hcju/isabelleps huggingface dataset:
simIChainFresh: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and xvec :: "name list"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and C :: "'e" assumes "eqvt Rel"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "\<And>\<alpha> Q'. \<lbrakk>\<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; bn \<alpha> \<sharp>* P; bn \<alpha> \<sharp>* Q; bn \<alpha> \<sharp>* \<Psi>; bn \<alpha> \<sharp>* subject \<alpha>; distinct (bn \<alpha>); bn \<alpha> \<sharp>* C; xvec \<sharp>* \<alpha>; xvec \<sharp>* Q'\<rbrakk> \<Longrightarrow> \<exists>P'. \<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P' \<and> (\<Psi>, P', Q') \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
topological_tendstoD: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter"   and S :: "'b set" assumes "class.topological_space open"   and "filterlim f (topological_space.nhds open l) F"   and "open S"   and "l \<in> S" shows "\<forall>\<^sub>F x in F. f x \<in> S"
right_total_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_total R" shows "right_total (list_all2 R)"
cases: fixes a :: "db" assumes "val a" obtains n :: "nat" where "a = DBNum n"   | "a = DBff"   | "a = DBtt"   | e :: "db" where "a = DBAbsN e__"   | e :: "db" where "a = DBAbsV e"
Fract_less_one_iff: fixes b :: "'b"   and a :: "'b" assumes "(0::'b) < b" shows "(Fract a b < 1) = (a < b)"
decomp: shows "{a..b} = {x \<in> {a..b}. x \<le> 0} \<union> (\<Union>i. {inverse (real (Suc i))..} \<inter> {a..b})"
low_pow: shows "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
pair_measure_countable: fixes S1 :: "'c set"   and S2 :: "'d set" assumes "countable S1"   and "countable S2" shows "count_space S1 \<Otimes>\<^sub>M count_space S2 = count_space (S1 \<times> S2)"
in_hom_def: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Right_a.in_hom f a b \<equiv> a \<in> Right_a.domains f \<and> b \<in> Right_a.codomains f"
splitAt_def: fixes c :: "'c"   and as :: "'c list" shows "splitAt c as \<equiv> splitAtRec c [] as"
fds_abs_converges_def: fixes f :: "'b fds"   and s :: "'b" shows "fds_abs_converges f s = summable (\<lambda>n. norm (fds_nth f n / nat_power n s))"
IH: fixes S :: "'a set" assumes "finite S"   and "n < card S" shows "wellorder_class.enumerate S n \<in> S"
TickLeftSync: fixes A :: "'a event set"   and t :: "'a event list"   and s :: "'a event list" assumes "tick \<in> A \<and> front_tickFree t \<and> s setinterleaves (([tick], t), A)" shows "s = t \<and> last t = tick"
coeff_0_reflect_poly: fixes p :: "'a poly" shows "coeff (reflect_poly p) 0 = lead_coeff p"
sorted_wrt_iff_nth_less: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "sorted_wrt P xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> P (xs ! i) (xs ! j))"
le_trans_equiv_rsp: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and v :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t v = le_trans u v"
rel_spmf_pos_distr: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_spmf A OO rel_spmf B \<le> rel_spmf (A OO B)"
y1_prop: shows "q < y1 \<and> (\<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y1}. d * x\<^sup>2 + e * x + f < 0)"
invertible_left_inverse: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = (\<exists>B. B ** A = mat (1::'b))"
continuous_exp: fixes x :: "ordinal" assumes "0 < x" shows "continuous ((**) x)"
uverts_edge_ugraph: fixes es :: "nat set set" shows "uverts (edge_ugraph es) = S_verts"
hyps: shows "is_unit (fst (echelon_form_of_upt_k_det bezout (1::'a, A) k))"
sup_distl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<squnion> z) = x \<cdot> y \<squnion> x \<cdot> z"
H_def: shows "H \<equiv> complex_of_real (1 / sqrt 2) \<cdot>\<^sub>m mat 2 2 (\<lambda>(i, j). if i \<noteq> j then 1 else if i = 0 then 1 else - 1)"
less_le_not_le: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(x <s y) = (x \<le>s y \<and> \<not> y \<le>s x)"
acc_point_range_imp_convergent_subsequence: fixes l :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "\<forall>U. l \<in> U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)" shows "\<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
part_equivp_fractrel: 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" assumes "class.idom times one plus zero minus uminus" shows "part_equivp (idom.fractrel times zero)"
assms_2: shows "(\<lambda>n. sum u {..<n} / real n) \<longlonglongrightarrow> 0"
join_return': fixes N :: "'c measure"   and M :: "'c measure" assumes "sets N = sets M" shows "join (distr M (subprob_algebra N) (return N)) = M"
orthogonal_complement_def: fixes W :: "'b set" shows "orthogonal_complement W = {x. \<forall>y\<in>W. orthogonal y x}"
twoCisTypeII_8: shows "(\<lambda>y. twoC (y, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
assms_4: shows "g1 ` T \<subseteq> U"
reduced_word_forI: fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a"   and "\<And>bs. \<lbrakk>bs \<in> lists A; sum_list bs = a\<rbrakk> \<Longrightarrow> order.greater_eq (length bs) (length as)" shows "reduced_word_for A a as"
t: shows "Ide t \<and> Arr t \<and> Dom t = t \<and> Cod t = t \<and> ide \<lbrace>t\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<rbrace> \<and> \<lbrace>t\<^bold>\<down>\<rbrace> \<in> hom \<lbrace>t\<rbrace> \<lbrace>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<rbrace>"
preprm_unit_commutes: fixes a :: "'b"   and b :: "'b" shows "preprm_unit a b =p preprm_unit b a"
rep_eq: fixes x :: "('d, 'e) fmap" shows "fmdom' x = dom (fmlookup x)"
valid_Not: fixes P :: "'state"   and x :: "('a, 'pred, 'act) formula" shows "P \<Turnstile> Formula.Not x = (\<not> P \<Turnstile> x)"
comp_fun_commute_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "comp_fun_commute f \<equiv> \<forall>y x. f y \<circ> f x = f x \<circ> f y"
induct: fixes x :: "tm"   and P :: "tm \<Rightarrow> bool" assumes "coding_tm x"   and "\<And>x. \<exists>i. x = ORD_OF i \<Longrightarrow> P x"   and "\<And>x y. \<lbrakk>coding_tm x; P x; coding_tm y; P y\<rbrakk> \<Longrightarrow> P (HPair x y)" shows "P x"
subgraph_imp_subverts: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G" shows "verts H \<subseteq> verts G"
get_set_eq: fixes r :: "'a array"   and x :: "'a list"   and h :: "heap" shows "Array.get (Array.set r x h) r = x"
widen_PrimT: fixes G :: "prog"   and x :: "prim_ty"   and T :: "ty" assumes "G\<turnstile>PrimT x\<preceq>T" shows "\<exists>y. T = PrimT y"
\<psi>_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and ba :: "'a \<times> 'a" assumes "hom_functor C S \<phi>" shows "hom_functor.\<psi> C \<phi> ba = inv_into {f. partial_magma.in_hom C f (fst ba) (snd ba)} (\<phi> ba)"
unat_word_ariths_4: shows "unat 0 = 0 mod 2 ^ LENGTH('b)"
normalize_quot_div_unit_left: fixes u :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_unit u" shows "normalize_quot (x div u, y) = (fst (normalize_quot (x, y)) div u, snd (normalize_quot (x, y)))"
configuration_rev_step: fixes q :: "'b"   and x :: "nat"   and n :: "nat" assumes "q \<noteq> q\<^sub>0"   and "x \<in> configuration q (Suc n)" obtains q' :: "'b" where "q = \<delta> q' (w n)" and "x \<in> configuration q' n"
mem_diff: shows "\<forall>x. var_asm_not_written mds x \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x \<and> dma mem\<^sub>1 x = dma mem\<^sub>2 x"
dense_repr_monom_closed: fixes K :: "'a set"   and p :: "'a list"   and t :: "'a \<times> nat" assumes "subring K R"   and "set p \<subseteq> K"   and "t \<in> set (dense_repr p)" shows "local.monom (fst t) (snd t) \<in> carrier (K [X])"
in_set_chopD: fixes x :: "'b"   and ys :: "'b list"   and d :: "nat"   and xs :: "'b list" assumes "x \<in> set ys"   and "ys \<in> set (chop d xs)" shows "x \<in> set xs"
group_hom_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
lsum_gt_0: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat" assumes "al \<noteq> []"   and "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a" shows "0 < lsum f al"
interleaving_Nil_implies_eq1: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "interleavings xs ys zs"   and "xs = []" shows "ys = zs"
square_integrable_trigonometric_set: fixes i :: "nat" shows "trigonometric_set i square_integrable {- pi..pi}"
first: shows "(\<Sum>j<i. d j * base ^ j) < base ^ i"
cases: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and a :: "'a stream" assumes "(\<phi> suntil \<psi>) a" obtains \<omega> :: "'a stream" where "a = \<omega>__" and "\<psi> \<omega>__"   | \<omega> :: "'a stream" where "a = \<omega>" and "\<phi> \<omega>" and "(\<phi> suntil \<psi>) (stl \<omega>)"
closeSubstE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<sigma> :: "(name list \<times> 'a list) list" assumes "(\<Psi>, P, Q) \<in> closeSubst Rel"   and "wellFormedSubst \<sigma>" shows "(\<Psi>, P[<\<sigma>>], Q[<\<sigma>>]) \<in> Rel"
fps_freshmans_dream: fixes p :: "nat" assumes "prime p" shows "((1 + fps_X) ^ p, 1 + fps_X ^ p) \<in> fpsmodrel (int p)"
path_root_acyclic: fixes r :: "'a"   and x :: "'a" assumes "path_root r x"   and "x ; r = 0" shows "is_acyclic x"
bi_unique_rel_set_bij_betw: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set" assumes "bi_unique R"   and "rel_set R A B" shows "\<exists>f. bij_betw f A B \<and> (\<forall>x\<in>A. R x (f x))"
inj_on_fst_map_to_set: fixes m :: "'e \<Rightarrow> 'f option" shows "inj_on fst (map_to_set m)"
the_inv_into_onto: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "inj_on f A" shows "the_inv_into A f ` f ` A = A"
rotate1_rotate_swap: fixes n :: "nat"   and xs :: "'c list" shows "rotate1 (rotate n xs) = rotate n (rotate1 xs)"
Suc_2: shows "Lfilled (Suc i) lholed (vs @ [$Br (Suc i + k)]) LI"
set_union_code: fixes xs :: "'g list"   and ys :: "'g list" shows "set xs \<union> set ys = set (xs @ ys)"
Arg2pi_eq_pi: fixes z :: "complex" shows "(Arg2pi z = pi) = (z \<in> \<real> \<and> Re z < 0)"
BNTr8: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier D \<rightarrow> carrier D"   and "a \<in> carrier D"   and "\<forall>x\<in>carrier D. x \<preceq> f x" shows "\<Union> (WWa D f a) \<in> WWa D f a"
reachable_points_to: fixes y :: "'g"   and x :: "'g"   and s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and m :: "'h" assumes "(x points_to y) s"   and "mut_m.reachable m x s" shows "mut_m.reachable m y s"
3: shows "is_principal_up (F \<inter> \<up>x \<inter> (G \<inter> \<up>x))"
exhaust_sel: fixes gpv :: "('a, 'b, 'c) gpv" obtains "gpv = GPV (the_gpv gpv)"
mem_single_l_span: fixes A :: "'b set"   and h :: "'a"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "h \<in> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "f \<in> {j. j \<le> n} \<rightarrow> {h}"   and "l_comb R M n s f \<in> linear_span R M A {h}" shows "\<exists>a\<in>A. l_comb R M n s f = a \<cdot>\<^sub>s h"
mset_le_asym: fixes M :: "'a multiset"   and N :: "'a multiset"   and P :: "bool" assumes "M < N"   and "\<not> P \<Longrightarrow> N < M" shows "P"
NT_ND: fixes c :: "'c event list"   and P :: "'c process" assumes "c \<notin> T P" shows "c \<notin> D P"
esssup_add: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> ereal"   and g :: "'b \<Rightarrow> ereal" shows "esssup M (\<lambda>x. f x + g x) \<le> esssup M f + esssup M g"
axioms_1: fixes R :: "('h \<times> 'i) set"   and A :: "('h, 'j) graph_rec_scheme"   and B :: "('i, 'k) graph_rec_scheme" assumes "simulation R A B" shows "graph A"
op_fmap_empty_def: shows "op_fmap_empty \<equiv> fmempty"
Path_is_List: fixes h :: "'c \<Rightarrow> 'c ref"   and b :: "'c ref"   and Ps :: "'c list"   and a :: "'c" assumes "Path h b Ps (Ref a)"   and "a \<notin> set Ps" shows "List (h(a := Null)) b (Ps @ [a])"
Zero_not_Suc: fixes m :: "nat" shows "0 \<noteq> Suc m"
P_k_2: shows "hd (tl P_k) \<notin> Q.second_vertices"
ovalid_grab_asm: fixes G :: "bool"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "G \<Longrightarrow> ovalid P f Q" shows "ovalid (\<lambda>s. G \<and> P s) f Q"
match_mono: fixes \<Delta> :: "'state \<Rightarrow> 'a \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool"   and s :: "'state"   and s1 :: "'state"   and vl1 :: "'value list"   and a :: "'act"   and ou :: "'out"   and s' :: "'state"   and vl' :: "'a"   and \<Delta>' :: "'state \<Rightarrow> 'a \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool" assumes "match \<Delta> s s1 vl1 a ou s' vl'"   and "\<And>s vl s1 vl1. \<Delta> s vl s1 vl1 \<Longrightarrow> \<Delta>' s vl s1 vl1" shows "match \<Delta>' s s1 vl1 a ou s' vl'"
apply_bind: fixes f :: "'b \<Rightarrow> 'a pred"   and xq :: "'b seq" shows "pred_of_seq (apply f xq) = bind (pred_of_seq xq) f"
summable_c: shows "summable (\<lambda>n. norm (fds_nth f n) / real n powr c)"
lm100: fixes x :: "'b"   and Y :: "'b set"   and X :: "'b set" assumes "x \<in> Y - X" shows "0 \<in> (X <|| Y) `` {x}"
mirror_D: fixes r :: "('c \<times> 'c) set"   and \<tau> :: "'c list"   and \<sigma> :: "'c list"   and \<sigma>' :: "'c list"   and \<tau>' :: "'c list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "D r \<sigma> \<tau> \<tau>' \<sigma>'"
set_restriction_Un: fixes f :: "'a set \<Rightarrow> 'a set"   and A :: "'a set"   and B :: "'a set" assumes "set_restriction f" shows "f (A \<union> B) = f A \<union> f B"
assign_local_def: fixes upd :: "('e list \<Rightarrow> 'e list) \<Rightarrow> 'd control_state_scheme \<Rightarrow> 'd control_state_scheme"   and rhs :: "'d control_state_scheme \<Rightarrow> 'e" shows "assign_local upd rhs = assign (\<lambda>\<sigma>. (upd \<circ> map_hd) (\<lambda>_. rhs \<sigma>) \<sigma>)"
sortedP_Cons: fixes xa :: "real \<times> real"   and x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "ccw'.sortedP xa (x # xs)" obtains "\<forall>xb\<in>set xs. ccw' xa x xb" and "ccw'.sortedP xa xs"
pair_next: fixes ls :: "nat"   and key :: "nat" shows "g_pair ls key = ls \<or> c_tl (g_pair ls key) = ls"
valid_states_snapshot: fixes PROB :: "(('a, 'c) fmap \<times> ('a, 'c) fmap) set"   and s :: "('a, 'c) fmap" shows "valid_states (prob_proj PROB (prob_dom (snapshot PROB s))) = valid_states (snapshot PROB s)"
cycle_edges_def: fixes p :: "'a list" shows "cycle_edges p = zip p (rotate1 p)"
retract_of_empty_1: fixes S :: "'a set" shows "({} retract_of S) = (S = {})"
arr_dom: fixes f :: "'b" assumes "D.arr f" shows "D.arr (D.dom f)"
all: shows "\<forall>nx\<in>set ns'. nx \<noteq> n'"
3: shows "\<forall>S\<in>\<S>. wf\<^sub>s\<^sub>t (wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (\<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])) (dual\<^sub>s\<^sub>t S)"
sorted_merge_adj: fixes xss :: "'c list list" assumes "\<forall>xs\<in>set xss. sorted xs" shows "\<forall>xs\<in>set (merge_adj xss). sorted xs"
comp_params_index_pair: fixes x :: "'a"   and y :: "'a" assumes "{x, y} \<subseteq> \<V>"   and "x \<noteq> y" shows "int (\<B>\<^sup>C index {x, y}) = \<b> + \<Lambda> - 2 * (\<Lambda> * (\<v> - 1) div (\<k> - 1))"
simps_5: fixes v :: "'b list"   and va :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list"   and S :: "('a, 'b) strand_step list" shows "ik\<^sub>s\<^sub>t (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S) = ik\<^sub>s\<^sub>t S"
option_while'_Some: fixes ra :: "'a"   and r' :: "'a"   and s :: "'s" assumes "(Some ra, Some r') \<in> option_while' (\<lambda>a. C a s) (\<lambda>a. B a s)" shows "(Some (ra, s), Some (r', s)) \<in> whileLoop_results C (\<lambda>a. gets_the (B a))"
inv_check_left_moving_on_leftmost_no_Oc: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and list :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_check_left_moving_on_leftmost (as, lm) (s, list, Oc # r) ires = False"
assms_3: shows "Y \<in> obj\<^bsub>C\<^esub>"
simps_8: fixes p :: "('a, 'b) form"   and s :: "'a FOL_Fitting.term"   and k :: "nat" shows "Forall p[s/k] = Forall (p[liftt s/Suc k])"
partList_sub_equi: fixes s :: "'b set"   and ts :: "'b set list" shows "set (partList1 s ts) = set (difference_list_opt s ts) \<union> set (intersection_list_opt s ts)"
walls_betw_def: fixes X :: "'b set set"   and C :: "'b set"   and D :: "'b set" assumes "ThinChamberComplex X" shows "ThinChamberComplex.walls_betw X C D \<equiv> {H \<in> \<Union>(f, g)\<in>ThinChamberComplex.foldpairs X. {{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X}}. separated_by H C D}"
part_closed_open: fixes X :: "'a set" assumes "I \<circ> K \<circ> I = I"   and "closed X" shows "open X"
rep_eq: fixes x :: "'b permutation" shows "(\<rightarrow>) (- x) = the_inv ((\<rightarrow>) x)"
hyps: assumes "\<not> is_final (steps0 c A n)" shows "steps0 c (A |+| B) n = steps0 c A n"
unsigned_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (numeral n) = semiring_1.of_nat one plus zero (take_bit LENGTH('b) (numeral n))"
Wbis_matchT_M_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>w t" shows "matchT_M t s"
rep_eq: fixes x :: "'a poly"   and xa :: "'a poly" shows "coeff (x - xa) = (\<lambda>n. coeff x n - coeff xa n)"
ideals_def: shows "ideals = {X. X \<noteq> {} \<and> downclosed_set X \<and> directed X}"
IH: fixes x2a :: "('a, 'c) Term.term" assumes "x2a \<in> set T"   and "t \<sqsubseteq> x2a \<cdot> \<theta>" shows "(\<exists>s'. s' \<sqsubseteq> x2a \<and> t = s' \<cdot> \<theta>) \<or> (\<exists>x\<in>fv x2a. t \<sqsubset> \<theta> x)"
less_numeral_extra_4: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
execn_to_execn_subseteq_guards: fixes c :: "('d, 'e, 'f) com"   and c' :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and s :: "('d, 'f) xstate"   and n :: "nat"   and t :: "('d, 'f) xstate" assumes "c \<subseteq>\<^sub>g c'"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c',s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
ideal_generated_eq_left_ideal: fixes A :: "'a set" shows "ideal_generated A = left_ideal_generated A"
mult_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a + b + c = a + (b + c)"
fract_poly_dvd: fixes p :: "'b poly"   and q :: "'b poly" assumes "p dvd q" shows "map_poly to_fract p dvd map_poly to_fract q"
simps_1: fixes x21 :: "tree"   and x22 :: "tree"   and y21 :: "tree"   and y22 :: "tree" shows "(Branching x21 x22 = Branching y21 y22) = (x21 = y21 \<and> x22 = y22)"
Int_absorb: fixes A :: "'h set" shows "A \<inter> A = A"
get_S_def: fixes psr :: "word32" shows "get_S psr \<equiv> if psr AND 128 = 0 then 0 else 1"
nr'_def: shows "nr' \<equiv> (\<pi>\<^sub>2 s, \<pi>\<^sub>3 nr, \<pi>\<^sub>4 nr, \<pi>\<^sub>5 nr, \<pi>\<^sub>6 nr, \<pi>\<^sub>7 nr)"
qFreshAbs_imp_ex_qAFreshAbs2: fixes V :: "'d set"   and A :: "('e, 'f, 'varSort, 'var, 'g) qAbs"   and xs :: "'varSort"   and x :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "finite V"   and "qGoodAbs A"   and "qFreshAbs xs x A"   and "qFreshAbs ys y A" shows "\<exists>A'. qGoodAbs A' \<and> A $= A' \<and> qAFreshAbs xs x A' \<and> qAFreshAbs ys y A'"
elem_less_size: fixes e :: "exp0"   and es :: "exp0 list" assumes "e \<in> set es" shows "size_exp' e \<le> size_list size_exp' es"
nth_drop: fixes n :: "nat"   and xs :: "'c list"   and i :: "nat" assumes "n \<le> length xs" shows "drop n xs ! i = xs ! (n + i)"
comp_right_subdist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y \<sqinter> z) \<le> x * y \<sqinter> x * z"
preserves_dom: fixes f :: "'a" assumes "F.A.arr f" shows "F.B.dom (F f) = F (F.A.dom f)"
cond_bind_spmf_fst_def: fixes p :: "'h spmf"   and f :: "'h \<Rightarrow> 'i spmf"   and x :: "'i" shows "cond_bind_spmf_fst p f x = cond_bind_spmf p (map_spmf (\<lambda>b. (b, ())) \<circ> f) ({x} \<times> UNIV)"
mult2_s_eq_mult2_s_alt: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "ns O s \<subseteq> s"   and "refl ns"   and "trans s" shows "mult2_alt_s ns s = mult2_s ns s"
cases: fixes a1 :: "'a list"   and a2 :: "'a list"   and a3 :: "'a list" assumes "interleavings a1 a2 a3" obtains ys :: "'a list" where "a1 = []" and "a2 = ys__" and "a3 = ys__"   | xs :: "'a list" where "a1 = xs__" and "a2 = []" and "a3 = xs__"   | xs :: "'a list" and ys :: "'a list" and zs :: "'a list" and x :: "'a" where "a1 = x # xsa__" and "a2 = ysa__" and "a3 = x # zs__" and "interleavings xsa__ ysa__ zs__"   | xs :: "'a list" and ys :: "'a list" and zs :: "'a list" and y :: "'a" where "a1 = xs" and "a2 = y # ys" and "a3 = y # zs" and "interleavings xs ys zs"
AE_ball_countable: fixes X :: "'a set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'b measure" assumes "countable X" shows "(AE x in M. \<forall>y\<in>X. P x y) = (\<forall>y\<in>X. AE x in M. P x y)"
assms_3: shows "\<forall>\<^sub>F x in F. g x \<in> {l x..u x}"
finite_measure_Diff': fixes A :: "'h set"   and p :: "'h spmf"   and B :: "'h set" assumes "A \<in> sets (measure_spmf p)"   and "B \<in> sets (measure_spmf p)" shows "Sigma_Algebra.measure (measure_spmf p) (A - B) = Sigma_Algebra.measure (measure_spmf p) A - Sigma_Algebra.measure (measure_spmf p) (A \<inter> B)"
simps: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "('c, 'd) rose_tree\<^sub>m"   and a2 :: "('c, 'd) rose_tree\<^sub>m" shows "blinding_of_tree h bo a1 a2 = (\<exists>t1 t2. a1 = Tree\<^sub>m t1 \<and> a2 = Tree\<^sub>m t2 \<and> blinding_of_blindable (map_prod h (map (hash_tree h))) (rel_prod bo (list_all2 (blinding_of_tree h bo))) t1 t2)"
prime_factorization_lcm: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "prime_factorization (lcm a b) = prime_factorization a \<union># prime_factorization b"
distinct_2: fixes x2 :: "agent"   and x1 :: "agent" shows "publK x2 \<noteq> sharK x1"
prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<le>p ys) = (\<exists>zs. ys = xs \<cdot> zs)"