hcju/isabelleps huggingface dataset:
qAFreshAll_qSwapAll_id: fixes zs :: "'d"   and z1 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and z2 :: "'e"   and A :: "('f, 'g, 'd, 'e, 'h) qAbs" shows "(qAFresh zs z1 X \<and> qAFresh zs z2 X \<longrightarrow> X #[[z1 \<and> z2]]_zs = X) \<and> (qAFreshAbs zs z1 A \<and> qAFreshAbs zs z2 A \<longrightarrow> A $[[z1 \<and> z2]]_zs = A)"
simps: fixes pat :: "[:'a:]"   and usvs :: "[:'a:] \<times> [:'a:]" shows "rep2\<cdot>pat\<cdot>usvs = Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs)"
associated_iff_dvd: fixes a :: "'a"   and b :: "'a" shows "(normalize a = normalize b) = (a ddvd b)"
Gauss_Jordan_det_P_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "Gauss_Jordan_det_P A = Gauss_Jordan_upt_k_det_P A (ncols A - 1)"
mat_assoc_test_11: fixes A :: "complex mat"   and n :: "nat"   and B :: "complex mat"   and C :: "complex mat"   and D :: "complex mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n"   and "D \<in> carrier_mat n n" shows "trace (A * B * C * D) = trace (C * D * A * B)"
single_valuedD: fixes r :: "('e \<times> 'f) set"   and x :: "'e"   and y :: "'f"   and z :: "'f" assumes "single_valued r"   and "(x, y) \<in> r"   and "(x, z) \<in> r" shows "y = z"
comm_ring_hom_axioms: shows "comm_ring_hom (\<lambda>a. [:a:])"
incomplete_design_axioms_def: fixes point_set :: "'b set"   and u_block_size :: "int" shows "incomplete_design_axioms point_set u_block_size \<equiv> u_block_size < int (card point_set)"
append_Nil2: fixes xs :: "'b list" shows "append xs [] = xs"
P18: shows "Bet D E'' C \<and> (E'' = A \<or> A Out E'' E')"
insert_Diff1: fixes x :: "'e"   and B :: "'e set"   and A :: "'e set" assumes "x \<in> B" shows "insert x A - B = A - B"
SR_implies_R_for_modified_view: fixes ES :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>' :: "'b V_rec" assumes "BSPTaxonomyDifferentCorrections ES \<V>"   and "SR \<V> Tr\<^bsub>ES\<^esub>"   and "\<V>' = \<lparr>V = V\<^bsub>\<V>\<^esub> \<union> N\<^bsub>\<V>\<^esub>, N = {}, C = C\<^bsub>\<V>\<^esub>\<rparr>" shows "R \<V>' Tr\<^bsub>ES\<^esub>"
subterm_cl_subst: fixes x :: "'b trm"   and C :: "'b literal set"   and \<sigma> :: "('b \<times> 'b trm) list" assumes "x \<in> subterms_of_cl C" shows "x \<lhd> \<sigma> \<in> subterms_of_cl (subst_cl C \<sigma>)"
length_nths: fixes xs :: "'a list"   and I :: "nat set" shows "length (nths xs I) = card {i. i < length xs \<and> i \<in> I}"
simps_1: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" shows "fw m n 0 = fwi m n 0 n n"
awalk_to_path_no_neg_cyc_cost: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and f :: "'b \<Rightarrow> real" assumes "awalk u p v"   and "\<nexists>w c. awalk w c w \<and> w \<in> set (awalk_verts u p) \<and> awalk_cost f c < 0" shows "awalk_cost f (awalk_to_apath p) \<le> awalk_cost f p"
add_eq: fixes x :: "nat"   and y :: "nat" shows "x \<oplus>\<^bsub>R\<^esub> y = (x + y) mod n"
LIM_def: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and a :: "'a" shows "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)"
hyps: assumes "distinct xs"   and "sorted_wrt rel xs" shows "sorted_wrt (\<lambda>x y. rel x y \<and> x \<noteq> y) xs"
cardeq_rcoset: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and g :: "'c" assumes "Group.group G"   and "finite (carrier G)"   and "M \<subseteq> carrier G"   and "g \<in> carrier G" shows "card (M #>\<^bsub>G\<^esub> g) = card M"
swap_def: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set" shows "X #[x \<and> y]_xs = asTerm (pick X #[[x \<and> y]]_xs)"
simplex_in_max: fixes y :: "'b set" assumes "y \<in> Y" shows "\<exists>x. codomain.chamber x \<and> order.greater_eq x y"
ResComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<equiv>\<^sub>s <\<nu>y><\<nu>x>P"
countable_Diff: fixes A :: "'f set"   and B :: "'f set" assumes "countable A" shows "countable (A - B)"
DirProds_comm_group_imp_comm_groups: fixes G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme"   and I :: "'d set"   and i :: "'d" assumes "comm_group (DirProds G I)"   and "i \<in> I" shows "comm_group (G i)"
atomize_all: fixes P :: "'f \<Rightarrow> bool" shows "(\<And>x. P x) \<equiv> \<forall>x. P x"
PDG_BS_w_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and S :: "'g set" assumes "WeakControlDependencePDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit" shows "WeakControlDependencePDG.PDG_BS_w sourcenode targetnode valid_edge Def Use Exit S \<equiv> PDG.PDG_BS sourcenode targetnode valid_edge Def Use (StrongPostdomination.weak_control_dependence sourcenode targetnode valid_edge Exit) S"
upper_float_interval: fixes prec :: "nat"   and X :: "float interval" shows "upper (sqrt_float_interval prec X) = ub_sqrt prec (upper X)"
Abs_type: fixes \<Gamma> :: "binding list"   and S :: "POPLmark.type"   and s :: "trm"   and T :: "POPLmark.type"   and U :: "POPLmark.type"   and U' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> (\<lambda>:S. s) : T"   and "\<Gamma> \<turnstile> T <: U \<rightarrow> U'" obtains S' :: "POPLmark.type" where "\<Gamma> \<turnstile> U <: S" and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'" and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'"
drop_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list"   and x :: "'b" assumes "n < m" shows "drop m (xs[n := x]) = drop m xs"
simplex_convex_hull: fixes C :: "'b set"   and n :: "int" assumes "\<not> affine_dependent C \<and> int (card C) = n + 1" shows "n simplex convex hull C"
closed_injective_image_subspace: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "subspace s"   and "bounded_linear f"   and "\<forall>x\<in>s. f x = (0::'c) \<longrightarrow> x = (0::'b)"   and "closed s" shows "closed (f ` s)"
machine_star_implies_cbv_star_ctx: fixes e :: "lam"   and Es :: "ctx list"   and e' :: "lam"   and Es' :: "ctx list" assumes "<e,Es> \<mapsto>* <e',Es'>" shows " Es\<down>\<lbrakk>e\<rbrakk> \<longrightarrow>cbv* Es'\<down>\<lbrakk>e'\<rbrakk>"
intro: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "sub_rel_of g subs"   and "wf_sub_rel subs" shows "wf_sub_rel_of g subs"
idiv_infinity: fixes n :: "enat" assumes "0 < n" shows "\<infinity> div n = \<infinity>"
Inf_counterex_reducing: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Inf :: "'a inference set"   and I_of :: "'a set \<Rightarrow> 'a set"   and N :: "'a set"   and D :: "'a" assumes "counterex_reducing_inference_system Bot entails Inf I_of"   and "N \<inter> Bot = {}"   and "D \<in> N"   and "\<not> entails (I_of N) {D}"   and "\<And>C. \<lbrakk>C \<in> N; \<not> entails (I_of N) {C}\<rbrakk> \<Longrightarrow> D \<le> C" shows "\<exists>\<iota>\<in>Inf. prems_of \<iota> \<noteq> [] \<and> main_prem_of \<iota> = D \<and> set (side_prems_of \<iota>) \<subseteq> N \<and> entails (I_of N) (set (side_prems_of \<iota>)) \<and> \<not> entails (I_of N) {concl_of \<iota>} \<and> concl_of \<iota> < D"
inf_le2: fixes x :: "'a upred"   and y :: "'a upred" shows "y \<sqsubseteq> (x \<and> y)"
2: shows "tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>s\<^sub>t ` {dual\<^sub>s\<^sub>t \<A>}))"
sorted_antimono_suffix: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys"   and "sorted ys" shows "sorted xs"
add_eq_conv_ex: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and N :: "'b multiset" shows "(a, M = b, N) = (M = N \<and> a = b \<or> (\<exists>K. M = b, K \<and> N = a, K))"
act3: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a = \<tau> \<or> (\<exists>d. a = i:deliver(d)) \<longrightarrow> U (\<sigma> i) (\<sigma>' i))"
wf_digraphI_app_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "wf_digraph (app_iso hom (app_iso hom G))"
Diff_partition: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "A \<union> (B\<setminus>A) = B"
refl_onI: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<^bsub>r\<^esub>\<preceq> x" shows "refl_on A r"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "infinite A" shows "prod g A = (1::'d)"
leg_simps_1: shows "C.arr H\<mu>H\<nu>\<pi>.cod.leg0"
CondRedsThrow: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>throw a,s',b'\<rangle>"
ide_hcomp: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.src f = Span.trg g" shows "Span.ide (f \<circ> g)"
pred_spmf_exec_gpv: fixes A :: "'d \<Rightarrow> bool"   and C :: "'e \<Rightarrow> bool"   and gpv :: "('d, 'e, 'f) gpv"   and S :: "'g \<Rightarrow> bool"   and callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and s :: "'g" assumes "pred_gpv A C gpv"   and "pred_fun S (pred_fun C (pred_spmf (pred_prod (\<lambda>_. True) S))) callee"   and "S s" shows "pred_spmf (pred_prod A S) (exec_gpv callee gpv s)"
simps_1: fixes a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
essential_def: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a" shows "essential G B S x = (\<exists>p. \<exists>y\<in>B. path G x p y \<and> (x \<noteq> y \<longrightarrow> (\<forall>z\<in>set p. z = x \<or> z \<notin> S)))"
DistinctNetsAllowAllow: fixes b :: "(int \<times> int \<times> protocol) set set"   and c :: "(int \<times> int \<times> protocol) set set"   and poo :: "protocol \<times> int"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "(int \<times> int \<times> protocol) set set"   and d :: "(int \<times> int \<times> protocol) set set"   and po :: "protocol \<times> int" assumes "AllowPortFromTo b c poo \<in> set p"   and "AllowPortFromTo a d po \<in> set p"   and "allNetsDistinct p"   and "dom (Cp (AllowPortFromTo b c poo)) \<inter> dom (Cp (AllowPortFromTo a d po)) \<noteq> {}" shows "b = a \<and> c = d \<and> poo = po"
infinitesimal_quad: fixes D :: "real"   and C :: "real"   and A :: "real"   and B :: "real"   and At :: "atomUni"   and x :: "real" assumes "D \<noteq> 0"   and "0 \<le> C" shows "(\<exists>y'>(A + B * sqrt C) / D. \<forall>x\<in>{(A + B * sqrt C) / D<..y'}. aEvalUni At x) = evalUni (substInfinitesimalQuadraticUni A B C D At) x"
borderE: fixes xs :: "'a list"   and ys :: "'a list" assumes "border xs ys" obtains "prefix xs ys" and "suffix xs ys"
lprod_list: fixes ah :: "'b list"   and at :: "'b list"   and bh :: "'b list"   and bt :: "'b list"   and R :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b" assumes "(ah @ at, bh @ bt) \<in> lprod R"   and "(a, b) \<in> R \<or> a = b" shows "(ah @ a # at, bh @ b # bt) \<in> lprod R"
scale_eq_0_iff: fixes a :: "real"   and x :: "'a" shows "(a *\<^sub>R x = (0::'a)) = (a = 0 \<or> x = (0::'a))"
compose_eq: fixes x :: "'d"   and A :: "'d set"   and g :: "'f \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'f" assumes "x \<in> A" shows "(g \<circ> f \<down> A) x = g (f x)"
lookup_monom_mult_zero: fixes c :: "'c"   and p :: "'d \<Rightarrow>\<^sub>0 'c"   and t :: "'d" shows "lookup (MPoly_Type_Class.punit.monom_mult c (0::'d) p) t = c * lookup p t"
poly_of_focus_flat_pm_of_poly: fixes p :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) poly"   and x :: "'a" assumes "range (poly.coeff p) \<subseteq> P[- {x}]" shows "poly_of_focus x (flat_pm_of_poly x p) = p"
wfactors_ee_cong_l: fixes as :: "'a list"   and bs :: "'a list"   and b :: "'a" assumes "essentially_equal G as bs"   and "wfactors G bs b"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "wfactors G as b"
pr_index_enumerator_increase1: fixes n :: "nat"   and m :: "nat" shows "pr_index_enumerator n m < pr_index_enumerator (n + 1) m"
symbSS'_def: shows "symbSS' = SymSystem initS localPrecSS' localRelSS'"
subset_refl: fixes A :: "'x set" shows "A \<subseteq> A"
trans: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry"   and c :: "'d DBMEntry" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
reachable_refl: fixes v :: "'a" assumes "v \<in> verts G" shows "v \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"
pasting_lemma_locally_finite: fixes X :: "'b topology"   and I :: "'c set"   and T :: "'c \<Rightarrow> 'b set"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" assumes "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>V. openin X V \<and> x \<in> V \<and> finite {i \<in> I. T i \<inter> V \<noteq> {}}"   and "\<And>i. i \<in> I \<Longrightarrow> closedin X (T i)"   and "\<And>i. i \<in> I \<Longrightarrow> continuous_map (subtopology X (T i)) Y (f i)"   and "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> topspace X \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"   and "\<And>x. x \<in> topspace X \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x" shows "continuous_map X Y g"
simps_4: fixes m :: "nat"   and n :: "nat"   and xs :: "nat list"   and ux :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((recf.Id m n, xs, ux) # fs, rv) = (fs, Some (xs ! n))"
unity_root_sum_2: fixes k :: "nat"   and n :: "int" assumes "1 \<le> k"   and "\<not> int k dvd n" shows "unity_root_sum k n = 0"
typ_of_eta_expand: fixes f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ" assumes "typ_of f = Some (\<tau> \<rightarrow> \<tau>')" shows "typ_of (Abs \<tau> (f $ Bv 0)) = Some (\<tau> \<rightarrow> \<tau>')"
terminal_arr_unique: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "terminal_arr f"   and "terminal_arr f'" shows "f = f'"
intersect_num_in_set: fixes point_set :: "'c set"   and block_collection :: "'c set multiset"   and b1 :: "'c set"   and b2 :: "'c set" assumes "incidence_system point_set block_collection"   and "b1 \<in># block_collection"   and "b2 \<in># remove1_mset b1 block_collection" shows "b1 |\<inter>| b2 \<in> incidence_system.intersection_numbers block_collection"
is_fl_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<And>a b. \<lbrakk>a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>; b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>\<rbrakk> \<Longrightarrow> \<FF>\<lparr>ArrMap\<rparr> `\<^sub>\<circ> Hom \<AA> a b = Hom \<BB> (\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>) (\<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>)" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>u\<^sub>l\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
play_def: fixes e :: "'a llist" shows "play e \<equiv> llength e = 2 * N"
eq: shows "\<forall>a b. b < mod_type_class.from_nat k \<longrightarrow> A $ a $ b = B $ a $ b"
comp_eq_id_dest: fixes a :: "'e \<Rightarrow> 'd"   and b :: "'c \<Rightarrow> 'e"   and c :: "'c \<Rightarrow> 'd"   and v :: "'c" assumes "a \<circ> b = id \<circ> c" shows "a (b v) = c v"
attach_wiring_simps: fixes f :: "'m \<Rightarrow> 'i"   and g :: "'j \<Rightarrow> 'n" shows "attach_wiring (f, g) = map_fun id (map_fun f (map_gpv (map_prod g id) id))"
pdg_vsubset_index_pdigraph_base: fixes J :: "V" assumes "J \<subseteq>\<^sub>\<circ> J \<union>\<^sub>\<circ> K" shows "pdigraph_base \<alpha> J \<AA>"
intros_2: fixes F :: "'b formula" shows "((F \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> F \<in> AX1"
sigma_prime_power_two: fixes n :: "nat" shows "sigma (2 ^ n) = 2 ^ (n + 1) - 1"
C_to_CValue_chain_take: shows "chain C_to_CValue_take"
execute_swap_1: fixes i :: "nat"   and h :: "heap"   and a :: "'a array"   and x :: "'a" assumes "i < Array.length h a" shows "execute (Array.swap i x a) h = Some (Array.get h a ! i, Array.update a i x h)"
ord_rec_Seq_succ_iff: fixes k :: "hf"   and T :: "hf"   and G :: "hf \<Rightarrow> hf"   and s :: "hf"   and z :: "hf" assumes "Ord k"   and "k \<noteq> 0" shows "ord_rec_Seq T G s (succ k) z = (\<exists>s' y. ord_rec_Seq T G s' k y \<and> z = G y \<and> s = insf s' k y)"
ring_id_seq_comp: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "ring_id \<circ> s = s"
u_simps_5: shows "cod (tab\<^sub>0 r) = tab\<^sub>0 r"
Int_Un_eq_3: fixes S :: "'e set"   and T :: "'e set" shows "S \<union> S \<inter> T = S"
one_aux: shows "f (Suc 0) = (1::'a)"
index_zero_implies_same_group_unique: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "block_collection index ps = 0" shows "\<exists>!G. G \<in> groups \<and> ps \<subseteq> G"
pow_ge_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "SN_Orders.class.ordered_semiring_1 one times plus zero less_eq"   and "less_eq zero a" shows "less_eq zero (power.power one times a n)"
less_eq_list_cons_inv: fixes x :: "'a"   and xs :: "'a list"   and yys :: "'a list" assumes "local.less_eq_list (x # xs) yys" shows "\<exists>y ys. yys = y # ys \<and> (x < y \<or> x = y \<and> local.less_eq_list xs ys)"
exp_plus_inverse_exp: fixes x :: "real" shows "2 \<le> exp x + inverse (exp x)"
bind_map_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf"   and g :: "'b \<Rightarrow> 'a pmf" shows "bind_pmf (map_pmf f M) g = bind_pmf M (\<lambda>x. g (f x))"
onorm_triangle: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "bounded_linear f"   and "bounded_linear g" shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
assms: shows "maximal_literal (subst_lit L \<sigma>) (subst_cl C \<sigma>)"
SUP_PInfty: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ereal" assumes "\<And>n. \<exists>i\<in>A. ereal (real n) \<le> f i" shows "Sup (f ` A) = \<infinity>"
weak_Nullstellensatz_aux_2: fixes X :: "'c set"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and x :: "'c" assumes "finite X"   and "F \<subseteq> P[insert x X]"   and "X \<subseteq> {..<x}"   and "1 \<notin> ideal F"   and "ideal F \<inter> P[{x}] \<subseteq> {0}" obtains a :: "'d" where "1 \<notin> ideal (poly_eval (\<lambda>_. monomial a 0) ` focus {x} ` F)"
incseq_SucI: fixes X :: "nat \<Rightarrow> 'a" assumes "\<And>n. X n \<le> X (Suc n)" shows "incseq X"
Rep_combine_matrix: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix"   and j :: "nat"   and i :: "nat" assumes "f (0::'c) (0::'d) = (0::'b)" shows "Rep_matrix (combine_matrix f A B) j i = f (Rep_matrix A j i) (Rep_matrix B j i)"
inner_prod_distrib_right: fixes a :: "complex Matrix.vec"   and b :: "complex Matrix.vec"   and c :: "complex" assumes "dim_vec a = dim_vec b" shows "\<langle>a|c \<cdot>\<^sub>v b\<rangle> = c * \<langle>a|b\<rangle>"
igSubstIPresIGWlsSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIPresIGWlsSTR MOD \<equiv> \<forall>ys Y y s X. igWls MOD s (igSubst MOD ys Y y X) = (igWls MOD (asSort ys) Y \<and> igWls MOD s X)"
refines_def: fixes G :: "'b program"   and F :: "'b program"   and X :: "'b program set" shows "G refines F wrt X \<equiv> \<forall>H. F ok H \<and> G ok H \<and> F \<squnion> H \<in> welldef \<inter> X \<longrightarrow> G \<squnion> H \<in> welldef \<inter> X"
antisymp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "antisymp r = (\<forall>x y. r x y \<longrightarrow> r y x \<longrightarrow> x = y)"
changes_itv_smods_congr: fixes a :: "real"   and a' :: "real"   and b' :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" assumes "a < a'"   and "a' < b'"   and "b' < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0"   and "\<forall>p\<in>set (smods p q). \<forall>x. a < x \<and> x \<le> a' \<or> b' \<le> x \<and> x < b \<longrightarrow> poly p x \<noteq> 0" shows "changes_itv_smods a b p q = changes_itv_smods a' b' p q"
carrier_direct_limE: fixes X :: "('c set \<times> 'd) set" assumes "X \<in> stfx.carrier_direct_lim" obtains U :: "'c set" and s :: "'d" where "U \<in> {U. is_open\<^sub>Y U \<and> f x \<in> U}" and "s \<in> \<O>\<^sub>Y U" and "X = stfx.class_of U s"
shuffle_to_top_range: fixes Y :: "'x set"   and d :: "'d" shows "Field (shuffle_to_top Y d) \<subseteq> {x. Xd x = d}"
per_root_border: fixes r :: "'b list"   and w :: "'b list" assumes "\<^bold>|r\<^bold>| < \<^bold>|w\<^bold>|"   and "r \<noteq> \<epsilon>"   and "w \<le>p r \<cdot> w" shows "r\<inverse>\<^sup>>w \<le>b w"
orthogonal_complement_closed_subspace: fixes A :: "'b set" shows "closed_csubspace (orthogonal_complement A)"
ZFfunExt: fixes f :: "ZF"   and g :: "ZF" assumes "|dom|f = |dom|g"   and "|cod|f = |cod|g"   and "isZFfun f"   and "isZFfun g"   and "\<And>x. x |\<in>| |dom|f \<Longrightarrow> f |@| x = g |@| x" shows "f = g"
univ_poly_units': fixes K :: "'a set"   and p :: "'a list" assumes "subfield K R" shows "(p \<in> Units (K [X])) = (p \<in> carrier (K [X]) \<and> p \<noteq> [] \<and> degree p = 0)"
g_eq_Some: fixes x :: "'a \<times> 'b"   and y :: "'a" shows "(g x = Some y) = (P x \<and> y = fst x)"
last_prefixes: fixes xs :: "'c list" shows "last (prefixes xs) = xs"
countably_additive_def: fixes M :: "'a set set"   and f :: "'a set \<Rightarrow> ennreal" shows "countably_additive M f = (\<forall>A. range A \<subseteq> M \<longrightarrow> disjoint_family A \<longrightarrow> \<Union> (range A) \<in> M \<longrightarrow> (\<Sum>i. f (A i)) = f (\<Union> (range A)))"
F_def: shows "F = (\<lambda>i. f \<circ> T ^^ i)"
valid_path_mirror: assumes "valid_path (- g)" shows "valid_path (- (- g))"
that: fixes s1 :: "'a" assumes "s1 \<in> S"   and "s1 \<cdot> (s' \<cdot> r - s \<cdot> r') = \<zero>" shows "thesis"
degen_simps: fixes cs :: "('b \<Rightarrow> bool) list"   and a :: "'b"   and k :: "nat" shows "degen cs (a, k) = (length cs \<le> k \<or> (cs ! k) a)"
simps: fixes a1 :: "'a"   and a2 :: "'a"   and \<delta> :: "('a, 'b) ta_rule set" shows "((a1, a2) \<in> f_succ \<delta>) = (\<exists>q l qs q'. a1 = q \<and> a2 = q' \<and> q \<rightarrow> l qs \<in> \<delta> \<and> q' \<in> set qs)"
delta: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and S :: "'b set"   and a :: "'b"   and b :: "'b \<Rightarrow> 'a" assumes "comm_monoid_set f z"   and "finite S" shows "comm_monoid_set.F f z (\<lambda>k. if k = a then b k else z) S = (if a \<in> S then b a else z)"
shift_map_keys_pprod_def: shows "shift_map_keys_pprod = pprod.shift_map_keys"
B_proj: fixes l :: "'lbl" shows "\<forall>b\<in>set (proj l B). \<exists>a\<in>set (proj l A). \<exists>\<delta>. b = a \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"