hcju/isabelleps huggingface dataset:
hrp_imp_refl: fixes RR :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)" shows "hrp_imp RR RR"
SD_preorderD_3: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and q :: "'b pmf"   and p :: "'b pmf"   and x :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "SD le q p"   and "x \<in> carrier" shows "measure_pmf.prob q (preferred_alts le x) \<le> measure_pmf.prob p (preferred_alts le x)"
map_of_concat': assumes "k \<in> dom (map_of (l ! bounded_hashcode_nat (length l) k)) \<and> bounded_hashcode_nat (length l) k < length l" shows "k \<in> dom (map_of (concat l))"
eliminate_entries_gen_zero: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and mminus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ttimes :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and v :: "integer \<Rightarrow> 'b"   and I :: "nat"   and J :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "eliminate_entries_gen_zero mminus ttimes z v A I J $$ (i, j) = (if v (integer_of_nat i) = z \<or> i = I then A $$ (i, j) else mminus (A $$ (i, j)) (ttimes (v (integer_of_nat i)) (A $$ (I, j))))"
le: shows "\<parallel>gso i\<parallel>\<^sup>2 \<le> \<parallel>h\<parallel>\<^sup>2"
hd_suffixes: fixes xs :: "'c list" shows "hd (suffixes xs) = []"
inject_4: fixes x51 :: "val"   and x52 :: "val"   and y51 :: "val"   and y52 :: "val" shows "(<|x51, x52|> = <|y51, y52|>) = (x51 = y51 \<and> x52 = y52)"
path_independent_on_consistency_on: fixes A :: "'b set"   and f :: "'b set \<Rightarrow> 'b set" assumes "path_independent_on A f" shows "consistency_on A f"
subst_def: fixes f :: "'a \<Rightarrow> 'b option"   and mem :: "'a \<Rightarrow> 'b" shows "subst f mem = (\<lambda>x. case f x of None \<Rightarrow> mem x | Some v \<Rightarrow> v)"
prj0_dup: fixes a :: "'a" assumes "ide a" shows "\<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<d>[a] = a"
inv_resl_neg': fixes y :: "'a"   and x :: "'a" shows "inv_resl (-' x) (-' y) = inv_resr x y"
IMP_PRE_def: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool" shows "IMP_PRE P1 P2 \<equiv> \<forall>x. P1 x \<longrightarrow> P2 x"
unreach_l: shows "\<not> s\<turnstile> l reachable_from x"
Macaulay_list_def: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.extended_ord.punit.Macaulay_list ps = filter (\<lambda>p. p \<noteq> 0) (mat_to_polys (pp_pm.extended_ord.punit.Keys_to_list ps) (row_echelon (pp_pm.extended_ord.punit.Macaulay_mat ps)))"
valid_cfg_cont: fixes cfg :: "'s cfg"   and s :: "'s" assumes "cfg \<in> valid_cfg"   and "s \<in> set_pmf (action cfg)" shows "cont cfg s \<in> valid_cfg"
F_def: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "Cauchy_Binet.F n m = {f \<in> {0..<n} \<rightarrow> {0..<m}. \<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i}"
assms_2: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1 \<in> H"   and "h2 \<in> H" shows "h1 \<otimes> h2 = h2 \<otimes> h1"
conj_absorb: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "conj x x = x"
wt: shows "wt_step r \<top> step \<tau>s"
WT_gpvD_1: fixes \<Gamma> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv"   and out :: "'d"   and c :: "('f, 'd, 'e) rpv" assumes "\<Gamma> \<turnstile>g gpv \<surd>"   and "IO out c \<in> set_spmf (the_gpv gpv)" shows "out \<in> outs_\<I> \<Gamma>"
lemma_2_5_11: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "less_eq (times a b) (inf a b)"
subst_compose2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Z :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var" assumes "good X"   and "good Y"   and "good Z"   and "ys \<noteq> zs \<or> y \<noteq> z"   and "fresh ys y Z" shows "X #[Y / y]_ys #[Z / z]_zs = X #[Z / z]_zs #[Y #[Z / z]_zs / y]_ys"
Collect_cong: fixes P :: "'q \<Rightarrow> bool"   and Q :: "'q \<Rightarrow> bool" assumes "\<And>x. P x = Q x" shows "{x. P x} = {x. Q x}"
length_append_singleton: fixes xs :: "'a list"   and x :: "'a" shows "length (xs @ [x]) = Suc (length xs)"
evalC_surjective: fixes b :: "nat"   and n :: "nat" shows "\<exists>n'\<in>hbase (Suc (Suc b)). evalC (Suc (Suc b)) n' = n"
lt_eq_TT_iff: fixes x :: "'a"   and y :: "'a" shows "(lt\<cdot>x\<cdot>y = TT) = (compare\<cdot>x\<cdot>y = LT)"
retract_of_space_imp_subset: fixes S :: "'b set"   and X :: "'b topology" assumes "S retract_of_space X" shows "S \<subseteq> topspace X"
Dom_dom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Dom (local.dom f) = Dom f"
size_wbt_eq_size: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.wbt \<Delta>1 \<Delta>2 t" shows "size_wbt t = size t"
hd_map: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []" shows "hd (map f xs) = f (hd xs)"
wf_digraph_del_vert: fixes u :: "'a" shows "wf_digraph (del_vert u)"
d_def: shows "d = (\<lambda>n. 1 + (2 / 3) ^ (n + 1))"
relcl_trans: fixes a :: "'b"   and b :: "'b"   and L :: "'b set set"   and c :: "'b" assumes "(a, b) \<in> relcl L"   and "(b, c) \<in> relcl L"   and "lattice L" shows "(a, c) \<in> relcl L"
map_dual_func2: shows "\<partial>\<^sub>F id = id"
load_packed_def: fixes sx :: "sx"   and m :: "mem"   and n :: "nat"   and off :: "nat"   and lp :: "nat"   and l :: "nat" shows "load_packed sx m n off lp l = map_option (sign_extend sx l) (load m n off lp)"
subgroup_def: fixes A :: "'a set" shows "subgroup A \<equiv> (0::'a) \<in> A \<and> (\<forall>a\<in>A. \<forall>b\<in>A. a + b \<in> A) \<and> (\<forall>a\<in>A. - a \<in> A)"
nat_lcm_def: fixes x :: "nat"   and y :: "nat" shows "nat_lcm x y = x * y div nat_gcd x y"
unfolding_next_step_equivalent: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set" assumes "w \<Turnstile>\<^sub>n Unf \<phi>[X]\<^sub>\<nu>" shows "suffix 1 w \<Turnstile>\<^sub>n af_letter \<phi> (w 0)[X]\<^sub>\<nu>"
ltl_prop_entailment_monotonI: fixes S :: "'b ltln set"   and \<phi> :: "'b ltln"   and S' :: "'b ltln set" assumes "S \<Turnstile>\<^sub>P \<phi>"   and "S \<subseteq> S'" shows "S' \<Turnstile>\<^sub>P \<phi>"
starrel_iff: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "((X, Y) \<in> starrel) = (\<forall>\<^sub>F n in \<U>. X n = Y n)"
wf_list_graph_def: fixes G :: "'b list_graph" shows "wf_list_graph G = (distinct (nodesL G) \<and> distinct (edgesL G) \<and> wf_list_graph_axioms G)"
comparator_sum_simps_1: fixes comp\<^sub>'\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> order"   and comp\<^sub>'\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> order"   and x :: "'c"   and y :: "'c" shows "comparator_sum comp\<^sub>'\<^sub>a comp\<^sub>'\<^sub>b (Inl x) (Inl y) = comp\<^sub>'\<^sub>a x y"
finite_enumerate_regs: fixes g :: "vname gexp" shows "finite (GExp.enumerate_regs g)"
pr_in_hom_2: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>pr1 a b : local.prod a b \<rightarrow> a\<guillemotright>"
equiv'_def: shows "A_B.equiv' \<equiv> \<lambda>a b. a \<sim> b \<and> PA a \<and> PB b"
pdevs_domain_eq_pdevs_of_list: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'b)" shows "pdevs_domain (pdevs_of_list xs) = {0..<length xs}"
abs_conv_abscissa_mono: fixes g :: "'b fds"   and f :: "'b fds" assumes "\<And>s. fds_abs_converges g s \<Longrightarrow> fds_abs_converges f s" shows "abs_conv_abscissa f \<le> abs_conv_abscissa g"
bvars_subst_ident: fixes S :: "('d, 'c) strand_step list"   and \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "bvars\<^sub>s\<^sub>t S = bvars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
take_below: fixes n :: "nat"   and x :: "[:'a:]" shows "slist_take n\<cdot>x \<sqsubseteq> x"
A: shows "\<delta>' = dist_code p\<^sub>0 p\<^sub>1"
finsum_dim: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b vec"   and n :: "nat" assumes "finite A"   and "f \<in> A \<rightarrow> carrier_vec n" shows "dim_vec (finsum (module_vec TYPE('b) n) f A) = n"
rHom_aHom: fixes f :: "'c \<Rightarrow> 'd"   and A :: "('c, 'e) Ring_scheme"   and R :: "('d, 'f) Ring_scheme" assumes "f \<in> rHom A R" shows "f \<in> aHom A R"
Derivation_swap_single_mid_to_front: fixes i :: "nat"   and j :: "nat"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and r :: "symbol \<times> symbol list"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "i < j"   and "derivation_ge D j"   and "Derivation a (D @ [(i, r)] @ E) b" shows "Derivation a ((i, r) # derivation_shift D 1 (length (snd r)) @ E) b"
a1: shows "low_equal s1 s2 \<and> t1 = mem_mod_w32 10 a bm data s1 \<and> t2 = mem_mod_w32 10 a bm data s2"
eq_fin_rev: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "rev w\<^sub>1 =\<^sub>F rev w\<^sub>2"
msubstneg_nb: fixes p :: "fm"   and t :: "tm"   and c :: "poly" assumes "islin p"   and "tmbound0 t" shows "bound0 (msubstneg p c t)"
compute_\<ff>s_def: fixes s :: "'b list" shows "compute_\<ff>s s = (let \<ff>s = replicate (length s + 1) 0; i = 0; j = 1 in WHILE\<^sub>T\<^bsup>I_out_cb s\<^esup> (\<lambda>(\<ff>s, uu_, j). j < length \<ff>s) (\<lambda>(\<ff>s, i, j). WHILE\<^sub>T\<^bsup>I_in_cb s j\<^esup> (\<lambda>i. 0 < i \<and> s ! (i - 1) \<noteq> s ! (j - 1)) (\<lambda>i. ASSERT (i - 1 < length \<ff>s) \<bind> (\<lambda>_. Let (\<ff>s ! (i - 1)) RETURN)) i \<bind> (\<lambda>i. let i = i + 1 in ASSERT (j < length \<ff>s) \<bind> (\<lambda>_. let \<ff>s = \<ff>s[j := i]; j = j + 1 in RETURN (\<ff>s, i, j)))) (\<ff>s, i, j) \<bind> (\<lambda>(\<ff>s, uu_, uu_). RETURN \<ff>s))"
list_emb_Nil: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list" shows "list_emb P [] ys"
jumpF_poly_bot_def: fixes q :: "real poly"   and p :: "real poly" shows "jumpF_poly_bot q p = jumpF (\<lambda>x. poly q x / poly p x) at_bot"
that: fixes P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P''"   and "insertAssertion (extractFrame Q) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame P'') \<Psi>"   and "\<Psi> \<rhd> P'' \<longmapsto> \<alpha> \<prec> P'" shows "thesis"
bigo: shows "(\<lambda>x. A x - ln x) \<in> O(\<lambda>x. 1)"
lincomb_obtain_same_length_coeffs: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "\<exists>ss. set ss \<subseteq> UNIV \<and> length ss = length ms \<and> take (length rs) ss = take (length ms) rs \<and> rs \<bullet>\<cdot> ms = ss \<bullet>\<cdot> ms"
dghm_cn_comp_components_4: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>HomCod\<rparr> = \<GG>\<lparr>HomCod\<rparr>"
undo_factorize_def: fixes cfs :: "rat \<times> (nat \<times> nat) list"   and signs :: "rat list" shows "undo_factorize cfs signs = squash (case cfs of (c, fs) \<Rightarrow> c * (\<Prod>(f, pow)\<leftarrow>fs. signs ! f ^ Suc pow))"
n_rules_def: shows "n_rules = flat (smap (\<lambda>r. map (Pair r) (consequent r)) rules)"
powr_closureI: fixes f :: "'a \<Rightarrow> real"   and p :: "real" shows "(\<lambda>x. f x powr p) \<in> powr_closure f"
eq: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = order.Eq) = (x = y)"
step_cjump_false: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and n :: "nat"   and d :: "'dyn"   and H :: "'henv"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.ICJump n"   and "is_false d" shows "Sinca.step (State F H (Frame f pc (d # \<Sigma>) # st)) (State F H (Frame f (Suc pc) \<Sigma> # st))"
that_1: assumes "Y = (\<lambda>_. \<bottom>)" shows "thesis"
arg_apps_Cons: fixes s :: "('a, 'b) tm"   and sa :: "('a, 'b) tm"   and ss :: "('a, 'b) tm list" shows "arg (apps (App s sa) ss) = last (sa # ss)"
V_def: shows "V \<equiv> {v. \<forall>x\<in>X. (0::'b) \<le> v x}"
simps_10: fixes v :: "int"   and va :: "Cooper.num" shows "not (Dvd v va) = NOT (Dvd v va)"
iso_inv_iso: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.iso f" shows "Maps.iso (Maps.inv f)"
nz: shows "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'a)"
inducts: fixes s :: "'a + 'b"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "setrp s x"   and "\<And>x. s = Inr x \<Longrightarrow> P x" shows "P x"
type_definition_Domainp': fixes a :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_equality a" shows "Transfer.Rel a (Domainp T) (\<lambda>x. x \<in> A)"
cycle_conv: fixes p :: "'b list" shows "cycle p = (\<exists>u. awalk u p u \<and> distinct (tl (awalk_verts u p)) \<and> distinct p \<and> p \<noteq> [])"
polynomial_of_mset_def: fixes \<phi> :: "char list \<Rightarrow> nat"   and p :: "(char list multiset \<times> int) multiset" assumes "poly_embed \<phi>" shows "poly_embed.polynomial_of_mset \<phi> p = \<Sum>\<^sub># (poly_embed.mononom_of_vars \<phi> `# p) 0"
assms_4: shows "\<psi>2 \<in> fmla"
funion_absorb2: fixes B :: "'c fset"   and A :: "'c fset" assumes "B |\<subseteq>| A" shows "A |\<union>| B = A"
convergent_const: fixes c :: "'b" shows "convergent (\<lambda>n. c)"
OF_match_linear_not_noD: fixes \<gamma> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and oms :: "('c, 'b) flow_entry_match list"   and p :: "'d" assumes "OF_match_linear \<gamma> oms p \<noteq> NoAction" shows "\<exists>ome. ome \<in> set oms \<and> \<gamma> (ofe_fields ome) p"
r_sub_prim: shows "prim_recfn 2 r_sub"
empty_not_insert: fixes a :: "'a"   and A :: "'a set" shows "\<not> {} = insert a A"
select_result_I: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and P :: "'o \<Rightarrow> bool" assumes "h \<turnstile> ok f"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> P x" shows "P |h \<turnstile> f|\<^sub>r"
chamber_distance_def: fixes C :: "'b set"   and D :: "'b set" shows "codomain.chamber_distance C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. codomain.gallery (C # Cs @ [D]))))"
assms_2: fixes res1 :: "('a, 'b) resource"   and res2 :: "('c, 'd) resource" assumes "X res1 res2" shows "rel_fun A (rel_spmf (rel_prod B (\<lambda>res1 res2. X res1 res2 \<or> rel_resource A B res1 res2))) (run_resource res1) (run_resource res2)"
simps_3: fixes m :: "nat" shows "Partition (Suc m) 0 = 0"
I_vals_nonempty: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I" shows "v ` I \<noteq> {}"
enum_ex_fun_def: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool" shows "enum_ex P = ex_n_lists (\<lambda>bs. P (the \<circ> map_of (zip enum bs))) (length enum)"
conditional_mutual_information_def: fixes b :: "real"   and MX :: "'e measure"   and MY :: "'f measure"   and MZ :: "'g measure"   and X :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'e"   and Y :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'f"   and Z :: "'b \<times> 'c \<times> 'd \<Rightarrow> 'g" shows "P.conditional_mutual_information b MX MY MZ X Y Z \<equiv> P.mutual_information b MX (MY \<Otimes>\<^sub>M MZ) X (\<lambda>x. (Y x, Z x)) - P.mutual_information b MX MZ X Z"
On_CTRL_Init_def: shows "On_CTRL_Init = ''TunerMode''"
mat_ge_def: fixes A :: "'a mat"   and B :: "'a mat" shows "(A \<ge>\<^sub>m B) = (\<forall>i<dim_row A. \<forall>j<dim_col A. B $$ (i, j) \<le> A $$ (i, j))"
apex_def: shows "HH\<mu>\<nu>\<pi>.cod.apex \<equiv> C.dom HH\<mu>\<nu>\<pi>.cod.leg0"
bind_returns_heap_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and h'' :: "'a" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"
flowsto_def: fixes X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and CX :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and X1 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set" shows "flowsto X0 T CX X1 = (\<forall>(x0, d0)\<in>X0. \<exists>h\<in>T. h \<in> existence_ivl0 x0 \<and> (flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1 \<and> (\<forall>h'\<in>{0<--<h}. (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX))"
BisT_prefix: shows "BisT \<subseteq> RetrT BisT"
eq1: shows "\<one>\<^bsub>homology_group (int p) (nsphere p)\<^esub> = \<one>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub>"
bidel_arc_same_face_axioms: shows "bidel_arc_same_face G M a"
outside_inside: fixes S :: "'b set" shows "outside S = - (S \<union> inside S)"
ahm_\<alpha>_def: fixes hm :: "('a, 'b) hashmap" shows "ahm_\<alpha> hm = ArrayHashMap_Impl.ahm_\<alpha> (impl_of hm)"
poly_pCons': fixes a :: "'a"   and q :: "'a poly" shows "poly (pCons a q) = (\<lambda>x. a + x * poly q x)"
det_dim_zero: fixes A :: "'b mat" assumes "A \<in> carrier_mat 0 0" shows "det A = (1::'b)"
equ_3: shows "subtopology (usphere n) (equator n) = nsphere (n - Suc 0)"
WRP_graph_aux_def: shows "WRP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>WRP_aux s x y. x1 = (x, y) \<and> x2 = SyntaxN.Ex s (SeqWRP (Var s) x y) \<and> atom s \<sharp> (x, y))"
sums_zero: shows "(\<lambda>n. 0::'b) sums (0::'b)"
rb_aux_is_RB_upt: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and rword_strict :: "'c \<times> ('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow> 'c \<times> ('d \<Rightarrow>\<^sub>0 'e) \<Rightarrow> bool"   and dgrad :: "'d \<Rightarrow> nat"   and z :: "nat"   and u :: "'c" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_strict_rewrite_ord pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs rword_strict"   and "dickson_grading dgrad"   and "distinct fs"   and "0 \<notin> set fs" shows "qpm_inf_term.is_RB_upt pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs dgrad (qpm_inf_term.rword rword_strict) (set (fst (fst (qpm_inf_term.rb_aux pair_of_term term_of_pair ord ord_term ord_term_strict fs rword_strict (([], qpm_inf_term.Koszul_syz_sigs pair_of_term term_of_pair ord ord_term fs, map Inr [0..<length fs]), z))))) u"
f2: shows "user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s1))) = user_reg_val (fst (fst (get_curr_win () s2))) 0 (snd (fst (get_curr_win () s2)))"
fun_pair_eq: fixes d :: "('fun, 'd) Term.term \<times> ('fun, 'd) Term.term"   and d' :: "('fun, 'd) Term.term \<times> ('fun, 'd) Term.term" assumes "pair d = pair d'" shows "d = d'"
finite_card_of_iff_card2: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B" shows "(|A| \<le>o |B|) = (card A \<le> card B)"
m_closed: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> y \<in> carrier (K [X])"
prefix_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) prefix prefix"
seq_mono_left: fixes c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(c\<^sub>0::'b::refinement_lattice) ; (d::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice) ; d"
ge_eq_refl: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "(=) \<le> R" shows "R x x"
E: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<phi> :: "'b \<times> 'b \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and a :: "'b"   and e :: "'c" assumes "yoneda_lemma_fixed_e C S \<phi> F a e" shows "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)"
Un_absorb: fixes A :: "'b set" shows "A \<union> A = A"
R13_1: shows "pmf (sds R13) a = 1 / 2"
renaming_preserves_literal_order: fixes atom_ordering :: "('a \<times> 'a) set"   and L1 :: "'a Literal"   and L2 :: "'a Literal"   and A :: "'a set" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.literal_ordering atom_ordering L1 L2" shows "propositional_atoms.literal_ordering atom_ordering (propositional_atoms.rename_literal A L1) (propositional_atoms.rename_literal A L2)"
rbt_inter_with_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "ord.rbt_inter_with less f = ord.rbt_inter_with_key less (\<lambda>_. f)"
before_dist_eq_snd: fixes vs :: "'a list"   and r2 :: "'a"   and r1 :: "'a" assumes "distinct vs" shows "(r2 \<in> set (snd (splitAt r1 vs))) = before vs r1 r2"
recurrent_subset_meas_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.recurrent_subset T A \<in> sets M"
Int_ac_1: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
distributive_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "(b + c) \<cdot> a = b \<cdot> a + c \<cdot> a"
intersect_mult_set_block_subset_card: fixes blv :: "'a set"   and p :: "'a set" assumes "blv \<in># \<B>"   and "p \<in># (\<Sum>b2\<in>#remove1_mset blv \<B>. mset_set {y. y \<subseteq> blv \<inter> b2 \<and> card y = 2})" shows "card p = 2"
measurable_iff_sets: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" shows "(f \<in> M \<rightarrow>\<^sub>M N) = (f \<in> space M \<rightarrow> space N \<and> sets (vimage_algebra (space M) f N) \<subseteq> sets M)"
l_inv: fixes G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "Group.group G"   and "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
not_msb_from_less: fixes v :: "'a word" assumes "v < 2 ^ (LENGTH('a) - 1)" shows "\<not> msb v"
matches_simps_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d"   and m' :: "'c match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m m') a p = matches \<gamma> m' a p"
borel_open: fixes A :: "'c set" assumes "open A" shows "A \<in> sets borel"
ground_sub_of_denotts: fixes E :: "char list \<Rightarrow> hterm"   and ts :: "fterm list" shows "ground\<^sub>t\<^sub>s (ts \<cdot>\<^sub>t\<^sub>s sub_of_denot E)"
eq_fps_cos: fixes a :: "'a fps"   and c :: "'a" assumes "fps_nth a 0 = (1::'a)"   and "fps_nth a 1 = (0::'a)"   and "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)" shows "fps_cos c = a"