hcju/isabelleps huggingface dataset:
that: fixes c :: "real" assumes "\<And>x. norm (pbernpoly n x) \<le> c" shows "thesis"
that: fixes L :: "real" assumes "0 < L"   and "\<And>x. x \<in> I \<Longrightarrow> dist (f x (v x)) (f x (w x)) \<le> L * dist (v x) (w x)" shows "thesis"
godel_code_great: fixes nl :: "nat list" shows "0 < godel_code nl"
rb_def: shows "rb = rec_nat rb0 (\<lambda>i rbi. SOME rbsi. (rbi, rbsi) \<in> b.E \<and> (ra (Suc i), rbsi) \<in> R)"
euclid_ext_aux_i: fixes ops :: "'c arith_ops_record"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "euclidean_ring_ops ops R" shows "(R ===> R ===> R ===> R ===> R ===> R ===> rel_prod (rel_prod R R) R) (arith_ops.euclid_ext_aux_i ops) euclid_ext_aux"
d_delta_arctan_upper_34_pos: fixes x :: "real" shows "diff_delta_arctan_upper_34 x < 0"
simple_function_compose: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "simple_function M f" shows "simple_function M (g \<circ> f)"
H_seq: fixes r :: "'a"   and y :: "'a"   and q :: "'a"   and p :: "'a"   and x :: "'a" assumes "H r y q"   and "H p x r" shows "H p (x \<cdot> y) q"
collinear_def: shows "collinear \<equiv> \<lambda>(Ax, Ay) (Bx, By) (Cx, Cy). (Ax - Bx) * (By - Cy) = (Ay - By) * (Bx - Cx)"
clauseTrueRemoveDuplicateLiterals: fixes c :: "Literal list"   and v :: "Literal list" shows "clauseTrue c v = clauseTrue (remdups c) v"
FreeUltrafilterNat_HFinite: fixes X :: "nat \<Rightarrow> 'a"   and u :: "real" assumes "\<forall>\<^sub>F n in \<U>. norm (X n) < u" shows "star_n X \<in> HFinite"
subset_image_iff: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(B \<subseteq> f ` A) = (\<exists>AA\<subseteq>A. B = f ` AA)"
If_cancel: fixes x :: "'a"   and b :: "tr" shows "If b then x else x = seq\<cdot>b\<cdot>x"
abort_Bleast_def: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "abort_Bleast S P = (LEAST x. x \<in> S \<and> P x)"
matchC_M_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_M theta d c = (\<forall>s t d' t'. s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> (\<exists>c' s'. (c, s) \<rightarrow>*c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta) \<or> (\<exists>s'. (c, s) \<rightarrow>*t s' \<and> s' \<approx> t' \<and> discr d'))"
erf_remainder_integral_bound: fixes x :: "real"   and n :: "nat" assumes "0 < x" shows "erf_remainder_integral n x \<le> exp (- x\<^sup>2) / x ^ (2 * n + 1)"
Sublists_conv_Prefixes: fixes A :: "'a list set" shows "Sublists A = Prefixes (Suffixes A)"
intro: fixes M :: "'c \<Rightarrow> 'd measure"   and I :: "'c set" assumes "product_sigma_finite M"   and "finite_product_sigma_finite_axioms I" shows "finite_product_sigma_finite M I"
word_log_esimps_1: fixes x :: "'a word" shows "x AND 0 = 0"
NOMATCH_def: fixes val :: "'c"   and pat :: "'c" shows "NOMATCH val pat \<equiv> True"
smap_szip_snd: fixes g :: "'e \<Rightarrow> 'c"   and s1 :: "'d stream"   and s2 :: "'e stream" shows "smap (\<lambda>x. g (snd x)) (szip s1 s2) = smap g s2"
monom_eq_1_iff: fixes c :: "'b"   and n :: "nat" shows "(monom c n = 1) = (c = (1::'b) \<and> n = 0)"
intros: fixes M :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and rinit :: "'o"   and rinit' :: "'p"   and S :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and IA :: "'s \<Rightarrow> 'v \<Rightarrow> bool"   and OA :: "'t \<Rightarrow> 'w \<Rightarrow> bool"   and E :: "'u \<Rightarrow> 'x \<Rightarrow> bool"   and rfunc_adv :: "'q \<Rightarrow> 's \<Rightarrow> (('t \<times> 'u list) \<times> 'q) spmf"   and rfunc_adv' :: "'r \<Rightarrow> 'v \<Rightarrow> (('w \<times> 'x list) \<times> 'r) spmf"   and IU :: "'y \<Rightarrow> 'aa \<Rightarrow> bool"   and OU :: "'z \<Rightarrow> 'ab \<Rightarrow> bool"   and rfunc_usr :: "'q \<Rightarrow> 'y \<Rightarrow> (('z \<times> 'u list) \<times> 'q) spmf"   and rfunc_usr' :: "'r \<Rightarrow> 'aa \<Rightarrow> (('ab \<times> 'x list) \<times> 'r) spmf" assumes "M rinit rinit'"   and "(S ===> IA ===> rel_spmf (rel_prod (rel_prod OA (list_all2 E)) S)) rfunc_adv rfunc_adv'"   and "(S ===> IU ===> rel_spmf (rel_prod (rel_prod OU (list_all2 E)) S)) rfunc_usr rfunc_usr'" shows "rel_rest' S E IA IU OA OU M (Rest rinit rfunc_adv rfunc_usr) (Rest rinit' rfunc_adv' rfunc_usr')"
F: shows "borel = sigma UNIV (F ` (UNIV \<times> Basis))"
list_all2_nths: fixes P :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and xs :: "'m list"   and A :: "nat set"   and ys :: "'n list" assumes "list_all2 P (nths xs A) (nths ys A)"   and "list_all2 P (nths xs (- A)) (nths ys (- A))" shows "list_all2 P xs ys"
mat_equal_list_lem: fixes A :: "'b mat"   and B :: "'b mat" assumes "dim_row A = dim_row B \<and> dim_col A = dim_col B \<and> mat_to_list A = mat_to_list B" shows "A = B"
distinct_1: fixes x2 :: "'b" shows "Ibottom \<noteq> Iup x2"
A: fixes Aa :: "real set" assumes "countable Aa" shows "Aa \<in> sets COCOUNT"
map_div_is_smult_inverse: fixes a :: "'b"   and p :: "'b poly" shows "map_poly (\<lambda>x. x / a) p = smult (inverse a) p"
map_of_remdups_wrt_rev': fixes xs :: "('b \<times> 'a) list"   and vs :: "'b list"   and k :: "'b" shows "map_of (remdups_wrt_rev fst xs vs) k = map_of (filter (\<lambda>x. fst x \<notin> set vs) xs) k"
cring_simprules_11: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> (y \<otimes>\<^bsub>R\<^esub> z)"
simps_3: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and q :: "'b" shows "\<xi> \<Turnstile>\<^sub>n prop\<^sub>n(q) = (q \<in> \<xi> 0)"
disjoint: fixes x :: "real" shows "\<exists>y. y \<in> T \<and> open_segment x y \<inter> T = {}"
bigo_const2: fixes c :: "'d" shows "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1::'d)"
ivl_disj_un_one_3: fixes l :: "'a"   and u :: "'a" assumes "l \<le> u" shows "{..l} \<union> {l<..u} = {..u}"
chop_subset1: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "N_Chop(i,j,k) \<longrightarrow> j \<le> i"
bi_total_alt_def: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
simps_1: fixes uu :: "'a" shows "(Infty \<le> Num uu) = False"
Domain_id_on: fixes R :: "('d \<times> 'f) set"   and S :: "('f \<times> 'e) set" shows "Domain (R O S) = Domain R \<inter> R\<inverse> `` Domain S"
one_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "\<one>\<^bsub>Ring_Divisibility.mult_of R\<^esub> = \<one>\<^bsub>R\<^esub>"
mul_ext_mono_2: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and P' :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys; fst (P x y)\<rbrakk> \<Longrightarrow> fst (P' x y)"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys; snd (P x y)\<rbrakk> \<Longrightarrow> snd (P' x y)"   and "snd (mul_ext P xs ys)" shows "snd (mul_ext P' xs ys)"
P2: shows "Bet E P E' \<and> Cong P E' P E"
vec_of_poly_pCons: fixes f :: "'b poly"   and a :: "'b" assumes "f \<noteq> 0" shows "vec_of_poly (pCons a f) = vec_of_poly f @\<^sub>v vec_of_list [a]"
is_balanced_def: shows "is_balanced \<equiv> \<exists>A B. A \<subseteq> {i. i < 2 ^ n} \<and> B \<subseteq> {i. i < 2 ^ n} \<and> card A = 2 ^ (n - 1) \<and> card B = 2 ^ (n - 1) \<and> (\<forall>x\<in>A. f x = 0) \<and> (\<forall>x\<in>B. f x = 1)"
inline1_Pause: fixes callee :: "'e \<Rightarrow> 'i \<Rightarrow> ('h \<times> 'e, 'f, 'g) gpv"   and out :: "'i"   and c :: "('d, 'i, 'h) rpv"   and s :: "'e" shows "inline1 callee (Generative_Probabilistic_Value.Pause out c) s = the_gpv (callee s out) \<bind> (\<lambda>react. case react of Pure (x, s') \<Rightarrow> inline1 callee (c x) s' | IO out' c' \<Rightarrow> return_spmf (Inr (out', c', c)))"
card_Diff_singleton_if: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card (A - {x}) = (if x \<in> A then card A - 1 else card A)"
list_app_typeD: fixes e :: "nat \<Rightarrow> LambdaType.type"   and t :: "dB"   and ts :: "dB list"   and T :: "LambdaType.type" assumes "e \<turnstile> t \<degree>\<degree> ts : T" shows "\<exists>Ts. e \<turnstile> t : Ts \<Rrightarrow> T \<and> e \<tturnstile> ts : Ts"
t_mult_cnj: shows "t * gauss_cnj t = (of_int z)\<^sup>2"
prime_cong: fixes p :: "'a"   and p' :: "'a" assumes "prime G p"   and "p \<sim> p'"   and "p \<in> carrier G"   and "p' \<in> carrier G" shows "prime G p'"
adds_pp_sminus: fixes t :: "'a"   and v :: "'t" assumes "t adds\<^sub>p v" shows "t \<oplus> (v \<ominus> t) = v"
kim: shows "k ` ({0..1} \<times> {undefined}) \<subseteq> C"
disc_2: fixes xs :: "('a, 'b) tllist" assumes "\<not> is_TNil xs" shows "\<not> lnull (llist_of_tllist xs)"
case_prodE: fixes c :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and p :: "'f \<times> 'g" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'f" and y :: "'g" where "p = (x, y)" and "c x y"
exp_minus: fixes x :: "'d" shows "exp (- x) = inverse (exp x)"
weak_unit_self_composable_3: fixes a :: "'a" assumes "weak_unit a" shows "a \<star> a \<noteq> null"
fls_inverse_subdegree': fixes f :: "'a fls" assumes "inverse (fls_nth f (fls_subdegree f)) \<noteq> (0::'a)" shows "fls_subdegree (inverse f) = - fls_subdegree f"
simps_3: fixes n :: "nat"   and t :: "Term_Class.term" shows "wellformed' n (\<Lambda> t) = wellformed' (n + 1) t"
PfP_inference: fixes \<alpha>1 :: "tm"   and \<alpha>2 :: "tm"   and \<beta> :: "tm" shows "{PfP \<alpha>1, PfP \<alpha>2, ModPonP \<alpha>1 \<alpha>2 \<beta> OR ExistsP \<alpha>1 \<beta> OR SubstP \<alpha>1 \<beta>} \<turnstile> PfP \<beta>"
new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('c, 'a, 'b, 'j option) RDocument_scheme \<Rightarrow> 'l" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
pr_id1_1: shows "(\<lambda>x. x) \<in> PrimRec1"
countable_basis_at_decseq: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.first_countable_topology open" obtains A :: "nat \<Rightarrow> 'b set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>S. \<lbrakk>open S; x \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. A i \<subseteq> S"
lm096: fixes f :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'b set" assumes "runiq f"   and "X \<subseteq> f\<inverse> `` Y" shows "f `` X \<subseteq> Y"
pred_inject_2: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e"   and aa :: "'e list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
inorder_delete: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (RBT_Set2.delete x t) = del_list x (Tree2.inorder t)"
hv_props_3: shows "\<gamma> = \<H> \<union> \<V>"
arr_dom_iff_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "partial_magma.arr C (partial_magma.dom C f) = partial_magma.arr C f"
Gmult_flincomb_comm: fixes g :: "'g"   and vs :: "'v list"   and as :: "'f list" assumes "g \<in> G"   and "set vs \<subseteq> V" shows "g *\<cdot> as \<bullet>\<sharp>\<cdot> vs = as \<bullet>\<sharp>\<cdot> map ((*\<cdot>) g) vs"
simps_1: shows "polynomial [] = \<zero>\<^bsub>function_ring (carrier R) R\<^esub>"
group_comm_groupI: assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<oplus> y = y \<oplus> x" shows "comm_group (add_monoid G)"
rep_eq: fixes x :: "'d ccsubspace"   and xa :: "'d ccsubspace" shows "(x < xa) = (space_as_set x \<subset> space_as_set xa)"
sees_methods_fun: fixes Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and mthd' :: "ty list \<times> ty \<times> char list list \<times> expr" assumes "(Cs, mthd) \<in> MethodDefs P C M"   and "(Cs, mthd') \<in> MethodDefs P C M" shows "mthd = mthd'"
zero_below: fixes qs :: "rat poly list"   and x1 :: "real" assumes "has_no_zeros (sign_vec qs x1)" shows "\<forall>x2<x1. sign_vec qs x1 \<noteq> sign_vec qs x2 \<longrightarrow> (\<exists>r<x1. x2 \<le> r \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
bell01_index_2: shows "|\<beta>\<^sub>0\<^sub>1\<rangle> $$ (1, 0) = complex_of_real (1 / sqrt 2)"
listset_ConsI: fixes y :: "'a"   and x :: "'a set"   and ys' :: "'a list"   and xs :: "'a set list"   and ys :: "'a list" assumes "y \<in> x"   and "ys' \<in> listset xs"   and "ys = y # ys'" shows "ys \<in> listset (x # xs)"
below_bottom_iff: fixes x :: "'b" shows "(x \<sqsubseteq> \<bottom>) = (x = \<bottom>)"
igFreshIGVar_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) Iteration.model_scheme" shows "igFreshIGVar MOD \<equiv> \<forall>ys y xs x. ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igFresh MOD ys y (igVar MOD xs x)"
deg_p2: fixes t :: "'x \<Rightarrow>\<^sub>0 'a" assumes "t \<in> keys p2" shows "deg_pm t < poly_deg p"
eq_game_refl: fixes G :: "game" shows "eq_game G G"
zr: fixes z :: "real" assumes "0 < z"   and "z < 1" shows "z * r \<le> y"
basis_in_ex: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "\<exists>B. basis_in \<E> B"
that: fixes K :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite K"   and "K \<subseteq> L"   and "K \<subseteq> T"   and "disjnt K S"   and "continuous_on (T - K) g"   and "g ` (T - K) \<subseteq> rel_frontier U"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
oreach_equiv_obeh_states: fixes S :: "('b, 'a) spec" shows "oreach S = \<Union> (set ` obeh S)"
simps_10: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "'a" shows "(v;; va \<le> SKIP {vb}) = False"
Pool_2: fixes w :: "nat \<Rightarrow> real" assumes "balanced_net m2" shows "shared_weight_net (insert_weights True m2 w)"
IH: shows "\<exists>I J K. N = I + J \<and> z = I + K \<and> J \<noteq> {#} \<and> (\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r)"
wt_subst_trm: fixes t :: "('fun, 'var) Term.term"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "\<And>v. v \<in> fv t \<Longrightarrow> \<Gamma> (Var v) = \<Gamma> (\<theta> v)" shows "\<Gamma> t = \<Gamma> (t \<cdot> \<theta>)"
coplanar_perm_9: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar B C D A"
orderI: fixes a :: "'a"   and b :: "'a" assumes "a = max a b" shows "b \<le> a"
vwalk_join_assoc': fixes p :: "'a list"   and q :: "'a list"   and r :: "'a list" assumes "p \<noteq> []"   and "q \<noteq> []" shows "(p \<oplus> q) \<oplus> r = p \<oplus> q \<oplus> r"
stake_preimage_measurable: fixes n :: "nat"   and M :: "bool stream measure"   and p :: "real"   and x :: "bool stream" assumes "Suc 0 \<le> n"   and "M = bernoulli_stream p" shows "{w \<in> space M. stake n w = stake n x} \<in> sets M"
collapse: fixes nat :: "nat" assumes "nat \<noteq> 0" shows "Suc (nat.pred nat) = nat"
eventually_ge_at_top: shows "eventually ((\<le>) (nat \<lceil>x0\<rceil>)) sequentially"
assms_2: shows "{b, G.face_cycle_succ b} \<inter> {a, a'} = {}"
open_map_into_subtopology: fixes X :: "'c topology"   and X' :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "open_map X X' f"   and "f ` topspace X \<subseteq> S" shows "open_map X (subtopology X' S) f"
has_domain_iff_arr: fixes f :: "(bool, bool list) concrete_category.arr" shows "(domains f \<noteq> {}) = arr f"
ideal_def: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "ideal I R \<equiv> additive_subgroup I R \<and> ring R \<and> ideal_axioms I R"
NotInQFrozenQReachability: fixes cfg1 :: "('p, 'v, 's) configuration"   and Q :: "'p set"   and cfg2 :: "('p, 'v, 's) configuration"   and p :: "'p" assumes "qReachable cfg1 Q cfg2"   and "p \<notin> Q" shows "states cfg1 p = states cfg2 p"
cblinfun_image_INF_leq: fixes U :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and V :: "'c \<Rightarrow> 'b ccsubspace" shows "U *\<^sub>S \<Sqinter> (range V) \<le> (INF i. U *\<^sub>S V i)"
ax_Abrupts_4: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and x :: "stmt" shows "G,A\<turnstile>{P\<leftarrow>\<diamondsuit> \<and>. Not \<circ> normal} .x. {P}"
is_Ind_def: fixes x :: "hf" shows "is_Ind x \<equiv> \<exists>m. Ord m \<and> x = \<langle>htuple 6, m\<rangle>"
ground\<^sub>l_ground_fatom: fixes l :: "fterm literal" assumes "ground\<^sub>l l" shows "get_atom l \<in> ground_fatoms"
cexpr_sem_expr_rf_to_cexpr: fixes e :: "expr"   and \<sigma> :: "nat \<Rightarrow> val" assumes "randomfree e" shows "cexpr_sem \<sigma> (expr_rf_to_cexpr e) = expr_sem_rf \<sigma> e"
m: shows "0 < the (eval r_result1 [Suc (max n t), i, f \<triangleright> n])"
max_path_2: shows "\<forall>n. \<forall>w\<in>E. \<not> is_trail n (w # max_path) v'"
case_optionE: fixes P :: "bool"   and Q :: "'f \<Rightarrow> bool"   and x :: "'f option" assumes "case x of None \<Rightarrow> P | \<lfloor>y\<rfloor> \<Rightarrow> Q y" obtains "x = None" and "P"   | y :: "'f" where "x = \<lfloor>y\<rfloor>" and "Q y"
free_Abelian_group_universal: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int"   and S :: "'c set" assumes "f ` S \<subseteq> carrier (free_Abelian_group T)" obtains h :: "('c \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'b \<Rightarrow>\<^sub>0 int" where "h \<in> hom (free_Abelian_group S) (free_Abelian_group T)" and "\<And>x. x \<in> S \<Longrightarrow> h (frag_of x) = f x"
repeat_mset_not_empty: fixes n :: "nat"   and A :: "'b multiset" assumes "0 < n"   and "A \<noteq> {#}" shows "repeat_mset n A \<noteq> {#}"
right2_def: shows "right2 \<equiv> prod.snd (prod.snd (prod.snd (prod.snd (fix_syn (\<lambda>(root2, op2, rep2, left2, right2). (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>Null))) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_scons\<cdot>v\<cdot>(\<Lambda> u us. Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us))))), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> us v. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>Null)) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> us va. match_scons\<cdot>va\<cdot>(\<Lambda> v vs. Fixrec.succeed\<cdot>(rep2\<cdot>pat\<cdot>(us :@ [:v:], vs)))))))))))"
subset_in_close: fixes S' :: "'b fm set"   and S :: "'b fm set"   and x :: "'b fm set"   and C :: "'b fm set set" assumes "S' \<subseteq> S"   and "S \<union> x \<in> C" shows "S' \<union> x \<in> close C"
T_nonTickFree_imp_decomp: fixes t :: "'a event list"   and P :: "'a process" assumes "t \<in> T P"   and "\<not> tickFree t" shows "\<exists>s. t = s @ [tick]"
circline_equation_x_axis: fixes z :: "complex" shows "(of_complex z \<in> circline_set x_axis) = (z = cnj z)"
sup_Un_eq: fixes R :: "'a set"   and S :: "'a set" shows "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
simps_18: fixes v :: "nat" shows "not (Closed v) = NOT (Closed v)"
wf_equallength: fixes \<Gamma> :: "binding list"   and T :: "POPLmark.type"   and \<Gamma>' :: "binding list" assumes "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T"   and "map is_TVarB \<Gamma>' = map is_TVarB \<Gamma>" shows "\<Gamma>' \<turnstile>\<^sub>w\<^sub>f T"
mono: fixes x :: "'b set \<times> 'b set" shows "spmf.mono_body (\<lambda>resample. case x of (A, B) \<Rightarrow> spmf_of_set A \<bind> (\<lambda>x. if x \<in> B then return_spmf x else curry resample A B))"
lincomb_list_Cons: fixes c :: "nat \<Rightarrow> 'a"   and v :: "'a vec"   and vs :: "'a vec list" shows "n.lincomb_list c (v # vs) = c 0 \<cdot>\<^sub>v v + n.lincomb_list (c \<circ> Suc) vs"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta>, m) \<in> derivable R* \<longrightarrow> ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable R*"
nonzero_of_real_divide: fixes y :: "real"   and x :: "real" assumes "y \<noteq> 0" shows "of_real (x / y) = of_real x / of_real y"
next_free_or: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "next_free (\<phi> or\<^sub>p \<psi>) = (next_free \<phi> \<and> next_free \<psi>)"
Act_eq_iff: fixes \<alpha>1 :: "'d"   and x1 :: "('b, 'c, 'd) formula"   and \<alpha>2 :: "'d"   and x2 :: "('b, 'c, 'd) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (Act\<^sub>\<alpha> \<alpha>1 (Rep_formula x1) = Act\<^sub>\<alpha> \<alpha>2 (Rep_formula x2))"
wqo_on_imp_good: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and f :: "nat \<Rightarrow> 'b" assumes "wqo_on P A"   and "\<forall>i. f i \<in> A" shows "good P f"
aperiodicI_pair: fixes K1 :: "'a \<Rightarrow> 'a pmf"   and C1 :: "'a set"   and K2 :: "'b \<Rightarrow> 'b pmf"   and C2 :: "'b set" assumes "MC_syntax.aperiodic K1 C1"   and "MC_syntax.aperiodic K2 C2" shows "MC_syntax.aperiodic (MC_pair.Kp K1 K2) (C1 \<times> C2)"
hom_boundary_default: fixes c :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set"   and p :: "int"   and X :: "'b topology"   and S :: "'b set" assumes "c \<notin> carrier (relative_homology_group p X S)" shows "hom_boundary p X S c = \<one>\<^bsub>homology_group (p - 1) (subtopology X S)\<^esub>"
Rep_inject: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b"   and y :: "'b" assumes "type_definition Rep Abs A" shows "(Rep x = Rep y) = (x = y)"
IH: fixes B :: "Core.term"   and lev :: "nat" assumes "list_all (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (idx, ty) \<Rightarrow> \<lambda>t. typ_of t = Some ty) b) xs"   and "\<not> loose_bvar B lev" shows "fold (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (idx, ty) \<Rightarrow> \<lambda>t B. subst_bv1 (bind_fv2 (idx, ty) lev B) lev t) b) xs B = fold (\<lambda>single. subst_term [single]) xs B"
DA0'_split: shows "DA0' = D0' + negs (mset As0')"
cbspec: fixes A :: "'d cset"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "cBall A P"   and "cin x A" shows "P x"
val_pos: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "0 \<le> val_Zp x"
ordertype_WW_2: shows "ordertype UNIV (lenlex less_than) \<le> \<omega> \<up> \<omega>"
lnth_Suc_LCons: fixes x :: "'a"   and xs :: "'a llist"   and n :: "nat" shows "lnth (LCons x xs) (Suc n) = lnth xs n"
functor_comp_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and G :: "('b, 'h, 'd, 'i, 'f, 'j, 'k) Functor_scheme" assumes "F \<approx>>;;; G" obtains "Functor F" and "Functor G" and "CatDom G = CatCod F"
servTicket_authentic: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "B \<noteq> Tgs"   and "B \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>Ta authK. 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> Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"
perm_1: fixes pi :: "(name \<times> name) list"   and x1 :: "name" shows "pi \<bullet> InputS x1 = InputS (pi \<bullet> x1)"