hcju/isabelleps huggingface dataset:
gso_oc_projection_span_1: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "fs.gs.gso i = gs.oc_projection (gs.span (fs.gs.gso ` {0..<i})) (map of_int_hom.vec_hom fs ! i)"
summable_Cauchy_product: fixes a :: "nat \<Rightarrow> 'b"   and b :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>k. norm (a k))"   and "summable (\<lambda>k. norm (b k))" shows "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
comp_cond_left_distr: fixes b :: "'b upred"   and P :: "('b, 'd) urel"   and Q :: "('b, 'd) urel"   and R :: "('d, 'c) urel" shows "(P \<triangleleft> b \<triangleright>\<^sub>r Q) ;; R = P ;; R \<triangleleft> b \<triangleright>\<^sub>r Q ;; R"
filter_diff_mset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" shows "filter_mset P (M - N) = filter_mset P M - filter_mset P N"
rank_Union: fixes A :: "V set" shows "rank (\<Squnion> A) = \<Squnion> (rank ` A)"
preserves_inv: fixes f :: "'a" assumes "local.iso f" shows "local.inv f \<otimes> \<I> = local.inv (f \<otimes> \<I>)"
cardXY: shows "card (X \<union> Y) = card X + card Y"
lambda_one: shows "(\<lambda>x. x) = (*) (1::'c)"
mat_cnj_sm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_cnj (k *\<^sub>s\<^sub>m A) = cnj k *\<^sub>s\<^sub>m mat_cnj A"
distinct_2: fixes name' :: "name"   and trm' :: "trm"   and stack' :: "stack" shows "[name']trm'\<ggreater>stack' \<noteq> stack.Id"
orbits: shows "pmf (sds R27) c = pmf (sds R27) b"
SS: shows "homotopic_with (\<lambda>x. True) X X (g1 \<circ> g2 \<circ> (f2 \<circ> f1)) id"
edge_succ_cyclic: fixes v :: "'a" assumes "v \<in> verts H"   and "out_arcs H v \<noteq> {}" shows "cyclic_on (edge_succ HM) (out_arcs H v)"
comp_arr_dom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a" assumes "category C"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a" shows "C f a = f"
word_eq_iff_signed: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and v :: "'b Word.word"   and w :: "'b Word.word" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(v = w) = (ring_1.signed one plus zero minus v = ring_1.signed one plus zero minus w)"
map_poly_hom_add: fixes p :: "'c poly poly"   and q :: "'c poly poly" shows "map_poly (pCons (0::'c)) (p + q) = map_poly (pCons (0::'c)) p + map_poly (pCons (0::'c)) q"
vlrestriction_vdomain: shows "r \<restriction>\<^sup>l\<^sub>\<circ> \<D>\<^sub>\<circ> r = r"
parallel_resource1: fixes real_resource :: "nat \<Rightarrow> ('h + 'i, 'j + 'k) resource"   and ideal_resource :: "nat \<Rightarrow> ('l + 'i, 'm + 'k) resource"   and sim :: "nat \<Rightarrow> ('h, 'j, 'l, 'm) converter"   and \<I>_real :: "nat \<Rightarrow> ('h, 'j) \<I>"   and \<I>_ideal :: "nat \<Rightarrow> ('l, 'm) \<I>"   and \<I>_common :: "nat \<Rightarrow> ('i, 'k) \<I>"   and bound :: "nat \<Rightarrow> enat"   and lossless :: "bool"   and w :: "nat \<Rightarrow> ('l \<Rightarrow> 'h) \<times> ('j \<Rightarrow> 'm)"   and \<I>_res :: "nat \<Rightarrow> ('n, 'o) \<I>"   and res :: "nat \<Rightarrow> ('n, 'o) resource" assumes "constructive_security real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common bound lossless w"   and "\<And>\<eta>. \<I>_res \<eta> \<turnstile>res res \<eta> \<surd>"   and "\<And>\<eta>. lossless \<Longrightarrow> lossless_resource (\<I>_res \<eta>) (res \<eta>)" shows "constructive_security (\<lambda>\<eta>. parallel_resource1_wiring \<rhd> res \<eta> \<parallel> real_resource \<eta>) (\<lambda>\<eta>. parallel_resource1_wiring \<rhd> res \<eta> \<parallel> ideal_resource \<eta>) sim \<I>_real \<I>_ideal (\<lambda>\<eta>. \<I>_res \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) bound lossless w"
w_def_3: shows "\<forall>w'. w \<rightarrow>\<^sub>e\<^sub>m\<^sub>b w' \<longrightarrow> \<not> subst \<rho> w' \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
web_flow: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "current \<Gamma> f"   and "\<And>x. \<lbrakk>x \<in> \<^bold>V\<^bsub>\<Gamma>\<^esub>; x \<notin> A \<Gamma>; x \<notin> B \<Gamma>\<rbrakk> \<Longrightarrow> KIR f x" shows "web_flow \<Gamma> f"
contI2: fixes f :: "'b \<Rightarrow> 'c" assumes "monofun f"   and "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))" shows "cont f"
matrix_vector_mult_iarray_def: fixes A :: "'b iarray iarray"   and x :: "'b iarray" shows "A *iv x = IArray.of_fun (\<lambda>i. \<Sum>j = 0..<IArray.length x. A !! i !! j * x !! j) (nrows_iarray A)"
\<phi>J_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and Ops :: "('b \<Rightarrow> 'b) set"   and tenc :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<psi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv enc Ops tenc F encF N ssap" shows "Jeroslow_Diagonalization.\<phi>J var subst tenc encF ssap \<psi> \<equiv> subst \<psi> (Jeroslow_Diagonalization.tJ tenc encF ssap \<psi>) (Generic_Syntax.Variable var 10)"
unbordered_border: fixes w :: "'b list" assumes "bordered w" shows "\<exists>x. x \<le>b w \<and> \<not> bordered x"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "semilattice f" shows "semilattice_fset f"
app: fixes n :: "nat"   and t :: "term"   and u :: "term" shows "wellformed' n (app t u) = (wellformed' n t \<and> wellformed' n u)"
fun_eq_iff: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f = g) = (\<forall>x. f x = g x)"
prefixes_init_psi: fixes j :: "nat"   and t :: "nat" shows "\<psi> j \<triangleright> (e_length (local.prefixes j (Suc t)) - 1) = local.prefixes j (Suc t)"
Rep_nat_int_inject: fixes x :: "nat_int"   and y :: "nat_int" shows "(Rep_nat_int x = Rep_nat_int y) = (x = y)"
WhileCondThrow: fixes 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and c :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>while (e) c,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
conjunction_def: fixes A :: "prop"   and B :: "prop" shows "(PROP A &&& PROP B) \<equiv> (\<And>C. (\<lbrakk>PROP A; PROP B\<rbrakk> \<Longrightarrow> PROP C) \<Longrightarrow> PROP C)"
start_steps_star: fixes A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and r :: "bool list"   and w :: "'b list" shows "((start (RegExp2NAe.star A), r) \<in> NAe.steps (RegExp2NAe.star A) w) = (w = [] \<and> r = start (RegExp2NAe.star A) \<or> (True # start A, r) \<in> NAe.steps (RegExp2NAe.star A) w)"
quasi_measure_preservingE_1: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> quasi_measure_preserving M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
changes_R_smods_0_2: fixes p :: "real poly" shows "changes_R_smods p 0 = 0"
finite_span_representation_bounded: fixes B :: "'d set" assumes "finite B"   and "independent B" shows "\<exists>D>0. \<forall>\<psi> b. \<bar>representation B \<psi> b\<bar> \<le> norm \<psi> * D"
sp: shows "automaton.trans A = seqp_sos \<Gamma>"
1: shows "PreFunctor (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"
chine_is_induced_map: shows "f.is_induced_map f.chine"
simps_1: fixes A :: "'a list set" shows "A ^^ 0 = {[]}"
indep_vars_cong: fixes I :: "'d set"   and J :: "'d set"   and X :: "'d \<Rightarrow> ennreal \<Rightarrow> 'e"   and Y :: "'d \<Rightarrow> ennreal \<Rightarrow> 'e"   and M' :: "'d \<Rightarrow> 'e measure"   and N' :: "'d \<Rightarrow> 'e measure"   and i :: "'b" assumes "I = J"   and "\<And>i. i \<in> I \<Longrightarrow> X i = Y i"   and "\<And>i. i \<in> I \<Longrightarrow> M' i = N' i" shows "Y.indep_vars i M' X I = Y.indep_vars i N' Y J"
simple_bochner_integrableI_bounded: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_function M f"   and "\<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M < \<infinity>" shows "simple_bochner_integrable M f"
unfoldE_2: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "Q \<leadsto><weakBisim> P"
simps_16: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list" shows "funs_term (Fun x21 x22) = insert x21 (\<Union> (funs_term ` set x22))"
fref_compI: shows "([P]\<^sub>f R1 \<rightarrow> R2) O ([Q]\<^sub>f S1 \<rightarrow> S2) \<subseteq> [\<lambda>x. Q x \<and> (\<forall>y. (y, x) \<in> S1 \<longrightarrow> P y)]\<^sub>f R1 O S1 \<rightarrow> R2 O S2"
analz_mono: fixes G :: "msg set"   and H :: "msg set" assumes "G \<subseteq> H" shows "analz G \<subseteq> analz H"
conc_only_var_assign_not_visible_abs: fixes mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" shows "\<forall>v\<^sub>C e. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> mem\<^sub>A_of mem\<^sub>C = mem\<^sub>A_of (mem\<^sub>C(v\<^sub>C := e))"
image_inv_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "inv f ` f ` A = A"
row_mat_mult_index: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'c list list"   and nc :: "nat"   and m2 :: "'c list list"   and i :: "nat"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr n m1"   and "mat n nc m2"   and "i < nr" shows "row (mat_multI ze pl ti nr m1 m2) i = map (\<lambda>j. scalar_prodI ze pl ti (row m1 i) (col m2 j)) [0..<nc]"
not_self_fresh: fixes x :: "'c" shows "atom x \<sharp> x = False"
r_cancel_one: fixes x :: "nat \<Rightarrow> 'a"   and a :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "a \<in> carrier P" shows "(a \<oplus>\<^bsub>P\<^esub> x = x) = (a = \<zero>\<^bsub>P\<^esub>)"
minimal_unsat_core_def: fixes I :: "'b set"   and ics :: "('b \<times> constraint) list" shows "minimal_unsat_core I ics = (I \<subseteq> fst ` set ics \<and> (\<nexists>v. (I, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics) \<and> (distinct_indices ics \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics)))"
x_plus_helpers_11: fixes x :: "nat" shows "14 + x = Suc (x + 13)"
inf_Seq: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e" shows "AlternativeSmallStep.inf \<Gamma> (Seq c1 c2 # cs) css (Normal s) = AlternativeSmallStep.inf \<Gamma> (c1 # c2 # cs) css (Normal s)"
vimage_snd: fixes A :: "'b set" shows "snd -` A = UNIV \<times> A"
0: shows "Domain a \<subseteq> N - {seller} \<and> \<Union> (Range a) \<subseteq> \<Omega>"
list_crsp_simp2: fixes lm1 :: "nat list"   and n :: "nat"   and lm2 :: "nat list" assumes "abc_list_crsp (lm1 @ 0 \<up> n) lm2" shows "abc_list_crsp lm1 lm2"
simps_5: fixes x1 :: "nat" shows "(_Exit_) \<noteq> (_ x1 _)"
con_rews_1: fixes cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "CFn\<cdot>cfun \<noteq> \<bottom>"
A: fixes x :: "nat" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
l1: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)"   and "\<not> QuadRes (int p) a"   and "x \<in> S1 p" shows "x \<noteq> f_1 p a x"
a_gt_sqrt: shows "\<forall>\<^sub>F n in sequentially. c (n + 1) < c n \<longrightarrow> real_of_int (a n) + (sqrt (real n) - 2) < real_of_int (a (n + 1))"
vector_space_on_def: fixes S :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "vector_space_on S scale \<equiv> module_on S scale"
cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "\<Theta>[F](f) = \<Theta>[F](g)"
simps_1: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and i :: "nat"   and R\<^sub>i :: "nat set" shows "pnet np \<langle>i; R\<^sub>i\<rangle> = \<langle>i : np i : R\<^sub>i\<rangle>"
span_base: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "a \<in> S" shows "a \<in> pmdl S"
valofn_less_2p: fixes x :: "('a, 'b) IEEE.float"   and e :: "nat" assumes "IEEE.exponent x < e" shows "valofn x < 2 ^ e / 2 ^ bias TYPE(('a, 'b) IEEE.float)"
eval_append: fixes A :: "atomUni fmUni"   and B :: "atomUni fmUni list"   and x :: "real" shows "(\<exists>l\<in>set (A # B). evalUni l x) = (evalUni A x \<or> (\<exists>l\<in>set B. evalUni l x))"
bisims1_xcp_Some_not_caught: fixes P :: "'a prog"   and h :: "'heap"   and es :: "'addr expr1 list"   and vs :: "'addr val list"   and a :: "'addr"   and es' :: "'addr expr1 list"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and f :: "String.literal \<Rightarrow> String.literal \<Rightarrow> ty list \<Rightarrow> ty \<Rightarrow> 'a \<Rightarrow> 'b"   and pc' :: "nat"   and d :: "nat" assumes "P,es,h \<turnstile> (map Val vs @ Throw a # es', xs) [\<leftrightarrow>] (stk, loc, pc, \<lfloor>a\<rfloor>)" shows "match_ex_table (compP f P) (cname_of h a) (pc' + pc) (compxEs2 es pc' d) = None"
step_15: shows "InvariantVarsQ (getQ state') F0 Vbl"
grd_def: fixes S :: "'d \<Rightarrow> 'c" shows "grd S = - S \<bottom>"
v2: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "a \<sharp> \<Gamma>" shows "valid ((a, T) # \<Gamma>)"
bdd_below_setge: fixes A :: "'a set" shows "bdd_below A = (\<exists>a. a <=* A)"
simps_1: fixes x :: "Preliminaries.var" shows "vars (Var x) = {x}"
fresh_4: fixes a :: "name"   and x3 :: "name"   and x1 :: "name"   and x2 :: "pi" shows "a \<sharp> x3<x1>.x2 = (a \<sharp> x3 \<and> a \<sharp> [x1].x2)"
iso_tuple_fst_def: fixes isom :: "('a, 'b, 'c) tuple_isomorphism" shows "iso_tuple_fst isom = fst \<circ> repr isom"
ub_Suc: fixes n :: "nat"   and i :: "nat"   and k :: "nat"   and x :: "float" shows "ub (Suc n) i k x = float_plus_up prec (rapprox_rat prec 1 (int k)) (- float_round_down prec (x * lb n (F i) (G i k) x))"
face_distance_eq_chamber_distance_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and D :: "'b set"   and z :: "'b set" assumes "ChamberComplexMorphism X Y f"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D"   and "C \<noteq> D"   and "order.greater_eq C z"   and "ChamberComplex.face_distance Y (f ` z) (f ` D) = ChamberComplex.face_distance X z D"   and "ChamberComplex.face_distance X z D = ChamberComplex.chamber_distance X C D" shows "ChamberComplex.face_distance Y (f ` z) (f ` D) = ChamberComplex.chamber_distance Y (f ` C) (f ` D)"
size_Diff_singleton: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "size (M - {#x#}) = size M - 1"
eval_m: shows "eval (Mn 2 (Cn 3 r_not [r_result])) [i, x] \<down>"
filterlim_iff: fixes f :: "'c \<Rightarrow> 'd"   and F2 :: "'d filter"   and F1 :: "'c filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
latestAct_def: fixes c :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" shows "\<langle>c \<leftarrow> t\<rangle>\<^bsub>n\<^esub> = (GREATEST n'. latestAct_cond c t n n')"
simple_match_to_of_match_generates_prereqs: fixes m :: "32 simple_match"   and r :: "of_match_field set"   and ifs :: "char list list" assumes "simple_match_valid m"   and "r \<in> set (simple_match_to_of_match m ifs)" shows "all_prerequisites r"
power_add: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (m + n) = times (power.power one times a m) (power.power one times a n)"
P_app_t: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a"   and t :: "'b" assumes "P (app u\<^sub>1 u\<^sub>2) t" shows "\<exists>t\<^sub>1 t\<^sub>2. t = app t\<^sub>1 t\<^sub>2 \<and> P u\<^sub>1 t\<^sub>1 \<and> P u\<^sub>2 t\<^sub>2"
JHS_Tr0: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and g :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g" shows "cmp_rfn G r f s g \<in> wcsr_rfns G r f s"
linear_span_image: fixes f :: "'d \<Rightarrow> 'e"   and S :: "'d set" assumes "clinear f" shows "cspan (f ` S) = f ` cspan S"
monom_mult_unfold_2: fixes f :: "'a poly"   and n :: "nat" shows "f * monom (1::'a) n = monom_mult n f"
nu_star1: fixes x :: "'a" assumes "\<And>x y z. x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z" shows "x\<^sup>\<star> \<le> \<nu> x\<^sup>\<star> \<cdot> (1\<^sub>\<sigma> + \<tau> x)"
it_OK: fixes iti :: "('\<sigma> \<Rightarrow> bool) \<Rightarrow> ('xa \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>) \<Rightarrow> '\<sigma> \<Rightarrow> '\<sigma>" assumes "set_iterator_genord iti S0 R" shows "P (iti c fa \<sigma>0)"
\<phi>'_def: shows "\<phi>' = af \<phi> (prefix k w)"
terminal_def: fixes b :: "'a" shows "A.terminal b \<equiv> A.ide b \<and> (\<forall>a. A.ide a \<longrightarrow> (\<exists>!f. \<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>))"
Suc: shows "bound (a * of_nat n ^ deg) \<le> bound a * of_nat n ^ deg"
empty: fixes g :: "'d \<Rightarrow> 'a" shows "sum_0 g {} = (0::'a)"
f_Exec_State_Idle_replicate_NoMsg_output: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and n :: "nat" assumes "State_Idle localState output_fun trans_fun (localState c)"   and "output_fun c = NoMsg" shows "output_fun (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c) = NoMsg"
ccw_def: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" shows "ccw p q r = (ccw' p q r \<or> coll p q r \<and> (psi p q r \<or> psi q r p \<or> psi r p q))"
eval_Var_q: fixes e :: "(name, hf) finfun"   and i :: "name" shows "\<lbrakk>\<guillemotleft>Var i\<guillemotright>\<rbrakk>e = q_Var i"
Worder_sym: fixes S :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme" assumes "Worder S"   and "Worder T"   and "ord_equiv S T" shows "ord_equiv T S"
simps: fixes j :: "nat" shows "local.quadratic_hensel_loop j = (if j \<le> 1 then (p, S1, T1, D1, H1) else if even j then case local.quadratic_hensel_loop (j div 2) of (q, S, T, D, H) \<Rightarrow> let qq = q * q in case local.quadratic_hensel_step q S T D H of (S', T', D', H') \<Rightarrow> (qq, S', T', D', H') else case local.quadratic_hensel_loop (j div 2 + 1) of (q, S, T, D, H) \<Rightarrow> case local.quadratic_hensel_step q S T D H of (S', T', D', H') \<Rightarrow> let qq = q * q; pj = qq div p; down = poly_mod.Mp pj in (pj, down S', down T', down D', down H'))"
perp_foot_cross_ratio_formula: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "a \<noteq> b" shows "cross_ratio (endpoint_in_S a b) (endpoint_in_S b a) (perp_foot c (proj2_line_through a b)) a = (cosh_dist b c * sqrt (exp_2dist a b) - cosh_dist a c) / (cosh_dist a c * exp_2dist a b - cosh_dist b c * sqrt (exp_2dist a b))"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "ide a"   and "ide a'"   and "\<guillemotleft>g : L a' \<rightarrow> L a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow> a\<guillemotright> \<and> L f = g"
induct: fixes P :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) list \<Rightarrow> bool"   and a0 :: "'a filter"   and a1 :: "('a \<Rightarrow> real) list" assumes "\<And>F f g gs. P F (g # gs) \<Longrightarrow> P F (f # g # gs)"   and "\<And>F f. P F [f]"   and "\<And>F. P F []" shows "P a0 a1"
assms: shows "{$$} \<turnstile> v, vP, Lam x vV' : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"
dgrad_p_set_le_spoly: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "dickson_grading d" shows "dgrad_p_set_le d {spoly p q} {p, q}"
tauChainResChainPres: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<Longrightarrow>\<^sup>^\<^sub>\<tau> \<lparr>\<nu>*xvec\<rparr>P'"
less_1: fixes y :: "nat"   and Xs :: "'b set llist"   and Ys :: "'b set llist" assumes "y < i"   and "emb Xs Ys"   and "\<not> lfinite Xs"   and "\<not> lfinite Ys"   and "\<forall>j\<ge>y. x \<in> lnth Ys j" shows "\<forall>j\<ge>y. x \<in> lnth Xs j"
run_shift_elim: fixes A :: "('c, 'd) nba"   and w :: "'c stream"   and V :: "(nat \<times> 'd) set"   and r :: "(nat \<times> 'd) list"   and s :: "(nat \<times> 'd) stream"   and p :: "nat \<times> 'd" assumes "grun A w V (r @- s) p" obtains "gpath A w V r p" and "grun A w V s (gtarget r p)"
dom_combination_dom_union: fixes \<tau> :: "'a \<Rightarrow> 'b option"   and \<tau>' :: "'a \<Rightarrow> 'b option" shows "dom (\<tau>;;\<tau>') = dom \<tau> \<union> dom \<tau>'"
distinct_67: fixes x7 :: "'a OCL_Types.type"   and x8 :: "'a OCL_Types.type" shows "Bag x7 \<noteq> Sequence x8"
\<Pi>_def: shows "\<Pi> = mk_ps_chain \<lbrace> initial_ps2 S G \<rbrace> \<pi>"
neutr_eq_iff: fixes a :: "'a"   and b :: "'a" shows "(0 = a + b) = (a = 0 \<and> b = 0)"
finite_measure_distr: fixes f :: "'c \<Rightarrow> 'a"   and M' :: "'a measure" assumes "f \<in> M \<rightarrow>\<^sub>M M'" shows "finite_measure (distr M M' f)"
INT_absorb: fixes k :: "'b"   and I :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "k \<in> I" shows "A k \<inter> \<Inter> (A ` I) = \<Inter> (A ` I)"
path_split_1: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n (as @ a # as') n'" shows "CFGExit.path n as (src a)"
act_def: fixes I :: "'d \<Rightarrow> bool" shows "act I \<equiv> \<lambda>_ _. I"
exhaust: fixes y :: "'c list" obtains "y = []"   | x21 :: "'c" and x22 :: "'c list" where "y = x21 # x22"
dual_ex: fixes P :: "'c dual \<Rightarrow> bool" shows "(\<exists>x. P (dual x)) = (\<exists>x'. P x')"
set_i_take_subset: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "set (f \<Down> n) \<subseteq> range f"