hcju/isabelleps huggingface dataset:
subresultant_index_mat: fixes i :: "nat"   and F :: "'b poly"   and J :: "nat"   and G :: "'b poly"   and j :: "nat" assumes "i < degree F - J + (degree G - J)"   and "j < degree F - J + (degree G - J)" shows "subresultant_mat J F G $$ (i, j) = (if j < degree G - J then if i = degree F - J + (degree G - J) - 1 then monom (1::'b) (degree G - J - 1 - j) * F else [:coeff_int F (int (degree F) - int i + int j):] else let jj = j - (degree G - J) in if i = degree F - J + (degree G - J) - 1 then monom (1::'b) (degree F - J - 1 - jj) * G else [:coeff_int G (int (degree G) - int i + int jj):])"
monoid_add_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_add_hom hom \<equiv> zero_hom hom \<and> semigroup_add_hom hom"
that: fixes p :: "real \<Rightarrow> 'b" assumes "polynomial_function p"   and "pathstart p = pathstart g"   and "pathfinish p = pathfinish g"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> norm (p t - g t) < e" shows "thesis"
f_def: shows "f \<equiv> \<lambda>x. (d - c) / (b - a) * x + (c - a * ((d - c) / (b - a)))"
Dag_unique_ex_conj_simp: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and P :: "dag \<Rightarrow> bool" assumes "Dag p l r t" shows "(\<exists>t. Dag p l r t \<and> P t) = P t"
swap: shows "F2 = F1[i := F1 ! (i - 1), i - 1 := F1 ! i]"
nfoldli_refine: fixes li :: "'c list"   and l :: "'d list"   and S :: "('c \<times> 'd) set"   and si :: "'e"   and s :: "'f"   and R :: "('e \<times> 'f) set"   and ci :: "'e \<Rightarrow> bool"   and c :: "'f \<Rightarrow> bool"   and fi :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "(li, l) \<in> \<langle>S\<rangle>list_rel"   and "(si, s) \<in> R"   and "(ci, c) \<in> R \<rightarrow> bool_rel"   and "\<And>xi x si s. \<lbrakk>(xi, x) \<in> S; (si, s) \<in> R; c s\<rbrakk> \<Longrightarrow> fi xi si \<le> \<Down> R (f x s)" shows "nfoldli li ci fi si \<le> \<Down> R (nfoldli l c f s)"
module_hom_def: fixes R :: "('j, 'k) ring_scheme"   and M :: "('j, 'h, 'l) module_scheme"   and N :: "('j, 'i, 'm) module_scheme" shows "module_hom R M N = {f \<in> carrier M \<rightarrow> carrier N. (\<forall>m1 m2. m1 \<in> carrier M \<and> m2 \<in> carrier M \<longrightarrow> f (m1 \<oplus>\<^bsub>M\<^esub> m2) = f m1 \<oplus>\<^bsub>N\<^esub> f m2) \<and> (\<forall>r m. r \<in> carrier R \<and> m \<in> carrier M \<longrightarrow> f (r \<odot>\<^bsub>M\<^esub> m) = r \<odot>\<^bsub>N\<^esub> f m)}"
pareto_efficient: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "scf R \<inter> pareto_losers R = {}"
gauss_i_times_gauss_i_times: fixes x :: "gauss_int" shows "\<i>\<^sub>\<int> * (\<i>\<^sub>\<int> * x) = - x"
exhaust: fixes y :: "variable" obtains x1 :: "char" where "y = RVar x1"   | x2 :: "char" where "y = DVar x2"
mask_eq_take_bit_minus_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "mask n = take_bit n (uminus one)"
sturm_squarefree'_adjacent_roots: fixes p :: "real poly"   and i :: "nat"   and x :: "real" assumes "p \<noteq> 0"   and "i < length (sturm_squarefree' p) - 1"   and "poly (sturm_squarefree' p ! i) x = 0"   and "poly (sturm_squarefree' p ! (i + 1)) x = 0" shows "False"
linear_plus_norm: fixes f :: "'c \<Rightarrow> 'd"   and \<xi> :: "'c"   and x :: "'c" assumes "linear f" shows "norm (f \<xi>) \<le> max (norm (f (x + \<xi>))) (norm (f (x - \<xi>)))"
simps_5: fixes f1 :: "int \<Rightarrow> 'b"   and f2 :: "'b" shows "(case Any of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = f2"
Target_StepConf: fixes S :: "'d"   and TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "S \<in> Target TS" shows "S \<in> StepConf (HA ST) (Conf ST) TS"
qmdecl_simp: fixes sig :: "sig"   and C :: "qtname"   and m :: "methd" shows "qmdecl sig (C, m) = (C, sig, m)"
vec_first_carrier: fixes v :: "'b vec"   and n :: "nat" shows "vec_first v n \<in> carrier_vec n"
fst_Gauss_Jordan_in_ij_PA: fixes P :: "(('a, 'c) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" assumes "P ** B = A" shows "fst (Gauss_Jordan_in_ij_PA (P, A) i j) ** B = snd (Gauss_Jordan_in_ij_PA (P, A) i j)"
sup_inf_absorb: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "sup x (inf x y) = x"
uint16_div_def: fixes x :: "uint16"   and y :: "uint16" shows "uint16_div x y = (if y = 0 then undefined (div) x 0 else x div y)"
intro: fixes history :: "nat \<Rightarrow> 'c list" assumes "\<And>i. distinct (history i)" shows "node_histories history"
cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) shadow_root_ptr"   and y :: "(_) shadow_root_ptr" shows "(cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y) = (x = y)"
tran: shows "universal (\<lambda>x y. \<lfloor>(\<lambda>z. x r y \<and> y r z \<longrightarrow> x r z)\<rfloor>)"
degree_eq_degreen0: fixes p :: "poly" shows "Reflected_Multivariate_Polynomial.degree p = degreen p 0"
set_insert_list: fixes x :: "'b"   and xs :: "'b list" shows "set (insert_list x xs) = insert x (set xs)"
is_prefix_def: fixes u :: "'a list"   and v :: "'a list" shows "is_prefix u v = (\<exists>w. u @ w = v)"
induct: fixes P1 :: "('b, 'c, 'a) hp \<Rightarrow> bool"   and P2 :: "('b, 'c, 'a) formula \<Rightarrow> bool"   and hp :: "('b, 'c, 'a) hp" assumes "\<And>x. P1 ($\<alpha> x)"   and "\<And>x1 x2. P1 (x1 := x2)"   and "\<And>x1 x2. P1 (DiffAssign x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P1 (? x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (EvolveODE x1 x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 \<union>\<union> x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1 ;; x2)"   and "\<And>x. P1 x \<Longrightarrow> P1 x**"   and "\<And>x1 x2. P2 (Geq x1 x2)"   and "\<And>x1 x2. P2 ($\<phi> x1 x2)"   and "\<And>x. P2 x \<Longrightarrow> P2 (! x)"   and "\<And>x1 x2. \<lbrakk>P2 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 && x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Exists x1 x2)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (\<langle> x1 \<rangle> x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (InContext x1 x2)" shows "P1 hp"
lincomb_sum: fixes rs :: "'r list"   and ss :: "'r list"   and ms :: "'m list" assumes "set rs \<subseteq> R"   and "set ss \<subseteq> R"   and "set ms \<subseteq> M"   and "length rs \<le> length ss" shows "rs \<bullet>\<cdot> ms + ss \<bullet>\<cdot> ms = (map2 (+) rs ss @ drop (length rs) ss) \<bullet>\<cdot> ms"
integrable_negligible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "negligible S" shows "f integrable_on S"
set_zip_conv: fixes A :: "'e list"   and B :: "'d list" shows "(set (zip A B))\<inverse> = set (zip B A)"
minus_carrier_mat: fixes B :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'c mat" assumes "B \<in> carrier_mat nr nc" shows "A - B \<in> carrier_mat nr nc"
simps_1: fixes v :: "v"   and es :: "e list" shows "split_vals_e (($C v) # es) = (let (vs', y) = split_vals_e es in (v # vs', y))"
inject: fixes x1 :: "real"   and x2 :: "real"   and y1 :: "real"   and y2 :: "real" shows "(Complex x1 x2 = Complex y1 y2) = (x1 = y1 \<and> x2 = y2)"
Comp: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and c1 :: "stmt"   and Q :: "'b assn"   and c2 :: "stmt"   and R :: "'b assn" assumes "G,A\<turnstile>{Normal P} .c1. {Q}"   and "G,A\<turnstile>{Q} .c2. {R}" shows "G,A\<turnstile>{Normal P} .c1;; c2. {R}"
simps_6: fixes x :: "'sz"   and y :: "'sz"   and \<alpha> :: "('sf, 'sc, 'sz) hp"   and \<phi> :: "('sf, 'sc, 'sz) formula" shows "FUrename x y (\<langle> \<alpha> \<rangle> \<phi>) = (\<langle> PUrename x y \<alpha> \<rangle> FUrename x y \<phi>)"
CallRedsParams: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and v :: "val"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>]* \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>Val v\<bullet>M(es),s,b\<rangle> \<rightarrow>* \<langle>Val v\<bullet>M(es'),s',b'\<rangle>"
not_INFM_eq_1: fixes a :: "'b" shows "\<not> (\<exists>\<^sub>\<infinity>x. x = a)"
pdivides_iff: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subfield K R"   and "polynomial K p"   and "polynomial K q" shows "p pdivides q = p divides\<^bsub>K [X]\<^esub> q"
intro: fixes f :: "'b \<Rightarrow> 'a" assumes "surj f" shows "surjective f"
diamond_boundaries_reparam_map_def: shows "diamond_boundaries_reparam_map \<equiv> id"
Schroeder_Bernstein: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "inj_on g B"   and "g ` B \<subseteq> A" shows "\<exists>h. bij_betw h A B"
assms_2: shows "[[r\<rightarrow>0]u]\<^sub>\<R> = [[r'\<rightarrow>0]u]\<^sub>\<R>"
le_SPEC_bindI: fixes \<Phi> :: "'b \<Rightarrow> bool"   and x :: "'b"   and m :: "'c nres"   and f :: "'b \<Rightarrow> 'c nres" assumes "\<Phi> x"   and "m \<le> f x" shows "m \<le> SPEC \<Phi> \<bind> f"
assms: shows "\<langle>c #\<^bsub>n\<^esub>t\<rangle> < \<langle>c #\<^bsub>n'\<^esub>t\<rangle>"
simps_3: fixes ParamDefs :: "'h \<Rightarrow> 'g list" shows "lift_ParamDefs ParamDefs NewExit = []"
inj_ring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "inj_ring_hom hom \<equiv> ring_hom hom \<and> zero_hom_0 hom"
SucSuc: fixes m :: "nat" assumes "n = Suc (Suc m)" shows "P"
in_new_spairsD_3: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "Inl (a, b) \<in> set (new_spairs bs p)" shows "is_regular_spair p b"
t1_space_product_topology: fixes X :: "'a \<Rightarrow> 'b topology"   and I :: "'a set" shows "t1_space (product_topology X I) = (topspace (product_topology X I) = {} \<or> (\<forall>i\<in>I. t1_space (X i)))"
env_restr_def: fixes S :: "'c set"   and m :: "'c \<Rightarrow> 'd" shows "m f|` S = (\<lambda>x. if x \<in> S then m x else \<bottom>)"
rec: shows "dmu_array_row_main fs fi i dmus' (Suc j) = res"
pran_upd: fixes f :: "('b, 'a) pfun"   and k :: "'b"   and v :: "'a" shows "pran (f(k \<mapsto> v)\<^sub>p) = insert v (pran ((- {k}) \<lhd>\<^sub>p f))"
simps_2: fixes a :: "'b"   and h :: "'b"   and t :: "'b list" shows "firstPos a (h # t) = (if a = h then 0 else 1 + firstPos a t)"
simps_3: fixes c :: "'b"   and v :: "'b" shows "unify (Const c) (Var v) = Some [(v, Const c)]"
gt'_def: fixes y :: "'a"   and x :: "'a" shows "gt' y x = (y \<noteq> x \<and> (x, y) \<in> Ge')"
continuous_map_in_subtopology: fixes X :: "'a topology"   and X' :: "'b topology"   and S :: "'b set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_map X (subtopology X' S) f = (continuous_map X X' f \<and> f ` topspace X \<subseteq> S)"
pne14: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "pid1 \<noteq> pid4"
case_eq_if: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'e"   and sum :: "'f + 'g" shows "(case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (if isl sum then f1 (projl sum) else f2 (projr sum))"
f1: shows "rk {a, c, A, C, A', C', \<beta>} + rk {a, c, \<beta>} \<le> rk {a, c, A, C, \<beta>} + rk {a, c, A', C', \<beta>}"
intros_2: fixes xs :: "'a list"   and ys :: "'a list"   and r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "(xs, ys) \<in> genPrefix r"   and "(x, y) \<in> r" shows "(x # xs, y # ys) \<in> genPrefix r"
while_unfold: fixes b :: "'a upred"   and P :: "'a hrel" shows "while\<^sup>\<top> b do P od = P ;; while\<^sup>\<top> b do P od \<triangleleft> b \<triangleright>\<^sub>r II"
TypeOperationCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a expr"   and k :: "call_kind"   and op :: "typeop"   and \<sigma> :: "'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E TypeOperationCall a k op \<sigma> : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" where "\<Gamma> \<turnstile>\<^sub>E a : \<tau>'" and "typeop_type k op \<tau>' \<sigma> \<tau>"
d: shows "AE x in MA. 0 < real_cond_exp MA A.Invariants (\<lambda>x. real (phiA x)) x"
borel_measurable_diff: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x - g x) \<in> borel_measurable M"
finished_def: fixes rs :: "(name \<times> (Term_Class.term list \<times> pterm) fset) fset" shows "finished rs = fBall rs (\<lambda>(uu_, irs). arity irs = 0)"
write_sops_le: shows "\<forall>i\<in>\<Union> (fst ` write_sops sb). i < t"
i_set_cont_IN: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<in> i_set_cont_min"
assms_3: shows "normalize (\<Prod>i\<in>#M. i) = normalize (\<Prod>i\<in>#N. i)"
reduce_system_matrix_subsets_helper: fixes p :: "real poly"   and signs :: "rat list list"   and subsets :: "nat list list"   and S :: "nat list" assumes "p \<noteq> 0"   and "length signs \<le> length subsets"   and "\<forall>x. List.member S x \<longrightarrow> x < length subsets" shows "M_mat signs (take_indices subsets S) = take_rows_from_matrix (M_mat signs subsets) S"
that: fixes a :: "real" assumes "x = a \<cdot> v" shows "thesis"
radical_mult_coprime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "algebraic_semidom.coprime times one a b" shows "factorial_semiring.radical zero times one normalize (times a b) = times (factorial_semiring.radical zero times one normalize a) (factorial_semiring.radical zero times one normalize b)"
rel_transp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "transp R" shows "transp (rel_bset R)"
find_index_eq_iff: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and i :: "nat" shows "(find_index P xs = i) = (i \<le> length xs \<and> (\<forall>j<i. \<not> P (xs ! j)) \<and> (i < length xs \<longrightarrow> P (xs ! i)))"
H: shows "\<Gamma> \<turnstile> (\<lambda>:S. s) : T"
intro: fixes \<G> :: "'b ltl set"   and w :: "nat \<Rightarrow> 'b set"   and \<Sigma> :: "'b set set" assumes "Only_G \<G>"   and "range w \<subseteq> \<Sigma>"   and "finite \<Sigma>" shows "ltl_FG_to_rabin \<Sigma> \<G> w"
STAR_cos_Infinitesimal: fixes x :: "'a star" assumes "x \<in> Infinitesimal" shows "(*f* cos) x \<approx> 1"
map: shows "(y + x * x') mod q = (y + x * y') mod q"
bl_and_mask': fixes w :: "'b word"   and n :: "nat" shows "to_bl (w AND mask n) = replicate (LENGTH('b) - n) False @ drop (LENGTH('b) - n) (to_bl w)"
fmdrop_fset_fmdom: fixes A :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmdrop_fset (fmdom A) A = fmempty"
ideal_generated_mult_unit2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and u :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_ring_1 times one plus zero minus uminus"   and "dvd.dvd times u one" shows "ring.ideal_generated plus zero uminus times {times u a, times u b} = ring.ideal_generated plus zero uminus times {a, b}"
same_diag_def: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat" shows "??.Jordan_Normal_Form_Existence.same_diag n A B \<equiv> \<forall>i<n. A $$ (i, i) = B $$ (i, i)"
sq_mtx_diag_vec_mult: fixes f :: "'b \<Rightarrow> real"   and s :: "(real, 'b) vec" shows "(\<d>\<i>\<a>\<g> i. f i) *\<^sub>V s = (\<chi>i. f i * s $ i)"
eepi: fixes i :: "real" assumes "i \<in> k" shows "0 < ee (p i)"
cmp_map_def: fixes cmpk :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and cmpv :: "'b \<Rightarrow> 'b \<Rightarrow> comp_res" shows "cmp_map cmpk cmpv \<equiv> cmp_extend (Collect (finite \<circ> dom)) (cmp_img (linorder.sorted_list_of_map (comp2le cmpk)) (cmp_lex (cmp_prod cmpk cmpv)))"
d2: assumes "z_R I sign = - 1" shows "(\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> poly (prod_list (retrieve_polys qs (snd I))) x < 0"
intros_3: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> analz H" shows "Y \<in> analz H"
FSbImorph_igWlsBinp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'i, 'j, 'k) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'j option"   and h :: "'i \<Rightarrow> 'l"   and hA :: "'j \<Rightarrow> 'm"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'l, 'm, 'n) model_scheme" assumes "igWlsBinp MOD delta binp"   and "FSbImorph h hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
inf_Sec_def: fixes d1 :: "Sec"   and d2 :: "Sec" shows "inf d1 d2 = (if d1 = Low \<or> d2 = Low then Low else High)"
oalist_inv_update_by_raw: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and kv :: "'a \<times> 'c" assumes "oalist_inv xs" shows "oalist_inv (update_by_raw kv xs)"
wf_ps_triple_edgeI: fixes e :: "node fset \<times> command_gadget \<times> node fset"   and \<sigma>1 :: "node \<rightharpoonup>\<^sub>f topbot"   and \<sigma>2 :: "node \<rightharpoonup>\<^sub>f topbot" assumes "\<exists>\<sigma>. fst3 e |\<union>| thd3 e |\<inter>| fmdom \<sigma> = {||} \<and> \<sigma>1 = [ fst3 e |=> Bot ] ++\<^sub>f \<sigma> \<and> \<sigma>2 = [ thd3 e |=> Top ] ++\<^sub>f \<sigma>" shows "wf_ps_triple (\<sigma>1, Inr e, \<sigma>2)"
assms: shows "\<forall>k<e_length e. \<phi> i k \<down>"
mHom_compos: fixes L :: "('g, 'b, 'h) Module_scheme"   and N :: "('i, 'b, 'j) Module_scheme"   and f :: "'g \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'i" assumes "R module L"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N" shows "compos L g f \<in> mHom R L N"
field_potent_nonzero1: fixes x :: "'a"   and n :: "int" assumes "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "x\<^bsub>K\<^esub>\<^bsup>n\<^esup> \<noteq> \<zero>"
subset_comp_red_basis_aux: fixes ys :: "('t \<Rightarrow>\<^sub>0 'c) list"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "set ys \<subseteq> set (comp_red_basis_aux xs ys)"
ta_hb_consistent_convert_RA: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list"   and t :: "'thread_id"   and ln :: "'l \<Rightarrow>f nat" shows "ta_hb_consistent P E (llist_of (map (Pair t) (convert_RA ln)))"
DynPDG_cdep_edge_CFG_path_1: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>c\<^sub>d n'" shows "n -as\<rightarrow>* n'"
not_call_simps_3: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "not_call (p1 \<oplus> p2)"
prv_ldsj_imp_remdups: fixes \<phi>s :: "'fmla list" assumes "set \<phi>s \<subseteq> fmla" shows "prv (imp (ldsj \<phi>s) (ldsj (remdups \<phi>s)))"
spr_trans_aec: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "set (local.spr_trans (fst ec) (snd ec)) = rel_ext (\<lambda>uu_. \<exists>t' s. uu_ = (tFirst t', s) \<and> t' \<leadsto> s \<in> SPR.jkbpC \<and> spr_jview a t' = spr_jview a t)"
sup_inf_implies: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y) \<sqinter> (x \<leadsto> y) = y"
vbrelation_vintersectionI: fixes r :: "V"   and s :: "V" assumes "vbrelation r" shows "vbrelation (r \<inter>\<^sub>\<circ> s)"
cases: fixes x :: "real_alg_2" obtains r :: "rat" where "x = Rational r"   | n :: "nat" and rai :: "int poly \<times> rat \<times> rat" where "x = Irrational n rai"
ineq_conv1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "a < c" shows "b < c"
rQuot_inter: fixes a :: "'b"   and A :: "'b list set"   and B :: "'b list set" shows "rQuot a (A \<inter> B) = rQuot a A \<inter> rQuot a B"
rbt_comp_union_with_key: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_union_with_key c = ord.rbt_union_with_key (lt_of_comp c)"
weak_strict_trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "le a b"   and "b \<prec>[le] c" shows "a \<prec>[le] c"
length_col_mat_to_cols_list: fixes j :: "nat"   and A :: "complex Matrix.mat" assumes "j < dim_col A" shows "length (Matrix_Legacy.col (mat_to_cols_list A) j) = dim_row A"
s: fixes x :: "real" assumes "x \<in> {0..1} - s" shows "g differentiable at x"
errMOD_iwlsFSwSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSw MOD" shows "iwlsFSwSTR (errMOD MOD)"
simp_2: shows "karatsuba_main_i ops f n g m = (if n \<le> karatsuba_lower_bound \<or> m \<le> karatsuba_lower_bound then let ff = poly_of_list_i ops f in foldr (\<lambda>a p. plus_poly_i ops (smult_i ops a ff) (cCons_i ops (arith_ops_record.zero ops) p)) g [] else let n2 = n div 2 in if n2 < m then case split_at n2 f of (f0, f1) \<Rightarrow> case split_at n2 g of (g0, g1) \<Rightarrow> let p1 = karatsuba_main_i ops f1 (n - n2) g1 (m - n2); p2 = karatsuba_main_i ops (coeffs_minus_i ops f1 f0) n2 (coeffs_minus_i ops g1 g0) n2; p3 = karatsuba_main_i ops f0 n2 g0 n2 in plus_poly_i ops (monom_mult_i ops (n2 + n2) p1) (plus_poly_i ops (monom_mult_i ops n2 (plus_poly_i ops (minus_poly_i ops p1 p2) p3)) p3) else case split_at n2 f of (f0, f1) \<Rightarrow> let p1 = karatsuba_main_i ops f1 (n - n2) g m in Let (karatsuba_main_i ops f0 n2 g m) (plus_poly_i ops (monom_mult_i ops n2 p1)))"