hcju/isabelleps huggingface dataset:
mpoly_to_poly_def: fixes v :: "nat"   and p :: "'b mpoly" shows "mpoly_to_poly v p = Abs_poly (\<lambda>k. MPoly_Type.coeff p (Poly_Mapping.single v k))"
hfcomp_tcomp_pre: fixes g :: "'a \<Rightarrow> 'b nres"   and h :: "'c \<Rightarrow> 'd nres"   and Q :: "'c \<Rightarrow> bool"   and T :: "('a \<times> 'c) set"   and U :: "('b \<times> 'd) set"   and f :: "'e \<Rightarrow> 'f Heap"   and P :: "'a \<Rightarrow> bool"   and RR' :: "('a \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'e \<Rightarrow> assn)"   and S :: "'b \<Rightarrow> 'f \<Rightarrow> assn" assumes "(g, h) \<in> [Q]\<^sub>f T \<rightarrow> \<langle>U\<rangle>nres_rel"   and "(f, g) \<in> [P]\<^sub>a RR' \<rightarrow> S" shows "(f, h) \<in> [tcomp_pre Q T P]\<^sub>a hrp_comp RR' T \<rightarrow> hr_comp S U"
C: shows "ipurge_tr_rev_aux I D U (xs @ ys @ zs @ [z]) = ipurge_tr_rev_aux I D (sources_aux I D U [z]) (xs @ ys @ zs) @ ipurge_tr_rev_aux I D U [z]"
f: shows "f = fpxs_compose_power (fpxs_of_fls f') (1 / rat_of_nat s)"
simps_2: fixes l :: "'b tree"   and x :: "'b"   and b :: "bool"   and r :: "'b tree"   and t :: "'b tree" shows "subtract (Node l x b r) t = (case delete x t of None \<Rightarrow> None | Some t' \<Rightarrow> case subtract l t' of None \<Rightarrow> None | Some t'' \<Rightarrow> subtract r t'')"
factors_cong_unit: fixes u :: "'a"   and a :: "'a"   and as :: "'a list" assumes "u \<in> Units G"   and "a \<notin> Units G"   and "factors G as a"   and "set as \<subseteq> carrier G" shows "factors G (as[0 := as ! 0 \<otimes> u]) (a \<otimes> u)"
hyps: fixes t :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "emb_step_at p d t = u"   and "t \<noteq> u" shows "position_of t (p @ [d])"
res_eq: shows "Rrule_result EquivR j (SG ! i) = [(p # \<Gamma>, q # closeI \<Delta> j), (q # \<Gamma>, p # closeI \<Delta> j)]"
intros_6: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Empty], Undecided\<rangle> \<Rightarrow> Undecided"
span_scale: fixes x :: "'a"   and S :: "'a set"   and c :: "'a" assumes "x \<in> More_Modules.ideal S" shows "c * x \<in> More_Modules.ideal S"
delta_neg_sol: shows "delta_neg = dual (F (dual delta_pos, undual delta_neg))"
graph_plan_lemma_5: fixes s :: "('d, 'e) fmap"   and vs :: "'d set"   and s' :: "('d, 'e) fmap" assumes "fmrestrict_set (fmdom' s - vs) s = fmrestrict_set (fmdom' s' - vs) s'"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "s = s'"
the_eqI: fixes P :: "'a pred"   and x :: "'a" assumes "(THE x. eval P x) = x" shows "Predicate.the P = x"
next_def: fixes A :: "'d \<times> 'c \<times> 'e" shows "next A = fst (snd A)"
diag_mat_one: fixes n :: "nat" shows "diag_mat (1\<^sub>m n) = replicate n (1::'b)"
linear_inj_on_left_inverse: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear (*a) (*b) f"   and "inj_on f (vs1.span S)" shows "\<exists>g. range g \<subseteq> vs1.span S \<and> linear (*b) (*a) g \<and> (\<forall>x\<in>vs1.span S. g (f x) = x)"
assms_2: shows "\<bar>a\<bar> \<noteq> \<infinity>"
valid_ConsE: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and trn :: "('a, 'b, 'c) trans"   and tr :: "('a, 'b, 'c) trans list" assumes "IO_Automaton.valid step (trn # tr)" obtains "tr = []" and "IO_Automaton.validTrans step trn"   | "IO_Automaton.validTrans step trn" and "tgtOf trn = srcOf (hd tr)" and "IO_Automaton.valid step tr"
subringE_3: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subring H R" shows "\<one>\<^bsub>R\<^esub> \<in> H"
directed_empty: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "directed {} less_eq"
sum_of_powers: fixes m :: "nat"   and n :: "nat" shows "(\<Sum>k\<le>n. real k ^ m) = (bernpoly (Suc m) (real (n + 1)) - bernpoly (Suc m) 0) / real (m + 1)"
isDiagonal_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "isDiagonal_upt_k A k = (\<forall>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<and> (mod_type_class.to_nat a < k \<or> mod_type_class.to_nat b < k) \<longrightarrow> A $ a $ b = (0::'d))"
closedin_frontier_of: fixes X :: "'c topology"   and S :: "'c set" shows "closedin X (X frontier_of S)"
subcomplex_image: shows "order.greater_eq Y (f \<turnstile> X) \<and> SimplicialComplex (f \<turnstile> X)"
emeasure_count_space_finite: fixes X :: "'b set"   and A :: "'b set" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (count_space A) X = of_nat (card X)"
pow_zero: fixes x :: "nat" shows "(0::'a) < b ^ x"
is_lubD1: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a" assumes "class.po below"   and "po.is_lub below S x" shows "po.is_ub below S x"
exp_plus_2pin: fixes z :: "complex"   and n :: "int" shows "exp (z + \<i> * (of_int n * (complex_of_real pi * 2))) = exp z"
timpls_transformable_to_inv_3: fixes TI :: "('e set \<times> 'e set) list"   and f :: "('f, 'g, 'e) prot_fun"   and T :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and g :: "('f, 'g, 'e) prot_fun"   and S :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list" assumes "timpls_transformable_to TI (Fun f T) (Fun g S)"   and "f \<noteq> g" shows "is_Abs f \<and> is_Abs g \<and> (the_Abs f, the_Abs g) \<in> set TI"
Parallel_Strong_Soundness: fixes Ts :: "('b ann_com option \<times> 'b set) list"   and s :: "'b"   and Rs :: "('b ann_com option \<times> 'b set) list"   and t :: "'b"   and j :: "nat" assumes "(Parallel Ts, s) -P*\<rightarrow> (Parallel Rs, t)"   and "interfree Ts"   and "j < length Rs"   and "\<forall>i<length Ts. \<exists>c q. Ts ! i = (Some c, q) \<and> s \<in> pre c \<and> \<turnstile> c  q" shows "if OG_Hoare.com (Rs ! j) = None then t \<in> OG_Hoare.post (Ts ! j) else t \<in> pre (the (OG_Hoare.com (Rs ! j)))"
s'_def: fixes a :: "real"   and v :: "real"   and t :: "real" assumes "braking_movement a v" shows "braking_movement.s' a v t = (if t \<le> movement.t_stop a v then movement.p' a v t else 0)"
secret_object5_def: shows "\<gamma> = \<delta> \<squnion> - (\<delta> \<squnion> - \<delta>)"
simps: fixes a :: "'a list" shows "steps a = ((\<exists>x. a = [x]) \<or> (\<exists>x y xs. a = x # y # xs \<and> x \<rightarrow> y \<and> steps (y # xs)))"
Collect_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_fun A (=)) (rel_pred A) Collect Collect"
numeral_plus_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus n) one = numeral.numeral one plus (n + num.One)"
perm_type: fixes pi :: "(name \<times> name) list"   and T :: "ty" shows "pi \<bullet> T = T"
red_single_nonzero2: fixes p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and f :: "'a \<Rightarrow>\<^sub>0 'd"   and t :: "'a" assumes "punit.red_single p q f t" shows "f \<noteq> 0"
HOM_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'j) Module_scheme" shows "HOM\<^bsub>R\<^esub> M N = \<lparr>carrier = mHom R M N, pop = tOp_mHom R M N, mop = iOp_mHom R M N, zero = mzeromap M N, sprod = sprod_mHom R M N\<rparr>"
preserves_seq: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "C.VV.seq g f" shows "C.seq (fst g \<star>\<^sub>C snd g) (fst f \<star>\<^sub>C snd f)"
ref_two_step: fixes A :: "'a nres"   and R :: "('a \<times> 'b) set"   and B :: "'b nres"   and C :: "'b nres" assumes "A \<le> \<Down> R B"   and "B \<le> C" shows "A \<le> \<Down> R C"
otnaux: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators" assumes "first_bothNet x = first_bothNet y"   and "\<not> member DenyAll x"   and "\<not> member DenyAll y"   and "onlyTwoNets y"   and "onlyTwoNets x" shows "onlyTwoNets (x \<oplus> y)"
cs_ipd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and m :: "nat"   and k :: "nat" assumes "\<pi> m = ipd (\<pi> k)"   and "\<forall>n\<in>{k..<m}. \<pi> n \<noteq> ipd (\<pi> k)"   and "k < m" shows "cs\<^bsup>\<pi>\<^esup> m = filter (\<lambda>n. ipd n \<noteq> \<pi> m) (cs\<^bsup>\<pi>\<^esup> k) @ [\<pi> m]"
right_one: fixes x :: "'a" shows "x \<odot> one = x"
lucas_lehmer_test_def: fixes p :: "nat" shows "lucas_lehmer_test p = (2 < p \<and> 2 ^ p - 1 dvd gen_lucas_lehmer_sequence 4 (p - 2))"
splits_wrt_image_sum_list_snd_subset: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "splits_wrt (ps, qs) T F" shows "sum_list ` listset (map cone qs) \<subseteq> P[X]"
med_progress_Some: fixes r :: "fid_t \<Rightarrow> Runs.var set option"   and x :: "fid_t"   and s :: "Runs.var set" assumes "r x = Some s" shows "med_progress r x = Some (s \<inter> {xnx, xny, xgnx, xgny, xsk, xEnd})"
wf_measures: fixes fs :: "('c \<Rightarrow> nat) list" shows "wf (measures fs)"
mem_T_I: fixes T :: "real set"   and t :: "real" assumes "compact_interval T"   and "compact_interval.tmin T \<le> t"   and "t \<le> compact_interval.tmax T" shows "t \<in> T"
fps_of_ratfps_diff: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "fps_of_ratfps (x - y) = fps_of_ratfps x - fps_of_ratfps y"
M_prop: fixes k :: "nat" assumes "M < k" shows "(T ^^ k) ((T ^^ n) x) \<notin> A"
steps_append_single: fixes xs :: "'a list"   and x :: "'a" assumes "steps xs"   and "last xs \<rightarrow> x"   and "xs \<noteq> []" shows "steps (xs @ [x])"
pm_of_poly_plus: fixes x :: "'a"   and p :: "'b poly"   and q :: "'b poly" shows "pm_of_poly x (p + q) = pm_of_poly x p + pm_of_poly x q"
abs_Tree\<^sub>\<alpha>_def: shows "FL_Formula.abs_Tree\<^sub>\<alpha> \<equiv> quot_type.abs (=\<^sub>\<alpha>) FL_Formula.Tree\<^sub>\<alpha>.Abs_Tree\<^sub>\<alpha>"
S\<^sub>i: fixes x :: "i"   and y :: "i" assumes "SkolemizedAxiomsSetMcL" shows "E x \<^bold>\<and> E y \<^bold>\<leftarrow> E (x \<cdot> y)"
e_var_elim: fixes v :: "val"   and x :: "nat"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E (EVar x) \<rho>" obtains v' :: "val" where "lookup \<rho> x = Some v'" and "v \<sqsubseteq> v'"
vquot_dbtm_eq: fixes V :: "name set"   and t :: "dbtm"   and W :: "name set" assumes "atom ` V \<inter> supp t = atom ` W \<inter> supp t" shows "vquot_dbtm V t = vquot_dbtm W t"
iIN_iprev_nth: fixes a :: "nat"   and d :: "nat"   and n :: "nat" assumes "a \<le> d" shows "[n\<dots>,d] \<leftarrow> a = n + d - a"
tdec1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and d :: "dec"   and s2 :: "'b state"   and new_env :: "v sem_env" assumes "evaluate_dec ck [] env s1 d (s2, Rval new_env)" shows "evaluate_top ck env s1 (Tdec d) (s2, Rval new_env)"
ide_char': fixes a :: "'a" shows "C.ide a = (C.arr a \<and> (C.dom a = a \<or> C.cod a = a))"
count_notin2: fixes xs :: "'a list"   and x :: "'a" assumes "count_list xs x = 0" shows "x \<notin> set xs"
subset_Zorn: fixes A :: "'d set set" assumes "\<And>C. subset.chain A C \<Longrightarrow> \<exists>U\<in>A. \<forall>X\<in>C. X \<subseteq> U" shows "\<exists>M\<in>A. \<forall>X\<in>A. M \<subseteq> X \<longrightarrow> X = M"
add_right_cancel: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.cancel_semigroup_add plus" shows "(plus b a = plus c a) = (b = c)"
h2: shows "(\<exists>a\<in>set (dnf \<phi>1). \<exists>a'\<in>set (dnf \<phi>2). f a \<and> f a') = ((\<exists>a\<in>set (dnf \<phi>1). f a) \<and> (\<exists>a\<in>set (dnf \<phi>2). f a))"
hash_def: fixes t :: "Syntax.term" shows "hash t = hash_real (map_fixed undefined [] t)"
simps: fixes s :: "(nat \<times> nat list \<times> form) list"   and f :: "form" shows "subst s f = subst' (filter (\<lambda>(v, s). v \<in> fv f) s) f"
foldseq_almostzero: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and j :: "nat"   and s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<forall>x. f (0::'a) x = x"   and "\<forall>x. f x (0::'a) = x"   and "\<forall>i. i \<noteq> j \<longrightarrow> s i = (0::'a)" shows "foldseq f s n = (if j \<le> n then s j else (0::'a))"
ldrop_eSuc_LCons: fixes n :: "enat"   and x :: "'c"   and xs :: "'c llist" shows "ldrop (eSuc n) (LCons x xs) = ldrop n xs"
commute: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "signed.min a b = signed.min b a"
nAct_def: fixes c :: "'id"   and n :: "enat"   and t :: "cnf llist" shows "\<langle>c #\<^bsub>n\<^esub>t\<rangle> \<equiv> llength (\<pi>\<^bsub>c\<^esub>ltake n t)"
m: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> m) = meets p q"
proc: shows "sh C = \<lfloor>(sfs, Processing)\<rfloor>"
of_int_power: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and z :: "int"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (z ^ n) = power.power one times (ring_1.of_int one plus zero minus z) n"
ns_ns_mul_ext_union_compat: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and C :: "'a multiset"   and D :: "'a multiset" assumes "(A, B) \<in> ns_mul_ext ns s"   and "(C, D) \<in> ns_mul_ext ns s" shows "(A + C, B + D) \<in> ns_mul_ext ns s"
rtrancl_into_rtrancl: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set"   and c :: "'d" assumes "(a, b) \<in> r\<^sup>*"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>*"
map_simp_ide: fixes a :: "'c" assumes "C.ide a" shows "\<eta>'GoG\<epsilon>'.map a = (\<eta>'.map \<circ> G) a \<cdot>\<^sub>D (G \<circ> \<epsilon>'.map) a"
abs_venv_union: fixes ve1 :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option"   and ve2 :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "|ve1 ++ ve2| \<lessapprox> |ve1| \<union>. |ve2|"
induct: fixes x1 :: "pi"   and x2 :: "pi"   and P :: "pi \<Rightarrow> pi \<Rightarrow> bool" assumes "x1 \<equiv>\<^sub>s x2"   and "\<And>Pa. P Pa Pa"   and "\<And>Pa Q. \<lbrakk>Pa \<equiv>\<^sub>s Q; P Pa Q\<rbrakk> \<Longrightarrow> P Q Pa"   and "\<And>Pa Q R. \<lbrakk>Pa \<equiv>\<^sub>s Q; P Pa Q; Q \<equiv>\<^sub>s R; P Q R\<rbrakk> \<Longrightarrow> P Pa R"   and "\<And>Pa Q. P (Pa \<oplus> Q) (Q \<oplus> Pa)"   and "\<And>Pa Q R. P ((Pa \<oplus> Q) \<oplus> R) (Pa \<oplus> Q \<oplus> R)"   and "\<And>Pa. P (Pa \<oplus> \<zero>) Pa"   and "\<And>Pa Q. P (Pa \<parallel> Q) (Q \<parallel> Pa)"   and "\<And>Pa Q R. P ((Pa \<parallel> Q) \<parallel> R) (Pa \<parallel> Q \<parallel> R)"   and "\<And>Pa. P (Pa \<parallel> \<zero>) Pa"   and "\<And>a Pa. P ([a\<frown>a]Pa) Pa"   and "\<And>x. P (<\<nu>x>\<zero>) \<zero>"   and "\<And>x y Pa. P (<\<nu>x><\<nu>y>Pa) (<\<nu>y><\<nu>x>Pa)"   and "\<And>x Pa Q. P (<\<nu>x>(Pa \<oplus> Q)) (<\<nu>x>Pa \<oplus> <\<nu>x>Q)"   and "\<And>x Pa Q. x \<sharp> Pa \<Longrightarrow> P (<\<nu>x>(Pa \<parallel> Q)) (Pa \<parallel> <\<nu>x>Q)"   and "\<And>x a y Pa. \<lbrakk>x \<noteq> a; x \<noteq> y\<rbrakk> \<Longrightarrow> P (<\<nu>x>a<y>.Pa) (a<y>.(<\<nu>x>Pa))"   and "\<And>x a b Pa. \<lbrakk>x \<noteq> a; x \<noteq> b\<rbrakk> \<Longrightarrow> P (<\<nu>x>a{b}.Pa) (a{b}.(<\<nu>x>Pa))"   and "\<And>x Pa. P (<\<nu>x>\<tau>.Pa) (\<tau>.(<\<nu>x>Pa))"   and "\<And>Pa. P (!Pa) (Pa \<parallel> !Pa)" shows "P x1 x2"
exp_add_not_zero_imp: fixes m :: "nat"   and n :: "nat" assumes "(2::'a) ^ (m + n) \<noteq> (0::'a)" shows "(2::'a) ^ m \<noteq> (0::'a)"
snd_if_conv: fixes P :: "bool"   and A :: "'b"   and B :: "'a"   and C :: "'b"   and D :: "'a" shows "snd (if P then (A, B) else (C, D)) = (if P then B else D)"
wf_greater_bounded: fixes N :: "nat" shows "wf (greater_bounded N)"
prv_imp_cnj5: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi>1 :: "'fmla"   and \<chi>2 :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi>1 \<in> fmla"   and "\<chi>2 \<in> fmla"   and "prv (imp \<phi>1 \<chi>1)"   and "prv (imp \<phi>2 \<chi>2)" shows "prv (imp \<phi>1 (imp \<phi>2 (cnj \<chi>1 \<chi>2)))"
hyps: fixes P :: "pi"   and Q :: "pi" assumes "depth P + depth Q \<le> n"   and "valid P"   and "valid Q"   and "uhnf P"   and "uhnf Q"   and "P \<sim> Q" shows "P \<equiv>\<^sub>e Q"
existsA: fixes x :: "nat" assumes "x \<in> A" shows "\<exists>y. P x y"
Collect_mono_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "order.greater_eq (Collect Q) (Collect P) = (\<forall>x. P x \<longrightarrow> Q x)"
index_eq_iff: fixes xs :: "'e list"   and x :: "'e"   and i :: "nat" shows "(index xs x = i) = (i \<le> length xs \<and> (\<forall>j<i. xs ! j \<noteq> x) \<and> (i < length xs \<longrightarrow> xs ! i = x))"
Poly_snoc: fixes xs :: "'a list"   and x :: "'a" shows "Poly (xs @ [x]) = Poly xs + monom x (length xs)"
ANR_imp_closed_neighbourhood_retract: fixes S :: "'a set"   and U :: "'a set" assumes "ANR S"   and "closedin (top_of_set U) S" obtains V :: "'a set" and W :: "'a set" where "openin (top_of_set U) V" and "closedin (top_of_set U) W" and "S \<subseteq> V" and "V \<subseteq> W" and "S retract_of W"
injective_conv_univalent: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.injective x = matrix_stone_relation_algebra.univalent (x\<^sup>t)"
fresh_2: fixes a :: "name"   and x2 :: "trm"   and x1 :: "trm" shows "a \<sharp> \<langle>x2, x1\<rangle> = (a \<sharp> x2 \<and> a \<sharp> x1)"
subset_invariant_def: fixes A :: "bool list bdd list \<times> bool list" shows "subset_invariant A = (\<lambda>(bdd, qss). bddh (length (fst A)) bdd)"
ideD_1: fixes a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.ide a" shows "S.VV.arr a"
exp_ln_iff: fixes x :: "real" shows "(exp (ln x) = x) = (0 < x)"
rev_subsetD: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "c \<in> A"   and "A \<subseteq> B" shows "c \<in> B"
nn_integral_spmf_neq_top: fixes p :: "'c spmf" shows "(\<Sum>\<^sup>+ x. ennreal (spmf p x)) \<noteq> \<top>"
leI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder less_eq less"   and "\<not> less x y" shows "less_eq y x"
INF_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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 bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (f ` {}) = top"
chine_eq_apex: shows "\<mu>\<nu>_\<pi>.chine = \<mu>\<nu>_\<pi>.apex"
imp_igSwapInpIPresIGWlsInpSTR: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "igSwapIPresIGWlsSTR MOD" shows "igSwapInpIPresIGWlsInpSTR MOD"
gt_imp_vars: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "vars s \<subseteq> vars t"
comp_arr_dom: fixes f :: "'b"   and a :: "'b" assumes "\<phi>.B.arr f"   and "\<phi>.B.dom f = a" shows "f \<cdot>\<^sub>B a = f"
Sups_upper: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> X" shows "x \<le> \<Squnion> X"
det3_self23: fixes a :: "real \<times> real"   and b :: "real \<times> real" shows "coll a b b"
r_delay_aux_def: shows "r_delay_aux \<equiv> Pr 2 (r_constn 1 0) (Cn 4 r_ifz [Cn 4 r_result1 [Cn 4 r_length [recf.Id 4 3], recf.Id 4 2, Cn 4 r_take [Cn 4 S [recf.Id 4 0], recf.Id 4 3]], recf.Id 4 1, Cn 4 S [recf.Id 4 0]])"
order_refl: fixes less_eq :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and less :: "'q \<Rightarrow> 'q \<Rightarrow> bool"   and x :: "'q" assumes "class.preorder less_eq less" shows "less_eq x x"
Un_insert_left: fixes a :: "'a"   and B :: "'a set"   and C :: "'a set" shows "Set.insert a B \<union> C = Set.insert a (B \<union> C)"
perm_rem_simps_2: fixes f :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "bij f"   and "f y = x" shows "perm_rem x f y = f x"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F" shows "category A"
pfun_graph_minus: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "pfun_graph (f - g) = pfun_graph f - pfun_graph g"
coG_dtor_o_corec: fixes s :: "'e \<Rightarrow> ('f, ('f coG + 'e) coF) coG0" shows "dtor_coG \<circ> dtor_corec_coG s = map_pre_coG id (case_sum id (dtor_corec_coG s)) \<circ> s"
Nonce_def: fixes N :: "nat" shows "Nonce N = Abs_Msg (msgrel `` {NONCE N})"
D_eq: shows "D = {x. \<exists>y\<in>A. x = y * a mod int p \<and> y * a mod int p \<le> (int p - 1) div 2}"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_ordered (IF v t e) = (Ball (ifex_var_set t \<union> ifex_var_set e) ((<) v) \<and> ifex_ordered t \<and> ifex_ordered e)"
gno: fixes x :: "'b" assumes "x \<in> span ((+) (- b) ` T)" shows "norm (g x) = norm x"
csqrt_real: fixes x :: "complex" assumes "is_real x" shows "0 \<le> Re x \<and> ccsqrt x = cor (sqrt (Re x)) \<or> Re x < 0 \<and> ccsqrt x = \<i> * cor (sqrt (- Re x))"
nempty'': shows "{} \<notin> (\<lambda>a. vote_set vs {a}) ` (Q \<inter> voters vs)"
list_all_list_mem: fixes P :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "list_all P l = (\<forall>e. ListMem e l \<longrightarrow> P e)"
snoc: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "ext_snoc ext" shows "ext gt (xs @ [x]) xs"
less_le: fixes x :: "'b"   and y :: "'b" shows "(x < y) = (x \<le> y \<and> x \<noteq> y)"
integral_abs_bound: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "\<bar>integral\<^sup>L M f\<bar> \<le> LINT x|M. \<bar>f x\<bar>"
vec_lambda_beta: fixes g :: "'d \<Rightarrow> 'c"   and i :: "'d" shows "vec_lambda g $h i = g i"
sorted_butlast: fixes xs :: "'a list" assumes "xs \<noteq> []"   and "sorted xs" shows "sorted (butlast xs)"
divides_mult_r: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "a \<otimes> c divides b \<otimes> c = a divides b"
deriv_n_def: fixes d :: "nat \<times> nat \<times> nat" shows "deriv_n d = fst d"
elims: fixes x :: "'a"   and xa :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "prepend_queue_t x xa = y" obtains a :: "'a" where "x = a__" and "xa = ([], [])" and "y = ([], [a__])"   | a :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = aa__" and "xa = (fs__, l # ls)" and "y = (aa__ # fs__, l # ls)"   | a :: "'a" and f :: "'a" and fs :: "'a list" where "x = a" and "xa = (f # fs, [])" and "y = undefined"
surjective: fixes r :: "('a, 'b, 'c) automaton_scheme" shows "r = \<lparr>init = init r, trans = automaton.trans r, \<dots> = automaton.more r\<rparr>"
singleton: fixes r :: "('a \<times> 'a) set"   and linear_cf :: "'a set \<Rightarrow> 'a set"   and x :: "'a"   and X :: "'a set" assumes "linear_cf r linear_cf" shows "(x \<in> linear_cf X) = (linear_cf X = {x})"
to_gba_alt_accept: fixes m :: "'b" shows "Automata.gba.accept (to_gba_ext m) = accept"
assms_2: shows "c * c\<^sup>T \<le> forest_components h"
find_remove_def: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "find_remove P xs = (case find P xs of None \<Rightarrow> (None, xs) | Some x \<Rightarrow> (Some x, remove1 x xs))"