hcju/isabelleps huggingface dataset:
chain_tllist_ord_tSup: fixes A :: "('d, 'b) tllist set"   and xs :: "('d, 'b) tllist" assumes "Complete_Partial_Order.chain local.tllist_ord A"   and "xs \<in> A" shows "local.tllist_ord xs (local.tSup A)"
ab: fixes i :: "'a" assumes "i \<in> Basis" shows "a \<bullet> i \<le> b \<bullet> i"
pth_9_1: fixes c :: "real"   and x :: "'c"   and z :: "'c"   and d :: "real" shows "c *\<^sub>R x + z + d *\<^sub>R x \<equiv> (c + d) *\<^sub>R x + z"
card: fixes y :: "nat" assumes "y \<in> S - {x}" shows "card (derangements (S - {x, y})) = count_derangements n"
set_choose_def: fixes s :: "'b set" shows "set_choose s \<equiv> SOME x. x \<in> s"
eval_quot_dbfm_ignore: fixes e :: "(name, hf) finfun"   and A :: "dbfm"   and e' :: "(name, hf) finfun" shows "\<lbrakk>quot_dbfm A\<rbrakk>e = \<lbrakk>quot_dbfm A\<rbrakk>e'"
zero_less_power2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (power.power one times a 2) = (a \<noteq> zero)"
Fvars_fm_simps_8: fixes a :: "name"   and A :: "fm" shows "Fvars (SyntaxN.Ex a A) = Fvars A - {a}"
two_comparisons_into_case_order_11: fixes x :: "'b"   and y :: "'b"   and P :: "'d"   and R :: "'d"   and Q :: "'d" shows "(if x = y then P else if c1.le y x then R else Q) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
rbt_lookup_rbt_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and k :: "'b"   and v :: "'c" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.rbt_lookup less (ord.rbt_insert less k v t) = ord.rbt_lookup less t(k \<mapsto> v)"
invh_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (join l r) \<and> bheight (join l r) = bheight l"
suminf_eq_lim: fixes f :: "nat \<Rightarrow> 'c" shows "suminf f = lim (\<lambda>n. sum f {..<n})"
cspp_def: fixes m :: "'node"   and S :: "'node SDG_node set"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" shows "cspp m S fs \<equiv> csppa m S 0 fs"
simps_3: fixes v :: "'a"   and va :: "'a" shows "first_destNet (DenyAllFromTo v va) = destNet (DenyAllFromTo v va)"
module_over_zeroring: fixes M :: "('f, 'a, 'g) Module_scheme" assumes "zeroring R"   and "R module M" shows "carrier M = {\<zero>\<^bsub>M\<^esub>}"
\<omega>consistentStd2_implies_consistent: assumes "\<omega>consistentStd2" shows "consistent"
has_bochner_integral_FTC_Icc: fixes a :: "real"   and b :: "real"   and F :: "real \<Rightarrow> 'c"   and f :: "real \<Rightarrow> 'c" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (F has_vector_derivative f x) (at x within {a..b})"   and "continuous_on {a..b} f" shows "has_bochner_integral lborel (\<lambda>x. indicat_real {a..b} x *\<^sub>R f x) (F b - F a)"
agop_gop: shows "(\<cdot>\<^bsub>b_ag A\<^esub>) = (\<plusminus>)"
rep_eq: fixes x :: "'e \<Rightarrow> 'g"   and xa :: "'h \<Rightarrow> 'f"   and xb :: "('g, 'h) \<I>" shows "Rep_\<I> (map_\<I> x xa xb) = (\<lambda>x. xa ` Rep_\<I> xb (x x))"
sp_Nil_in_extends_imp: fixes v :: "'a \<times> nat"   and v' :: "'a \<times> nat"   and subs' :: "(('a \<times> nat) \<times> 'a \<times> nat) set"   and v1 :: "'a \<times> nat"   and v2 :: "'a \<times> nat" assumes "extends g subs (v, v') subs'"   and "SubRel.subpath g v1 [] v2 subs'" shows "SubRel.subpath g v1 [] v2 subs = (v1 \<noteq> v \<or> v2 \<noteq> v')"
val_ab: shows "valof (a - b) = valof (zerosign (if is_zero a \<and> is_zero b \<and> sign a \<noteq> sign b then sign a else 0) (IEEE.round To_nearest (valof a - valof b)))"
is_offending_flows_min_set_def: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "('c \<times> 'c) set"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar f G nP \<equiv> SecurityInvariant_withOffendingFlows.is_offending_flows sinvar f G nP \<and> (\<forall>(e1, e2)\<in>f. \<not> sinvar (FiniteGraph.add_edge e1 e2 (FiniteGraph.delete_edges G f)) nP)"
is_dghm_op': fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<AA>' :: "V"   and \<BB>' :: "V"   and \<alpha>' :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<AA>' = op_dg \<AA>"   and "\<BB>' = op_dg \<BB>"   and "\<alpha>' = \<alpha>" shows "op_dghm \<FF> : \<AA>' \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>'\<^esub> \<BB>'"
cod_simps_3: shows "dom_\<nu>.cod.leg1 = doms.\<nu>.leg1"
fps_shift_fps_shift: fixes m :: "nat"   and n :: "nat"   and f :: "'a fps" shows "fps_shift (m + n) f = fps_shift m (fps_shift n f)"
at_in_sub_branch: fixes i :: "'c"   and ps :: "('b, 'c) fm list"   and a :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and f :: "'c \<Rightarrow> 'd" assumes "p at i in' set ((ps, a) # branch)" shows "sub f p at f i in' set ((sub_list f ps, f a) # sub_branch f branch)"
DSourcesA8_L0: shows "DSources level0 sA8 = {sA7, sA9}"
appendS_defined: fixes xs :: "('b, 'c) Stream"   and ys :: "('b, 'd) Stream" assumes "xs \<noteq> \<bottom>"   and "ys \<noteq> \<bottom>" shows "appendS\<cdot>xs\<cdot>ys \<noteq> \<bottom>"
simps: fixes it_has_next :: "'b \<Rightarrow> bool Heap"   and it_next :: "'b \<Rightarrow> ('c \<times> 'b) Heap"   and l_prepend :: "'c \<Rightarrow> 'a \<Rightarrow> 'a Heap"   and it :: "'b"   and l :: "'a" shows "to_list_ga_rec it_has_next it_next l_prepend it l = it_has_next it \<bind> (\<lambda>b. if b then it_next it \<bind> (\<lambda>(x, it). l_prepend x l \<bind> to_list_ga_rec it_has_next it_next l_prepend it) else return l)"
rel_Grp: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "rel_option (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_option x \<subseteq> A} (map_option f)"
rtrancl_on_rtranclI: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> rtrancl_on F r" shows "(a, b) \<in> r\<^sup>*"
widen_PrimT2_strong: fixes G :: "prog"   and S :: "ty"   and x :: "prim_ty" assumes "G\<turnstile>S\<preceq>PrimT x" shows "S = PrimT x"
bij_betw_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A' :: "'d set" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "bij_betw f A A' = bij_betw g A A'"
huntington_3_pp: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> - y) \<squnion> - (- x \<squnion> y) = - - x"
Q_def: shows "Q = (THE Q. Q \<in> subgroups_of_size q)"
inorder_delete: fixes t :: "'b tree23"   and x :: "'b" assumes "complete t"   and "Sorted_Less.sorted (inorder t)" shows "inorder (delete x t) = del_list x (inorder t)"
hyps_5: assumes "s \<subseteq> r"   and "F \<subseteq> G" shows "(y, b) \<in> rtrancl_on G r"
ImplLeadsto_simple: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> F \<leadsto> G"
simps_2: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list"   and n :: "nat" shows "map_index f (x # xs) n = f x n # map_index f xs (Suc n)"
higher_differentiable_on_cong: fixes S :: "'a set"   and T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "open S"   and "S = T"   and "\<And>x. x \<in> T \<Longrightarrow> f x = g x" shows "higher_differentiable_on S f n = higher_differentiable_on T g n"
a_insert_node_def: fixes ptr :: "(_) object_ptr"   and new_child :: "(_) node_ptr"   and reference_child_opt :: "(_) node_ptr option" shows "a_insert_node ptr new_child reference_child_opt = Heap_Error_Monad.bind (get_child_nodes ptr) (\<lambda>children. set_child_nodes ptr (insert_before_list new_child reference_child_opt children))"
reduced_ifex: fixes b :: "'c bexp" shows "reduced (Boolean_Expression_Example.ifex_of b) {}"
state_sim_sym: fixes s :: "'b"   and A :: "('b \<Rightarrow> bool) set"   and t :: "'b" assumes "s ~A~ t" shows "t ~A~ s"
length_bound: fixes emBits :: "nat"   and maskedDB :: "bit list"   and a :: "bit list"   and b :: "bit list" assumes "emBits mod 8 \<noteq> 0"   and "8 \<le> length maskedDB" shows "length (remzero (maskedDB_zero maskedDB emBits @ a @ b)) \<le> length (maskedDB @ a @ b) - (8 - emBits mod 8)"
mulex1_empty: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset" shows "mulex1 P M {#} = False"
M_cong_7_mod_8: shows "[M = 7] (mod 8)"
Inf_utrans_def: fixes S :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) set" shows "Inf_utrans S = (if S = {} then \<lambda>P s. 1 else Inf_trans S)"
elims: fixes x :: "'a"   and xa :: "'a list \<times> 'a list"   and y :: "'a list \<times> 'a list" assumes "replace_last_t x xa = y" obtains a :: "'a" where "x = a__" and "xa = ([], [])" and "y = ([], [])"   | a :: "'a" and fs :: "'a list" and l :: "'a" and ls :: "'a list" where "x = aa__" and "xa = (fs, l # ls)" and "y = (fs, aa__ # ls)"   | a :: "'a" and v :: "'a" and va :: "'a list" where "x = a" and "xa = (v # va, [])" and "y = undefined"
assms: shows "limitin X (\<lambda>i. f (i + k)) l sequentially"
llist_of_tllist_lappendt: fixes xs :: "'a llist"   and tr :: "('a, 'b) tllist" shows "llist_of_tllist (lappendt xs tr) = lappend xs (llist_of_tllist tr)"
If_else_absorb: fixes c :: "tr"   and e :: "'a"   and t :: "'a" assumes "c = \<bottom> \<Longrightarrow> e = \<bottom>"   and "c = TT \<Longrightarrow> e = t" shows "If c then t else e = e"
max_reg_def: fixes g :: "vname aexp" shows "AExp.max_reg g = (let regs = AExp.enumerate_regs g in if regs = {} then None else Some (Max regs))"
prems_3: shows "atom ` (p \<bullet> V) \<sharp>* V"
finite_ipoly_roots: fixes p :: "int poly" assumes "p \<noteq> 0" shows "finite {x. poly (real_of_int_poly p) x = 0}"
preserves_hom: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "CC.in_hom f a b" shows "\<guillemotleft>T f : T a \<rightarrow> T b\<guillemotright>"
obligation1': fixes s :: "(nat list \<times> (bool list \<times> nat list + nat list)) pmf"   and Da :: "(nat list \<times> bool list \<times> nat list) pmf"   and Db :: "(nat list \<times> nat list) pmf"   and h :: "nat list"   and qs :: "nat list" assumes "map_pmf con s = Sum_pmf (8 / 10) Da Db" shows "Partial_Cost_Model.config'_rand (COMB h) s qs = map_pmf noc (Sum_pmf (8 / 10) (Partial_Cost_Model.config'_rand BIT Da qs) (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) Db qs))"
Fejer_kernel_continuous_strong: fixes n :: "nat" shows "continuous_on {- (2 * pi)<..<2 * pi} (Fejer_kernel n)"
subs: shows "{0<--<t + r / 2} \<subseteq> T"
class_to_eq_linorder: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "eq_linorder (dflt_cmp less_eq less)"
absorb_iff2: fixes a :: "'h"   and b :: "'h" shows "(a \<le> b) = (max a b = b)"
ct_list_eq_commute: fixes xs :: "ct list"   and ys :: "ct list" assumes "ct_list_eq xs ys" shows "ct_list_eq ys xs"
Un_commute: fixes A :: "'b set"   and B :: "'b set" shows "(A::'b set) \<union> (B::'b set) = B \<union> A"
comm_ring_1_axioms: shows "class.comm_ring_1 (*) (1::'a) (+) (0::'a) (-) uminus"
assms: shows "r\<inverse>\<^sub>\<bullet> \<subseteq>\<^sub>\<circ> s"
induct: fixes x :: "'state"   and P :: "'state \<Rightarrow> bool" assumes "reachNT x"   and "P istate"   and "\<And>trn. \<lbrakk>reachNT (srcOf trn); P (srcOf trn); step (srcOf trn) (actOf trn) = (outOf trn, tgtOf trn); \<not> T trn\<rbrakk> \<Longrightarrow> P (tgtOf trn)" shows "P x"
simps: fixes B :: "ty"   and C :: "ty"   and X :: "ctrm set" shows "ORRIGHT1 (B OR C) X = {<b>:OrR1 <a>.M b |a b M. fic (OrR1 <a>.M b) b \<and> <a>:M \<in> X}"
eucl_of_list_inner: fixes i :: "'b"   and xs :: "real list" assumes "i \<in> Basis"   and "length xs = DIM('b)" shows "eucl_of_list xs \<bullet> i = xs ! index Basis_list i"
setmult_rcos_assoc: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H <#> (K #> x) = H <#> K #> x"
a: shows "a \<in> g \<cdot>| Class \<one>"
assms_1: shows "sgnx (poly p) (at_right a) \<noteq> sgnx (poly p) (at_left b)"
iT_Plus_cut_greater2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>> (t + k) = I \<down>> t \<oplus> k"
elims_2: fixes x :: "'a ext0"   and xa :: "'a ext0" assumes "x \<le> xa" obtains uu_ :: "'a ext0" where "xa__ = ext0.Bot" and "xa = uu_"   | x :: "'a" and y :: "'a" where "xa__ = ext0.Val x" and "xa = ext0.Val y" and "x \<le> y"   | uw_ :: "'a" where "xa__ = ext0.Val uw_" and "xa = ext0.Top"   | "xa__ = ext0.Top" and "xa = ext0.Top"
Lp_D_4: fixes p :: "real"   and f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "0 < p"   and "f \<in> lspace M (ennreal p)" shows "eNorm (\<LL> (ennreal p) M) f = ennreal ((LINT x|M. \<bar>f x\<bar> powr p) powr (1 / p))"
cases: fixes x :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping" obtains nt :: "nat" and gc :: "nat" and maskL :: "bool list" and maskR :: "bool list" and data_prev :: "(nat \<times> 'a option list set) queue" and data_in :: "(nat \<times> 'a option list set) queue" and tuple_in :: "('a option list, nat) mapping" and tuple_since :: "('a option list, nat) mapping" where "x = (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)"
RR: shows "(\<lambda>(b, k). (Inum (a # bs) b, k)) ` R = (\<lambda>(b, k). (Inum (a # bs) b, k)) ` set (\<rho> p)"
triv_subgroup: shows "subgroup {\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>} (G\<lparr>carrier := H\<rparr>)"
f_borel_Suc_expl_cond_expect: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure"   and w :: "bool stream" assumes "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "g -` {g w} \<in> sets (F n)"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z"   and "0 < p"   and "p < 1" shows "expl_cond_expect M g f w = p * f (pseudo_proj_True n w) + (1 - p) * f (pseudo_proj_False n w)"
topological_space_generate_topology: fixes S :: "'c set set" shows "class.topological_space (generate_topology S)"
assms_3: shows "f ` U\<^sub>M\<^sub>_\<^sub>1 = U\<^sub>M\<^sub>_\<^sub>2"
conj_residuatedI2: fixes f :: "'a \<Rightarrow> 'a" assumes "\<exists>g. conjugation_pair g f" shows "residuated f"
dioph_is_node_def: fixes ks :: "int list"   and l :: "int"   and m :: "int" shows "dioph_is_node ks l m = (\<bar>m\<bar> \<le> max \<bar>l\<bar> (sum_list (map abs ks)))"
OUTfromVCorrect1_data8: shows "OUTfromVCorrect1 data8"
word_shiftl_add_distrib: fixes x :: "'a word"   and y :: "'a word"   and n :: "nat" shows "x + y << n = (x << n) + (y << n)"
inter: shows "{0..<length p} \<inter> {d} = {d}"
comp_def: fixes f :: "'d::type \<Rightarrow> 'c::type"   and g :: "'b::type \<Rightarrow> 'd::type" shows "(f::'d::type \<Rightarrow> 'c::type) \<circ> (g::'b::type \<Rightarrow> 'd::type) = (\<lambda>x::'b::type. f (g x))"
d_delta_lnexp_cf7_nonneg: fixes x :: "real" assumes "0 < numer_cf7 x"   and "0 < numer_cf7 (- x)" shows "0 \<le> diff_delta_lnexp_cf7 x"
assms_2: shows "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> n'"
distinct_4: fixes x3 :: "char list list" shows "Called x3 \<noteq> No_ics"
single_eq_add_mset: fixes a :: "'k"   and b :: "'k"   and M :: "'k multiset" shows "({#a#} = add_mset b M) = (b = a \<and> M = {#})"
simps_2: fixes uu :: "'a \<Rightarrow> bool" shows "uu \<Turnstile> \<bottom> = False"
incseq_SucD: fixes A :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "incseq A" shows "A i \<le> A (Suc i)"
in_domains_char: fixes a :: "('o, 'a) arr"   and f :: "('o, 'a) arr" shows "(a \<in> domains f) = ((f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f)) \<and> a = MkIde (Dom f))"
bij_betw_from_nat_into: fixes S :: "'d set" assumes "countable S"   and "infinite S" shows "bij_betw (from_nat_into S) UNIV S"
Fourier_sum_offset_unpaired: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and n :: "nat" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<Sum>k\<le>2 * n. Fourier_coefficient f k * trigonometric_set k t) = (\<Sum>k\<le>n. Fourier_coefficient (\<lambda>x. f (x + t)) (2 * k) * trigonometric_set (2 * k) 0)"
sfeq: shows "\<forall>n w. prices Mkt x n w = val_process Mkt pf2 n w"
zero_def: fixes x :: "'b"   and R :: "('b \<Rightarrow> 'c) set" shows "zero x R \<equiv> \<forall>u\<in>R. u x = (0::'c)"
diagonal_step_PQ_PA: fixes P :: "(('b, 'c) vec, 'c) vec"   and Q :: "(('b, 'd) vec, 'd) vec"   and A :: "(('b, 'd) vec, 'c) vec"   and i :: "nat"   and k :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and p :: "'b"   and q :: "'b"   and u :: "'b"   and v :: "'b"   and d :: "'b" assumes "(P, Q) = diagonal_step_PQ A i k bezout"   and "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)" shows "P ** A = row_add (interchange_rows (row_add A (from_nat k) (from_nat i) p) (from_nat i) (from_nat k)) (from_nat k) (from_nat i) (- v)"
is_extensional: fixes f :: "'d" assumes "\<not> D.arr f" shows "D.\<ll>' f = D.null"
u_m_other: shows "\<forall>a. a \<notin> u_owns \<longrightarrow> u_m a = m a"
reusables_neq_Nil_if_well_shaped_and_overlapping: fixes t :: "'c tree"   and w :: "nat \<times> nat" assumes "well_shaped t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "fst w \<le> r t" shows "reusables t w \<noteq> []"
r_zero: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "x + 0\<^sub>v n = x"
simps_5: fixes f1 :: "nat \<Rightarrow> 'c"   and f2 :: "'c" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
assms: fixes i :: "'a" assumes "i \<in> K" shows "0 < r i \<and> r i \<le> B"
triangulize_closed: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "triangulize sel_fun mf A = (r', A')"   and "A \<in> carrier_mat m n" shows "A' \<in> carrier_mat m n"
lw_1: shows "(\<lambda>n. w n / real n) \<longlonglongrightarrow> lw"
Am_ub3: fixes \<alpha> :: "real"   and \<beta> :: "real"   and t :: "'b tree" assumes "Splay_Analysis \<alpha> \<beta>"   and "bst t" shows "Splay_Analysis.Am \<alpha> \<beta> t \<le> log \<alpha> (real (size1 t)) + 1"
dghm_ArrMap_vdomain: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
length_update_matchF: fixes n :: "nat"   and I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and rels :: "event_data option list set list"   and nt :: "nat"   and aux :: "(nat \<times> event_data option list set list \<times> event_data option list set) list" shows "length (update_matchF n I mr mrs rels nt aux) = Suc (length aux)"
nat_less_power_trans: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n < 2 ^ (m - k)"   and "k \<le> m" shows "2 ^ k * n < 2 ^ m"
h_inv_bij: shows "bij_betw (inv_into (carrier R) h) (carrier S) (carrier R)"
closure_def: fixes f :: "'a \<Rightarrow> 'a" shows "closure f \<equiv> (\<forall>x. x \<le> f x) \<and> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y) \<and> (\<forall>x. f (f x) = f x)"
rbt_less_prop: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and k :: "'c"   and t :: "('c, 'd) rbt" shows "ord.rbt_less less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less x k)"
simps_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and v :: "'b"   and xs :: "('a \<times> 'b) list" shows "ListUpdateE f ((x, v) # xs) = (ListUpdateE f xs)(x := v)"
szip_smap_fst: fixes xs :: "'c stream"   and ys :: "'d stream" shows "smap fst (xs ||| ys) = xs"
t_stop_zero: fixes a :: "real"   and v :: "real" assumes "braking_movement a v"   and "movement.t_stop a v = 0" shows "v = 0"
basic_div_exp: fixes x :: "'b"   and n :: "nat"   and y :: "'b" assumes "to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract" shows "to_fract (basic_div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"
intro: fixes s\<^sub>0 :: "'a"   and lstep :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'a" assumes "df_system.is_fair_run s\<^sub>0 lstep s" shows "fair_run_axioms s\<^sub>0 lstep s"
rank_of_eq_card_basis_in: fixes \<E> :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> B" shows "rank_of \<E> = card B"
weak_equivalence_implies_weak_bisimilarity: fixes P :: "'state"   and Q :: "'state" assumes "S_transform.weakly_logically_equivalent P Q" shows "P \<approx>\<cdot>\<^sub>S Q"
Predict_\<pi>_fix: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "Predict k (\<pi> k T I) = \<pi> k T I"
underS_Field: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "underS r a \<subseteq> Field r"
D_While_If: fixes b :: "bexp"   and c :: "com" shows "D (WHILE b DO c) = D (IF b THEN c;; WHILE b DO c ELSE SKIP)"
sinks_aux_subset_dom: fixes U :: "'b set"   and V :: "'b set"   and I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "U \<subseteq> V" shows "sinks_aux I D U xs \<subseteq> sinks_aux I D V xs"
star_def: fixes x :: "'a" shows "x\<^sup>\<star> = top"
wf_graph_remove_edges: fixes V :: "'c set"   and E :: "('c \<times> 'c) set"   and X :: "('c \<times> 'c) set" assumes "wf_graph \<lparr>nodes = V, edges = E\<rparr>" shows "wf_graph \<lparr>nodes = V, edges = E - X\<rparr>"
upper_asymptotic_density_Union: fixes A :: "nat \<Rightarrow> nat set" assumes "summable (\<lambda>n. upper_asymptotic_density (A n))" shows "\<exists>B. upper_asymptotic_density B \<le> (\<Sum>n. upper_asymptotic_density (A n)) \<and> (\<forall>n. \<exists>N. A n \<inter> {N..} \<subseteq> B)"
gram_schmidt_sub_eq: fixes n :: "nat"   and us :: "'b vec list"   and ws :: "'b vec list" shows "rev (gram_schmidt_sub n us ws) = rev us @ gram_schmidt_sub2 n us ws"
square_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "square R S T U = (\<forall>x y. R x y \<longrightarrow> (\<forall>z. S x z \<longrightarrow> (\<exists>u. T y u \<and> U z u)))"
le_rem_id_map_2: fixes w :: "'a list" shows "map le_rem_s (map ENV w) = w"
distinct_rotate: fixes n :: "nat"   and xs :: "'d list" shows "distinct (List.rotate n xs) = distinct xs"
pow_nat_mod_ord: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier G"   and "ord a \<noteq> 0" shows "a [^] n = a [^] (n mod ord a)"
Inf_subdistl_iso: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 f :: "'b \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "f \<circ> Inf \<le> Inf \<circ> (`) f" shows "order.mono less_eq f"