>2 "
simps_2: fixes x :: "('a, 'b, 'c) prog"   and xs :: "('a, 'b, 'c) prog list" shows "iterate_M (x # xs) = x \<bind> (\<lambda>_. iterate_M xs)"
simps: fixes p :: "rat poly" shows "roots_of_rat_poly_main p = (let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then rat_roots2 p else case rational_root_test p of None \<Rightarrow> [] | Some x \<Rightarrow> x # roots_of_rat_poly_main (p div [:- x, 1:]))"
assms_13: shows "\<forall>y. (y \<in> R_NQ) = (wf_tuple n J y \<and> (\<forall>(A, X)\<in>NQ_pos. restrict A y \<in> X) \<and> (\<forall>(A, X)\<in>NQ_neg. restrict A y \<notin> X))"
prems_7: shows "set (clause # Wl') \<inter> set newWl = {}"
stateSpace_def: fixes ss :: "('a, 'b) fmap set"   and vs :: "'a set" shows "stateSpace ss vs \<equiv> \<forall>s. s \<in> ss \<longrightarrow> fmdom' s = vs"
path_components_eq_connected_components_of: fixes X :: "'c topology" assumes "locally_path_connected_space X" shows "path_components_of X = connected_components_of X"
H2_J_1: fixes P :: "'b alpha_d_scheme \<times> 'b alpha_d_scheme \<Rightarrow> bool" assumes "P is H2 healthy" shows "\<forall>A. case A of (A, A') \<Rightarrow> P (A, A'\<lparr>ok := False\<rparr>) \<longrightarrow> P (A, A'\<lparr>ok := True\<rparr>)"
non_Nil: shows "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"
r_result1_saturating': fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and v :: "nat"   and d :: "nat" assumes "eval r_result1 [t, i, x] \<down>= Suc v" shows "eval r_result1 [t + d, i, x] \<down>= Suc v"
f': fixes t :: "real"   and x :: "'a" assumes "t \<in> T"   and "x \<in> X" shows "(f t has_derivative blinfun_apply (f' (t, x))) (at x)"
update_convs_2: fixes locals' :: "locals \<Rightarrow> locals"   and heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "locals_update locals' \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = \<lparr>heap = heap, locals = locals' locals, \<dots> = more\<rparr>"
word_add_rbl: fixes v :: "'b word"   and vbl :: "bool list"   and w :: "'b word"   and wbl :: "bool list" assumes "to_bl v = vbl"   and "to_bl w = wbl" shows "to_bl (v + w) = rev (rbl_add (rev vbl) (rev wbl))"
iterate_step_valid: fixes stack :: "(recf \<times> nat list \<times> nat list) list"   and t :: "nat"   and rv :: "nat option" assumes "valid stack" shows "valid (fst (iterate t step (stack, rv)))"
image_f_inv_f: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'b set" assumes "surj f" shows "f ` inv f ` A = A"
assms_3: shows "\<forall>\<^sub>F x in net. b \<le> f x \<bullet> i"
strict_mono_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "(f x = f y) = (x = y)"
abs_eq: fixes x :: "'a set" shows "Sup (dual ` x) = dual (Inf x)"
codomains_def: fixes f :: "'b" shows "C'.codomains f \<equiv> {b. C'.ide b \<and> C' b f \<noteq> C'.null}"
step: fixes s :: "'a net_state"   and t :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state"   and t' :: "'a net_state" assumes "SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I"   and "P s t"   and "(SubnetS s t, a, SubnetS s' t') \<in> automaton.trans (pnet np (p1 \<parallel> p2))"   and "I a" shows "P s' t'"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> kerbIV" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
size_ffilter_card: fixes f :: "'b \<Rightarrow> bool"   and s :: "'b fset" shows "size (ffilter f s) = card {e \<in> fset s. f e}"
cont_case_prod_apply_iff: fixes luba :: "('b \<times> 'c) set \<Rightarrow> 'b \<times> 'c"   and orda :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'd"   and y :: "'e" shows "cont luba orda lubb ordb (\<lambda>x. (case x of (x, xa) \<Rightarrow> f x xa) y) = cont luba orda lubb ordb (\<lambda>(a, b). f a b y)"
hyps_3: fixes X :: "'a multiset" assumes "X' = add_mset x X" obtains z :: "'a" and Z :: "'a multiset" where "Y' = add_mset z Z" and "(x, z) \<in> ns" and "(X, Z) \<in> multpw ns"
linI: fixes y :: "'b"   and a :: "real"   and x :: "'b" assumes "y = a \<cdot> x" shows "y \<in> lin x"
in_set_simps_4: fixes P :: "'a \<Rightarrow> bool" shows "Ball (set []) P = True"
natural_isomorphism_axioms: shows "natural_isomorphism A B G H \<psi>"
supp: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b" assumes "GroupHom G T" shows "supp T \<subseteq> G"
return_node_def: fixes n :: "'b" shows "return_node n \<equiv> \<exists>a a'. valid_edge a \<and> n = targetnode a \<and> valid_edge a' \<and> a \<in> get_return_edges a'"
subst0_bound0: fixes p :: "fm"   and t :: "MIR.num" assumes "qfree p"   and "numbound0 t" shows "bound0 (subst0 t p)"
strict_implies_order: fixes b :: "'b::order"   and a :: "'b::order" assumes "(b::'b::order) < (a::'b::order)" shows "(b::'b::order) \<le> (a::'b::order)"
induct: fixes P :: "(name \<times> trm) list \<Rightarrow> name \<Rightarrow> bool"   and a0 :: "(name \<times> trm) list"   and a1 :: "name" assumes "\<And>x. P [] x"   and "\<And>y e \<theta> x. (x \<noteq> y \<Longrightarrow> P \<theta> x) \<Longrightarrow> P ((y, e) # \<theta>) x" shows "P a0 a1"
i_set_mult_mod_0_div: fixes I :: "nat set"   and k :: "nat"   and d :: "nat" assumes "I \<in> i_set_mult k"   and "k mod d = 0" shows "I \<oslash> d \<in> i_set_mult (k div d)"
sas_plus_equivalent_to_strips: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) \<pi>" shows "SAS_Plus_Semantics.is_parallel_solution_for_problem \<Psi> (\<phi>\<^sub>P\<inverse> \<Psi> \<pi>)"
subset_eq: fixes A :: "'g set"   and B :: "'g set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. x \<in> B)"
subterms_const: fixes f :: "'a" shows "subterms (Fun f []) = {Fun f []}"
hindered_by_into_hindered: fixes \<Gamma> :: "('a, 'b) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" shows "hindered \<Gamma>"
mset_msort: fixes xs :: "'c list" shows "mset (msort xs) = mset xs"
\<R>\<^sub>3_mem_eq: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
r_appendD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and rs2 :: "'c rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> rs1 @ rs2 \<Rightarrow>\<^sub>r t" shows "\<exists>s. \<Gamma>,\<gamma>,p\<turnstile> rs1 \<Rightarrow>\<^sub>r s"
AKcryptSK_Says: fixes authK :: "nat"   and servK :: "nat"   and S :: "agent"   and A :: "agent"   and X :: "msg"   and evs :: "event list" shows "AKcryptSK authK servK (Says S A X # evs) = (S = Tgs \<and> (\<exists>B tt. X = \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, tt\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, tt\<rbrace>\<rbrace>) \<or> AKcryptSK authK servK evs)"
tiling_domino_01: fixes t :: "(nat \<times> nat) set" assumes "t \<in> tiling domino" shows "card (evnodd t 0) = card (evnodd t 1)"
All2_SUCC_E': fixes H :: "fm set"   and u :: "tm"   and t :: "tm"   and i :: "name"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> u EQ SUCC t"   and "atom i \<sharp> t"   and "\<forall>C\<in>H. atom i \<sharp> C"   and "insert (All2 i t A) (insert (A(i::=t)) H) \<turnstile> B" shows "insert (All2 i u A) H \<turnstile> B"
a: shows "{i. i = a \<and> i < dim_row A} = {a}"
even_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b)"
ordIso_finite_Field: fixes r :: "('b \<times> 'b) set"   and s :: "('c \<times> 'c) set" assumes "r =o s" shows "finite (Field r) = finite (Field s)"
unique_class: fixes a :: "'a" assumes "a \<in> A" shows "\<exists>!b. b \<in> B \<and> a \<in> b"
set_child_nodes_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> Shadow_DOM.set_child_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
UNIF_idemp: fixes ss :: "('c \<Rightarrow> ('d, 'c) Term.term) list"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF ss E E'" shows "compose ss \<circ>\<^sub>s compose ss = compose ss"
weakBisimParPresAuxSym: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>R :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi"   and A\<^sub>R :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q"   and "extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle>"   and "A\<^sub>R \<sharp>* \<Psi>"   and "A\<^sub>R \<sharp>* P"   and "A\<^sub>R \<sharp>* Q" shows "\<Psi> \<rhd> R \<parallel> P \<approx> R \<parallel> Q"
w_def_2: shows "subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
forall_intro_vars: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and Hs :: "Core.term list"   and B :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,set Hs \<turnstile> B" shows "\<Theta>,set Hs \<turnstile> forall_intro_vars B Hs"
map_butlast: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "map f (butlast xs) = butlast (map f xs)"
valid1: shows "digit_encode (nths ds rmodes) r \<lhd> nths ds rmodes"
iT_Div_Min: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}" shows "iMin (I \<oslash> k) = iMin I div k"
in_sets_PiFI: fixes X :: "('d \<Rightarrow>\<^sub>F 'e) set"   and J :: "'d set"   and S :: "'d \<Rightarrow> 'e set"   and I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure" assumes "X = Pi' J S"   and "J \<in> I"   and "\<And>i. i \<in> J \<Longrightarrow> S i \<in> sets (M i)" shows "X \<in> sets (Pi\<^sub>F I M)"
Join_def: fixes F :: "'c program"   and G :: "'c program" shows "F \<squnion> G = mk_program (Init F \<inter> Init G, Acts F \<union> Acts G, AllowedActs F \<inter> AllowedActs G)"
is_cnf_encode_initial_state: fixes \<Pi> :: "'b strips_problem" assumes "is_valid_problem_strips \<Pi>" shows "is_cnf (\<Phi>\<^sub>I \<Pi>)"
f_def: fixes i :: "'n" shows "f i = (if i = j then 1::'a else (0::'a))"
dropWhile_strip_while_commute: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "dropWhile P (strip_while Q xs) = strip_while Q (dropWhile P xs)"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = r\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
continuous_map_componentwise: fixes X :: "'c topology"   and Y :: "'d \<Rightarrow> 'e topology"   and I :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" shows "continuous_map X (product_topology Y I) f = (f ` topspace X \<subseteq> extensional I \<and> (\<forall>k\<in>I. continuous_map X (Y k) (\<lambda>x. f x k)))"
HAStates_CompFun_notmem: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto"   and SA :: "('a, 'b, 'c) seqauto" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)" shows "S \<notin> States SA"
vdiff_vinsert_left_nin: fixes a :: "V"   and B :: "V"   and A :: "V" assumes "a \<notin>\<^sub>\<circ> B" shows "vinsert a A -\<^sub>\<circ> B = vinsert a (A -\<^sub>\<circ> B)"
rel_interior_cball: fixes S :: "'b set" shows "rel_interior S = {x \<in> S. \<exists>e>0. cball x e \<inter> affine hull S \<subseteq> S}"
pre_cross_edge_def: fixes u :: "'v"   and v :: "'v"   and s0 :: "'s"   and s :: "'s" shows "pre_cross_edge u v s0 s \<equiv> post_get_pending u (Some v) s0 s \<and> gds_is_discovered gds v s \<and> gds_is_finished gds v s"
gState_progress_rel_rtrancl_absorb: fixes prog :: "program"   and g :: "'b gState_scheme" assumes "gState_inv prog g" shows "(gState_progress_rel prog)\<^sup>* `` {g} = gState_progress_rel prog `` {g}"
wf: shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"
gcd_0_0: shows "gcd (0::'a) (0::'a) = (0::'a)"
col_mk_poly_mk_poly2: fixes A :: "'b mat" assumes "0 < dim_col A" shows "col (mk_poly A) (dim_col A - 1) = mk_poly2 A"
prems_3: shows "position_of t ((a # p) @ [d])"
that: fixes bl :: "'a set" assumes "bl \<in># \<B> \<and> b \<subseteq> bl" shows "thesis"
tl_take: fixes n :: "nat"   and xs :: "'a list" shows "tl (take n xs) = take (n - 1) (tl xs)"
composition_closed: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M"   and "b \<in> M" shows "composition a b \<in> M"
def_scheme': fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" shows "f X Y = (\<lambda>\<tau>. if (\<delta> X) \<tau> = true \<tau> \<and> d\<^sub>y Y \<tau> = true \<tau> then g (X \<tau>) (Y \<tau>) else invalid \<tau>)"