hcju/isabelleps huggingface dataset:
replicating_expectation_finite: fixes N :: "bool stream measure"   and pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and pyf :: "bool stream \<Rightarrow> real"   and matur :: "nat" assumes "risk_neutral_prob N"   and "filt_equiv G M N"   and "replicating_portfolio pf pyf matur"   and "\<forall>n. \<forall>asset\<in>support_set pf. finite (prices Mkt asset n ` space M)"   and "\<forall>n. \<forall>asset\<in>support_set pf. finite (pf asset n ` space M)"   and "viable_market Mkt"   and "sets (G 0) = {{}, space M}"   and "pyf \<in> borel_measurable (G matur)" shows "fair_price Mkt (integral\<^sup>L N (discounted_value r (\<lambda>m. pyf) matur)) pyf matur"
inf_compl_bot_left2: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> (- x \<cdot> y) = 0"
NatTrans: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and f :: "'i"   and X :: "'g"   and Y :: "'g" assumes "NatTransP \<eta>"   and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y" shows "NTDom \<eta> ## f ;;\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> $$ Y) = \<eta> $$ X ;;\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f)"
path_connected_Times: fixes s :: "'b set"   and t :: "'c set" assumes "path_connected s"   and "path_connected t" shows "path_connected (s \<times> t)"
Torder_axioms_def: fixes D :: "('h, 'i) Order_scheme" shows "Torder_axioms D \<equiv> \<forall>a b. a \<in> carrier D \<longrightarrow> b \<in> carrier D \<longrightarrow> a \<preceq>\<^bsub>D\<^esub> b \<or> b \<preceq>\<^bsub>D\<^esub> a"
mod_by_0: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and modulo :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and a :: "'m" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a zero = a"
faulty2: fixes t :: "lam"   and t' :: "lam"   and x :: "name" assumes "t \<rightarrow> t'"   and "x \<sharp> t" shows "x \<sharp> t'"
totient_le: fixes n :: "nat" shows "totient n \<le> n"
sep_disj_fun_def: fixes m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd" shows "m1 ## m2 \<equiv> domain m1 \<inter> domain m2 = {}"
compactin_imp_closedin: fixes X :: "'c topology"   and S :: "'c set" assumes "Hausdorff_space X"   and "compactin X S" shows "closedin X S"
cmod_eq_Re: fixes z :: "complex" assumes "Im z = 0" shows "cmod z = \<bar>Re z\<bar>"
ChiRel_ChiPlus_OneAncestor: fixes T :: "'d"   and U :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "(T, U) \<in> ChiPlus A"   and "T \<noteq> S"   and "(S, U) \<in> ChiRel A" shows "(T, S) \<in> ChiPlus A"
f_nth_derivative_is_poly: fixes k :: "nat" shows "higher_differentiable_on {0<..} f k \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
br_run_sim: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool"   and r :: "nat \<Rightarrow> 'a" assumes "R = br \<alpha> I"   and "a.is_run r" shows "b.is_run (\<alpha> \<circ> r)"
out_var_indep: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "x \<bowtie> y" shows "$x\<acute> \<bowtie> $y\<acute>"
elims: fixes x :: "nat list \<times> form"   and y :: "nat set" assumes "fv_schema x = y" obtains ps :: "nat list" and f :: "form" where "x = (ps, f)" and "y = fv f - set ps"
take_self: fixes w :: "'b list" shows "take \<^bold>|w\<^bold>| w = w"
rbt_bulkload_is_rbt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "('a \<times> 'b) list" assumes "class.linorder less_eq less" shows "ord.is_rbt less (ord.rbt_bulkload less xs)"
length_enum_CARD: shows "length enum_class.enum = CARD('b)"
conventional: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'val"   and x :: "'node"   and v' :: "'val" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<notin> set (tl ns)"   and "v \<in> allDefs g n"   and "v \<in> allUses g m"   and "x \<in> set (tl ns)"   and "v' \<in> allDefs g x" shows "var g v' \<noteq> var g v"
weight_map_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf" shows "weight_spmf (map_spmf f p) = weight_spmf p"
sel_1: fixes x2 :: "'c" shows "un_Skip (Skip x2) = x2"
iso_codim_map: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> X"   and "y \<in> X" shows "card (f ` x - f ` y) = card (x - y)"
msum_comm: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule K N1 V"   and "LinearCombinations.submodule K N2 V" shows "subspace_sum N1 N2 = subspace_sum N2 N1"
bn_fresh_Act: fixes \<alpha> :: "'d"   and x :: "('b, 'c, 'd) formula" shows "bn \<alpha> \<sharp>* Act \<alpha> x"
rep_in_hom: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and r :: "'b"   and \<rho> :: "'b"   and f :: "'b"   and g :: "'b" assumes "tabulation_data_with_T0 V H \<a> \<i> src trg r \<rho> f g" shows "partial_magma.in_hom V (adjunction_in_bicategory.trnr\<^sub>\<epsilon> V H \<a> \<i> src f (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f) (chosen_right_adjoints.some_counit V H \<a> \<i> src trg f) r \<rho>) (H g (chosen_right_adjoints.some_right_adjoint V H \<a> \<i> src trg f)) r"
1: shows "interpret_floatarith ra vs # vs \<in> aforms_err e (a # VS)"
closed_cball: fixes x :: "'b"   and e :: "real" shows "closed (cball x e)"
lang_dual_final_dual: fixes r :: "'a atom rexp_dual"   and \<Sigma> :: "'a list"   and n :: "nat" shows "final_dual r = ([] \<in> lang_dual \<Sigma> n r)"
LocalSecretsComposition_neg1_s: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and s :: "Secrets" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionLoc PQ"   and "\<not> ine P (sE s)"   and "\<not> ine Q (sE s)"   and "sKS s \<notin> LocalSecrets P"   and "sKS s \<notin> LocalSecrets Q" shows "sKS s \<notin> LocalSecrets PQ"
polynomial_in_carrier: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "polynomial K p" shows "set p \<subseteq> carrier R"
system_iff: fixes Q :: "'c mat"   and v :: "'c vec" assumes "dim_row Q = dim_col Q"   and "dim_row Q = dim_vec v" shows "(Q\<^sup>T *\<^sub>v v = v) = ((Q\<^sup>T - 1\<^sub>m (dim_row Q)) *\<^sub>v v = 0\<^sub>v (dim_vec v))"
valid_prefix_ipset_from_netmask_ipset_from_cidr: fixes pfx :: "'b prefix_match" shows "ipset_from_netmask (pfxm_prefix pfx) (~~ pfxm_mask pfx) = ipset_from_cidr (pfxm_prefix pfx) (pfxm_length pfx)"
le_left_mono: fixes x :: "'c"   and y :: "'c"   and a :: "'c" assumes "x \<le> y" shows "y \<le> a \<longrightarrow> x \<le> a"
max2_equals1: fixes a :: "'b"   and b :: "'b" assumes "a \<in> Field s"   and "b \<in> Field s" shows "(s.max2 a b = a) = ((b, a) \<in> s)"
c_merge_all: fixes key :: "'b \<Rightarrow> 'c"   and xss :: "'b list list" shows "int (Mergesort_Complexity.c_merge_all key xss) \<le> int (length (concat xss)) * \<lceil>log 2 (real (length xss))\<rceil>"
GC_processing_step: fixes N :: "('a literal multiset \<times> label) set"   and C :: "'a literal multiset" shows "N \<union> {(C, New)} \<leadsto>GC N \<union> {(C, Processed)}"
bitmagic_zeroLast_leq_or1Last: fixes a :: "'a word"   and len :: "nat"   and x :: "nat"   and y :: "nat" shows "a && (mask len << x - len) \<le> a || mask (y - len)"
causal_delivery: fixes m2 :: "'msgid \<times> 'op"   and j :: "nat"   and m1 :: "'msgid \<times> 'op" assumes "Deliver m2 \<in> set (history j)"   and "hb m1 m2" shows "Deliver m1 \<sqsubset>\<^sup>j Deliver m2"
eval_FinA: fixes G :: "prog"   and s :: "st"   and a :: "abopt"   and c :: "stmt"   and s' :: "state" shows "G\<turnstile>Norm s \<midarrow>FinA a c\<rightarrow> s' = False"
1: shows "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2"
underS_E: fixes i :: "'c"   and R :: "('c \<times> 'c) set"   and j :: "'c" assumes "i \<in> underS R j" shows "i \<noteq> j \<and> (i, j) \<in> R"
Bounded_mat_def: fixes b :: "'b" shows "Bounded_mat b = {A. \<forall>i<dim_row A. \<forall>j<dim_col A. \<bar>A $$ (i, j)\<bar> \<le> b}"
connected_eq_connected_components_eq: fixes s :: "'c set" shows "connected s = (\<forall>c\<in>components s. \<forall>c'\<in>components s. c = c')"
dem: shows "rfr_disc_equity_market M G Mkt r risk_free_asset"
reduce_all_def: fixes D :: "'a literal list" shows "reduce_all D = map (apfst (reduce [D] []))"
total_quasi_ordered_set_axioms: shows "total_quasi_ordered_set A r"
brouwer_surjective: fixes T :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and x :: "'a" assumes "compact T"   and "convex T"   and "T \<noteq> {}"   and "continuous_on T f"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> T\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> T"   and "x \<in> S" shows "\<exists>y\<in>T. f y = x"
validtI: fixes \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and c :: "('d, 'f, 'e) com"   and P :: "'d set"   and F :: "'e set"   and Q :: "'d set"   and A :: "'d set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"   and "\<And>s. s \<in> P \<Longrightarrow> \<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
bitxor_zero: fixes y :: "bit" shows "(\<zero> \<oplus>\<^sub>b y) = y"
assms_6: shows "finite X \<longrightarrow> b < card X"
is_natural_transformation: shows "natural_transformation A B F K \<rho>\<sigma>_\<tau>.map"
\<delta>_prod_sng2_def: fixes \<delta>1 :: "('a, 'c) ta_rule set"   and r :: "('b, 'c) ta_rule" shows "\<delta>_prod_sng2 \<delta>1 r \<equiv> case r of q2 \<rightarrow> l qs2 \<Rightarrow> {r_prod q1 \<rightarrow> l qs1 r |q1 qs1. length qs1 = length qs2 \<and> q1 \<rightarrow> l qs1 \<in> \<delta>1}"
altc_nondet_def: fixes A :: "'c cset"   and f :: "'c \<Rightarrow> ('a, 'm) nondetT" shows "altc_nondet A f = NondetT (mergec A (run_nondet \<circ> f))"
underS_ofilter: fixes a :: "'a" shows "r.ofilter (r.underS a)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a" assumes "category C"   and "dual_category C"   and "replete_set_category S"   and "set_valued_functor (dual_category.comp C) S F"   and "yoneda_functor_fixed_object C S \<phi> a" shows "yoneda_lemma C S \<phi> F a"
HComplex_minus: fixes x :: "real star"   and y :: "real star" shows "- HComplex x y = HComplex (- x) (- y)"
assms: fixes n :: "nat" shows "upper_asymptotic_density (A n) = 0"
y_eq: shows "y = [0..<nrows_iarray (matrix_to_iarray A)] ! i"
cring_simprules_22: shows "- (0::'c) = (0::'c)"
op_inter_ivl_def: shows "op_inter_ivl = (\<inter>)"
ptrm_fvs_finite: fixes X :: "'b ptrm" shows "finite (ptrm_fvs X)"
unat_gt_0: shows "(0 < unat (- 1)) = (- 1 \<noteq> 0)"
div_mod_int_poly_def: fixes p :: "int poly"   and q :: "int poly" shows "div_mod_int_poly p q = (if q = 0 then None else let n = degree q; qn = coeff q n in fold_coeffs (div_int_poly_step q) p (Some (0, 0)))"
simps_3: fixes a :: "'a"   and uu :: "('b, 'a) Digit"   and uv :: "('b, 'a) FingerTreeStruc"   and uw :: "('b, 'a) Digit" shows "FingerTreeStruc.gmft (Deep a uu uv uw) = a"
concurrent_ops_commute_def: fixes xs :: "('a \<times> operation) list" shows "hb.concurrent_ops_commute xs \<equiv> \<forall>x y. {x, y} \<subseteq> set xs \<longrightarrow> hb.concurrent x y \<longrightarrow> interp_msg x \<rhd> interp_msg y = interp_msg y \<rhd> interp_msg x"
lconjseq_alternating_order2_repeats': fixes s :: "'a"   and t :: "'a"   and k :: "nat"   and m :: "nat" assumes "s + s = (0::'a)"   and "t + t = (0::'a)"   and "(s + t) +^ k = (0::'a)" shows "map (lconjby (sum_list (alternating_list k s t))) (lconjseq (if gcd_nat.greater_eq k 2 then alternating_list m s t else alternating_list m t s)) = lconjseq (alternating_list m s t)"
write_state_reg_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_state_reg_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
cs_select_id: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" assumes "is_path \<pi>"   and "\<pi> k \<noteq> local.return" shows "(\<pi> \<exclamdown> cs\<^bsup>\<pi>\<^esup> k) = k"
fresh_list_nil: fixes a :: "'d" shows "a \<sharp> []"
marker_last: shows "Marker = last (msgs (S t (j + 1)) cid) \<and> msgs (S t (j + 1)) cid \<noteq> []"
add_col_sub_index_row: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat"   and l :: "nat"   and a :: "'b"   and k :: "nat" assumes "i < dim_row A"   and "i < dim_col A"   and "j < dim_row A"   and "j < dim_col A"   and "l < dim_row A" shows "add_col_sub_row a k l A $$ (i, j) = (if i = k \<and> j = l then A $$ (i, j) + a * A $$ (i, i) - a * a * A $$ (j, i) - a * A $$ (j, j) else if i = k \<and> j \<noteq> l then A $$ (i, j) - a * A $$ (l, j) else if i \<noteq> k \<and> j = l then A $$ (i, j) + a * A $$ (i, k) else A $$ (i, j))"
single_Diff_lessThan: fixes k :: "'b" shows "{k} - {..<k} = {k}"
extremum: fixes a :: "'c word" shows "a \<le> - 1"
integrable_distr_eq: fixes g :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and f :: "'e \<Rightarrow> 'f" assumes "g \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> borel_measurable N" shows "integrable (distr M N g) f = integrable M (\<lambda>x. f (g x))"
dict_swf: fixes i :: "indi" assumes "dictator (swf f) i" shows "dict f i"
norm_bound_def: fixes A :: "'a mat"   and b :: "real" shows "norm_bound A b \<equiv> \<forall>i j. i < dim_row A \<longrightarrow> j < dim_col A \<longrightarrow> norm (A $$ (i, j)) \<le> b"
rolling_rule_ltr: fixes g :: "'b \<rightarrow> 'a"   and f :: "'a \<rightarrow> 'b" shows "fix\<cdot>(g oo f) \<sqsubseteq> g\<cdot>(fix\<cdot>(f oo g))"
MIN'_1: shows "Q_\<alpha> Qi u = enat d"
update_convs_1: fixes getF' :: "Literal list list \<Rightarrow> Literal list list"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a" shows "getF_update getF' \<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> = \<lparr>getF = getF' getF, getM = getM, \<dots> = more\<rparr>"
cols_def: fixes A :: "'a mat" shows "cols A = map (col A) [0..<dim_col A]"
dg_1_components_2: fixes \<aa> :: "V"   and \<ff> :: "V" shows "dg_1 \<aa> \<ff>\<lparr>Arr\<rparr> = ZFC_in_HOL.set {\<ff>}"
sum_subset_iff: fixes A :: "V"   and B :: "V"   and C :: "V"   and D :: "V" shows "(A \<Uplus> B \<le> C \<Uplus> D) = (A \<le> C \<and> B \<le> D)"
part_equiv_refl': fixes R :: "('a \<times> 'a) set"   and x :: "'a" assumes "part_equiv R"   and "x \<in> Domain R" shows "(x, x) \<in> R"
subtype_x_Collection: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> Collection \<sigma>" obtains \<tau>' :: "'a OCL_Types.type" where "\<tau> = Collection \<tau>'" and "\<tau>' \<sqsubset> \<sigma>"   | "\<tau> = Set \<sigma>"   | "\<tau> = OrderedSet \<sigma>"   | "\<tau> = Bag \<sigma>"   | "\<tau> = Sequence \<sigma>"
bi_unique_alt_def2: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "bi_unique R = (R ===> R ===> (=)) (=) (=)"
un_assoc: fixes i :: "nat_int"   and j :: "nat_int"   and k :: "nat_int" shows "consec i j \<and> consec j k \<longrightarrow> (i \<squnion> j) \<squnion> k = i \<squnion> (j \<squnion> k)"
dualization_rew_4: fixes A :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd" shows "\<not> \<diamond>\<langle>A\<rangle>_v \<equiv> \<box>[\<not> A]_v"
Re_sin_pos: fixes z :: "complex" assumes "0 < Re z"   and "Re z < pi" shows "0 < Re (sin z)"
unbounded_line_def: fixes a :: "'a"   and b :: "'a" shows "unbounded_line a b = {x. \<exists>u. x = (1 - u) *\<^sub>R a + u *\<^sub>R b}"
axioms_2: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL" shows "Supraclassical_logic_axioms Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"
elims: fixes x :: "cospan_shape.Arr"   and y :: "cospan_shape.Arr" assumes "cospan_shape.Cod x = y" obtains "x = cospan_shape.AA" and "y = cospan_shape.AA"   | "x = cospan_shape.BB" and "y = cospan_shape.BB"   | "x = cospan_shape.TT" and "y = cospan_shape.TT"   | "x = cospan_shape.AT" and "y = cospan_shape.TT"   | "x = cospan_shape.BT" and "y = cospan_shape.TT"   | "x = cospan_shape.Null" and "y = cospan_shape.Null"
Str_def: fixes s :: "char list" shows "EFSM.Str s \<equiv> value.Str (String.implode s)"
paths: shows "path A (w || r1) p1 \<and> path B (w || r2) p2"
dagger_def: fixes M :: "complex mat" shows "M\<^sup>\<dagger> \<equiv> mat (dim_col M) (dim_row M) (\<lambda>(i, j). cnj (M $$ (j, i)))"
Left_equals_Right_coset: fixes g :: "'a" assumes "g \<in> G" shows "g \<cdot>| K = K |\<cdot> g"
Inter_subset: fixes A :: "'c set set"   and B :: "'c set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B"   and "A \<noteq> {}" shows "\<Inter> A \<subseteq> B"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "local.less_eq_list (v # va) [] = False"
domain_finmap_of: fixes inds :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite inds" shows "domain (finmap_of inds f) = inds"
cases: fixes x :: "'b extB" obtains "x = extB.Bot"   | uu_ :: "'b" where "x = extB.Val uu_"
trm_simps_3: fixes a :: "'a" assumes "ide a" shows "cod \<t>[a] = \<one>"
find_closest_pair_dist: fixes ps :: "(int \<times> int) list"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and \<delta> :: "real"   and l :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and C\<^sub>0 :: "int \<times> int"   and C\<^sub>1 :: "int \<times> int" assumes "sorted_snd ps"   and "distinct ps"   and "set ps = ps\<^sub>L \<union> ps\<^sub>R"   and "0 \<le> \<delta>"   and "\<forall>p\<in>set ps. real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R"   and "dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> \<delta>"   and "(C\<^sub>0, C\<^sub>1) = find_closest_pair (c\<^sub>0, c\<^sub>1) ps" shows "sparse (dist (case C\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (set ps)"
State_Idle_imp_exists_state_change2: fixes localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and n :: "nat" assumes "\<not> State_Idle localState output_fun trans_fun (localState c)"   and "State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>n\<^esup> c))" shows "\<exists>i<n. (\<forall>j\<le>i. \<not> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>i\<^esup> c))) \<and> (\<forall>j\<le>n. i < j \<longrightarrow> State_Idle localState output_fun trans_fun (localState (f_Exec_Comp trans_fun NoMsg\<^bsup>j\<^esup> c)))"
nz': shows "f \<noteq> 0 \<or> g \<noteq> 0"
representation_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and basis :: "'b set"   and v :: "'b"   and r :: "'a" assumes "Modules.module scale"   and "\<not> module.dependent scale basis"   and "v \<in> module.span scale basis" shows "module.representation scale basis (scale r v) = (\<lambda>b. r * module.representation scale basis v b)"
holomorphic_ln_Gamma: fixes A :: "complex set" assumes "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}" shows "ln_Gamma holomorphic_on A"
WHILEIT_unfold: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and x :: "'c" shows "WHILE\<^sub>T\<^bsup>I\<^esup> b f x = (if I x then if b x then f x \<bind> WHILE\<^sub>T\<^bsup>I\<^esup> b f else RETURN x else FAIL)"
simps_1: fixes l :: "nat" shows "isFresh l [] = True"
nat_pow_distrib: fixes x :: "int \<times> int"   and y :: "int \<times> int"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<otimes>\<^bsub>R\<^esub> y) [^]\<^bsub>R\<^esub> n = x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> y [^]\<^bsub>R\<^esub> n"
Norm_zero: fixes N :: "'b quasinorm" shows "Norm N (0::'b) = 0"
strong_confluentp_into_semiconfluentp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "strong_confluentp r" shows "semiconfluentp r"
normNum_nilpotent_aux: fixes x :: "int \<times> int" assumes "isnormNum x" shows "normNum x = x"
shows_list_arctic_def: fixes as :: "arctic list" shows "shows_list as = showsp_list shows_prec 0 as"
QP: shows "Q' * Q * (P * P') = 1\<^sub>m n"
fresh_prod: fixes a :: "'v"   and x :: "'w"   and y :: "'x" shows "a \<sharp> (x, y) = (a \<sharp> x \<and> a \<sharp> y)"
neg_lt: fixes x :: "'a"   and y :: "'a" shows "neg\<cdot>(lt\<cdot>x\<cdot>y) = le\<cdot>y\<cdot>x"
effect_newI: fixes a :: "'a array"   and h' :: "heap"   and n :: "nat"   and x :: "'a"   and h :: "heap" assumes "(a, h') = Array.alloc (replicate n x) h" shows "effect (Array.new n x) h h' a"
abs_add_one_gt_zero: 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 x :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (plus one (abs x))"
order_asym: fixes R :: "('a \<times> 'a) set" assumes "trans R" shows "asym R = irrefl R"
Per_cases: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C \<or> Tarski_neutral_dimensionless.Per Bet Cong C B A" shows "Tarski_neutral_dimensionless.Per Bet Cong A B C"
invertible_transpose_mult: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "invertible (Finite_Cartesian_Product.transpose A ** A)"
diff_zero_right: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "(0::'b) - x = - x"
card_has: fixes S :: "'a set"   and n :: "nat" assumes "card S = n" shows "has n S"
sel_4: fixes x1 :: "'b"   and x2 :: "'c"   and x3 :: "nat"   and x4 :: "('b, 'c) SkewBinomialTree list" shows "children (Node x1 x2 x3 x4) = x4"
sup_inf_absorb: fixes 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 x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "sup x (inf x y) = x"
ed_ou_witness_card: shows "card {f. CK_nf_pos f} = 4"
meet_in_def: fixes l :: "Lines"   and m :: "Lines"   and P :: "Points" shows "meet_in l m P \<equiv> incid P l \<and> incid P m"