hcju/isabelleps huggingface dataset:
coprime_lead_coeff_factor_2: fixes p :: "int"   and f :: "int poly"   and g :: "int poly" assumes "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff (f * g)) p" shows "comm_monoid_mult_class.coprime (lead_coeff g) p"
unique_factorization_m_smultD: fixes m :: "int"   and d :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and di :: "int" assumes "poly_mod_2 m"   and "poly_mod.unique_factorization_m m (Polynomial.smult d f) (c, fs)"   and "poly_mod.M m (di * d) = 1" shows "poly_mod.unique_factorization_m m f (c * di, fs)"
fls_deriv_mult: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_deriv (f * g) = f * fls_deriv g + fls_deriv f * g"
project_Constrains_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "F \<squnion> project h (reachable (extend h F \<squnion> G)) G \<in> A Co B" shows "extend h F \<squnion> G \<in> extend_set h A Co extend_set h B"
angle_inverse: fixes A :: "'a"   and C :: "'a"   and B :: "'a"   and D :: "'a" assumes "between (A, C) B"   and "A \<noteq> B"   and "B \<noteq> C" shows "angle A B D = pi - angle C B D"
runit_in_hom: fixes a :: "'c local.term set" assumes "ide a" shows "\<guillemotleft>\<r>[a] : a \<otimes> \<I> \<rightarrow> a\<guillemotright>"
diff_gt_0_iff_gt: fixes a :: "'a"   and b :: "'a" shows "((0::'a) < a - b) = (b < a)"
traces_RUN_sub: fixes A :: "'b set" shows "{s. set s \<subseteq> ev ` A} \<subseteq> T (RUN A)"
simps_3: fixes m :: "nat"   and n :: "nat"   and xs :: "nat list" shows "eval_wellf (recf.Id m n) xs \<down>= xs ! n"
zip_with_altdef': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "zip_with f xs ys = map (\<lambda>i. f (xs ! i) (ys ! i)) [0..<min (length xs) (length ys)]"
fmeasurable_def: fixes M :: "'c measure" shows "fmeasurable M = {A \<in> sets M. emeasure M A < \<infinity>}"
simps_2: fixes uw :: "nat"   and ux :: "nat" shows "inl OpMul [DNum uw, DNum ux] = Some OpMulNum"
bilinear_map_def: fixes f :: "'e \<times> 'f \<Rightarrow> 'g"   and R :: "('h, 'i) Ring_scheme"   and M1 :: "('e, 'h, 'j) Module_scheme"   and M2 :: "('f, 'h, 'j) Module_scheme"   and N :: "('g, 'h, 'j) Module_scheme" shows "bilinear_map f R M1 M2 N = (f \<in> M1 \<times>\<^sub>c M2 \<rightarrow> carrier N \<and> f \<in> extensional (M1 \<times>\<^sub>c M2) \<and> (\<forall>x1\<in>carrier M1. \<forall>x2\<in>carrier M1. \<forall>y\<in>carrier M2. f (x1 \<plusminus>\<^bsub>M1\<^esub> x2, y) = f (x1, y) \<plusminus>\<^bsub>N\<^esub> f (x2, y)) \<and> (\<forall>x\<in>carrier M1. \<forall>y1\<in>carrier M2. \<forall>y2\<in>carrier M2. f (x, y1 \<plusminus>\<^bsub>M2\<^esub> y2) = f (x, y1) \<plusminus>\<^bsub>N\<^esub> f (x, y2)) \<and> (\<forall>x\<in>carrier M1. \<forall>y\<in>carrier M2. \<forall>r\<in>carrier R. f (r \<cdot>\<^sub>s\<^bsub>M1\<^esub> x, y) = r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (x, y) \<and> f (x, r \<cdot>\<^sub>s\<^bsub>M2\<^esub> y) = r \<cdot>\<^sub>s\<^bsub>N\<^esub> f (x, y)))"
det_swap_cols: fixes A :: "'b mat"   and k :: "nat"   and n :: "nat" assumes "A \<in> carrier_mat (k + n) (k + n)" shows "det A = (- (1::'b)) ^ (k * n) * det (mat (k + n) (k + n) (\<lambda>(i, j). A $$ (i, if j < k then j + n else j - k)))"
set_conv_nth: fixes xs :: "'a list" shows "set xs = {xs ! i |i. i < length xs}"
dd0_lt: shows "\<forall>D'\<in>DD0. D' < D"
awalk_def: fixes a :: "'a \<times> nat \<times> 'a"   and u :: "'a"   and p :: "('a \<times> nat \<times> 'a) list"   and v :: "'a" shows "H.awalk a u p v \<equiv> u \<in> verts (add_arc a) \<and> set p \<subseteq> arcs (add_arc a) \<and> H.cas u p v"
h: shows "map_poly complex_of_real q = r * map_poly complex_of_real p"
power2_nat_le_imp_le: fixes m :: "nat"   and n :: "nat" assumes "m\<^sup>2 \<le> n" shows "m \<le> n"
prod_prime_factors': fixes n :: "nat" assumes "0 < n" shows "divisor_count n = (\<Prod>p\<in>prime_factors n. Suc (multiplicity p n))"
degree_add_eq_left: fixes q :: "'b poly"   and p :: "'b poly" assumes "degree q < degree p" shows "degree (p + q) = degree p"
iT_Minus_iSince_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<S> t2 k \<ominus> I. Q t2) = (P (k - t1). t1 \<U> t2 I \<down>\<le> k. Q (k - t2))"
lt_monomial: fixes c :: "'d"   and t :: "'t" assumes "c \<noteq> (0::'d)" shows "lt (monomial c t) = t"
m: shows "ys = map (\<lambda>x. the (f x)) xs \<and> (\<forall>x\<in>set xs. f x \<noteq> None)"
assms_3: shows "component_of_term ` keys s \<subseteq> {0..<length bs}"
BinOp: fixes e\<^sub>1 :: "expr"   and bop :: "bop"   and e\<^sub>2 :: "expr"   and s :: "char list \<Rightarrow> val option" shows "\<lbrakk>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2\<rbrakk>s = (case \<lbrakk>e\<^sub>1\<rbrakk>s of None \<Rightarrow> None | Some v\<^sub>1 \<Rightarrow> case \<lbrakk>e\<^sub>2\<rbrakk>s of None \<Rightarrow> None | Some x \<Rightarrow> binop bop v\<^sub>1 x)"
length_removeKey: fixes w :: "'c"   and ps :: "('c \<times> 'd) list" shows "|removeKey w ps| \<le> |ps|"
head_if: fixes g :: "nat \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat" shows "prod g {m..n} = (if n < m then 1::'c else prod g {m..<n} * g n)"
between_midpoint_1: fixes a :: "'b"   and b :: "'b" shows "between (a, b) (midpoint a b)"
sgn_sgn: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "sgn (sgn a) = sgn a"
succ_tableau_valuated: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"   and "\<nabla> s" shows "\<nabla> s'"
minimal_redex_def: fixes p :: "indices list"   and t :: "'a trm"   and C :: "'a eclause"   and S :: "'a eclause set"   and t' :: "'a trm" shows "minimal_redex p t C S t' = (\<forall>q s. (q, p) \<in> pos_ord C t' \<longrightarrow> subterm t q s \<longrightarrow> trm_rep s S = s)"
bij_betw_inv_into_left: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and A' :: "'c set"   and a :: "'b" assumes "bij_betw f A A'"   and "a \<in> A" shows "inv_into A f (f a) = a"
strip_simp: fixes F :: "'f set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e" shows "strip F \<Gamma> p = map_option (strip_guards F) (\<Gamma> p)"
equiv_trans_comm: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "equiv_trans t u = equiv_trans u t"
p_def: shows "p = nfa_steps A (nfa_startnode A) w"
3: shows "\<forall>i\<in>{0..<k}. i \<le> strict_from_inj k g i"
real_root_minus: fixes n :: "nat"   and x :: "real" shows "root n (- x) = - root n x"
simplex_furthest_le_exists: fixes S :: "'b set"   and a :: "'b" assumes "finite S" shows "\<forall>x\<in>convex hull S. \<exists>y\<in>S. norm (x - a) \<le> norm (y - a)"
simps_1: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list" shows "interpret h TrueF xs = True"
generate_nat_pow: fixes a :: "'a" assumes "ord a \<noteq> 0"   and "a \<in> carrier G" shows "generate G {a} = {a [^] k |k. k \<in> {0..ord a - 1}}"
hcomp_simps_2: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "hseq \<nu> \<mu>" shows "trg (\<nu> \<star> \<mu>) = trg \<nu>"
strip_all_single_form: fixes B :: "Core.term"   and \<tau> :: "typ" assumes "strip_all_single_var B = Some \<tau>" shows "Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> (strip_all_single_body B) = B"
simps_1: fixes uu :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and uv :: "'c set" shows "rmshadow uu [] uv = []"
consistent_in_R2: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" assumes "\<psi> \<in> \<R>\<^sup>2" shows "consistent \<psi> \<in> \<R>\<^sup>2"
awalk_app_isoI: fixes G :: "('e, 'f) pre_digraph"   and u :: "'e"   and p :: "'f list"   and v :: "'e"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.awalk G u p v"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph.awalk (app_iso hom G) (iso_verts hom u) (map (iso_arcs hom) p) (iso_verts hom v)"
induct: fixes P :: "poly \<Rightarrow> bool"   and a0 :: "poly" assumes "\<And>c n p. P p \<Longrightarrow> P (CN c n p)"   and "\<And>n. P (C n)"   and "\<And>v. P (Bound v)"   and "\<And>v va. P (Add v va)"   and "\<And>v va. P (Sub v va)"   and "\<And>v va. P (Mul v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Pw v va)" shows "P a0"
dgrad_p_set_leI_Un: fixes d :: "'a \<Rightarrow> nat"   and F1 :: "('t \<Rightarrow>\<^sub>0 'f) set"   and G :: "('t \<Rightarrow>\<^sub>0 'f) set"   and F2 :: "('t \<Rightarrow>\<^sub>0 'f) set" assumes "dgrad_p_set_le d F1 G"   and "dgrad_p_set_le d F2 G" shows "dgrad_p_set_le d (F1 \<union> F2) G"
diff_eq_0_iff_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a - b = 0) = (a < \<top> \<and> a \<le> b)"
connex_iff_semiconnex_reflexive: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "connex A less_eq = (semiconnex A less_eq \<and> reflexive A less_eq)"
assms_4: shows "\<forall>\<^sub>F x in at x within T. f x \<in> S"
rw: shows "(\<Sum>i\<le>Suc b. monom (g i) i) = monom (g (Suc b)) (Suc b) + (\<Sum>i\<le>b. monom (g i) i)"
Eval_monom: fixes r :: "'a"   and n :: "nat" assumes "r \<in> carrier R" shows "eval R S h s (monom P r n) = h r \<otimes>\<^bsub>S\<^esub> s [^]\<^bsub>S\<^esub> n"
prems_2: shows "cupcake_match_result cenv v0 (pe # pes) err_v = Rval (e, p, env')"
translate_minus: fixes x :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "ccw' (x - a) (x - b) (x - c) = ccw' (- a) (- b) (- c)"
removeAll_less: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "length (removeAll x xs) < length xs"
kronecker_product_def: fixes A :: "'b mat"   and B :: "'b mat" shows "kronecker_product A B = (let ra = dim_row A; ca = dim_col A; rb = dim_row B; cb = dim_col B in mat (ra * rb) (ca * cb) (\<lambda>(i, j). A $$ (i div rb, j div cb) * B $$ (i mod rb, j mod cb)))"
aHom_add: fixes F :: "('d, 'e) aGroup_scheme"   and G :: "('f, 'g) aGroup_scheme"   and f :: "'d \<Rightarrow> 'f"   and a :: "'d"   and b :: "'d" assumes "aGroup F"   and "aGroup G"   and "f \<in> aHom F G"   and "a \<in> carrier F"   and "b \<in> carrier F" shows "f (a \<plusminus>\<^bsub>F\<^esub> b) = f a \<plusminus>\<^bsub>G\<^esub> f b"
a_inv_closed: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "additive_subgroup H G"   and "x \<in> H" shows "\<ominus>\<^bsub>G\<^esub> x \<in> H"
iprev_closed: fixes n :: "nat"   and I :: "nat set" assumes "n \<in> I" shows "iprev n I \<in> I"
caseassm: shows "strip_while ((=) (0::'r)) as \<bullet>\<cdot> ms = as \<bullet>\<cdot> ms"
ext_inject: fixes verts :: "'c set"   and arcs :: "'d set"   and tail :: "'d \<Rightarrow> 'c"   and head :: "'d \<Rightarrow> 'c"   and more :: "'e"   and verts' :: "'c set"   and arcs' :: "'d set"   and tail' :: "'d \<Rightarrow> 'c"   and head' :: "'d \<Rightarrow> 'c"   and more' :: "'e" shows "(\<lparr>pre_digraph.verts = verts, arcs = arcs, tail = tail, head = head, \<dots> = more\<rparr> = \<lparr>pre_digraph.verts = verts', arcs = arcs', tail = tail', head = head', \<dots> = more'\<rparr>) = (verts = verts' \<and> arcs = arcs' \<and> tail = tail' \<and> head = head' \<and> more = more')"
p_residue_def: fixes p :: "int"   and m :: "nat"   and n :: "int" assumes "padic_integers p" shows "padic_integers.p_residue p m n \<equiv> residue (p ^ m) n"
intros_21: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and n :: "nat"   and c :: "i32"   and mem' :: "mem"   and vs :: "v list" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "mem_size m = n"   and "mem_grow m (nat_of_int c) = mem'" shows "\<lparr>s;vs;[$C ConstInt32 c, $Grow_memory]\<rparr> \<leadsto>_ i \<lparr>s\<lparr>s.mem := (s.mem s)[j := mem']\<rparr>;vs;[$C ConstInt32 (int_of_nat n)]\<rparr>"
AllI: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "\<And>x. [\<phi> x in v]" shows "[\<^bold>\<forall>x. \<phi> x in v]"
order_prod: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly"   and c :: "'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> coprime (f x) (f y)" shows "order c (prod f A) = (\<Sum>x\<in>A. order c (f x))"
project_coord_def: fixes x :: "'a"   and b :: "'a"   and n :: "real" shows "project_coord x b n = (\<Sum>i\<leftarrow>Basis_list. (if i = b then n else if i = - b then - n else x \<bullet> i) *\<^sub>R i)"
valid_cfg_state_in_S: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "state cfg \<in> S"
simps: fixes a1 :: "('e \<times> 'f) list"   and a2 :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g" shows "((a1, a2) \<in> c_failures step out s\<^sub>0) = (a1 = [] \<and> a2 = {(x, p). p \<noteq> out s\<^sub>0 x} \<or> (\<exists>xps uu_ s x. a1 = xps @ [(x, out s x)] \<and> a2 = {(y, p). p \<noteq> out (step s x) y} \<and> (xps, uu_) \<in> c_failures step out s\<^sub>0 \<and> s = foldl step s\<^sub>0 (map fst xps)) \<or> (\<exists>xps Y X. a1 = xps \<and> a2 = X \<and> (xps, Y) \<in> c_failures step out s\<^sub>0 \<and> X \<subseteq> Y))"
ca: fixes e :: "db" assumes "evalDdb e\<cdot>env_empty_db \<noteq> \<bottom>"   and "closed e" shows "\<exists>V. e \<Down> V"
i: shows "ipath (mkTree (stl (trim rs s)) s') (stl steps)"
split: fixes P :: "'j \<Rightarrow> bool"   and f :: "'k \<Rightarrow> 'l \<Rightarrow> 'j"   and prod :: "'k \<times> 'l" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<forall>x1 x2. prod = (x1, x2) \<longrightarrow> P (f x1 x2))"
sum_representation_eq: fixes basis :: "'b set"   and v :: "'b"   and B :: "'b set" assumes "independent basis"   and "v \<in> span basis"   and "finite B"   and "basis \<subseteq> B" shows "(\<Sum>b\<in>B. representation basis v b *\<^sub>R b) = v"
sorted_Max_Cons: fixes l :: "'a list"   and a :: "'a" assumes "l \<noteq> []"   and "sorted (a # l)" shows "Max (set (a # l)) = Max (set l)"
VectorSpace_fSubspace: fixes W :: "'v set" assumes "RModule UNIV (\<sharp>\<cdot>) W \<and> W \<subseteq> V" shows "VectorSpace (\<sharp>\<cdot>) W"
list_all2_same: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" shows "list_all2 P xs xs = (\<forall>x\<in>set xs. P x x)"
simps: fixes P :: "'a prog"   and a1 :: "ty"   and a2 :: "ty" shows "P \<turnstile> a1 \<le> a2 = ((\<exists>T. a1 = T \<and> a2 = T) \<or> (\<exists>C D. a1 = Class C \<and> a2 = Class D \<and> P \<turnstile> C \<preceq>\<^sup>* D) \<or> (\<exists>C. a1 = NT \<and> a2 = Class C) \<or> (\<exists>A. a1 = NT \<and> a2 = A\<lfloor>\<rceil>) \<or> (\<exists>A. a1 = A\<lfloor>\<rceil> \<and> a2 = Class Object) \<or> (\<exists>A B. a1 = A\<lfloor>\<rceil> \<and> a2 = B\<lfloor>\<rceil> \<and> P \<turnstile> A \<le> B))"
non_overlapping_extension3: fixes P :: "'b set set"   and elem :: "'b"   and Q :: "'b set set" assumes "is_non_overlapping P"   and "elem \<notin> \<Union> P"   and "Q \<in> coarser_partitions_with elem P" shows "is_non_overlapping Q"
pointsTo: shows "\<forall>l l'. pointsTo l \<noteq> Link l'"
closedin_closed_eq: fixes S :: "'a set"   and T :: "'a set" assumes "closed S" shows "closedin (top_of_set S) T = (closed T \<and> T \<subseteq> S)"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "cod (g \<cdot> f) = cod g"
is_weak_ranking_singleton: fixes x :: "'b set" shows "is_weak_ranking [x] = (x \<noteq> {})"
covector_bot_closed: fixes top :: "'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b" assumes "class.bounded_idempotent_semiring top one times sup less_eq less bot" shows "times top bot = bot"
correctCompositionDiffLevelsA31: shows "correctCompositionDiffLevels sA31"
align_lem_and: fixes xs :: "bool list"   and n :: "nat"   and m :: "nat"   and ys :: "bool list" assumes "length xs = n + m"   and "length ys = n + m"   and "drop m xs = replicate n False"   and "take m ys = replicate m False" shows "map2 (\<and>) xs ys = replicate (n + m) False"
Nonce_secrecy_lemma: fixes P :: "bool"   and X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "P \<longrightarrow> X \<in> analz (G \<union> H) \<longrightarrow> X \<in> analz H" shows "P \<longrightarrow> (X \<in> analz (G \<union> H)) = (X \<in> analz H)"
steps_cong: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> nat"   and g :: "'a \<Rightarrow> nat"   and l :: "nat"   and l' :: "nat"   and u :: "nat"   and v :: "nat" assumes "A = B"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "l = l'" shows "steps A f l u v = steps B g l' u v"
not_lindepD: fixes S :: "'a vec set"   and A :: "'a vec set"   and f :: "'a vec \<Rightarrow> 'a" assumes "M.lin_indpt S"   and "finite A"   and "A \<subseteq> S"   and "f \<in> A \<rightarrow> carrier class_ring"   and "M.lincomb f A = \<zero>\<^bsub>M m\<^esub>" shows "f \<in> A \<rightarrow> {\<zero>\<^bsub>class_ring\<^esub>}"
sum_closed: fixes A :: "'q set"   and X :: "'r set"   and f :: "'r \<Rightarrow> 'q" assumes "ring_closed A"   and "\<And>x. x \<in> X \<Longrightarrow> f x \<in> A" shows "sum f X \<in> A"
execn_strip: shows "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
reduce_basis_hybrid_def: shows "reduce_basis_hybrid = (if enable_external_lll_solver then reduce_basis_external else reduce_basis)"
insert_neq_0: fixes a :: "V"   and X :: "V set" shows "(set (insert a X) = 0) = (\<not> small X)"
assms_3: shows "pmdl (set gs) = pmdl (set (init_syzygy_list bs))"
simps_1: fixes uu :: "char list \<Rightarrow> int option" shows "Fold.fold SKIP uu = SKIP"
distinct_3: fixes x3 :: "num" shows "num.One \<noteq> num.Bit1 x3"
diff1: shows "k-smooth_on (apply_chart c1 ` (U \<inter> W)) (apply_chart c3 \<circ> inv_chart c1)"
simps_2: fixes f :: "nat \<Rightarrow> 'b"   and x :: "char list"   and e :: "aexp" shows "annotate f (x ::= e) = x ::= e {f 0}"
less_le_trans: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
simps_5: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> and\<^sub>n \<psi> = (\<xi> \<Turnstile>\<^sub>n \<phi> \<and> \<xi> \<Turnstile>\<^sub>n \<psi>)"
arc_in_sup_arc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "arc x"   and "arc y"   and "x \<le> z \<squnion> y" shows "x \<le> z \<or> x \<le> y"
ps2_P_vars1': shows "ps2_P.vars1' = vars1"
norm_octo_squared: fixes x :: "octo" shows "(norm x)\<^sup>2 = (Ree x)\<^sup>2 + (Im1 x)\<^sup>2 + (Im2 x)\<^sup>2 + (Im3 x)\<^sup>2 + (Im4 x)\<^sup>2 + (Im5 x)\<^sup>2 + (Im6 x)\<^sup>2 + (Im7 x)\<^sup>2"
ASSERT_def: shows "ASSERT \<equiv> iASSERT RETURN"
assms_6: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
cod_comp: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "seq g f" shows "cod (COMP g f) = cod g"
hyps_1: shows "((es, s), es'', s'') \<in> Reds P E"
pre_mangoldt_def: fixes d :: "nat" shows "pre_mangoldt d = (if primepow d then aprimedivisor d else 1)"
Ag_ind_carrier: fixes A :: "('c, 'd) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and D :: "'e set" assumes "aGroup A"   and "bij_to f (carrier A) D" shows "carrier (Ag_ind A f) = f ` carrier A"
l4_13: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.Cong3 Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Col Bet A' B' C'"
list_update_code_2: fixes x :: "'k"   and xs :: "'k list"   and y :: "'k" shows "(x # xs)[0 := y] = y # xs"
subst_cls_poss: fixes AA :: "'a multiset"   and \<sigma> :: "'s" shows "poss AA \<cdot> \<sigma> = poss (AA \<cdot>am \<sigma>)"
float_distinct_finite_2: fixes a :: "('a, 'b) IEEE.float" shows "\<not> (is_infinity a \<and> is_finite a)"
augmenting_path_imp_shortest: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "\<exists>p. cf.isShortestPath s p t"
nat_pow_Suc: fixes x :: "'a"   and n :: "nat" shows "x [^] Suc n = x [^] n \<otimes> x"
fst_gb_schema_dummy_dgrad_p_set_le_init: fixes d :: "'a \<Rightarrow> nat"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "dickson_grading d"   and "struct_spec sel ap ab compl" shows "dgrad_p_set_le d (fst ` set (fst (gb_schema_dummy data D (ab gs [] bs (snd data)) (ap gs [] [] bs (snd data))))) (fst ` (set gs \<union> set bs))"
assms_2: assumes "x = None" shows "fn \<le> \<Down> R fn'"
positively_invariant_iff: fixes M :: "'a set" shows "positively_invariant M = ((\<Union>x\<in>M. flow0 x ` (existence_ivl0 x \<inter> {0..})) \<subseteq> M)"
in_components_subset: fixes c :: "'b set"   and s :: "'b set" assumes "c \<in> components s" shows "c \<subseteq> s"
bunch_of_lemmata_about_matches_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m1 :: "'c match_expr"   and m2 :: "'c match_expr"   and a :: "action"   and p :: "'d" shows "matches \<gamma> (MatchAnd m1 m2) a p = (matches \<gamma> m1 a p \<and> matches \<gamma> m2 a p)"
closedin_Euclidean_imp_closed: fixes m :: "nat"   and S :: "(nat \<Rightarrow> real) set" assumes "closedin (Euclidean_space m) S" shows "closed S"
fps_unit_factor_mult: fixes f :: "'b fps"   and g :: "'b fps" shows "unit_factor (f * g) = unit_factor f * unit_factor g"
le_any_nat_inv: fixes v :: "val"   and n :: "nat" assumes "v \<sqsubseteq> VNat n" obtains "v = VNat n"
iall_mp: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" assumes "\<box> t I. P t \<longrightarrow> Q t"   and "\<box> t I. P t" shows "\<box> t I. Q t"
char_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier (G\<lparr>carrier := H\<rparr>)"   and "b \<in> carrier (G\<lparr>carrier := H\<rparr>)" shows "\<chi> (a \<otimes>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub> b) = \<chi> a * \<chi> b"
prefixes_length_mono: fixes j :: "nat"   and t :: "nat" shows "e_length (local.prefixes j t) < e_length (local.prefixes j (Suc t))"
"1_2": fixes x2 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0"   and "find_adds fs (lt p) = Some x2" shows "(red (set fs))\<^sup>*\<^sup>* (tail p - monom_mult (lc p / lc x2) (lp p - lp x2) (tail x2)) (trd_aux fs (tail p - monom_mult (lc p / lc x2) (lp p - lp x2) (tail x2)) r - r)"
gco_approx_finite: fixes prog :: "'a program_scheme" shows "finite (gco_approx prog)"
spr_simInit: fixes iobs :: "'cobs \<times> 'as option"   and a :: "'a" assumes "iobs \<in> envObs a ` set envInit" shows "spr_simAbs (local.spr_simInit a iobs) = spr_sim ` rel_ext (\<lambda>t'. t' \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jviewInit a iobs)"
minus_closed': shows "\<forall>x\<in>U. \<forall>y\<in>U. x -\<^sub>o\<^sub>w y \<in> U"
linear_split_alt: fixes xs :: "('a \<times> 'b) list"   and x :: "'b" shows "linear_split xs x = (takeWhile (\<lambda>(uu_, s). s < x) xs, dropWhile (\<lambda>(uu_, s). s < x) xs)"