hcju/isabelleps huggingface dataset:
letter_eq: fixes alphabet1 :: "'b list"   and init1 :: "'c \<Rightarrow> 'd"   and delta1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and accept1 :: "'d \<Rightarrow> bool"   and wellformed1 :: "'d \<Rightarrow> bool"   and Language1 :: "'d \<Rightarrow> 'b language"   and wf1 :: "'c \<Rightarrow> bool"   and Lang1 :: "'c \<Rightarrow> 'b language"   and alphabet2 :: "'e list"   and init2 :: "'f \<Rightarrow> 'g"   and delta2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and accept2 :: "'g \<Rightarrow> bool"   and wellformed2 :: "'g \<Rightarrow> bool"   and Language2 :: "'g \<Rightarrow> 'e language"   and wf2 :: "'f \<Rightarrow> bool"   and Lang2 :: "'f \<Rightarrow> 'e language"   and letter_eq :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'b"   and b :: "'e" assumes "DAs alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1 alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2 letter_eq"   and "letter_eq a b" shows "(a \<in> set alphabet1) = (b \<in> set alphabet2)"
update_zmultiset_commute: fixes M :: "'c zmultiset"   and t' :: "'c"   and n' :: "int"   and t :: "'c"   and n :: "int" shows "update_zmultiset (update_zmultiset M t' n') t n = update_zmultiset (update_zmultiset M t n) t' n'"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<nu>' \<star> \<mu>')"
distinct_pos_imaginary_parts_3: fixes p :: "int poly" shows "distinct (map real_of_3 (pos_imaginary_parts_3 p))"
fresh_star_insert: fixes a :: "atom"   and as :: "atom set"   and x :: "'b" shows "insert a as \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
list_of_pdevs_def: fixes x :: "'a pdevs" shows "list_of_pdevs x = map (\<lambda>i. (i, pdevs_apply x i)) (rev (sorted_list_of_set (pdevs_domain x)))"
not_frequently_False: fixes F :: "'b filter" shows "\<not> (\<exists>\<^sub>F x in F. False)"
Compl_disjoint2: fixes A :: "'a set" shows "- A \<inter> A = {}"
row_space_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "row_space A = vec.span (rows A)"
hyps: fixes x :: "real" shows "tan (x + real n * pi) = tan x"
throw_def: fixes a' :: "val"   and x :: "abopt" shows "throw a' x = abrupt_if True (Some (Xcpt (Loc (the_Addr a')))) (np a' x)"
coprimeI: fixes a :: "'b"   and b :: "'b" assumes "\<And>c. \<lbrakk>c dvd a; c dvd b\<rbrakk> \<Longrightarrow> is_unit c" shows "algebraic_semidom_class.coprime a b"
le_fin_word: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list"   and p :: "'state" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"   and "path w\<^sub>2 p" shows "path w\<^sub>1 p"
fminus_fsubset: fixes A :: "'b fset"   and B :: "'b fset" shows "A |-| B |\<subseteq>| A"
obtain_complete: obtains TAC :: "(nat set option, 'a) tree_automaton_rec" where "ta_lang TAC = ta_lang ta.complementTA" and "complete_tree_automaton TAC A"
at_most_one_mset_mset_diff: fixes a :: "'a"   and M :: "'a multiset" assumes "a \<notin># M - {#a#}" shows "set_mset (M - {#a#}) = set_mset M - {a}"
wt_sym_def: fixes n :: "nat" shows "wt_sym n = 1"
inter_sctn_spec_def: fixes X :: "'c set"   and sctn :: "'c sctn" shows "inter_sctn_spec X sctn = SPEC (\<lambda>R. X \<inter> plane_of sctn \<subseteq> R \<and> R \<subseteq> plane_of sctn)"
seE_fringe5: fixes re :: "('d \<times> nat) edge"   and prb :: "('d, 'e, 'f) pre_RedBlack"   and prb' :: "('d, 'e, 'f) pre_RedBlack"   and c' :: "('e, 'f) conf" assumes "ui_edge re \<in> edges (black prb) \<and> (src re \<in> red_vertices prb \<and> tgt re \<notin> red_vertices prb \<and> red prb' = add_edge (red prb) re) \<and> src re \<notin> subsumees (subs prb) \<and> se (confs prb (src re)) (labelling (black prb) (ui_edge re)) c' \<and> prb' = \<lparr>red = red prb', black = black prb, subs = subs prb, init_conf = init_conf prb, confs = (confs prb)(tgt re := c'), marked = (marked prb)(tgt re := marked prb (src re)), strengthenings = strengthenings prb\<rparr>"   and "\<not> ui_edge ` out_edges (red prb') (src re) \<subset> out_edges (black prb) (fst (src re))"   and "out_edges (black prb) (fst (tgt re)) = {}" shows "fringe prb' = fringe prb - {src re}"
le_neq_trans: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># b"   and "a \<noteq> b" shows "a \<subset># b"
factor_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>f ys = (\<exists>ps ss. ys = ps \<cdot> xs \<cdot> ss)"
destrci_def: fixes n :: "nat"   and bdd :: "bddi" shows "destrci n bdd \<equiv> case n of 0 \<Rightarrow> return FD | Suc 0 \<Rightarrow> return TD | Suc (Suc p) \<Rightarrow> pm_pthi (dpmi bdd) p \<bind> (\<lambda>(v, t, e). return (IFD v t e))"
assms: shows "(a, b) \<in> \<langle>R\<rangle>list_wset_rel"
not_Infinitesimal_mult: fixes x :: "'a star"   and y :: "'a star" assumes "x \<notin> Infinitesimal"   and "y \<notin> Infinitesimal" shows "x * y \<notin> Infinitesimal"
reflp_mono: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "reflp R"   and "\<And>x y. R x y \<longrightarrow> Q x y" shows "reflp Q"
domain_Lambda: fixes A :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "HOLZF.Domain (Lambda A f) = A"
r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier H'"   and "x \<in> carrier H'" shows "H #>\<^bsub>H'\<^esub> x \<subseteq> carrier H'"
polysub: fixes bs :: "'a list"   and p :: "poly"   and q :: "poly" shows "\<lparr>p -\<^sub>p q\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> - \<lparr>q\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
frac_cancel: fixes s1 :: "'a"   and r :: "'a"   and s :: "'a" assumes "s1 \<in> S"   and "(r, s) \<in> R \<times> S" shows "(s1 \<cdot> r) / (s1 \<cdot> s) = r / s"
leg0_simps_4: shows "local.dom t\<^sub>0 = t\<^sub>0"
decl_eval\<^sub>w: fixes mu :: "var\<^sub>C ModeUpd"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>Skip@[mu], mds, mem\<rangle>\<^sub>C \<langle>Stop, C.update_modes mu mds, mem\<rangle>\<^sub>C"
positive_same_outer_prod: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "positive (outer_prod v v)"
simps_1: fixes x11 :: "name"   and x12 :: "name \<Rightarrow> pi noption"   and y11 :: "name"   and y12 :: "name \<Rightarrow> pi noption" shows "(BoundOutputR_Rep x11 x12 = BoundOutputR_Rep y11 y12) = (x11 = y11 \<and> x12 = y12)"
ih: fixes y :: "nat"   and e :: "nat" assumes "y < n" shows "(local.encode e y = 0) = (y = 0)"
hd_map: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []" shows "hd (map f xs) = f (hd xs)"
fempty_ffilter: fixes A :: "'a fset" shows "ffilter (\<lambda>_. False) A = {||}"
a_b': shows "Derives1 a i q (b1 @ snd q @ u @ [fst p] @ a2)"
gcd_greatest: fixes gcd' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "is_gcd gcd'"   and "c dvd a"   and "c dvd b" shows "c dvd gcd' a b"
pow_mat_dim_1: fixes A :: "'b mat"   and k :: "nat" shows "dim_row (A ^\<^sub>m k) = dim_row A"
inf_sup_absorb: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"
a_get_scdom_component_def: fixes ptr :: "(_) object_ptr" shows "local.a_get_scdom_component ptr = Heap_Error_Monad.bind (get_owner_document ptr) (\<lambda>document. Heap_Error_Monad.bind (get_disconnected_nodes document) (\<lambda>disc_nodes. Heap_Error_Monad.bind (to_tree_order (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document)) (\<lambda>tree_order. Heap_Error_Monad.bind (map_M (to_tree_order \<circ> cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r) disc_nodes) (\<lambda>disconnected_tree_orders. return (tree_order @ concat disconnected_tree_orders)))))"
lappend_LNil2: fixes xs :: "'b llist" shows "lappend xs LNil = xs"
BIT_special_simps_6: shows "(- 1) BIT True = - 1"
max_subgraph_prop: fixes P :: "('a, 'b) pre_digraph \<Rightarrow> bool"   and x :: "('a, 'b) pre_digraph" assumes "max_subgraph P x" shows "P x"
insert_is_Un: fixes a :: "'d"   and A :: "'d set" shows "insert a A = {a} \<union> A"
dbl_inc_simps_4: shows "neg_numeral_class.dbl_inc (- (1::'a)) = - (1::'a)"
simps_2: fixes n :: "nat"   and b :: "bool"   and w :: "int" shows "bin_sc (Suc n) b w = of_bool (odd w) + 2 * bin_sc n b (w div 2)"
pderiv_of_nat: fixes n :: "nat" shows "pderiv (of_nat n) = 0"
mods_with_nats: fixes w :: "nat"   and v :: "nat"   and b :: "nat"   and a :: "nat" assumes "w < v"   and "v * b mod a = w * b mod a" shows "(v - w) * b mod a = 0"
IH: assumes "\<forall>P\<in>B. card P \<le> k" shows "card (\<Union> B) \<le> card B * k"
iso_ozero_empty: fixes r :: "('d \<times> 'd) set" shows "(r =o ozero) = (r = {})"
Rep_hierauto_select: fixes HA :: "('b, 'c, 'a) hierauto" shows "(HAInitValue HA, SAs HA, HAEvents HA, CompFun HA) \<in> hierauto"
abs_majoritiesE: fixes E :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "card E < card S + card T"   and "S \<subseteq> E"   and "T \<subseteq> E"   and "finite E" obtains p :: "'b" where "p \<in> S" and "p \<in> T"
While_\<tau>execI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and e' :: "'addr expr1" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (while (e) e') h s s'"
degree_prod_eq_sum_degree: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b poly" assumes "\<forall>i\<in>A. f i \<noteq> 0" shows "Polynomial.degree (prod f A) = (\<Sum>i\<in>A. Polynomial.degree (f i))"
sin_Taylor_1_lower_bound: fixes x :: "real" assumes "0 \<le> x" shows "x - x ^ 3 / 6 \<le> sin x"
supporting_hyperplane_relative_frontier: fixes S :: "'b set"   and x :: "'b" assumes "convex S"   and "x \<in> closure S"   and "x \<notin> rel_interior S" obtains a :: "'b" where "a \<noteq> (0::'b)" and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y" and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y"
enat_ord_code_2: fixes m :: "nat"   and n :: "nat" shows "(enat m < enat n) = (m < n)"
free_tv_of_substitutions_extend_to_types: fixes n :: "nat"   and t :: "typ"   and S :: "nat \<Rightarrow> typ" assumes "n \<in> free_tv t" shows "free_tv (S n) \<subseteq> free_tv ($ S t)"
sets_T: fixes s :: "'a cfg" shows "MC.T.events s = sets (stream_space (count_space UNIV))"
bind_writer_simps_1: fixes f :: "'f \<rightarrow> 'd\<cdot>'e writer" shows "\<bottom> \<bind> f = \<bottom>"
successive_step_rev: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" assumes "successive P (xs @ [x])"   and "\<not> P x" shows "successive P xs"
Quadratic_Reciprocity_Jacobi: fixes p :: "int"   and q :: "int" assumes "coprime p q"   and "2 < p"   and "2 < q"   and "odd p"   and "odd q" shows "Jacobi p q * Jacobi q p = (- 1) ^ nat ((p - 1) div 2 * ((q - 1) div 2))"
sees_method_idemp: fixes P :: "'a prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'a"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> D sees M, b :  Ts\<rightarrow>T = m in D"
Q2'_a_def: shows "Q2'_a \<equiv> [: \<lambda>(stk, mrk). {(stk', mrk'). root \<noteq> nil \<and> stk' = [root] \<and> mrk' = mrk \<union> {root}} :]"
union_single_eq_member: fixes x :: "'c"   and M :: "'c multiset"   and N :: "'c multiset" assumes "add_mset x M = N" shows "x \<in># N"
match_ex_table_shift_pcD: fixes P :: "'a prog"   and C :: "String.literal"   and pc :: "nat"   and n :: "nat"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and pc' :: "nat"   and d :: "nat" assumes "match_ex_table P C pc (shift n xt) = \<lfloor>(pc', d)\<rfloor>" shows "n \<le> pc \<and> n \<le> pc' \<and> match_ex_table P C (pc - n) xt = \<lfloor>(pc' - n, d)\<rfloor>"
path_in_nodes: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<sigma> :: "'b \<Rightarrow> 'c"   and k :: "nat" assumes "IFC prob" shows "IFC_def.path prob \<sigma> k \<in> IFC_def.nodes prob"
xs: shows "xs = take j xs @ xs ! j # drop (Suc j) xs"
rev_eq_flow: fixes y :: "'a"   and t :: "real" shows "flow0 y t = rev.flow0 y (- t)"
select_convs_4: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getC :: "Literal list"   and more :: "'a" shows "getC \<lparr>getF = getF, getM = getM, getConflictFlag = getConflictFlag, getC = getC, \<dots> = more\<rparr> = getC"
eucl_truncate_up: fixes x :: "'b"   and e :: "nat" shows "x \<le> eucl_truncate_up e x"
atLeastatMost_subset_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a..b} \<subseteq> {c..d}) = (\<not> a \<le> b \<or> c \<le> a \<and> b \<le> d)"
prod_as_plus: fixes X :: "'d \<Longrightarrow> 'f"   and Y :: "'e \<Longrightarrow> 'g" shows "X \<times> Y = X ;\<^sub>L fst\<^sub>L +\<^sub>L Y ;\<^sub>L snd\<^sub>L"
sup1CI: fixes B :: "'a \<Rightarrow> bool"   and x :: "'a"   and A :: "'a \<Rightarrow> bool" assumes "\<not> B x \<Longrightarrow> A x" shows "(A \<squnion> B) x"
sum_image_mset_mono: fixes A :: "'a multiset"   and B :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "A \<subseteq># B" shows "\<Sum>\<^sub># (image_mset f A) \<le> \<Sum>\<^sub># (image_mset f B)"
set_integral_snd: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "set_integrable (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f" shows "set_lebesgue_integral (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f = LBINT y:B. (LBINT x:A. f (x, y))"
arr_char: fixes f :: "('a, 'a) CC.arr" shows "CC.arr f = (f \<noteq> CC.Null \<and> CC.Dom f \<in> Collect ide \<and> CC.Cod f \<in> Collect ide \<and> CC.Map f \<in> hom (CC.Dom f) (CC.Cod f))"
list_rel_def: fixes R :: "('a \<times> 'c) set" shows "\<langle>R\<rangle>list_rel \<equiv> {(l, l'). list_all2 (\<lambda>x x'. (x, x') \<in> R) l l'}"
int_simps_3: shows "\<turnstile> \<not> #False = #True"
initState_Spy: shows "initState Spy = Key ` invKey ` pubK ` bad \<union> Key ` invKey ` pubSK ` bad \<union> Key ` shrK ` bad \<union> Key ` range pubK \<union> Key ` range pubSK"
closest_pair_simps: fixes ps :: "(int \<times> int) list" assumes "1 < length ps" shows "closest_pair ps = (let (uu_, c\<^sub>0, c\<^sub>1) = closest_pair_rec (mergesort fst ps) in (c\<^sub>0, c\<^sub>1))"
pnormal_last_length: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times"   and "0 < length p"   and "last p \<noteq> zero" shows "semiring_0.pnormal zero p"
swap_cbox_Pair: fixes c :: "'b"   and a :: "'a"   and d :: "'b"   and b :: "'a" shows "prod.swap ` cbox (c, a) (d, b) = cbox (a, c) (b, d)"
ptensor_mat_dim_row: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_row (tensor_P m1 m2) = ps2_P.d0"
intros: fixes q :: "'a"   and l :: "'b"   and qs :: "'a list"   and \<delta> :: "('a, 'b) ta_rule set"   and q' :: "'a" assumes "q \<rightarrow> l qs \<in> \<delta>"   and "q' \<in> set qs" shows "(q, q') \<in> f_succ \<delta>"
kernel_four_block_0_mat: fixes A :: "'b mat"   and B :: "'b mat"   and n :: "nat"   and m :: "nat"   and D :: "'b mat" assumes "A = four_block_mat B (0\<^sub>m n m) (0\<^sub>m m n) D"   and "B \<in> carrier_mat n n"   and "D \<in> carrier_mat m m" shows "vectorspace.dim class_ring (module_vec TYPE('b) (n + m)\<lparr>carrier := mat_kernel A\<rparr>) = vectorspace.dim class_ring (module_vec TYPE('b) n\<lparr>carrier := mat_kernel B\<rparr>) + vectorspace.dim class_ring (module_vec TYPE('b) m\<lparr>carrier := mat_kernel D\<rparr>)"
r_inv_ex: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)" shows "\<exists>y\<in>carrier (UP INTEG). x \<oplus>\<^bsub>UP INTEG\<^esub> y = \<zero>\<^bsub>UP INTEG\<^esub>"
TyAll_canonical: fixes v :: "trm"   and X :: "tyvrs"   and T\<^sub>2 :: "ty"   and T\<^sub>1 :: "ty" assumes "[] \<turnstile> v : (\<forall>X<:T\<^sub>1. T\<^sub>2)"   and "val v" shows "\<exists>X t S. v = (\<lambda>X<:S. t)"
flow_initial_time: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> S" shows "flow t0 x0 t0 = x0"
map_spmf_cong: fixes p :: "'h spmf"   and q :: "'h spmf"   and f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "map_spmf f p = map_spmf g q"
invar_NodeR: fixes l :: "('c \<times> nat) tree"   and x :: "'c"   and n :: "nat"   and r :: "('c \<times> nat) tree"   and r' :: "('c \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "n = lvl r + 1"   and "invar r'"   and "lvl r' = lvl r" shows "invar \<langle>l, (x, n), r'\<rangle>"
rev_map: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list" shows "rev (map f xs) = map f (rev xs)"
list_all3_empty: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "list_all3 P [] [] []"
elims: fixes x :: "V list"   and xa :: "nat list"   and y :: "V" assumes "Cantor_sum x xa = y" obtains ms :: "nat list" where "x = []" and "xa = ms__" and "y = 0"   | \<alpha> :: "V" and \<alpha>s :: "V list" where "x = \<alpha>'__ # \<alpha>s__" and "xa = []" and "y = 0"   | \<alpha> :: "V" and \<alpha>s :: "V list" and m :: "nat" and ms :: "nat list" where "x = \<alpha> # \<alpha>s" and "xa = m # ms" and "y = \<omega> \<up> \<alpha> * ord_of_nat m + Cantor_sum \<alpha>s ms"
eventually_rev_mp: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter"   and Q :: "'d \<Rightarrow> bool" assumes "eventually P F"   and "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x" shows "eventually Q F"
bottom_closed: shows "\<^bold>\<bottom> \<in> carrier thy_order"
len_help2: shows "length (pivot_positions A) \<le> dim_row A"
a_inv_closed: fixes x :: "'a" assumes "x \<in> carrier K" shows "\<ominus>\<^bsub>K\<^esub> x \<in> carrier K"
nullhomotopic_from_contractible_space: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" assumes "continuous_map X Y f"   and "contractible_space X" obtains c :: "'c" where "homotopic_with (\<lambda>h. True) X Y f (\<lambda>x. c)"
verit_comp_simplify1_3: fixes b' :: "'b"   and a' :: "'b" shows "(\<not> b' \<le> a') = (a' < b')"
no_loops: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "fst e \<noteq> snd e"
distinct_266: fixes x13 :: "Syntax.term"   and x12 :: "Syntax.term" shows "Roll x13 \<noteq> Prj2 x12"
results_gpv_inline: fixes x :: "'a"   and s' :: "('b, 'c, 'd, 'e) converter"   and \<I>' :: "('d, 'e) \<I>"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c, 'd, 'e) converter"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> results_gpv \<I>' (inline run_converter gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<I>, \<I>' \<turnstile>\<^sub>C s \<surd>" shows "x \<in> results_gpv \<I> gpv \<and> \<I>, \<I>' \<turnstile>\<^sub>C s' \<surd>"
exec_mono: fixes s :: "state"   and c :: "com"   and m :: "nat"   and t :: "state"   and n :: "nat" assumes "s -c-m\<rightarrow> t"   and "m \<le> n" shows "s -c-n\<rightarrow> t"
assms_3: shows "zs \<noteq> ys \<or> z \<noteq> y1"
sub_one_not_zero: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subdomain H R" shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
walk_iarray_append: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list"   and x :: "'c"   and b :: "'b" shows "walk_iarray f (IArray (xs @ [x])) b = f (walk_iarray f (IArray xs) b) x"
insert_swap_set_eq: fixes i :: "nat"   and l :: "'a list"   and x :: "'a" assumes "i < length l" shows "insert (l ! i) (set (l[i := x])) = insert x (set l)"
divide_divide_eq_right: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a / (b / c) = a * c / b"
is_mut_def: fixes tg :: "tg" shows "is_mut tg = (tg_mut tg = T_mut)"
set_mset_mono: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "A \<subseteq># B" shows "set_mset A \<subseteq> set_mset B"
prefs_from_table_map_of: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list"   and i :: "'c" assumes "prefs_from_table_wf agents alts xss"   and "i \<in> agents" shows "prefs_from_table xss i = of_weak_ranking (the (map_of xss i))"
gb_schema_dummy_not_Nil: fixes ps :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" assumes "struct_spec sel ap ab compl"   and "ps \<noteq> []" shows "gb_schema_dummy data D bs ps = (let sps = sel gs bs ps (snd data); ps0 = ps -- sps; aux = compl gs bs ps0 sps (snd data); remcomps = fst data - count_const_lt_components (fst aux) in if remcomps = 0 then (full_gb (gs @ bs), D) else let (hs, data') = add_indices aux (snd data) in gb_schema_dummy (remcomps, data') (D \<union> (set hs \<times> (set gs \<union> set bs \<union> set hs) \<union> set (ps -- sps) -\<^sub>p set (ap gs bs ps0 hs data'))) (ab gs bs hs data') (ap gs bs ps0 hs data'))"
C: shows "prv (imp (ldsj \<phi>1s) (ldsj \<phi>1s'))"
fst_pairs_VLambda: fixes A :: "V"   and f :: "V \<Rightarrow> V" shows "fst ` pairs (VLambda A f) = elts A"
jvmd_NormalE: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" assumes "P,t \<turnstile> Normal \<sigma> -ta-jvmd\<rightarrow> Normal \<sigma>'" obtains xcp :: "'addr option" and h :: "'heap" and f :: "'addr frame" and frs :: "'addr frame list" where "check P \<sigma>" and "(ta, \<sigma>') \<in> exec P t \<sigma>" and "\<sigma> = (xcp, h, f # frs)"
\<eta>_def: shows "\<phi>\<psi>.\<eta> \<equiv> transformation_by_components.map (\<cdot>\<^sub>D) (\<cdot>\<^sub>D) D.map \<phi>\<psi>.\<eta>o"
\<alpha>_def': shows "\<alpha> \<in> carrier Zp"
independent_bound_general: fixes S :: "('c, 'd) vec set" assumes "vec.independent S" shows "finite S \<and> card S \<le> vec.dim S"
right_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.right unapp t = (case unapp t of Some (x, r) \<Rightarrow> r)"
liftemb_eq: fixes below :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and liftemb :: "'e\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'e\<^sub>\<bottom>"   and liftdefl :: "'e itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'e \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'e"   and defl :: "'e itself \<Rightarrow> udom defl" assumes "class.domain below liftemb liftprj liftdefl emb prj defl" shows "liftemb = u_map\<cdot>emb"