hcju/isabelleps huggingface dataset:
subnetAux: fixes Dd :: "'a set"   and A :: "'a set"   and B :: "'a set" assumes "Dd \<inter> A \<noteq> {}"   and "A \<subseteq> B" shows "Dd \<inter> B \<noteq> {}"
hyps: fixes A :: "'a mat" assumes "A \<in> carrier_mat n m"   and "J < m" shows "swap_col_to_front A J = mat n m (\<lambda>a. case a of (i, j) \<Rightarrow> if j = 0 then A $$ (i, J) else if j \<le> J then A $$ (i, j - 1) else A $$ (i, j))"
simps_1: fixes a :: "'b" shows "BinomialHeapStruc.remove1Prio a [] = []"
sorted_wrt_nth_less: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and i :: "nat"   and j :: "nat" assumes "sorted_wrt P xs"   and "i < j"   and "j < length xs" shows "P (xs ! i) (xs ! j)"
inv_ide: fixes a :: "'a" assumes "C.ide a" shows "C.inv a = a"
icard_atMost: fixes u :: "nat" shows "icard {..u} = enat (Suc u)"
list_update_length: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and y :: "'b" shows "(xs @ x # ys)[length xs := y] = xs @ y # ys"
join_connection: fixes x :: "'c"   and y :: "'c" shows "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
is_extensional: fixes f :: "'a \<times> 'a" assumes "\<not> C.VV.arr f" shows "fst f \<star>\<^sub>C snd f = C.null"
sorted2: fixes x :: "'a"   and y :: "'a"   and zs :: "'a list" shows "local.sorted (x # y # zs) = (le x y \<and> local.sorted (y # zs))"
prime_harmonic_asymp_equiv: shows "prime_sum_upto (\<lambda>p. 1 / real p) \<sim>[at_top] (\<lambda>x. ln (ln x))"
less_top_def: fixes x :: "'a top"   and y :: "'a top" shows "(x < y) = (case x of top.Value x \<Rightarrow> case y of top.Value y \<Rightarrow> x < y | Top \<Rightarrow> True | Top \<Rightarrow> False)"
seq2: fixes c1 :: "('a, 'b) MWLsCom"   and m :: "'b \<Rightarrow> 'c"   and \<alpha> :: "('a, 'b) MWLsCom list"   and c1' :: "('a, 'b) MWLsCom"   and m' :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and c2 :: "('a, 'b) MWLsCom" assumes "((c1, m), \<alpha>, Some c1', m') \<in> MWLs_semantics.MWLsSteps_det E BMap" shows "((c1;c2, m), \<alpha>, Some (c1';c2), m') \<in> MWLs_semantics.MWLsSteps_det E BMap"
vimage2p_def: fixes f :: "'a \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> 'c" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
assms_13: shows "D = \<Union> (roots ` X (\<noteq>))"
is_array_list_def: fixes l :: "'b list" shows "is_array_list l \<equiv> \<lambda>(a, n). \<exists>\<^sub>Al'. a \<mapsto>\<^sub>a l' * \<up> (n \<le> length l' \<and> l = take n l' \<and> 0 < length l')"
spmf_rel_map1: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c spmf" shows "rel_spmf R (map_spmf f x) = rel_spmf (\<lambda>x. R (f x)) x"
Least_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" shows "ord.Least less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq x y))"
abs_eq: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "pmf_of_set S = embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
Jordan_curve: fixes c :: "real \<Rightarrow> complex" assumes "simple_path c"   and "pathfinish c = pathstart c" obtains inner :: "complex set" and outer :: "complex set" where "inner \<noteq> {}" and "open inner" and "connected inner" and "outer \<noteq> {}" and "open outer" and "connected outer" and "bounded inner" and "\<not> bounded outer" and "inner \<inter> outer = {}" and "inner \<union> outer = - path_image c" and "frontier inner = path_image c" and "frontier outer = path_image c"
diff_neq0_1: fixes a :: "'b"   and b :: "'b" assumes "a \<noteq> b" shows "a - b \<noteq> (0::'b)"
Fun_range_case_2: fixes M :: "('a, 'b) Term.term set"   and P :: "'a \<Rightarrow> ('a, 'b) Term.term list \<Rightarrow> bool" shows "(\<forall>f T. Fun f T \<in> M \<longrightarrow> P f T) = (\<forall>u\<in>M. is_Fun u \<longrightarrow> P (the_Fun u) (args u))"
rev_induct: fixes P :: "'f list \<Rightarrow> bool"   and xs :: "'f list" assumes "P []"   and "\<And>x xs. P xs \<Longrightarrow> P (xs @ [x])" shows "P xs"
rotate_def: fixes n :: "nat"   and dxs :: "'d dlist" shows "Dlist.rotate n dxs = Dlist (List.rotate n (list_of_dlist dxs))"
restrict_return_spmf: fixes x :: "'b"   and A :: "'b set" shows "return_spmf x \<upharpoonleft> A = (if x \<in> A then return_spmf x else return_pmf None)"
assms_3: fixes i :: "'i" assumes "i \<in> I" shows "integrable M (X i)"
if_raise_if_None: fixes b :: "bool"   and y :: "abopt"   and c :: "bool"   and x :: "xname" shows "((if b then y else raise_if c x y) = None) = ((c \<longrightarrow> b) \<and> y = None)"
present_def: fixes h :: "heap"   and a :: "'b array" shows "Array.present h a = (addr_of_array a < lim h)"
toList_fromList: fixes xs :: "'b list" shows "toList (fromList xs) = msort xs"
lfilter_lfind_3: fixes P :: "'b \<Rightarrow> bool"   and lst :: "'b llist" assumes "lfilter P lst \<noteq> LNil" shows "enat (lfind P lst) < llength lst"
restrict_compose_left: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'e" assumes "f ` S \<subseteq> T" shows "restrict (restrict g T \<circ> f) S = restrict (g \<circ> f) S"
assms: fixes a :: "atom"   and b :: "atom" shows "((a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> t) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> p \<bullet> t)"
paint_inv2: fixes t :: "('b, 'c) RBT_Impl.rbt"   and c :: "color" assumes "inv2 t" shows "inv2 (paint c t)"
ct: shows "r1.cond_action_ok (ls, (ts1, m1), ws, is) t ct"
assms_1: shows "\<forall>\<^sub>F x in at_top. f x = 0"
gfp_trans_def: fixes T :: "(('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "gfp_trans T = Sup_trans {t. (\<forall>P. unitary P \<longrightarrow> unitary (t P)) \<and> le_utrans t (T t)}"
simps_6: fixes x3 :: "'a"   and x1 :: "'a" shows "Goes_wrong x3 \<noteq> Terminates x1"
space_def: fixes M :: "'d measure" shows "space M = fst (Rep_measure M)"
top_eq_inf_iff: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "(top = inf x y) = (x = top \<and> y = top)"
len_match: shows "length (rows_to_keep (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) = dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
factorization_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and P :: "'b eclause"   and C :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.factorization trm_ord sel filter_trms P C \<sigma> k C' = (\<exists>L1 L2 L' t s u v Cl_P Cl_C trms_C. basic_superposition.eligible_literal trm_ord sel L1 P \<sigma> \<and> L1 \<in> cl_ecl P \<and> L2 \<in> cl_ecl P - {L1} \<and> Cl_C = cl_ecl C \<and> Cl_P = cl_ecl P \<and> basic_superposition.orient_lit_inst trm_ord L1 t s pos \<sigma> \<and> basic_superposition.orient_lit_inst trm_ord L2 u v pos \<sigma> \<and> t \<lhd> \<sigma> \<noteq> s \<lhd> \<sigma> \<and> t \<lhd> \<sigma> \<noteq> v \<lhd> \<sigma> \<and> ck_unifier t u \<sigma> k \<and> L' = Neg (Eq s v) \<and> (C = Ecl Cl_C trms_C \<and> trms_C = basic_superposition.get_trms filter_trms Cl_C (basic_superposition.dom_trms trm_ord Cl_C (subst_set (trms_ecl P \<union> proper_subterms_of t) \<sigma>)) k) \<and> Cl_C = subst_cl C' \<sigma> \<and> C' = Cl_P - {L2} \<union> {L'})"
mult_isol_var: fixes u :: "'a \<times> 'b"   and x :: "'a \<times> 'b"   and v :: "'a \<times> 'b"   and y :: "'a \<times> 'b" assumes "less_eq_prod u x"   and "less_eq_prod v y" shows "less_eq_prod (times_prod' u v) (times_prod' x y)"
sel_12: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "partner (Recv x31 x32 x33 x34 x35 x36) = x33"
assms_1: shows "(xs @ zs, ys @ qs) \<in> lexord r"
lifted_q: fixes q :: "'q" assumes "q \<in> Q" shows "standard_lifting Inf_FL Bot_G (Inf_G_q q) (entails_q q) (Red_I_q q) (Red_F_q q) Bot_FL (\<G>_F_L_q q) (\<G>_I_L_q q)"
sorted_wrt_less_sum_mono_lowerbound: fixes f :: "nat \<Rightarrow> 'b"   and ns :: "nat list" assumes "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y"   and "sorted_wrt (<) ns" shows "sum f {0..<length ns} \<le> sum_list (map f ns)"
gSubstAbsPresGWlsAbs_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gSubstAbsPresGWlsAbs MOD \<equiv> \<forall>ys Y' Y y us s A' A. isInBar (us, s) \<and> wls (asSort ys) Y' \<and> gWls MOD (asSort ys) Y \<and> wlsAbs (us, s) A' \<and> gWlsAbs MOD (us, s) A \<longrightarrow> gWlsAbs MOD (us, s) (gSubstAbs MOD ys Y' Y y A' A)"
Quotient3_msg: shows "Quotient3 (\<sim>) abs_msg rep_msg"
one: fixes bs :: "bit list" assumes "w = \<one> # bs" shows "P w"
distinct_11: fixes x11 :: "'a"   and x12 :: "'b" shows "x11 \<leftarrow> x12 \<noteq> Stop"
triangle_in_hom_1: shows "\<guillemotleft>(D.inv (unit (trg\<^sub>C f)) \<cdot>\<^sub>D F \<epsilon> \<cdot>\<^sub>D \<Phi> (f, g) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F f, F g, F f] \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)) : F f \<star>\<^sub>D src\<^sub>D (F f) \<Rightarrow>\<^sub>D trg\<^sub>D (F f) \<star>\<^sub>D F f\<guillemotright>"
fmapU_resT_def: shows "fmapU \<equiv> resT_map"
fst_scaleR: fixes r :: "real"   and A :: "'c \<times> 'd" shows "fst (r *\<^sub>R A) = r *\<^sub>R fst A"
T_append: fixes qs :: "'c list"   and as :: "(nat \<times> nat list) list"   and s :: "'c list"   and q :: "'c"   and a :: "nat \<times> nat list" assumes "length qs = length as" shows "T\<^sub>p s (qs @ [q]) (as @ [a]) = T\<^sub>p s qs as + t\<^sub>p (Partial_Cost_Model.steps s qs as) q a"
cases: fixes a :: "('fun, 'var) Term.term" assumes "public_ground_wf_term a" obtains f :: "'fun" and T :: "('fun, 'var) Term.term list" where "a = Fun f T" and "public f" and "arity f = length T" and "\<And>t. t \<in> set T \<Longrightarrow> public_ground_wf_term t"
termi: shows "\<Gamma>\<turnstile>call init p return c \<down> Normal s"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lookup (lowerPoly\<^sub>0 x xa xb) = (\<lambda>mon. lookup xb (lowerPowers x xa mon))"
inR2': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, G, H, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> G, H, F, \<Delta> \<down> n"
assms_1: shows "xs \<in> set_of_appr XS"
mono_sync_ref: fixes P :: "'a process"   and P' :: "'a process"   and Q :: "'a process"   and Q' :: "'a process"   and A :: "'a set" assumes "P \<sqsubseteq> P'"   and "Q \<sqsubseteq> Q'" shows "(P \<lbrakk>A\<rbrakk> Q) \<sqsubseteq> (P' \<lbrakk>A\<rbrakk> Q')"
degree_sum_notin: fixes x' :: "nat"   and B :: "'a mpoly"   and A :: "'a mpoly" assumes "x' \<notin> vars B" shows "MPoly_Type.degree (A + B) x' = MPoly_Type.degree A x'"
has_max_lideal: shows "\<exists>\<ww>. max_lideal \<ww> A addA multA zeroA oneA"
bound_intersect_2d_ud_segments_of_aform_le: fixes prec :: "nat"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and xl :: "real"   and m0 :: "real"   and M0 :: "real" assumes "bound_intersect_2d_ud prec X xl = Some (m0, M0)" shows "m0 \<le> M0"
mset_set_set_mset_empty_mempty: fixes D :: "'a multiset" shows "(mset_set (set_mset D) = {#}) = (D = {#})"
going_to_def: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'c filter" shows "f going_to F = filtercomap f F"
image2p_def: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "BNF_Greatest_Fixpoint.image2p f g R = (\<lambda>x y. \<exists>x' y'. R x' y' \<and> f x' = x \<and> g y' = y)"
w_reach_2: shows "w \<rightarrow>\<^sup>*\<^bsub>H\<^esub> u"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog" assumes "known_ptr = ShadowRootClass.known_ptr"   and "get_owner_document = l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node_si get_disconnected_nodes get_host" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node_si get_host get_owner_document"
col_conga_col: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "Col A B C"   and "A B C CongA D E F" shows "Col D E F"
bij: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "bij f"
lemLineEndpoint: fixes q :: "'a Point"   and p :: "'a Point" shows "q = p \<leadsto> from p to q"
th1: shows "(\<forall>T. openin (top_of_set S) T \<and> closedin (top_of_set S) T \<longrightarrow> T = {} \<or> T = S) = (\<nexists>t' t. closed t' \<and> t = S \<inter> t' \<and> t \<noteq> {} \<and> t \<noteq> S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
assms_2: shows "\<Gamma> \<turnstile> M : \<tau>"
homogeneous_monom_mult: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and c :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p" shows "homogeneous (MPoly_Type_Class.punit.monom_mult c t p)"
effect_lengthI: fixes h' :: "heap"   and h :: "heap"   and r :: "nat"   and a :: "'b array" assumes "h' = h"   and "r = Array.length h a" shows "effect (Array.len a) h h' r"
smult_commute_sg_nsg: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "H \<diamondop>\<^bsub>G\<^esub> N = N \<diamondop>\<^bsub>G\<^esub> H"
higher_deriv_ln_Gamma_complex_of_real: fixes x :: "real"   and j :: "nat" assumes "0 < x" shows "(deriv ^^ j) ln_Gamma (complex_of_real x) = complex_of_real ((deriv ^^ j) ln_Gamma x)"
bin_sign_lem: fixes n :: "nat"   and bin :: "int" shows "(bin_sign (signed_take_bit n bin) = - 1) = bit bin n"
ftv_ty_eqvt: fixes pi :: "(W.tvar \<times> W.tvar) list"   and T :: "ty" shows "pi \<bullet> ftv T = ftv (pi \<bullet> T)"
is_ground_cls_is_ground_on_var: fixes C :: "('a, 'b) Term.term literal multiset"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and v :: "'b" assumes "is_ground_cls (subst_cls C \<sigma>)"   and "v \<in> vars_clause C" shows "is_ground_atm (\<sigma> v)"
52520: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (- x \<sqinter> y) = - x \<sqinter> - y"
dom_locals_fvar_mono: fixes s :: "state"   and statDeclC :: "qtname"   and stat :: "bool"   and fn :: "vname"   and a :: "val" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (snd (fvar statDeclC stat fn a s))))"
pole_theorem: fixes g :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "g holomorphic_on S"   and "a \<in> interior S"   and "\<And>z. z \<in> S - {a} \<Longrightarrow> g z = (z - a) * f z" shows "(\<lambda>z. if z = a then deriv g a else f z - g a / (z - a)) holomorphic_on S"
uses'_def: fixes g :: "'a"   and n :: "'node" shows "i.uses' g n \<equiv> i.lookupDef g n ` uses2 g n"
oalist_inv_def: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" shows "ko_ntm.oalist_inv xs = ko.oalist_inv_raw (key_order_of_nat_term_order_inv (snd xs)) (fst xs)"
preserves_ide: fixes a :: "'b" assumes "B.ide a" shows "A.ide (F a)"
reduc_fair_imp_Liminf_reduc_sat: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and Ns :: "'a set llist" assumes "calculus Bot Inf entails Red_I Red_F"   and "chain (calculus.derive Red_F) Ns"   and "calculus.reduc_fair Inf Red_I Red_F Ns" shows "calculus.reduc_saturated Inf Red_I Red_F (Liminf_llist Ns)"
univalent_bot_closed: shows "coreflexive (bot[[bot]])"
coeff_add: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "poly.coeff (p + q) n = poly.coeff p n + poly.coeff q n"
univ_asTerm_alphaGood: fixes f :: "('c, 'd, 'varSort, 'var, 'e) qTerm \<Rightarrow> 'f"   and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "f respectsP alphaGood"   and "qGood X" shows "Equiv_Relation2.univ f (asTerm X) = f X"
lselect_lproject: fixes A :: "'b set"   and w :: "'b llist"   and s :: "nat set" shows "lproject A (lselect s w) = lselect (s \<inter> liset A w) w"
vs: shows "vs = getFrN (map snd txs) (map fst txs) [\<phi>] (length txs)"
fdisjnt_subset_left: fixes N' :: "'a fset"   and N :: "'a fset"   and M :: "'a fset" assumes "N' |\<subseteq>| N"   and "fdisjnt N M" shows "fdisjnt N' M"
cont_sp0: shows "continuous_on (affine hull S - {0::'a}) (surf \<circ> proj)"
case_prodI2: fixes p :: "'ap \<times> 'aq"   and c :: "'ap \<Rightarrow> 'aq \<Rightarrow> bool" assumes "\<And>a b. p = (a, b) \<Longrightarrow> c a b" shows "case p of (a, b) \<Rightarrow> c a b"
BSI1: shows "BSI \<V>1 Tr\<^bsub>ES1\<^esub>"
mem_WWa_inc_a: fixes W :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "W \<in> WWa D g a" shows "a \<in> W"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "rat" assumes "gs.\<mu> x xa = y" obtains i :: "nat" and j :: "nat" where "x = i" and "xa = j" and "y = (if j < i then map of_int_hom.vec_hom fs_init ! i \<bullet> gs.gso j / \<parallel>gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
fmlookup_restrict_fset: fixes A :: "'e fset"   and m :: "'e \<rightharpoonup>\<^sub>f 'd"   and x :: "'e" shows "fmlookup (fmrestrict_fset A m) x = (if x |\<in>| A then fmlookup m x else None)"
alpha_name_coname: fixes x :: "name"   and b :: "coname"   and M :: "trm"   and y :: "name"   and c :: "coname"   and N :: "trm"   and z :: "name"   and a :: "coname" assumes "[x].[b].M = [y].[c].N"   and "z \<sharp> (x, y, M, N)"   and "a \<sharp> (b, c, M, N)" shows "M = [(x, z)] \<bullet> [(b, a)] \<bullet> [(c, a)] \<bullet> [(y, z)] \<bullet> N"
\<Delta>\<Phi>_pass1: fixes hs :: "'b tree" assumes "hs \<noteq> \<langle>\<rangle>" shows "\<Phi> (pass\<^sub>1 hs) - \<Phi> hs \<le> 2 * log 2 (real (size hs)) - real (len hs) + 2"
dim_basis: fixes A :: "'e set" assumes "finite A"   and "W.basis A" shows "W.dim = card A"
SFAssThrow: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := throw e,s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
rel_mono_strong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "('d, 'f) optionT"   and y :: "('e, 'g) optionT"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_optionT' R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> phantom_optionT x; y1 \<in> phantom_optionT y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> set_optionT x; y2 \<in> set_optionT y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_optionT' R1a R2a x y"
foldr_node_correct: fixes f :: "'e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd"   and nd :: "('e, 'f) Node"   and \<sigma> :: "'d" shows "foldr_node f nd \<sigma> = List.foldr f (nodeToList nd) \<sigma>"
linear_comb0_1Tr: fixes H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat \<Rightarrow> 'a" assumes "H \<subseteq> carrier M" shows "s \<in> {j. j \<le> n} \<rightarrow> {\<zero>\<^bsub>R\<^esub>} \<and> m \<in> {j. j \<le> n} \<rightarrow> H \<longrightarrow> l_comb R M n s m = \<zero>"
total_ES2_C2_inter_Upsilon2: shows "EventSystems.total ES2 (C\<^bsub>\<V>2\<^esub> \<inter> \<Upsilon>\<^bsub>\<Gamma>2\<^esub>)"
no_outstanding_write_to_read_only_memory_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "no_outstanding_write_to_read_only_memory \<S> (t # ts)" shows "no_outstanding_write_to_read_only_memory \<S> ts"
open_imp_locally_path_connected: fixes S :: "'b set" assumes "open S" shows "locally path_connected S"
rev_exhaust: fixes xs :: "'a list" obtains "xs = []"   | ys :: "'a list" and y :: "'a" where "xs = ys @ [y]"
noetherian_alt: fixes x :: "'a" shows "Noetherian x = (\<forall>y. d y \<le>  |x\<rangle> y \<longrightarrow> d y = (zero_class.zero::'a))"
distinct_27: fixes x31 :: "nat"   and x32 :: "'e"   and x4 :: "nat" shows "unicast x31 x32 \<noteq> \<not>unicast x4"
arr_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" shows "arr (mkArr A B F) = (A \<subseteq> Univ \<and> |A| <o cardSuc (cmax |UNIV| natLeq) \<and> B \<subseteq> Univ \<and> |B| <o cardSuc (cmax |UNIV| natLeq) \<and> F \<in> A \<rightarrow> B)"
dg_Dom_vsubset_Vset: shows "\<CC>\<lparr>Dom\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
A: shows "CSP2 (R (r \<turnstile> p)) (a, b)"
Lim_transform_within_set_eq: fixes S :: "'b set"   and T :: "'b set"   and a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "\<forall>\<^sub>F x in at a. (x \<in> S) = (x \<in> T)" shows "(f \<longlongrightarrow> l) (at a within S) = (f \<longlongrightarrow> l) (at a within T)"
assms: shows "v \<in> \<Union> (set (map fv\<^sub>s\<^sub>n\<^sub>d S))"
distinct_sorted_list_of_fset: fixes S :: "'a fset" shows "distinct (sorted_list_of_fset S)"
cases: fixes a1 :: "Player"   and a2 :: "'a llist"   and a3 :: "'a \<Rightarrow> 'a" assumes "path_conforms_with_strategy a1 a2 a3" obtains p :: "Player" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pa__" and "a2 = LNil" and "a3 = \<sigma>'__"   | p :: "Player" and v :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = pb__" and "a2 = LCons v__ LNil" and "a3 = \<sigma>''__"   | v :: "'a" and p :: "Player" and w :: "'a" and \<sigma> :: "'a \<Rightarrow> 'a" and Ps :: "'a llist" where "a1 = pc__" and "a2 = LCons va__ (LCons w Ps__)" and "a3 = \<sigma>'''__" and "va__ \<in> VV pc__" and "w = \<sigma>'''__ va__" and "path_conforms_with_strategy pc__ (LCons w Ps__) \<sigma>'''__"   | v :: "'a" and p :: "Player" and Ps :: "'a llist" and \<sigma> :: "'a \<Rightarrow> 'a" where "a1 = p" and "a2 = LCons v Ps" and "a3 = \<sigma>" and "v \<notin> VV p" and "path_conforms_with_strategy p Ps \<sigma>"
cross3_simps_1: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" shows "a \<times> b \<equiv> vector [a $ 2 * b $ 3 - a $ 3 * b $ 2, a $ 3 * b $ 1 - a $ 1 * b $ 3, a $ 1 * b $ 2 - a $ 2 * b $ 1]"
q_def: shows "q = embed_pmf (\<lambda>z. if z = y then 0 else pmf p z / a)"
mask_nat_def: fixes n :: "nat" shows "mask n = 2 ^ n - 1"