hcju/isabelleps huggingface dataset:
eq1: shows "Imon (drop (Suc 0) l) M \<otimes> head l [^] (j - i) \<otimes> Ipol l (snd (mfactor P (MX (j - i) M))) \<otimes> head l [^] i = Imon (drop (Suc 0) l) M \<otimes> Ipol l (snd (mfactor P (MX (j - i) M))) \<otimes> (head l [^] (j - i) \<otimes> head l [^] i)"
neq_NilE: fixes l :: "'c list" assumes "l \<noteq> []" obtains x :: "'c" and xs :: "'c list" where "l = x # xs"
domain_very_strict: fixes x :: "'a" shows "(ar ar x = (zero_class.zero::'a)) = (x = (zero_class.zero::'a))"
fpxs_supp_altdef: fixes f :: "'b fpxs" shows "fpxs_supp f = {x. fpxs_nth f x \<noteq> (0::'b)}"
eint_ord_code_4: fixes m :: "int" shows "(eint m < \<infinity>) = True"
bind_pmf_const: fixes M :: "'c pmf"   and c :: "'b pmf" shows "M \<bind> (\<lambda>x. c) = c"
apply_ncrit_subset_aux: fixes ic :: "bool"   and p :: "'e"   and q :: "'f"   and c :: "'g \<Rightarrow> (bool \<times> 'e \<times> 'f) list \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ps :: "(bool \<times> 'g \<times> 'e \<times> 'f) list"   and acc :: "(bool \<times> 'e \<times> 'f) list" assumes "(ic, p, q) \<in> set (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps acc)" shows "(ic, p, q) \<in> set acc \<or> (\<exists>q_in_bs. (ic, q_in_bs, p, q) \<in> set ps)"
counit_simps_2: fixes a :: "'a" assumes "C.obj a" shows "D.arr (counit a)"
HPhase1or2Write_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv5_inner s q"   and "HInv4a s p"   and "p \<noteq> q" shows "HInv5_inner s' q"
merge_def: fixes x :: "'a"   and ss :: "(nat \<times> 'a) list"   and c :: "'a list"   and pc :: "nat" assumes "x \<in> A"   and "snd ` set ss \<subseteq> A" shows "merge c pc ss x = (if \<forall>(pc', s')\<in>set ss. pc' \<noteq> pc + 1 \<longrightarrow> s' \<sqsubseteq>\<^bsub>r\<^esub> c ! pc' then map snd (filter (\<lambda>(p', t'). p' = pc + 1) ss) \<Squnion>\<^bsub>f\<^esub> x else \<top>)"
disc_1: fixes x1 :: "'f" shows "is_Link (Link x1)"
vector_restrict_comp_conv: fixes x :: "'a"   and y :: "'a" assumes "vector x" shows "x \<sqinter> y \<le> x\<^sup>T * y"
weakCongruence_def: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "P \<doteq>\<^sub>c Q \<equiv> \<forall>\<Psi> \<sigma>. wellFormedSubst \<sigma> \<longrightarrow> \<Psi> \<rhd> P[<\<sigma>>] \<doteq> Q[<\<sigma>>]"
composite_simps_4: shows "cod (\<mu> \<star> \<nu>) = cod \<mu> \<star> cod \<nu>"
uminus_dual_def: shows "uminus \<equiv> map_fun undual dual uminus"
unique_decomposition: fixes K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "subfield K R"   and "R.independent K Us"   and "a \<in> R.Span K Us" shows "\<exists>!Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = R.combine Ks Us"
add_vars'_fv_pre: fixes t :: "Core.term"   and acc :: "(variable \<times> typ) list" shows "set (add_vars' t acc) = set acc \<union> fv t"
2: fixes y :: "('t \<Rightarrow>\<^sub>0 'b) set" assumes "y \<sqsupset>p z1" shows "y \<notin> (`) fst ` set ` fst ` Q"
exe_wf_theory_code: fixes \<Theta> :: "exetheory" shows "exe_wf_theory \<Theta> = exe_wf_theory' \<Theta>"
Abs_key_order_cases: fixes x :: "'a key_order" obtains y :: "'a \<Rightarrow> 'a \<Rightarrow> order" where "x = Abs_key_order y" and "y \<in> {compare. comparator compare}"
closure_approachableD: fixes x :: "'a"   and S :: "'a set"   and e :: "real" assumes "x \<in> closure S"   and "0 < e" shows "\<exists>y\<in>S. dist x y < e"
3: shows "h = ZFfun (|dom|h) (|cod|h) ((|@|) h)"
prod_algebraE: fixes A :: "('c \<Rightarrow> 'd) set"   and I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" assumes "A \<in> prod_algebra I M" obtains J :: "'c set" and E :: "'c \<Rightarrow> 'd set" where "A = prod_emb I M J (Pi\<^sub>E J E)" and "finite J" and "J \<noteq> {} \<or> I = {}" and "J \<subseteq> I" and "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)"
distinct_delete_filter: fixes xs :: "['b]"   and a :: "'b" assumes "distinct xs"   and "\<forall>x\<in>set xs. eq\<cdot>a\<cdot>x \<noteq> \<bottom>" shows "delete\<cdot>a\<cdot>xs = filter\<cdot>(\<Lambda> x. neq\<cdot>a\<cdot>x)\<cdot>xs"
Q_def: shows "Q = (\<lambda>bs lt UBI LBI UB LB UBI_upd UI LI LE GE s'. \<forall>I v. (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s bs \<longrightarrow> (\<forall>x c. LB s' x = Some c \<longrightarrow> LI s' x \<in> I \<longrightarrow> lt c (v x) \<or> c = v x) \<and> (\<forall>x c. UB s' x = Some c \<longrightarrow> UI s' x \<in> I \<longrightarrow> lt (v x) c \<or> v x = c))"
"1_2": shows "a \<notin> set v\<^sub>1"
Cauchy_convergent_iff: fixes X :: "nat \<Rightarrow> 'b" shows "Cauchy X = convergent X"
exhaust: fixes y :: "typ" obtains x11 :: "String.literal" and x12 :: "typ list" where "y = Ty x11 x12"   | x21 :: "variable" and x22 :: "String.literal set" where "y = Tv x21 x22"
ffold_empty: fixes fa :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d" assumes "comp_fun_commute fa" shows "ffold f z {||} = z"
deriv_consistency: assumes "infinite UNIV" shows "consistency {set G |G. \<not> G \<turnstile> FF}"
image_mset_filter_swap: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> bool"   and M :: "'d multiset" shows "image_mset f {#x \<in># M. P (f x)#} = filter_mset P (image_mset f M)"
derivation_A: fixes P :: "'a set set" assumes "Q.derivation P" shows "P \<subseteq> Q"
f_not_zero: shows "f \<noteq> \<zero>\<^bsub>P\<^esub>"
ring_subset_carrier: fixes x :: "'g"   and A :: "'g set"   and R :: "('g, 'h) partial_object_scheme" assumes "x \<in> A"   and "A \<subseteq> carrier R" shows "x \<in> carrier R"
coeff_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and m' :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (MPoly_Type.monom m a) m' = (a when m' = m)"
power_strict_increasing: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and n :: "nat"   and N :: "nat"   and a :: "'d" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n < N"   and "less one a" shows "less (power.power one times a n) (power.power one times a N)"
ex_fixed_selector: fixes H :: "nat set \<times> nat set set" assumes "uwellformed H"   and "finite_graph H" obtains f :: "nat set \<Rightarrow> nat set \<times> nat set set" where "is_fixed_selector H f"
bij_def: fixes f :: "'b \<Rightarrow> 'c" shows "bij f = (inj f \<and> surj f)"
\<psi>: shows "\<guillemotleft>\<psi> : F f \<rightarrow>\<^sub>B F f'\<guillemotright> \<and> B.iso \<psi>"
comparator_of_rep: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "comparator_of (rep_nat x) (rep_nat y) = linorder.comparator_of less x y"
intFF_def: shows "intFF \<equiv> InfModel.intFF arOf resOf intTI intFI"
a: shows "obj a \<and> Trg t = \<^bold>\<langle>a\<^bold>\<rangle>\<^sub>0"
f: fixes n :: "nat" assumes "n \<notin> N" shows "f n = (0::'a)"
compatible_modes_def: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>A set) list" shows "abs.compatible_modes mdss = (\<forall>i x. i < length mdss \<longrightarrow> (x \<in> (mdss ! i) AsmNoReadOrWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoReadOrWrite)) \<and> (x \<in> (mdss ! i) AsmNoWrite \<longrightarrow> (\<forall>j<length mdss. j \<noteq> i \<longrightarrow> x \<in> (mdss ! j) GuarNoWrite)))"
has_lock_upd_locks_implies_has_lock: fixes l :: "('d \<times> nat) option"   and t :: "'d"   and Ls :: "lock_action list"   and t' :: "'d" assumes "has_lock (upd_locks l t Ls) t'"   and "t \<noteq> t'" shows "has_lock l t'"
strip_while_snoc: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "strip_while P (xs @ [x]) = (if P x then strip_while P xs else xs @ [x])"
codomains_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C" shows "partial_magma.codomains C f \<equiv> {b. partial_magma.ide C b \<and> C b f \<noteq> partial_magma.null C}"
invertibleE: fixes u :: "'a \<Rightarrow> 'a"   and thesis :: "bool" assumes "invertible u"   and "\<And>v. \<lbrakk>compose S u v = identity S \<and> compose S v u = identity S; v \<in> S \<rightarrow>\<^sub>E S\<rbrakk> \<Longrightarrow> thesis"   and "u \<in> S \<rightarrow>\<^sub>E S" shows "thesis"
assms_5: shows "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
bit_sshiftr1_iff: fixes w :: "'b word"   and n :: "nat" shows "bit (sshiftr1 w) n = bit w (if n = LENGTH('b) - 1 then LENGTH('b) - 1 else Suc n)"
simps_2: shows "to_mini_formula \<bottom> = \<bottom>"
sq_mtx_scaleR_ith: fixes c :: "real"   and A :: "'b sq_mtx"   and i :: "'b" shows "(c *\<^sub>R A) $$ i = c *\<^sub>R A $$ i"
X_plus_closed: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" assumes "UP_cring R"   and "a \<in> carrier R" shows "X_poly_plus R a \<in> carrier (UP R)"
left_lesseq: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.left_complemented_monoid_algebra times inf imp_l less_eq less one" shows "less_eq a b = (imp_l a b = one)"
in_closureI: fixes x :: "'b"   and X :: "'b set" assumes "x \<in> X" shows "x \<in> closure X"
dropWhile_append2: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "dropWhile P (xs @ ys) = dropWhile P ys"
KBasic_9: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<^bold>\<not>\<phi> \<^bold>& \<^bold>\<not>\<psi>) \<^bold>\<rightarrow> \<^bold>\<box>(\<phi> \<^bold>\<equiv> \<psi>) in v]"
of_nat_le_hmset: fixes M :: "nat"   and N :: "nat" shows "(of_nat M \<le> of_nat N) = (M \<le> N)"
supp_def: fixes f :: "'a \<Rightarrow> real" shows "supp f = {x. f x \<noteq> 0}"
undual_inverse: fixes x :: "'d dual" shows "dual (undual x) = x"
inner_sum_left_Basis: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and b :: "'b"   and f :: "'b \<Rightarrow> real" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "b \<in> Basis" shows "inner (comm_monoid_add.sum plus zero (\<lambda>i. scaleR (f i) i) Basis) b = f b"
finite_enat_bounded: fixes A :: "enat set"   and n :: "nat" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n" shows "finite A"
assoc_naturality: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b"   and \<tau> :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "V (\<a> (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>) (partial_magma.cod V \<tau>)) (H (H \<mu> \<nu>) \<tau>) = V (H \<mu> (H \<nu> \<tau>)) (\<a> (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
OUT_h'_outside: fixes x :: "'v" assumes "x \<notin> \<^bold>V" shows "d_OUT h' x = 0"
atom_set_perm_eq: fixes as :: "atom set"   and p :: "perm" assumes "as \<sharp>* p" shows "p \<bullet> as = as"
intensional_simps_14: fixes P :: "'a \<Rightarrow> bool" shows "P \<longrightarrow> #False \<equiv> \<not> P"
rfn_compseries_iMTr3: fixes r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and j :: "nat"   and h :: "nat \<Rightarrow> 'a set" assumes "0 < r"   and "0 < s"   and "compseries G r f"   and "j \<le> r"   and "\<forall>i\<le>r. h (i * s) = f i" shows "h (j * s) = f j"
smcf_0_components_1: fixes \<AA> :: "V" shows "smcf_0 \<AA>\<lparr>ObjMap\<rparr> = []\<^sub>\<circ>"
composition_conseq2ls: fixes \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "L \<cdot>\<^sub>l\<^sub>s \<sigma>\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>\<^sub>2 = L \<cdot>\<^sub>l\<^sub>s (\<sigma>\<^sub>1 \<cdot> \<sigma>\<^sub>2)"
bn_count_le: fixes ns :: "nat list" shows "bn_count ns \<le> foldl (+) 0 ns"
fbox_one: fixes x :: "('a \<times> 'a) set" shows "wp Id x = rdom x"
atLeastAtMost_singleton: fixes a :: "'a" shows "{a..a} = {a}"
HC_intros_6: fixes G :: "'b formula"   and F :: "'b formula" shows "AX10 \<turnstile>\<^sub>H G \<^bold>\<rightarrow> F \<^bold>\<or> G"
P1_def: shows "\<P> \<equiv> {\<lambda>x. eval r [x] |r. recfn 1 r}"
Atm_ZObisT: fixes atm :: "'atom" assumes "compatAtm atm" shows "Atm atm \<approx>01T Atm atm"
assms_2: shows "h \<turnstile> f \<rightarrow>\<^sub>r x"
valid_path_no_deadends: fixes P :: "'a llist"   and i :: "nat" assumes "valid_path P"   and "enat (Suc i) < llength P" shows "\<not> deadend (P $ i)"
Cons_1: shows "oalist_inv_raw ((k', v') # xs)"
lim_4: fixes x :: "'a \<times> 'b"   and i :: "nat" assumes "x \<in> space (M1 \<Otimes>\<^sub>M M2)" shows "norm (s i x) \<le> 2 * norm (f x)"
rel_refl_strong: fixes x :: "('b, 'd, 'e) generat"   and R1a :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and R2a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R3a :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> generat_pures x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> generat_outs x \<Longrightarrow> R2a z2 z2"   and "\<And>z3. z3 \<in> generat_conts x \<Longrightarrow> R3a z3 z3" shows "rel_generat R1a R2a R3a x x"
minim_inField: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim B \<in> Field r"
fin_string_cons_def: shows "fin_string_cons \<equiv> map_fun id (map_fun Rep_fin_string Abs_fin_string) string_cons"
finpref_infsuff: fixes r :: "'b llist"   and A :: "'b set"   and t :: "'b llist" assumes "r \<in> finpref A t"   and "t \<in> A\<^sup>\<omega>" shows "t \<in> infsuff A r"
Beta_altdef: fixes a :: "'b"   and b :: "'b" shows "Beta a b = Gamma a * Gamma b * rGamma (a + b)"
likelihood_def: fixes s :: "'s"   and os :: "'t list" shows "likelihood s os = emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o\<^sub>0 xs \<omega>'. \<omega> = (s, o\<^sub>0) ## xs @- \<omega>' \<and> map snd xs = os}"
continuous_on_def: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
eq0: shows "f (\<Sum>v | r v \<noteq> (0::'a). r v *a v) = (0::'c)"
matrix_vector_mul_1: fixes g :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "linear (*s) (*s) g" shows "(*v) (matrix g) = g"
map_disc_iff: fixes f :: "'d \<Rightarrow> 'b"   and a :: "'d option" shows "(map_option f a = None) = (a = None)"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.dom f1, f2))"
distinct_lists_no_parent: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "local.CD.a_distinct_lists h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes" shows "\<nexists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
directedD: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "directed A r"   and "x \<in> A"   and "y \<in> A" shows "\<exists>z\<in>A. r x z \<and> r y z"
Quotient_total_abs_eq_iff: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and T :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "Quotient R Abs Rep T"   and "reflp R" shows "(Abs x = Abs y) = R x y"
hconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "P \<turnstile> h \<surd> \<equiv> (\<forall>a obj. h a = \<lfloor>obj\<rfloor> \<longrightarrow> P,h \<turnstile> obj \<surd>) \<and> preallocated h"
dirichlet_prod_neutral_sum: fixes n :: "nat" shows "dirichlet_prod_neutral n = (\<Sum>k = 1..n. unity_root n (int k))"
wt_tpOf_subst: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and T :: "'fsym trm" assumes "wtSB \<pi>"   and "wt T" shows "wt (subst \<pi> T) \<and> tpOf (subst \<pi> T) = tpOf T"
index_distinct_group_num_alt_def: 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 "size (filter_mset ((\<subseteq>) ps) (mset_set groups)) = 0" shows "int (block_collection index ps) = index"
axioms_1: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "comm_ring R addition multiplication zero unit" shows "ring R addition multiplication zero unit"
comp_ipresIGVar: fixes h :: "'l \<Rightarrow> 'm"   and MOD :: "('n, 'o, 'p, 'q, 'r, 's, 'l, 't, 'u) model_scheme"   and MOD' :: "('v, 'w, 'p, 'x, 'y, 's, 'm, 'z, 'aa) model_scheme"   and h' :: "'m \<Rightarrow> 'ab"   and MOD'' :: "('ac, 'ad, 'p, 'ae, 'af, 's, 'ab, 'ag, 'ah) model_scheme" assumes "ipresIGVar h MOD MOD'"   and "ipresIGVar h' MOD' MOD''" shows "ipresIGVar (h' \<circ> h) MOD MOD''"
cblinfun_compose_assoc: fixes A :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and C :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L C = A o\<^sub>C\<^sub>L (B o\<^sub>C\<^sub>L C)"
1: shows "rawpsubst (rawpsubst \<phi> (zip (map Var ws) (map snd txs))) (zip (map fst txs) ws) = rawpsubst (rawpsubst \<phi> (zip (map Var vs) (map snd txs))) (zip (map fst txs) vs)"
hyps_2: fixes y :: "'a"   and x :: "'a" shows "lexext_rev gt (y # ys) (x # xs) = (lexext_rev gt ys xs \<or> ys = xs \<and> gt y x)"
reducecoeff: fixes t :: "MIR.num"   and bs :: "real list" shows "real_of_int (numgcd t) * Inum bs (reducecoeff t) = Inum bs t"
real: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat" assumes "real_dcharacter n \<chi>" shows "\<chi> k \<in> \<real>"
L3_5i: fixes X :: "'b set"   and Y :: "'b set"   and c :: "'b com" assumes "X \<subseteq> Y" shows "SEM c X \<subseteq> SEM c Y"
R: shows "topspace X = U \<and> (\<forall>x\<in>U. openin X {x})"
power_diff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat"   and m :: "nat" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero"   and "n \<le> m" shows "power.power one times a (m - n) = divide (power.power one times a m) (power.power one times a n)"
rotate_downpos_kauff_mat: shows "kauff_mat (basic [cup, vert] \<circ> basic [vert, over]) = kauff_mat (basic [vert, cup] \<circ> basic [brick.under, vert])"
Fd: shows "Ftor NTDom (YFtorNT' C f) : Op C \<longrightarrow> SET"
compl: fixes A :: "'a set" assumes "A \<in> {A \<in> sigma_sets \<Omega> G. P A}" shows "\<Omega> - A \<in> {A \<in> sigma_sets \<Omega> G. P A}"
Fyz: shows "finite_entropy (T \<Otimes>\<^sub>M P) (\<lambda>x. (Y x, Z x)) Pyz"
igSubstIGVar1_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIGVar1 MOD \<equiv> \<forall>ys y Y xs x. igWls MOD (asSort ys) Y \<longrightarrow> ys \<noteq> xs \<or> y \<noteq> x \<longrightarrow> igSubst MOD ys Y y (igVar MOD xs x) = igVar MOD xs x"
meet_equality: fixes x :: "'a"   and y :: "'a"   and i :: "'a" assumes "is_inf x y i" shows "x \<sqinter> y = i"
assms_1: shows "A \<subseteq>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"
direct_subsumption: fixes s1 :: "nat"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and f :: "nat \<Rightarrow>f value option \<Rightarrow> bool"   and t1 :: "transition"   and t2 :: "transition" assumes "\<And>t c1 c2. \<lbrakk>obtains s1 c1 e1 0 <> t; obtains s2 c2 e2 0 <> t\<rbrakk> \<Longrightarrow> f c2"   and "\<And>c. f c \<Longrightarrow> subsumes t1 c t2" shows "directly_subsumes e1 e2 s1 s2 t1 t2"
absorb_iff1: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a \<subseteq># b) = (a \<inter># b = a)"
rbd_Id_pres: shows "bd\<^sub>\<R> Id = id"
mat_of_row_dim_1: fixes y :: "'a vec" shows "dim_row (mat_of_row y) = 1"
HMA_I_def: shows "HMA_I = (\<lambda>i a. i = Bij_Nat.to_nat a)"
real_cond_exp_mult: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. f x * g x)" shows "AEeq M (real_cond_exp M F (\<lambda>x. f x * g x)) (\<lambda>x. f x * real_cond_exp M F g x)"
simps_6: fixes f1 :: "('b, 'c) hd \<Rightarrow> 'a"   and f2 :: "('b, 'c) tm \<Rightarrow> ('b, 'c) tm \<Rightarrow> 'a"   and x21 :: "('b, 'c) tm"   and x22 :: "('b, 'c) tm" shows "(case App x21 x22 of Hd x \<Rightarrow> f1 x | App x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
\<sigma>'_def: fixes j :: "nat" shows "\<sigma>' j = (if j = i then the (fst (netgmap sr s') i) else \<sigma> j)"
atom_map_def: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a" assumes "class.bounded_lattice inf less_eq less sup bot top" shows "bounded_lattice.atom_map less_eq less bot x = {y. bounded_lattice.atom less bot y \<and> less_eq y x}"
infinite_arbitrarily_large: fixes A :: "'b set"   and n :: "nat" assumes "infinite A" shows "\<exists>B. finite B \<and> card B = n \<and> B \<subseteq> A"
measure_point_measure_singleton: fixes x :: "'a"   and A :: "'a set"   and X :: "'a \<Rightarrow> ennreal" assumes "x \<in> A" shows "Sigma_Algebra.measure (point_measure A X) {x} = enn2real (X x)"
range_eqI: fixes b :: "'d"   and f :: "'e \<Rightarrow> 'd"   and x :: "'e" assumes "b = f x" shows "b \<in> range f"
hs_get_roots_loop_locs_def: shows "hs_get_roots_loop_locs = prefixed ''hs_get_roots_loop''"
left_neutral: fixes a :: "'g" shows "(0::'g) + a = a"
DIM_prod: shows "DIM('a \<times> 'b) = DIM('a) + DIM('b)"