hcju/isabelleps huggingface dataset:
intros_10: fixes \<sigma>' :: "nat \<Rightarrow> 'c"   and i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and R :: "nat set"   and i' :: "nat"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set" assumes "\<sigma>' i = \<sigma> i" shows "((\<sigma>, NodeS i s R), connect(i', i), \<sigma>', NodeS i s (R \<union> {i'})) \<in> onode_sos S"
eucl_inf: fixes x :: "'a"   and y :: "'a" shows "inf x y = (\<Sum>i\<in>Basis. inf (x \<bullet> i) (y \<bullet> i) *\<^sub>R i)"
full_btrees_sharp: fixes k :: "nat"   and a :: "'b"   and h :: "nat" shows "nodes (full_node k a h) * (2 * k) = bound (2 * k) h"
height_def: fixes g :: "graph"   and v :: "nat" shows "height g v \<equiv> heights g ! v"
fMax_ge: fixes x :: "'m"   and A :: "'m fset" assumes "x |\<in>| A" shows "x \<le> fMax A"
Cons_1: fixes t :: "'a trm"   and u :: "'a trm"   and v :: "'a trm" assumes "subterm t q u" shows "\<exists>a. replace_subterm t q v a"
split_paired_all: fixes P :: "'c \<times> 'd \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
map_IInf_def: fixes F :: "'b upset \<Rightarrow> 'c upset" shows "map_IInf F = IInf \<circ> F \<circ> us"
interaction_bounded_by_Done: fixes "consider" :: "'m \<Rightarrow> bool"   and x :: "'n"   and n :: "enat" shows "interaction_bounded_by consider (Done x) n"
perm_length: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs <~~> ys" shows "length xs = length ys"
r'_def: shows "r' \<equiv> Cn 1 (r_lifz (r_const y) r) [Cn 1 r_eq [recf.Id 1 0, r_const x], recf.Id 1 0]"
intro: fixes is_open :: "'c set \<Rightarrow> bool"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>U. is_open U \<Longrightarrow> ring_homomorphism (fam_morphisms U) (\<FF> U) (add_str U) (mult_str U) (zero_str U) (one_str U) (\<FF>' U) (add_str' U) (mult_str' U) (zero_str' U) (one_str' U)"   and "\<And>U V x. \<lbrakk>is_open U; is_open V; V \<subseteq> U; x \<in> \<FF> U\<rbrakk> \<Longrightarrow> (\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x" shows "morphism_presheaves_of_rings_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str \<FF>' \<rho>' add_str' mult_str' zero_str' one_str' fam_morphisms"
c: shows "c = p\<^sup>2 + q\<^sup>2"
WhileTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and n :: "nat"   and s' :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> =n\<Rightarrow> t"
map_add_subsumed2: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m g" shows "Map.map_add g f = g"
sorted_wrt_dist_last_insort_key_eq: fixes p :: "(real, 'b) vec"   and ps :: "(real, 'b) vec list"   and q :: "(real, 'b) vec" assumes "sorted_wrt_dist p ps"   and "insort_key (\<lambda>q. dist q p) q ps \<noteq> ps @ [q]" shows "last (insort_key (\<lambda>q. dist q p) q ps) = last ps"
of_real_power: fixes x :: "real"   and n :: "nat" shows "of_real (x ^ n) = of_real x ^ n"
fully_faithful_functor_axioms: shows "fully_faithful_functor Left_a.comp Left_a.comp (H\<^sub>L a)"
linear_cmul: fixes f :: "'c \<Rightarrow> 'd"   and r :: "real"   and b :: "'c" assumes "linear f" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
that_2: assumes "k \<in> K"   and "x = v" shows "thesis"
gt_proper_sub: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "proper_sub s t" shows "t >\<^sub>t s"
fds_converges_cmult_right: fixes f :: "'b fds"   and s :: "'b"   and c :: "'b" assumes "fds_converges f s" shows "fds_converges (f * fds_const c) s"
function_smult_one: fixes f :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F" shows "\<one> \<odot>\<^bsub>F\<^esub> f = f"
ft: fixes x :: "real" assumes "\<bar>x - t\<bar> < d" shows "\<bar>f x - f t\<bar> \<le> M * \<bar>x - t\<bar> powr a"
interior_subset_rel_interior: fixes S :: "'b set" shows "interior S \<subseteq> rel_interior S"
inject: fixes x :: "nat"   and ya :: "nat" shows "(Location x = Location ya) = (x = ya)"
optimal_paths_min: fixes paths'' :: "'a list set"   and P_new'' :: "'a list" assumes "ProofStepInduct G v0 v1 paths'' P_new'' sep_size" shows "Digraph.distance (remove_vertex v0) (last P_new') v1 \<le> Digraph.distance (remove_vertex v0) (last P_new'') v1"
power_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a"   and n :: "nat" assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x ^ n \<le> y"
mult_nonpos_nonneg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ordered_semiring_0 plus zero less_eq less times"   and "less_eq a zero"   and "less_eq zero b" shows "less_eq (times a b) zero"
prime_p_Jacobi_eq_Legendre: fixes p :: "int"   and a :: "int" assumes "prime p" shows "Jacobi a p = Legendre a p"
application: fixes nr :: "nat"   and nc :: "nat" shows "\<forall>m1 m2. mat nr nc m1 \<and> mat nr nc m2 \<and> (\<forall>j<nc. \<forall>i<nr. m1 ! j ! i = m2 ! j ! i) \<longrightarrow> m1 = m2"
is_distinguishing_formula_eqvt: fixes x :: "('idx, 'pred, 'act) formula"   and P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "x distinguishes P from Q" shows "(p \<bullet> x) distinguishes (p \<bullet> P) from (p \<bullet> Q)"
valid_composed: fixes M :: "msg" assumes "M \<in> valid" shows "composed M"
forest_def: fixes E :: "'a uprod set" shows "forest E \<equiv> \<nexists>u p. decycle E u p"
uint16_shiftl_def: fixes x :: "uint16"   and n :: "integer" shows "uint16_shiftl x n = (if n < 0 \<or> 16 \<le> n then undefined push_bit x n else push_bit (nat_of_integer n) x)"
independent_strict_incl: fixes u :: "'a"   and Us :: "'a list" assumes "local.independent K (u # Us)" shows "Span K Us \<subset> Span K (u # Us)"
measurable_restrict_space_iff: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and c :: "'d"   and N :: "'d measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M"   and "c \<in> space N" shows "(f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N) = ((\<lambda>x. if x \<in> \<Omega> then f x else c) \<in> M \<rightarrow>\<^sub>M N)"
semilattice_axioms: shows "semilattice (\<iinter>)"
remove_comp_cls_val_process_Suc: fixes pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and Mkt :: "('d, 'c) discrete_market"   and y :: "'c" assumes "portfolio pf1" shows "\<forall>n w. cls_val_process Mkt (qty_rem_comp pf1 y) (Suc n) w = cls_val_process Mkt pf1 (Suc n) w - prices Mkt y (Suc n) w * pf1 y (Suc n) w"
dc2c_principal: shows "dc2c \<chi>\<^sub>0 = principal_char G"
inverse: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)"   and "f \<in> o[F](g)" shows "(\<lambda>x. inverse (g x)) \<in> o[F](\<lambda>x. inverse (f x))"
Cons: fixes x :: "'d"   and xs :: "'d list" shows "sum_list (x # xs) = x + sum_list xs"
bounded_hashcode_nat_bounds: assumes "1 < length l" shows "bounded_hashcode_nat (length l) k < length l"
assms_7: shows "(hs, data') = add_indices aux (snd data)"
dvd_iff_poly_eq_0: fixes c :: "'b"   and p :: "'b poly" shows "([:c, 1::'b:] dvd p) = (poly p (- c) = (0::'b))"
dom_mkIde: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "partial_magma.dom S (set_category_given_img.mkIde S img \<AA> A) = set_category_given_img.mkIde S img \<AA> A"
nb: shows "set (filter (\<lambda>v. s v \<noteq> None) (\<Psi>\<^sub>\<V>\<^sub>+)) = {v |v. v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+) \<and> s v \<noteq> None}"
Carmichael_number_gt_3: fixes n :: "nat" assumes "Carmichael_number n" shows "3 < n"
N_vector_top: fixes x :: "'a" shows "N (x * top) * top = x * top"
drop_Suc_Cons: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list" shows "drop (Suc n) (x # xs) = drop n xs"
assms_5: shows "\<not> formulaFalse (getF state) (elements (getM state))"
evenperm_def: fixes p :: "'b \<Rightarrow> 'b" shows "evenperm p = even (SOME n. swapidseq n p)"
int_Suc: fixes n :: "nat" shows "int (Suc n) = int n + 1"
that: fixes us :: "'b list"   and vs :: "'b list" assumes "xs = us @ vs"   and "filter P (map f us) = ys"   and "filter P (map f vs) = zs" shows "thesis"
neg_def: fixes \<phi> :: "'c" shows "B.neg \<phi> = imp \<phi> fls"
vector_inf_comp: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "times x top = x" shows "times (inf x y) z = inf x (times y z)"
unify_vars: fixes M :: "'b trm"   and N :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "unify_dom (M, N)"   and "unify M N = Some \<sigma>" shows "vars_of (t \<lhd> \<sigma>) \<subseteq> vars_of M \<union> vars_of N \<union> vars_of t"
qfree_cooper1_hd_coeffs1: fixes \<phi> :: "atom fm" assumes "qfree \<phi>" shows "qfree (qe_cooper\<^sub>1 (hd_coeffs1 \<phi>))"
thetaZOSeqI_ZObis: shows "thetaZOSeqI \<subseteq> ZObis"
tiny_dg_Dom_vdomain: shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
lim_sequentially: fixes X :: "nat \<Rightarrow> 'b"   and L :: "'b" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
prefix_trie_Lfs: fixes ks :: "bool list" shows "prefix_trie ks (Nd True (Lf, Lf)) = insert ks Lf"
lsu_comp_closed: shows "comp_closed_set (Fix \<nu>\<^sup>\<natural>)"
bufferd_val_filter_is_Write\<^sub>s\<^sub>b_eq_ext: fixes sb :: "'b memref list"   and a :: "nat" shows "buffered_val (filter is_Write\<^sub>s\<^sub>b sb) a = buffered_val sb a"
measurable_count_space_eq1: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and M :: "'b measure" shows "(f \<in> count_space A \<rightarrow>\<^sub>M M) = (f \<in> A \<rightarrow> space M)"
simps: fixes x :: "'a extN"   and y :: "'a extN" shows "x * y = x \<sqinter> y"
rep_eq: fixes x :: "'a interval"   and xa :: "'a interval" shows "bounds_of_interval (x + xa) = (case bounds_of_interval x of (a, b) \<Rightarrow> \<lambda>(c, d). (a + c, b + d)) (bounds_of_interval xa)"
contr: shows "verts c \<inter> verts d \<noteq> {}"
cmult_Nil: fixes c :: "'b" shows "c %* [] = []"
ord_eq_le_trans: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "c \<le> b" shows "c \<le> a"
assms_2: fixes w :: "'a stream" assumes "R w" shows "\<exists>u v x y. w = u @- x ## y ## v \<and> steps (u @ [x]) \<and> E x y \<and> R (y ## v)"
lt_row_to_poly_pivot_fun: fixes S :: "'t set"   and A :: "'c mat"   and f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "card S = dim_col A"   and "pivot_fun A f (dim_col A)"   and "i < dim_row A"   and "f i < dim_col A" shows "lt (mat_to_polys (pps_to_list S) A ! i) = pps_to_list S ! f i"
retraction_maps_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "retraction_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace Y. f (g x) = x)"
assms_3: fixes a :: "'c" assumes "a \<in> S" shows "f a = g a"
nth_Cons_Suc: fixes x :: "'g"   and xs :: "'g list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
eq_zero_imp_nonneg: fixes x :: "real" assumes "x = 0" shows "0 \<le> x"
basis_row_space_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "basis_row_space A = {row i (Gauss_Jordan A) |i. row i (Gauss_Jordan A) \<noteq> 0}"
cr_wn'': fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and j :: "nat"   and n :: "nat" assumes "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> m"   and "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"   and "\<not> m cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"   and "i \<le> j" shows "m \<le> n"
2: shows "\<^bold>\<lfloor>DOM f\<^bold>\<rfloor> = \<^bold>\<lfloor>COD f\<^bold>\<rfloor>"
strict_lunit: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "S.lunit f = f"
lc_set_nonzero: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "(0::'b) \<notin> punit.lc_set F"
inner_pdevs_def: fixes xs :: "'b pdevs list"   and ys :: "real list" shows "inner_pdevs xs ys = sum_pdevs (\<lambda>i. scaleR_pdevs (ys ! i) (xs ! i)) {..<length xs}"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "(case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
IN: fixes x :: "'v" shows "d_IN f (Inr x) = f (Inl x, Inr x)"
unity_root_k_0: fixes k :: "nat" shows "unity_root k 0 = 1"
drop_takefill: fixes k :: "nat"   and fill :: "'c"   and m :: "nat"   and w :: "'c list" shows "drop k (takefill fill (m + k) w) = takefill fill m (drop k w)"
inorder_split_maxD: fixes t :: "('c \<times> nat) tree"   and t' :: "('c \<times> nat) tree"   and a :: "'c" assumes "split_max t = (t', a)"   and "t \<noteq> \<langle>\<rangle>" shows "Tree2.inorder t' @ [a] = Tree2.inorder t"
phiUses_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and n :: "'node" shows "uninst.phiUses p g n \<equiv> \<Union>n'\<in>set (old.successors g n). \<Union>v'\<in>uninst.phiDefs p g n'. snd ` Set.filter (\<lambda>(n'', v). n'' = n) (set (zip (old.predecessors g n') (the (p g (n', v')))))"
pmf_positive: fixes x :: "'b"   and p :: "'b pmf" assumes "x \<in> set_pmf p" shows "0 < pmf p x"
moebius_pt_invert: fixes M :: "moebius"   and z1 :: "complex_homo"   and z2 :: "complex_homo" assumes "moebius_pt M z1 = z2" shows "moebius_pt (moebius_inv M) z2 = z1"
elims: fixes x :: "('c, 'd) chain"   and y :: "nat" assumes "chainlen x = y" obtains uu_ :: "'c" where "x = \<lbrace> uu_ \<rbrace>" and "y = 0"   | uv_ :: "'c" and uw_ :: "'d" and \<Pi> :: "('c, 'd) chain" where "x = \<lbrace> uv_ \<rbrace> \<cdot> uw_ \<cdot> \<Pi>" and "y = 1 + chainlen \<Pi>"
f_lemma_1: fixes p :: "nat"   and a :: "int"   and x :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)"   and "x \<in> S1 p" shows "f p a x = f p a (f_1 p a x)"
exec_n_drop_right: fixes c :: "instr list"   and P' :: "instr list"   and s :: "(char list \<Rightarrow> int) \<times> int list"   and n :: "nat"   and j :: "int"   and s' :: "(char list \<Rightarrow> int) \<times> int list" assumes "c @ P' \<turnstile> (0, s) \<rightarrow>^n (j, s')"   and "j \<notin> {0..<size c}" shows "\<exists>s'' i' k m. (if c = [] then s'' = s \<and> i' = 0 \<and> k = 0 else c \<turnstile> (0, s) \<rightarrow>^k (i', s'') \<and> i' \<in> exits c) \<and> c @ P' \<turnstile> (i', s'') \<rightarrow>^m (j, s') \<and> n = k + m"
implies_bot: fixes x :: "'c" shows "(if bot \<le> x then top else x) = top"
intros_2: fixes xs :: "['b]"   and x :: "'b" assumes "distinct xs"   and "x \<notin> set xs" shows "distinct (x : xs)"
l1: shows "\<forall>t i cid. cotrace init t \<and> Marker \<in> set (msgs (coS t i) cid) \<longrightarrow> (\<exists>j. enat j \<le> llength t \<and> i \<le> j \<and> Marker \<notin> set (msgs (coS t j) cid))"
a_subdist: fixes x :: "'a"   and y :: "'a" shows "ad (x + y) \<le> ad x"
unfold: shows "unfold = (\<Lambda> h v. Fixrec.run\<cdot>(match_bottom\<cdot>v\<cdot>(Fixrec.succeed\<cdot>\<bottom>) +++ Fixrec.succeed\<cdot>(case h\<cdot>v of Done \<Rightarrow> LNil | Skip\<cdot>s' \<Rightarrow> unfold\<cdot>h\<cdot>s' | Yield\<cdot>x\<cdot>s' \<Rightarrow> LCons\<cdot>x\<cdot>(unfold\<cdot>h\<cdot>s'))))"
fun_set_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R"   and "Range R = UNIV" shows "single_valued (\<langle>R\<rangle>fun_set_rel)"
degree_derivative: fixes p :: "real mpoly"   and x :: "nat" assumes "0 < MPoly_Type.degree p x" shows "MPoly_Type.degree p x = MPoly_Type.degree (derivative x p) x + 1"
dom_char: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "local.dom f = (if arr f then MkIde (Dom f) else null)"
IH: assumes "insert_ops ops"   and "insert_seq (Some ref) xs"   and "insert_ops xs"   and "set xs \<subseteq> set ops"   and "ref \<in> set (interp_ins ops)" shows "\<forall>a\<in>set (map fst xs). list_order ops ref a"
cases: fixes a1 :: "'a poly"   and a2 :: "'a poly"   and a3 :: "'a poly \<times> 'a poly" assumes "eucl_rel_poly a1 a2 a3" obtains x :: "'a poly" where "a1 = x__" and "a2 = 0" and "a3 = (0, x__)"   | y :: "'a poly" and x :: "'a poly" and q :: "'a poly" where "a1 = xa__" and "a2 = y__" and "a3 = (q__, 0)" and "y__ \<noteq> 0" and "xa__ = q__ * y__"   | y :: "'a poly" and r :: "'a poly" and x :: "'a poly" and q :: "'a poly" where "a1 = x" and "a2 = y" and "a3 = (q, r)" and "y \<noteq> 0" and "degree r < degree y" and "x = q * y + r"
block_design_axioms: shows "block_design \<V>' \<B>' \<k>"
l_one: fixes x :: "'a set" assumes "x \<in> carrier (ideals_set R)" shows "\<one>\<^bsub>ideals_set R\<^esub> \<otimes>\<^bsub>ideals_set R\<^esub> x = x"
length_mapAt: fixes vs :: "nat list"   and f :: "'a \<Rightarrow> 'a"   and xs :: "'a list" shows "|mapAt vs f xs| = |xs|"
raw_has_prod_def: fixes f :: "nat \<Rightarrow> 'b"   and M :: "nat"   and p :: "'b" shows "raw_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p \<and> p \<noteq> (0::'b)"
Compl_set_less_aux_Compl_iff: fixes ao :: "'a option"   and xs :: "'a list"   and ys :: "'a list" shows "Compl_set_less_aux ao xs ys = (Compl_set_less_eq_aux ao xs ys \<and> \<not> set_less_eq_aux_Compl ao ys xs)"
finite_field_ops_int: shows "field_ops (finite_field_ops_int p) mod_ring_rel"
\<alpha>: shows "\<alpha> \<le> ordertype A VWF"
simps_2: fixes \<mu> :: "'a" shows "E.Ide \<^bold>\<langle>\<mu>\<^bold>\<rangle> = B.ide \<mu>"
\<phi>_node_no_defs: fixes g :: "'g"   and \<phi> :: "'val"   and n :: "'node" assumes "unnecessaryPhi g \<phi>"   and "\<phi> \<in> allVars g"   and "var g \<phi> \<in> oldDefs g n" shows "defNode g \<phi> \<noteq> n"
cm_Euclidean_space_iff_continuous_on: fixes m :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and n :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" shows "continuous_map (subtopology (Euclidean_space m) S) (Euclidean_space n) f = (continuous_on (topspace (subtopology (Euclidean_space m) S)) f \<and> f ` topspace (subtopology (Euclidean_space m) S) \<subseteq> topspace (Euclidean_space n))"
einterpret_mvmult_fa: fixes Dn :: "nat"   and Dm :: "nat"   and A :: "floatarith list"   and B :: "floatarith list"   and vs :: "real list" assumes "Dn = CARD('a)"   and "Dm = CARD('b)"   and "length A = CARD('a) * CARD('b)"   and "length B = CARD('b)" shows "einterpret (mvmult_fa Dn Dm A B) vs = einterpret A vs *v einterpret B vs"
prime_elem_imp_coprime: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and n :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times p n" shows "algebraic_semidom.coprime times one p n"
ctr_simps_mem_Collect_eq: fixes a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
m: shows "2 ^ 7 * 73727 = 9437056"
reduction_subsets_def: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list"   and matr :: "rat mat" shows "reduction_subsets p qs signs subsets matr = take_indices subsets (rows_to_keep (reduce_mat_cols matr (solve_for_lhs p qs subsets matr)))"
IH: assumes "k < n" shows "nth_prime k \<le> p k"
app_strict: fixes x :: "'b" shows "\<bottom> x = \<bottom>"
face_of_halfspace_le: fixes F :: "'b set"   and a :: "'b"   and b :: "real" shows "(F face_of {x. a \<bullet> x \<le> b}) = (F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. a \<bullet> x \<le> b})"
distinct_36: fixes name1' :: "name"   and name2' :: "name"   and pi' :: "pi"   and pi :: "pi" shows "name1'<name2'>.pi' \<noteq> \<tau>.pi"
att_gdpr: shows "\<turnstile>[\<N>\<^bsub>(Igdpr, GDPR')\<^esub>, \<N>\<^bsub>(GDPR', sgdpr)\<^esub>] \<oplus>\<^sub>\<and>\<^bsup>(Igdpr, sgdpr)\<^esup>"
root_cycle_converse: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "point r"   and "x ; r \<noteq> 0" shows "x\<^sup>T ; r \<noteq> 0"
exhaust: fixes y :: "('l, 'm, 'n) prog" obtains x1 :: "'l" and x2 :: "'m" and x3 :: "'n" where "y = Prog x1 x2 x3"
source_nonempty: fixes x :: "'d \<Longrightarrow> 'e" assumes "weak_lens x" shows "\<exists>s. s \<in> \<S>\<^bsub>x\<^esub>"
le_of_key_order_alt: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "le_of_key_order ko x y = (key_compare ko x y \<noteq> Gt)"
IH: fixes V :: "'a set" assumes "wf'\<^sub>s\<^sub>s\<^sub>t V (S @ S')" shows "wf'\<^sub>s\<^sub>s\<^sub>t (V \<union> wfvarsoccs\<^sub>s\<^sub>s\<^sub>t S) S'"
is_universal: fixes a' :: "'a"   and \<chi>' :: "'b \<Rightarrow> 'a" assumes "D0.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow> a0\<guillemotright> \<and> D0.cones_map f \<pi>0 = \<chi>'"
config_rand: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init \<and> distinct (fst x) = distinct init \<and> length (fst x) = length init"
quicksort_is_sort: shows "quicksort = Sorting_Algorithms.sort"
simps: fixes a1 :: "('test, 'atom) com \<times> 'state"   and a2 :: "'state" shows "a1 \<rightarrow>t a2 = ((\<exists>atm s. a1 = (Atm atm, s) \<and> a2 = aval atm s) \<or> (\<exists>tst s c. a1 = (while tst do c, s) \<and> a2 = s \<and> \<not> tval tst s))"
raw_subst_consts': fixes t :: "'a" shows "abs_pred (\<lambda>a. \<forall>x. consts (subst a x) = consts a |\<union>| ffUnion (consts |`| fmimage x (frees a))) t"
triangle_greatest_le: fixes e :: "nat" shows "triangle (GREATEST y. triangle y \<le> e) \<le> e"
intro: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and val :: "'d \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<And>b s. val_ifex (ifex_of b) s = val b s"   and "\<And>b. reduced (ifex_of b) {}" shows "reduced_bdt_checkers ifex_of val"