hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b"   and a2 :: "'a list \<times> (nat \<Rightarrow> 'a option) \<times> 'b" assumes "step a1 a2" obtains ws :: "nat \<Rightarrow> 'a option" and i :: "nat" and t :: "'a" and ts :: "'a list" and s :: "'b" where "a1 = (t # tsa__, ws__, s__)" and "a2 = (tsa__, ws__(i__ \<mapsto> t), s__)" and "ws__ i__ = None" and "i__ < n"   | ws :: "nat \<Rightarrow> 'a option" and i :: "nat" and a :: "'a" and ts :: "'a list" and s :: "'b" where "a1 = (ts, ws, s)" and "a2 = (ts, ws(i := None), f a s)" and "ws i = Some a" and "i < n"
char_poly_matrix_def: fixes A :: "'b mat" shows "char_poly_matrix A = [:0::'b, 1::'b:] \<cdot>\<^sub>m 1\<^sub>m (dim_row A) + map_mat (\<lambda>a. [:- a:]) A"
jozsa_transform_coeff_is_zero: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "jozsa f n"   and "i < dim_row (jozsa.jozsa_transform f n) \<and> j < dim_col (jozsa.jozsa_transform f n)" shows "i \<noteq> j \<and> \<not> (i = j + 1 \<and> odd i) \<and> \<not> (i = j - 1 \<and> even i \<and> 1 \<le> j) \<longrightarrow> jozsa.jozsa_transform f n $$ (i, j) = 0"
Union_empty_conv: fixes A :: "'c set set" shows "(\<Union> A = {}) = (\<forall>x\<in>A. x = {})"
\<alpha>: shows "Derives1 [\<SS>] (fst d) (snd d) \<alpha> \<and> derives \<alpha> (terminals p)"
finite_Pow_iff: fixes A :: "'b set" shows "finite (Pow A) = finite A"
child_ix_right: fixes d :: "nat"   and p :: "(nat \<times> int) list" assumes "d < length p" shows "ix (child p right d) d = 2 * ix p d + 1"
SconsI: fixes M :: "('c, unit) Old_Datatype.node set"   and N :: "('c, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "Scons M N \<in> sexp"
tso_store_refs_def: shows "tso_store_refs = (\<lambda>s. \<Union> (store_refs ` set (sys_mem_store_buffers (mutator m) s)))"
send_insert_id: fixes i :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) insert_message" assumes "event_at i (Send (Insert m))" shows "I m = i"
l3_obs_iagreement: shows "oreach l3 \<subseteq> l2_iagreement"
perm_aux_fold: fixes pi :: "('d \<times> 'd) list"   and x :: "'c" shows "perm_aux pi x = pi \<bullet> x"
path_strategies_good: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and P :: "'c llist"   and n :: "nat" assumes "WellOrderedStrategies G S p good r"   and "lset P \<subseteq> S"   and "enat n < llength P" shows "WellOrderedStrategies.path_strategies good r P $ n \<in> good (P $ n)"
T_TAbs: fixes X :: "tyvrs"   and T\<^sub>1 :: "ty"   and \<Gamma> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "ty" assumes "TVarB X T\<^sub>1 # \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (\<lambda>X<:T\<^sub>1. t\<^sub>2) : (\<forall>X<:T\<^sub>1. T\<^sub>2)"
prio_emb_posI: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and prio :: "dir list \<Rightarrow> nat" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s" shows "prio_emb_pos prio t s \<noteq> [] \<and> position_of t (prio_emb_pos prio t s) \<and> emb_step_at' (prio_emb_pos prio t s) t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
Alist_inject: fixes x :: "('a \<times> 'b) list"   and y :: "('a \<times> 'b) list" assumes "x \<in> {xs. (distinct \<circ> list.map fst) xs}"   and "y \<in> {xs. (distinct \<circ> list.map fst) xs}" shows "(Alist x = Alist y) = (x = y)"
Q_def: shows "Q = Liminf_llist (lmap Q_of_state Sts)"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "seq g f" shows "g \<cdot> f \<otimes> \<I> = (g \<otimes> \<I>) \<cdot> (f \<otimes> \<I>)"
inv_into_def2: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" shows "inv_into A f x = (SOME y. y \<in> A \<and> f y = x)"
emeasure_eq_1_AE: fixes M :: "'a measure"   and S :: "'a set" assumes "prob_space M"   and "S \<in> sets M"   and "AE x in M. x \<in> S" shows "emeasure M S = 1"
preserves_reflects_arr: fixes f :: "'c local.term set" shows "arr (\<F>C.D f) = \<F>C.arr f"
is_tm_tdghm_axioms: shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>.\<^sub>t\<^sub>m \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
properfactor_prod_l: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "properfactor G a b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a (c \<otimes> b)"
simps: fixes a1 :: "'d set"   and a2 :: "'d \<Rightarrow> nat"   and a3 :: "'d \<Rightarrow> Regions.intv"   and a4 :: "('d \<times> 'd) set" shows "Regions.valid_region a1 a2 a3 a4 = (\<exists>X\<^sub>0 X I r k. a1 = X \<and> a2 = k \<and> a3 = I \<and> a4 = r \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d} \<and> refl_on X\<^sub>0 r \<and> trans r \<and> total_on X\<^sub>0 r \<and> (\<forall>x\<in>X. Regions.valid_intv (k x) (I x)))"
init_node_comp: fixes i :: "nat"   and np :: "('c, 'd seq_action) automaton"   and R\<^sub>i :: "nat set" shows "init (\<langle>i : np : R\<^sub>i\<rangle>) = {NodeS i s R\<^sub>i |s. s \<in> init np}"
IH2: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "wf_net_tree p2"   and "s \<in> reachable (pnet np p2) TT"   and "(s, \<tau>, s') \<in> automaton.trans (pnet np p2)" shows "\<exists>i\<in>net_ips s. (\<forall>j. j \<noteq> i \<longrightarrow> netmap s' j = netmap s j) \<and> net_ip_action np \<tau> i p2 s s'"
Cidr: fixes f :: "'b" assumes "f \<in> mor" shows "f ;; id (cod f) = f"
echelon_form_imp_upper_triagular: fixes A :: "(('a, 'b) vec, 'b) vec" assumes "echelon_form A" shows "upper_triangular A"
lead_coeff_monic_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p" shows "lead_coeff (p * q) = lead_coeff q"
minus_inv_sym_aux: fixes a :: "'a"   and b :: "'a" shows "(- {(a, b), (b, a)})\<inverse> = - {(a, b), (b, a)}"
coeff_minus: fixes p :: "'a poly"   and n :: "nat" shows "coeff (- p) n = - coeff p n"
l11_43_aux: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "A \<noteq> C"   and "Per B A C \<or> Obtuse B A C" shows "Acute A B C"
pointwise_adm: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "adm (\<lambda>x. P (fst x) (snd x))" shows "adm (\<lambda>m. pointwise P (fst m) (snd m))"
memb_correct: fixes s :: "('a, unit) HashMap.hashmap"   and x :: "'a" assumes "hs.invar s" shows "hs.memb x s = (x \<in> hs.\<alpha> s)"
b1: shows "bernoulli_pmf 1 = return_pmf True"
dual_additive_additive: fixes f :: "'a \<Rightarrow> 'a" assumes "dual_additive (uminus \<circ> f)" shows "additive f"
ordIso_ordLess_trans: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set"   and r'' :: "('d \<times> 'd) set" assumes "r =o r'"   and "r' <o r''" shows "r <o r''"
local_2: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and arb_return :: "t list option"   and arb_label :: "t list list" assumes "store_typing s \<S>"   and "\<S>\<bullet>\<C> \<turnstile> es : ts _> ts'"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ map typeof vs, label := arb_label, return := arb_return\<rparr>"   and "i < length (s_inst \<S>)" shows "store_typing s' \<S>"
assms: shows "((\<lambda>t. exp (t *\<^sub>R - s) * f t) has_integral L) {0..}"
pr_algo_cost_bound: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "Height_Bounded_Labeling c s t f l"   and "((f, l), p, f', l') \<in> trcl (Network.pr_algo_lts c t)" shows "sum_list (map (Network.cost_estimate c) p) \<le> 26 * (card (Graph.V c))\<^sup>2 * card (Graph.E c)"
eq_\<I>_comp_cong: fixes \<I>1 :: "('h, 'i) \<I>"   and \<I>2 :: "('j, 'k) \<I>"   and conv1 :: "('h, 'i, 'j, 'k) converter"   and conv1' :: "('h, 'i, 'j, 'k) converter"   and \<I>3 :: "('l, 'm) \<I>"   and conv2 :: "('j, 'k, 'l, 'm) converter"   and conv2' :: "('j, 'k, 'l, 'm) converter" assumes "\<I>1,\<I>2 \<turnstile>\<^sub>C conv1 \<sim> conv1'"   and "\<I>2,\<I>3 \<turnstile>\<^sub>C conv2 \<sim> conv2'" shows "\<I>1,\<I>3 \<turnstile>\<^sub>C conv1 \<odot> conv2 \<sim> conv1' \<odot> conv2'"
card_image_le: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite A" shows "card (f ` A) \<le> card A"
substs_preserve_dom_trm: fixes t :: "'a trm"   and C :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "dom_trm t C" shows "dom_trm (t \<lhd> \<sigma>) (subst_cl C \<sigma>)"
bnds_powr: fixes l :: "float"   and u :: "float"   and prec :: "nat"   and l1 :: "float"   and u1 :: "float"   and l2 :: "float"   and u2 :: "float"   and x :: "real"   and y :: "real" assumes "Some (l, u) = bnds_powr prec l1 u1 l2 u2"   and "x \<in> {real_of_float l1..real_of_float u1}"   and "y \<in> {real_of_float l2..real_of_float u2}" shows "x powr y \<in> {real_of_float l..real_of_float u}"
assms_1: fixes a :: "'a" assumes "a \<in> A" shows "a dvd b"
finfun_update_code_def: shows "finfun_update_code = finfun_update"
coprimeI: fixes a :: "'a"   and b :: "'a" assumes "\<And>c. \<lbrakk>c dvd a; c dvd b\<rbrakk> \<Longrightarrow> is_unit c" shows "coprime a b"
normalize_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize zero = zero"
add_point_def: fixes p :: "'a" shows "add_point p \<equiv> insert p \<V>"
eFreshAbs_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and y :: "'var" shows "eFreshAbs MOD ys y ERR"
counterexample_not_gu_condition: shows "\<nexists>R. view_partition P\<^sub>c id R \<and> weakly_future_consistent P\<^sub>c I\<^sub>c id R \<and> weakly_step_consistent P\<^sub>c id R \<and> locally_respects P\<^sub>c I\<^sub>c id R"
Succ_2: shows "length (AAs' \<cdot>aml \<eta>) = Suc n"
adapt: shows "P \<subseteq> {s. \<exists>Z. init s \<in> P' Z \<and> (\<forall>t. t \<in> Q' Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' Z \<longrightarrow> return s t \<in> A)}"
msort_msort2: fixes xs :: "'a list" shows "msort (\<le>) xs = msort2 (\<le>) xs"
ac_Zp_equation: fixes x :: "nat \<Rightarrow> int"   and k :: "nat"   and m :: "nat" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>"   and "0 < k"   and "m = nat (ord_Zp x)" shows "ac_Zp x k \<in> carrier (residue_ring (p ^ k)) \<and> ac_Zp x k * p ^ m = x (m + k)"
space_closed: fixes M :: "'c measure" shows "sets M \<subseteq> Pow (space M)"
hyps_free_acyclic: fixes v :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "path v v pth"   and "hyps_free pth" shows "pth = []"
deutsch_transform_alt_rep: shows "U\<^sub>f = V\<^sub>f"
iso_correct: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'a list list" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "[] \<notin> set Fs\<^sub>2"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)" shows "iso_test Fs\<^sub>1 Fs\<^sub>2 = Fs\<^sub>1 \<simeq> Fs\<^sub>2"
A': shows "adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D (F f) (F g) \<eta>' \<epsilon>'"
univ_rw: shows "UNIV = UNIV - {a} \<union> {a}"
c_aux: fixes i :: "nat" assumes "i \<le> N" shows "coeff p i = fpxs_compose_power (fpxs_of_fls (c_aux i)) (1 / rat_of_nat b)"
not_col_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col A B C \<and> A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C"
matrix_to_iarray_mat: fixes k :: "'c" shows "matrix_to_iarray (mat k) = mat_iarray k CARD('d)"
hyps: fixes i :: "'a" assumes "\<And>e. f n e = e"   and "\<And>e. f e n = e"   and "\<And>a b c. f a (f b c) = f (f a b) c" shows "foldl f i ww = f i (foldl f n ww)"
step: fixes y :: "'b"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "y \<in> orbit f x" shows "f y \<in> orbit f x"
emeasure_map_pmf: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c pmf"   and X :: "'b set" shows "emeasure (measure_pmf (map_pmf f M)) X = emeasure (measure_pmf M) (f -` X)"
pr_tuple_2: fixes f :: "'a"   and g :: "'a"   and a :: "'a"   and b :: "'a" assumes "span f g"   and "cod f = a"   and "cod g = b" shows "pr0 a b \<cdot> tuple f g = g"
gt_imp_ge: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and default :: "'b"   and x :: "'b"   and y :: "'b" assumes "order_pair gt default"   and "gt x y" shows "y \<le> x"
rel_interior_substd_simplex: fixes D :: "'a set" assumes "D \<subseteq> Basis" shows "rel_interior (convex hull insert (0::'a) D) = {x. (\<forall>i\<in>D. 0 < x \<bullet> i) \<and> sum ((\<bullet>) x) D < 1 \<and> (\<forall>i\<in>Basis. i \<notin> D \<longrightarrow> x \<bullet> i = 0)}"
ln_Gamma_series_complex_converges: fixes z :: "complex"   and d :: "real" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0"   and "0 < d"   and "\<And>n. n \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> d < cmod (z - of_int n)" shows "uniformly_convergent_on (ball z d) (\<lambda>n z. ln_Gamma_series z n)"
not_eq_extremum: fixes a :: "'a" shows "(a \<noteq> bot) = (bot < a)"
mset_eq_mset_set_imp_distinct: fixes A :: "'b set"   and xs :: "'b list" assumes "finite A"   and "mset_set A = mset xs" shows "distinct xs"
norm_iarray_def: fixes A :: "real iarray" shows "norm_iarray A = sqrt (A \<bullet>i A)"
ext: fixes g :: "'b"   and f :: "'b" assumes "g \<cdot>\<^sub>D f \<noteq> D.null" shows "D.seq g f"
Eq\<^sub>1E: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and G :: "\<Pi>\<^sub>1" assumes "[F \<^bold>=\<^sub>1 G in v]" shows "F = G"
equal_mapping_def: fixes m1 :: "('a, 'b) mapping"   and m2 :: "('a, 'b) mapping" shows "equal_class.equal m1 m2 = (\<forall>k. lookup m1 k = lookup m2 k)"
simps_2: fixes f :: "'c" shows "Diag \<^bold>\<langle>f\<^bold>\<rangle> = C.arr f"
worklist_tree_coll_aux_def: fixes succs :: "'c \<Rightarrow> 'c list"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a" shows "worklist_tree_coll_aux succs P f = worklist_tree succs (insert_mod2 P f)"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "Sublist.prefix b a"   and "Sublist.prefix c b" shows "Sublist.prefix c a"
lm038: fixes a :: "('c \<times> 'd set) set"   and b :: "('c \<times> 'd set) set" assumes "a \<in> partitionValuedUniverse" shows "a - b \<in> partitionValuedUniverse"
ln_approx_abs'': fixes x :: "real"   and x' :: "real"   and e :: "real"   and n :: "nat" assumes "1 < x"   and "ln_approx_aux1 x' e n x" shows "\<bar>ln x - x'\<bar> \<le> e"
power_not_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero"   and "a \<noteq> zero" shows "power.power one times a n \<noteq> zero"
unfold_plussub_lift2: fixes e\<^sub>1 :: "'d err"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c err"   and e\<^sub>2 :: "'e err" shows "e\<^sub>1 \<squnion>\<^bsub>lift2 f\<^esub> e\<^sub>2 = lift2 f e\<^sub>1 e\<^sub>2"
elims_1: fixes x :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and y :: "bool" assumes "SkewBinomialHeapStruc.rank_invar x = y" obtains "x = []" and "y"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" where "x = [t__]" and "y"   | t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" where "x = t # t' # bq" and "y = (SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
abelian_group_axioms_def: fixes G :: "('c, 'd) ring_scheme" shows "abelian_group_axioms G \<equiv> comm_group (add_monoid G)"
suntil_as_until: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(\<phi> suntil \<psi>) \<omega> = ((\<phi> until \<psi>) \<omega> \<and> ev \<psi> \<omega>)"
simps_3: fixes s :: "'b word"   and e :: "'b word"   and v :: "'b word \<times> 'b word"   and va :: "('b word \<times> 'b word) list" shows "l2wi ((s, e) # v # va) = RangeUnion (WordInterval s e) (l2wi (v # va))"
stirling_integral_continuous_on_complex: fixes m :: "nat"   and A :: "complex set" assumes "0 < m"   and "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}" shows "continuous_on A (stirling_integral m)"
unif_preserves_unifiable: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, E') \<in> unif"   and "unifiable (set_mset E)" shows "unifiable (set_mset E')"
tl_queue_rep: fixes q :: "'a queue" assumes "\<not> Optimized_MTL.is_empty q" shows "linearize (tl_queue q) = tl (linearize q)"
lzip_eq_LNil_conv: fixes xs :: "'c llist"   and ys :: "'d llist" shows "(lzip xs ys = LNil) = (xs = LNil \<or> ys = LNil)"
T2: shows "Col A B E \<and> A B Perp P E"
order_Aw: shows "order Aw = 2 * N_half"
cod_hcomp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data"   and \<nu> :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>"   and "partial_magma.arr (span_vertical_category.vcomp C) \<nu>"   and "span_vertical_category.src C \<nu> = span_vertical_category.trg C \<mu>" shows "partial_magma.cod (span_vertical_category.vcomp C) (span_bicategory.hcomp C prj0 prj1 \<nu> \<mu>) = span_bicategory.hcomp C prj0 prj1 (partial_magma.cod (span_vertical_category.vcomp C) \<nu>) (partial_magma.cod (span_vertical_category.vcomp C) \<mu>)"
tuple_def: fixes f :: "'a"   and g :: "'a" shows "\<langle>f, g\<rangle> \<equiv> if span f g then THE l. \<pp>\<^sub>1[cod f, cod g] \<cdot> l = f \<and> \<pp>\<^sub>0[cod f, cod g] \<cdot> l = g else null"
wf_action_translation_on_actionD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and A :: "nat set"   and f :: "nat \<Rightarrow> nat"   and a :: "nat" assumes "wf_action_translation_on E E' A f"   and "a \<in> A" shows "action_tid E a = action_tid E' (f a) \<and> action_obs E a \<approx> action_obs E' (f a)"
dual_assertion_disjunct: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "dual x \<in> disjunctive.disjunctive sup sup times"
bv_select_def: fixes w :: "bit list"   and i :: "nat" shows "bv_select w i = w ! (length w - 1 - i)"
subset_iff: fixes A :: "'d set"   and B :: "'d set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
vconn_implies_hpar_3: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D g = src\<^sub>D f"
sqr_cn: fixes P :: "pol" assumes "isnorm P" shows "isnorm (sqr P)"
p_max_eq: shows "delayed_safe_distance.other.p_max = other.s \<delta> - (other.s' \<delta>)\<^sup>2 / a\<^sub>o / 2"
simps_21: fixes vb :: "('e, 'f, 'g) Tree"   and v :: "'g"   and va :: "('e, 'f, 'g) Tree" shows "(tNot vb =\<^sub>\<alpha> tAct v va) = False"
HF_ax_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> HF_ax" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A"
r_def: shows "r = cnj (cinner x y) / cinner y y"
sb_simps2: fixes zs :: "'a"   and xs :: "'a"   and z2 :: "'b"   and x :: "'b"   and z1 :: "'b" assumes "zs \<noteq> xs \<or> z2 \<noteq> x" shows "x @xs[z1 / z2]_zs = x"
counit_simps_4: shows "S.src (S.inv (UP.unit (trg f)) \<cdot>\<^sub>S S.UP \<epsilon> \<cdot>\<^sub>S S.cmp\<^sub>U\<^sub>P (f, g)) = S.trg (S.UP f)"
simps_6: fixes r :: "mregex" shows "RPD (MStar r) = MTimesL (MStar r) (RPD r)"
col124__nos: fixes P :: "'p"   and Q :: "'p"   and B :: "'p"   and A :: "'p" assumes "Col P Q B" shows "\<not> P Q OS A B"
Uniq_D: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "Uniq P"   and "P a"   and "P b" shows "a = b"
RootExSem_def: fixes F :: "('d, 'e, 'f) seqauto set"   and G :: "'d \<Rightarrow> ('d, 'e, 'f) seqauto set option"   and C :: "'d set" shows "RootExSem F G C \<equiv> \<exists>!S. S \<in> States (Root F G) \<and> S \<in> C"
le_approx2T: fixes P :: "'b process"   and Q :: "'b process"   and s :: "'b event list" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "(s \<in> T Q) = (s \<in> T P)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) A.map F \<phi>'.map"
a_star_subst2: fixes M :: "trm"   and M' :: "trm"   and c :: "coname"   and y :: "name"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>a* M'" shows "M{c:=y.P} \<longrightarrow>\<^sub>a* M'{c:=y.P}"
induct: fixes P :: "nat option list \<Rightarrow> nat list \<Rightarrow> nat \<Rightarrow> bool list list \<Rightarrow> bool"   and a0 :: "nat option list"   and a1 :: "nat list"   and a2 :: "nat"   and a3 :: "bool list list" assumes "\<And>zs i T. P [] zs i T"   and "\<And>xs zs i T. P (mk_eqcl' xs i (Suc i) (length zs) T) (zs @ [i]) (Suc i) T \<Longrightarrow> P (None # xs) zs i T"   and "\<And>l xs zs i T. P xs zs (Suc i) T \<Longrightarrow> P (Some l # xs) zs i T" shows "P a0 a1 a2 a3"
h_plus_source_in: fixes i :: "nat"   and x :: "'v" shows "h_plus i (x, source \<Delta>) = 0"
mod_ring_inverse: shows "(pm.mod_ring_rel ===> pm.mod_ring_rel) (inverse_p m) inverse"
memoryless_on_cfg_onI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and f :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "Markov_Decision_Process K"   and "\<And>s. f s \<in> K s" shows "memoryless_on f s \<in> Markov_Decision_Process.cfg_on K s"
rsquarefree_imp_distinct_roots: fixes p :: "'a poly"   and xs :: "'a list" assumes "rsquarefree p"   and "mset xs = poly_roots p" shows "distinct xs"
invertible_left_inverse: fixes u :: "'a \<Rightarrow> 'a" assumes "symmetric.invertible u"   and "u \<in> Sym" shows "compose S (symmetric.inverse u) u = identity S"
t1_space: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.t1_space open"   and "x \<noteq> y" shows "\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
rw1: shows "map (\<lambda>x. IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) CARD('c)) [0..<CARD('b)] ! mod_type_class.to_nat a !! mod_type_class.to_nat j = A $ a $ j"
ex_bij_betw_finite_nat: fixes M :: "'b set" assumes "finite M" shows "\<exists>h. bij_betw h M {0..<card M}"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>xs. P xs []"   and "\<And>x xs y ys. \<lbrakk>P xs (y # ys); P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
LIMSEQ_SUP: fixes X :: "nat \<Rightarrow> 'c" assumes "incseq X" shows "X \<longlonglongrightarrow> Sup (range X)"