hcju/isabelleps huggingface dataset:
P_mod_diffxxx3: fixes t :: "'a"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and f :: "'a"   and gh :: "'a \<times> 'a"   and m :: "nat" assumes "Idomain S"   and "t \<in> carrier S"   and "t \<noteq> \<zero>\<^bsub>S\<^esub>"   and "maximal_ideal S (S \<diamondsuit>\<^sub>p t)"   and "PolynRg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y"   and "f \<in> carrier R"   and "gh \<in> carrier R \<times> carrier R"   and "deg R S X (fst gh) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh))"   and "deg R S X (snd gh) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh)) \<le> deg R S X f"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh))"   and "0 < deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd gh))"   and "rel_prime_pols R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh)) (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (snd gh))"   and "P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (f \<plusminus> -\<^sub>a fst gh \<cdot>\<^sub>r snd gh)"   and "0 < m" shows "\<exists>gh1. gh1 \<in> carrier R \<times> carrier R \<and> deg R S X (fst gh1) \<le> deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh1)) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (fst gh \<plusminus> -\<^sub>a fst gh1) \<and> deg R S X (snd gh1) + deg R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (erH R S X R' (S /\<^sub>r (S \<diamondsuit>\<^sub>p t)) Y (pj S (S \<diamondsuit>\<^sub>p t)) (fst gh1)) \<le> deg R S X f \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S m\<^esup>)) (snd gh \<plusminus> -\<^sub>a snd gh1) \<and> P_mod R S X (S \<diamondsuit>\<^sub>p (t^\<^bsup>S Suc m\<^esup>)) (f \<plusminus> -\<^sub>a fst gh1 \<cdot>\<^sub>r snd gh1)"
piecewise_C1_differentiable_on_def: fixes f :: "real \<Rightarrow> 'b"   and i :: "real set" shows "f piecewise_C1_differentiable_on i \<equiv> continuous_on i f \<and> (\<exists>S. finite S \<and> f C1_differentiable_on i - S)"
antisym: fixes x :: "(('a, 'b) T\<^sub>m, 'b T\<^sub>h, 'c, 'd) F\<^sub>m"   and y :: "(('a, 'b) T\<^sub>m, 'b T\<^sub>h, 'c, 'd) F\<^sub>m" assumes "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y"   and "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob y x"   and "x \<in> {x. set1_F\<^sub>m x \<subseteq> {x. set1_T\<^sub>m x \<subseteq> A} \<and> set3_F\<^sub>m x \<subseteq> B}" shows "x = y"
isCont_sum: fixes A :: "'c set"   and a :: "'d"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e" assumes "\<forall>i\<in>A. isCont (f i) a" shows "isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
edom_env_delete: fixes x :: "'a"   and m :: "'a \<Rightarrow> 'b" shows "edom (env_delete x m) = edom m - {x}"
filter_sort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "filter P (linorder.sort_key less_eq f xs) = linorder.sort_key less_eq f (filter P xs)"
continuous_onI: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x e. \<lbrakk>0 < e; x \<in> s\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e" shows "continuous_on s f"
Lp_zero_space: fixes p :: "real"   and M :: "'b measure" assumes "0 < p" shows "zero_space\<^sub>N (\<LL> (ennreal p) M) = {f \<in> borel_measurable M. AE x in M. f x = 0}"
subrelI: fixes r :: "('k \<times> 'l) set"   and s :: "('k \<times> 'l) set" assumes "\<And>x y. (x, y) \<in> r \<Longrightarrow> (x, y) \<in> s" shows "r \<subseteq> s"
subgraph_trans: fixes G :: "('e, 'f) pre_digraph"   and H :: "('e, 'f) pre_digraph"   and I :: "('e, 'f) pre_digraph" assumes "subgraph G H"   and "subgraph H I" shows "subgraph G I"
convert_new_thread_action_inv_1: fixes t :: "'a"   and x :: "'b"   and h :: "'c"   and f :: "'d \<Rightarrow> 'b"   and nta :: "('a, 'd, 'c) new_thread_action" shows "(NewThread t x h = convert_new_thread_action f nta) = (\<exists>x'. nta = NewThread t x' h \<and> x = f x')"
trivial_phis: fixes g :: "'g" shows "trivial_phis g = {(n, v). Mapping.lookup (phis g) (n, v) \<noteq> None \<and> trivial g v}"
B: shows "(r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sup>*] \<cdot> (\<rho> \<star> f\<^sup>*) = ((r \<star> \<theta>') \<cdot> \<a>[r, f, w] \<cdot> (\<rho> \<star> w)) \<cdot> \<nu> \<cdot> \<r>[r] \<cdot> (r \<star> \<epsilon>) \<cdot> \<a>[r, f, f\<^sup>*] \<cdot> (\<rho> \<star> f\<^sup>*)"
valid_return_list_valid_edges: fixes rs :: "'edge list"   and n :: "'node" assumes "valid_return_list rs n" shows "\<forall>r\<in>set rs. valid_edge r"
bisim1Sync14: fixes P :: "'b prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and a' :: "'addr" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v, Addr a'], xs, 7 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
intros_1: fixes n :: "nat" shows "VNat n \<sqsubseteq> VNat n"
lens_indep_sym: fixes x :: "'c \<Longrightarrow> 'd"   and y :: "'e \<Longrightarrow> 'd" assumes "x \<bowtie> y" shows "y \<bowtie> x"
cfSize_def: fixes cfile :: "CFile" shows "cfSize cfile \<equiv> fileSize cfile"
lexext_rev_mono_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and gt' :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"   and "lexext_rev gt ys xs" shows "lexext_rev gt' ys xs"
trace_parallel_plan_plan_prefix: fixes k :: "nat"   and I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" assumes "k < length (trace_parallel_plan_strips I \<pi>)" shows "trace_parallel_plan_strips I \<pi> ! k = execute_parallel_plan I (take k \<pi>)"
ccw'_sortedP_translateD: fixes x0 :: "real \<times> real"   and x :: "real \<times> real"   and g :: "'a \<Rightarrow> real \<times> real"   and xs :: "'a list" assumes "ccw'.sortedP x0 (map ((+) x \<circ> g) xs)" shows "ccw'.sortedP (x0 - x) (map g xs)"
outward_cmat_def: fixes H1 :: "complex \<times> complex \<times> complex \<times> complex"   and H2 :: "complex \<times> complex \<times> complex \<times> complex" shows "outward_cmat H1 H2 = (let (A1, B1, C1, D1) = H1; (A2, B2, C2, D2) = H2 in - Re B1 / Re A1 \<le> - Re B2 / Re A2)"
assms: shows "(c::'a::refinement_lattice) \<sqsubseteq> (d::'a::refinement_lattice)"
linorder_le_less_linear: fixes x :: "'b"   and y :: "'b" shows "x \<le> y \<or> y < x"
ssubst_intro: fixes sa :: "char list"   and t :: "sterm"   and pa :: "char list"   and p :: "sterm"   and n :: "nat"   and s :: "sterm" assumes "sa \<notin> FV t"   and "pa \<notin> FV t"   and "sa \<noteq> pa"   and "sa \<notin> FV p" shows "{n \<rightarrow> [s,p]} t = [sa \<rightarrow> s] [pa \<rightarrow> p] {n \<rightarrow> [Fvar sa,Fvar pa]} t"
interior_of_eq: fixes X :: "'c topology"   and S :: "'c set" shows "(X interior_of S = S) = openin X S"
ereal_le_divide_pos: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "0 < x"   and "x \<noteq> \<infinity>" shows "(y \<le> z / x) = (x * y \<le> z)"
iT_card_4: fixes r :: "nat"   and m :: "nat" shows "card [ r, mod m ] = (if m = 0 then Suc 0 else 0)"
merge_on_def: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" shows "merge_on A h bo m \<equiv> blinding_of_on UNIV h bo \<and> merge_on_axioms A h bo m"
1: shows " shift r x + y = Abs_real_int (left r + (x + y), right r + (x + y))"
fm_def: shows "fm = finmap_of (f ` J) \<circ> (\<lambda>g. compose (f ` J) g f')"
simps: fixes x :: "'a \<times> real"   and xs :: "('a \<times> real) msllist" shows "inverse_ms_aux (MSLCons x xs) = (let c' = inverse (fst x) in scale_shift_ms_aux (c', - snd x) (powser_ms_aux (msllist_of_msstream (mssalternate 1 (- 1))) (scale_shift_ms_aux (c', - snd x) xs)))"
hta_mem_def: fixes t :: "'a tree"   and H :: "('b, 'a, 'c) hashedTa_scheme" shows "hta_mem t H \<equiv> hta_mem' t (hta_ensure_idx_f H)"
simps_8: fixes \<phi> :: "formula"   and \<psi> :: "formula" shows "future_bounded (And \<phi> \<psi>) = (future_bounded \<phi> \<and> future_bounded \<psi>)"
assms_2: fixes i :: "nat" assumes "p i \<noteq> 0" shows "i \<le> n"
left_mult_one_mat: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "1\<^sub>m nr * A = A"
freeword_funlift_uminus_Abs_freeletter: fixes f :: "'c \<Rightarrow> 'b"   and s :: "'c" shows "freeword_funlift f (Abs_freeword [pairfalse s]) = - f s"
r_3: shows "prob {x. infdist x S = r} = 0"
brw_initial_abs: fixes \<Sigma> :: "('d \<Rightarrow> 'e tree option) \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<delta> :: "('d, 'e) ta_rule set" assumes "\<Sigma> \<in> brw_initial \<delta>" shows "brw_\<alpha> \<Sigma> \<in> br'_initial \<delta>"
simps_1: fixes P :: "'g \<Rightarrow> bool" shows "filter P [] = []"
pw_ASSERT_1: fixes \<Phi> :: "bool" shows "nofail (ASSERT \<Phi>) = \<Phi>"
order_prop: fixes x :: "'b"   and y :: "'b" shows "(x \<le> y) = (\<exists>z. sup x z = y)"
subgraph_semantics: fixes A :: "('b, 'c) labeled_graph"   and B :: "('b, 'c) labeled_graph"   and a :: "'c"   and b :: "'c"   and e :: "'b allegorical_term" assumes "subgraph A B"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>" shows "(a, b) \<in> :B:\<lbrakk>e\<rbrakk>"
awalkE: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" obtains "set (awalk_verts u p) \<subseteq> verts G" and "set p \<subseteq> arcs G" and "cas u p v" and "awhd u p = u" and "awlast u p = v"
mset_le_subtract: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "A \<subseteq># B" shows "A - C \<subseteq># B - C"
apply_matrix_def: fixes f :: "'b \<Rightarrow> 'c" shows "apply_matrix f \<equiv> \<lambda>A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
single_subset_iff: fixes a :: "'c"   and M :: "'c multiset" shows "({#a#} \<subseteq># M) = (a \<in># M)"
convg: shows "Lower_Semicontinuous.convex_on UNIV g"
Healthy_subset_member: fixes A :: "'d upred set"   and H :: "'d upred \<Rightarrow> 'd upred"   and P :: "'d upred" assumes "A \<subseteq> \<lbrakk>H\<rbrakk>\<^sub>H"   and "P \<in> A" shows "H P = P"
that: fixes p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list" assumes "p = p1 @ p2"   and "isPath s p1 u"   and "pathVertices_fwd s p1 = pv1 @ [u]"   and "isPath u p2 t"   and "pathVertices_fwd u p2 = u # pv2" shows "thesis"
rep_msg_def: shows "rep_msg \<equiv> quot_type.rep Rep_msg"
PAC_checker_l_step_def: shows "PAC_checker_l_step = (\<lambda>spec (st', \<V>, A) st. case st of Add x xa xb xc \<Rightarrow> full_normalize_poly (pac_res st) \<bind> (\<lambda>r. check_addition_l spec A \<V> (pac_src1 st) (pac_src2 st) (new_id st) r \<bind> (\<lambda>eq. let _ = eq in if \<not> is_cfailed eq then RETURN (merge_cstatus st' eq, \<V>, fmupd (new_id st) r A) else RETURN (eq, \<V>, A))) | Mult x xa xb xc \<Rightarrow> full_normalize_poly (pac_res st) \<bind> (\<lambda>r. full_normalize_poly (pac_mult st) \<bind> (\<lambda>q. check_mult_l spec A \<V> (pac_src1 st) q (new_id st) r \<bind> (\<lambda>eq. let _ = eq in if \<not> is_cfailed eq then RETURN (merge_cstatus st' eq, \<V>, fmupd (new_id st) r A) else RETURN (eq, \<V>, A)))) | Extension x xa xb \<Rightarrow> full_normalize_poly (([new_var st], - 1) # pac_res st) \<bind> (\<lambda>r. check_extension_l spec A \<V> (new_id st) (new_var st) r \<bind> (\<lambda>eq. if \<not> is_cfailed eq then RETURN (st', insert (new_var st) \<V>, fmupd (new_id st) r A) else RETURN (eq, \<V>, A))) | Del x \<Rightarrow> check_del_l spec A (pac_src1 st) \<bind> (\<lambda>eq. let _ = eq in if \<not> is_cfailed eq then RETURN (merge_cstatus st' eq, \<V>, fmdrop (pac_src1 st) A) else RETURN (eq, \<V>, A)))"
PiE_UNIV_domain: fixes T :: "'b \<Rightarrow> 'c set" shows "Pi\<^sub>E UNIV T = Pi UNIV T"
map_to_set_inverse: fixes m :: "'g \<Rightarrow> 'h option" shows "set_to_map (map_to_set m) = m"
removeFalseLiterals_def: fixes clause :: "Literal list"   and valuation :: "Literal list" shows "removeFalseLiterals clause valuation = filter (\<lambda>l. \<not> literalFalse l valuation) clause"
delete_text: fixes b :: "'a textbuffer" shows "get_text (delete b) = take (get_pos b - 1) (get_text b) @ drop (get_pos b) (get_text b)"
even_card_loops: fixes w :: "'a" shows "even (card (in_arcs H w \<inter> out_arcs H w))"
axioms_1: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" assumes "coretract G" shows "galois_connection G"
assms: shows "l dd\<^bsup>path \<sigma>,v\<^esup>\<rightarrow> k"
cball_subset_ball_iff: fixes a :: "'a"   and r :: "real"   and a' :: "'a"   and r' :: "real" shows "(cball a r \<subseteq> ball a' r') = (dist a a' + r < r' \<or> r < 0)"
sdrop_neg: fixes i :: "Integer"   and xs :: "[:'r:]" assumes "le\<cdot>i\<cdot>0 = TT" shows "sdrop\<cdot>i\<cdot>xs = xs"
P2_ideal_model'_def: fixes M :: "'a \<times> 'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" shows "P2_ideal_model' M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. let l = b0 \<otimes> inv (h0 [^] r) in P2_ideal_model_end (x0, x1) l ((h0, h1, a, b0, b1), s') \<A>3))))))"
3: shows "\<langle>z\<rangle>\<^sup>2 * \<langle>w\<rangle>\<^sup>2 - (cmod \<langle>z,w\<rangle>)\<^sup>2 = Re ((z1 * w2 - w1 * z2) * (cnj z1 * cnj w2 - cnj w1 * cnj z2))"
distinct_3: fixes name :: "name"   and trm1' :: "trm"   and trm2' :: "trm" shows "Var name \<noteq> App trm1' trm2'"
assms: shows "(c1 | c2, s) \<rightarrow>t s'"
is_strict_rewrite_ord_antisym: fixes rel :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and x :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and y :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "is_strict_rewrite_ord rel"   and "\<not> rel x y"   and "\<not> rel y x" shows "fst x = fst y"
valid_SDG_node_parent_Exit: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n"   and "parent_node n = Exit" shows "n = CFG_node Exit"
proj_setD_1: fixes y :: "'c"   and x :: "'c"   and A :: "'c set" assumes "y \<in> proj_set x A" shows "y \<in> A"
negligible_differentiable_vimage: fixes T :: "'a set"   and S :: "'a set"   and f' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "negligible T"   and "\<And>x. x \<in> S \<Longrightarrow> inj (f' x)"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)" shows "negligible {x \<in> S. f x \<in> T}"
lfp_lowerbound: fixes L :: "'a set"   and Glb :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "Complete_Lattice L Glb"   and "a \<in> L"   and "f a \<le> a" shows "Complete_Lattice.lfp L Glb f \<le> a"
hyps_3: assumes "Q \<noteq> Q''" shows "P \<oplus> Q \<Longrightarrow>\<^sub>\<tau> Q''"
subtensor_combine_tensor_prod: fixes As :: "'a tensor list"   and ds :: "nat list"   and B :: "'a tensor" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "local.tensor_prod_otimes (subtensor_combine ds As) B = subtensor_combine (ds @ dims B) (map (\<lambda>A. local.tensor_prod_otimes A B) As)"
assms_2: shows "bvars\<^sub>s\<^sub>s\<^sub>t T \<inter> subst_domain \<theta> = {}"
degree_map_poly_2: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly" assumes "f (lead_coeff p) \<noteq> (0::'a)" shows "degree (map_poly f p) = degree p"
f2r_mono: shows "\<forall>f g. \<R> \<circ> f = \<R> \<circ> g \<longrightarrow> f = g"
g: shows "n + 2 < max (n + 3) (max fft gft)"
carrier_topo: fixes B :: "'b set set" shows "carrier\<^bsub>topo B\<^esub> = \<Union> B"
sum_nonneg_eq_0_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> (0::'d) \<le> f x" shows "(sum f A = (0::'d)) = (\<forall>x\<in>A. f x = (0::'d))"
rebase_mult_eq: fixes a :: "'a mod_ring"   and a' :: "'a mod_ring" shows "(of_nat CARD('d) * a = of_nat CARD('d) * a') = (@a = @a')"
Icc_subset_Iic_iff: fixes l :: "'b multiset"   and h :: "'b multiset"   and h' :: "'b multiset" shows "(subset_mset.atLeastAtMost l h \<subseteq> subset_mset.atMost h') = (\<not> l \<subseteq># h \<or> h \<subseteq># h')"
inside_iter: fixes z :: "nat"   and n :: "nat" shows "inside z n = (0 < Iter pdec2 n z)"
content_smult: fixes c :: "'b"   and p :: "'b poly" shows "content (smult c p) = normalize c * content p"
take_all: fixes xs :: "'c list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
DSources_def: fixes i :: "AbstrLevelsID"   and C :: "CSet" shows "DSources i C \<equiv> {Z. \<exists>x. x \<in> IN C \<and> x \<in> OUT Z \<and> Z \<in> AbstrLevel i \<and> C \<in> AbstrLevel i}"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext ext"   and "ext_wf_axioms ext" shows "ext_wf ext"
subtopology_discrete_topology: fixes U :: "'b set"   and S :: "'b set" shows "subtopology (discrete_topology U) S = discrete_topology (U \<inter> S)"
m_comm: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (add_monoid G)"   and "y \<in> carrier (add_monoid G)" shows "x \<otimes>\<^bsub>add_monoid G\<^esub> y = y \<otimes>\<^bsub>add_monoid G\<^esub> x"
convex_bound_le: fixes x :: "'b"   and a :: "'b"   and y :: "'b"   and u :: "'b"   and v :: "'b" assumes "x \<le> a"   and "y \<le> a"   and "(0::'b) \<le> u"   and "(0::'b) \<le> v"   and "u + v = (1::'b)" shows "u * x + v * y \<le> a"
that: fixes k :: "real \<times> 'c \<Rightarrow> 'a" assumes "continuous_on ({0..1} \<times> U) k"   and "k ` ({0..1} \<times> U) \<subseteq> C"   and "\<And>y. y \<in> U \<Longrightarrow> k (0, y) = f y"   and "\<And>z. z \<in> {0..1} \<times> U \<Longrightarrow> h z = p (k z)" shows "thesis"
comp_eq_elim: fixes a :: "'f \<Rightarrow> 'e"   and b :: "'d \<Rightarrow> 'f"   and c :: "'g \<Rightarrow> 'e"   and d :: "'d \<Rightarrow> 'g" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
strictly_preserves_lunit: fixes a :: "'c local.term set" assumes "\<F>C.ide a" shows "F (\<F>C.lunit a) = \<l>\<^sub>D[F a]"
no_microsteps_sterms_refl: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "(\<nexists>q. p \<leadsto>\<^bsub>\<Gamma>\<^esub> q) = (sterms \<Gamma> p = {p})"
rep_eq: fixes x :: "('a, 'b) pfun"   and xa :: "('a, 'b) pfun" shows "Rep_pfun (x - xa) = Rep_pfun x -- Rep_pfun xa"
ve_dist: fixes e :: "'a"   and v :: "'a" assumes "e \<le> v * - v\<^sup>T"   and "vector v"   and "arc e" shows "(v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T = v * v\<^sup>T \<squnion> v * v\<^sup>T * e \<squnion> e\<^sup>T * v * v\<^sup>T \<squnion> e\<^sup>T * e"
is_document_ptr_kind_none: fixes ptr :: "(_) object_ptr" assumes "\<not> is_document_ptr_kind ptr" shows "cast ptr = None"
cpI2: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'd"   and P :: "('a state \<times> 'a state \<Rightarrow> 'e) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b"   and Q :: "('a state \<times> 'a state \<Rightarrow> 'e) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c" assumes "\<forall>X Y \<tau>. f X Y \<tau> = f (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) \<tau>"   and "cp P"   and "cp Q" shows "cp (\<lambda>X. f (P X) (Q X))"
rotate1_rotate_swap: fixes n :: "nat"   and xs :: "'d list" shows "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
AtSuc: fixes x :: "'a \<times> 'b \<times> 'c"   and n :: "nat" assumes "(case x of (x, y, z) \<Rightarrow> At x y z) n" shows "\<exists>x'. (case x' of (x, y, z) \<Rightarrow> At x y z) (Suc n) \<and> (case x' of (x', y', z') \<Rightarrow> \<lambda>(x, y, z). Follows x' y' z' x y z) x"
box_subset_cbox: fixes a :: "'b"   and b :: "'b" shows "box a b \<subseteq> cbox a b"
supp_atm_1: fixes x :: "tyvrs" shows "supp x = {x}"
Abs_assn_eqI_1: fixes P :: "heap \<times> nat set \<Rightarrow> bool"   and Pr :: "assn" assumes "\<And>h. P h = (h \<Turnstile> Pr)" shows "Abs_assn P = Pr"
plus_of_nat_plus_of_nat_hmset: fixes k :: "hmultiset"   and m :: "nat"   and n :: "nat" shows "k + of_nat m + of_nat n = k + of_nat (m + n)"
purgeIdle_append: fixes sl1 :: "('a, 'b, 'c) state list"   and sl2 :: "('a, 'b, 'c) state list" shows "GM_sec_model.purgeIdle (sl1 @ sl2) = GM_sec_model.purgeIdle sl1 @ GM_sec_model.purgeIdle sl2"
rel_Grp: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "rel_option (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_option x \<subseteq> A} (map_option f)"
isCont_LIM_compose2: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "isCont f a"   and "g \<midarrow>f a\<rightarrow> l"   and "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a" shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
g_def: shows "g = (\<lambda>y. SOME x. x \<in> X \<and> f x = y)"
0: shows "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP"
assms_2: shows "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1,b\<^sub>1\<rangle>"
is_natural_2: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "\<a>[fst (VVV.cod f), fst (snd (VVV.cod f)), snd (snd (VVV.cod f))] \<cdot> HoHV f = \<a>[fst f, fst (snd f), snd (snd f)]"
assms: shows "t \<in>\<^sub>A frontier M - frontier N"
card_1_singletonE: fixes A :: "'c set" assumes "card A = 1" obtains x :: "'c" where "A = {x}"
False: shows "\<not> (a = (0::'a) \<or> is_unit a)"
Re_infsetsum: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set" assumes "f abs_summable_on A" shows "Re (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Re (f x))"