hcju/isabelleps huggingface dataset:
subgroupE_3: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H H'"   and "a \<in> H" shows "inv\<^bsub>H'\<^esub> a \<in> H"
gcd_int_poly_def: fixes f :: "int poly"   and g :: "int poly" shows "gcd_int_poly f g = (if f = 0 then normalize g else if g = 0 then normalize f else let cf = content f; cg = content g; ct = gcd cf cg; ff = map_poly (\<lambda>x. x div cf) f; gg = map_poly (\<lambda>x. x div cg) g in if coprime_heuristic ff gg then [:ct:] else Polynomial.smult ct (gcd_poly_code_aux ff gg))"
iFROM_inter: fixes n' :: "nat"   and n :: "nat" assumes "n' \<le> n" shows "[n\<dots>] \<inter> [n'\<dots>] = [n\<dots>]"
bij_lens_equiv_id: fixes X :: "'c \<Longrightarrow> 'd" shows "bij_lens X = X \<approx>\<^sub>L \<Sigma>"
simps_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> (\<phi> or\<^sub>n \<psi>) w = \<F>\<G> \<phi> w \<union> \<F>\<G> \<psi> w"
dim_col_polys_to_mat: fixes ts :: "'d list"   and ps :: "('d \<Rightarrow>\<^sub>0 'c) list" shows "dim_col (polys_to_mat ts ps) = length ts"
pairwise_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S. x \<noteq> y \<longrightarrow> R x y)"
greaterThanLessThan_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "ord.greaterThanLessThan less l u \<equiv> ord.greaterThan less l \<inter> ord.lessThan less u"
simps_14: shows "set_tree \<langle>\<rangle> = {}"
map_F_rel_cong: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L3 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co1 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co2 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Co3 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra1 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2 :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra3 :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and xi :: "('p, 'r, 't, 'v, 'x, 'z, 'ab, 'ad, 'af, 'ah) F"   and yi :: "('q, 's, 'u, 'w, 'y, 'aa, 'ac, 'ae, 'ag, 'ah) F"   and L1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and x :: "'p \<Rightarrow> 'ai"   and y :: "'q \<Rightarrow> 'aj"   and L2' :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and xa :: "'r \<Rightarrow> 'ak"   and ya :: "'s \<Rightarrow> 'al"   and L3' :: "'am \<Rightarrow> 'an \<Rightarrow> bool"   and xb :: "'t \<Rightarrow> 'am"   and yb :: "'u \<Rightarrow> 'an"   and Co1' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool"   and xc :: "'v \<Rightarrow> 'ao"   and yc :: "'w \<Rightarrow> 'ap"   and Co2' :: "'aq \<Rightarrow> 'ar \<Rightarrow> bool"   and xd :: "'x \<Rightarrow> 'aq"   and yd :: "'y \<Rightarrow> 'ar"   and Co3' :: "'as \<Rightarrow> 'at \<Rightarrow> bool"   and xe :: "'z \<Rightarrow> 'as"   and ye :: "'aa \<Rightarrow> 'at"   and Contra1' :: "'au \<Rightarrow> 'av \<Rightarrow> bool"   and xf :: "'au \<Rightarrow> 'ab"   and yf :: "'av \<Rightarrow> 'ac"   and Contra2' :: "'aw \<Rightarrow> 'ax \<Rightarrow> bool"   and xg :: "'aw \<Rightarrow> 'ad"   and yg :: "'ax \<Rightarrow> 'ae"   and Contra3' :: "'ay \<Rightarrow> 'az \<Rightarrow> bool"   and xh :: "'ay \<Rightarrow> 'af"   and yh :: "'az \<Rightarrow> 'ag" assumes "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 xi yi"   and "\<And>x y. L1 x y \<Longrightarrow> L1' (x x) (y y)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. L3 x y \<Longrightarrow> L3' (xb x) (yb y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xc x) (yc y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xd x) (yd y)"   and "\<And>x y. Co3 x y \<Longrightarrow> Co3' (xe x) (ye y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xf x) (yf y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xg x) (yg y)"   and "\<And>x y. Contra3' x y \<Longrightarrow> Contra3 (xh x) (yh y)" shows "rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3' (map_F x xa xb xc xd xe xf xg xh xi) (map_F y ya yb yc yd ye yf yg yh yi)"
half_ts_def: shows "half_ts = Abs_freelist (pair_relator_halflist t s)"
emeasure_subadditive_countably: fixes f :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range f \<subseteq> sets M" shows "emeasure M (\<Union> (range f)) \<le> (\<Sum>i. emeasure M (f i))"
mult_mat_vec_split: fixes A :: "'b mat"   and n :: "nat"   and D :: "'b mat"   and m :: "nat"   and a :: "'b vec"   and d :: "'b vec" assumes "A \<in> carrier_mat n n"   and "D \<in> carrier_mat m m"   and "a \<in> carrier_vec n"   and "d \<in> carrier_vec m" shows "four_block_mat A (0\<^sub>m n m) (0\<^sub>m m n) D *\<^sub>v (a @\<^sub>v d) = A *\<^sub>v a @\<^sub>v D *\<^sub>v d"
image_iff: fixes z :: "'f"   and f :: "'g \<Rightarrow> 'f"   and A :: "'g set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
analz_disj_parts: fixes X :: "msg"   and H :: "msg set" shows "(X \<in> analz H \<or> X \<in> parts H) = (X \<in> parts H)"
wf_residual_networkD_3: assumes "res.wf_residual_network"   and "edge (residual_network f) (source (residual_network f)) (sink (residual_network f))" shows "False"
ring_simprules_22: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z) = y \<oplus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> z)"
l: assumes "set (xs @ us @ vs) \<subseteq> var"   and "set ts \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla" shows "length (getFrN (xs @ us @ vs) ts [\<phi>] (length xs)) = length xs"
rb_aux_inv_is_Groebner_basis: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat" assumes "hom_grading dgrad"   and "rb_aux_inv (bs, ss, ps)"   and "\<And>p0. p0 \<in> set ps \<Longrightarrow> j \<le> component_of_term (sig_of_pair p0)" shows "punit.is_Groebner_basis (rep_list ` set (filter (\<lambda>b. component_of_term (lt b) < j) bs))"
deadlockD3: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x1" assumes "r1.deadlock s"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" shows "\<forall>w. wset s t \<noteq> \<lfloor>PostWS w\<rfloor>"
inj_on_set_encode: shows "inj_on set_encode (Collect finite)"
z_hyp: shows "list_constr I (length qs) \<longrightarrow> z I sign = 1 \<or> z I sign = - 1"
map_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "\<alpha>.map f = (if VVV.arr f then \<a>[fst (VVV.cod f), fst (snd (VVV.cod f)), snd (snd (VVV.cod f))] \<cdot> HoHV f else null)"
plus_irr_def: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly" shows "plus_irr a b = (a + b) mod f"
greek_less_def: fixes r :: "('a \<times> 'a) set" shows "greek_less r = lfp (nest r)"
in_mkbot: fixes x :: "alt"   and y :: "alt"   and L :: "(alt \<times> alt) set"   and z :: "alt" shows "(x <\<^bsub>mkbot L z\<^esub> y) = (y \<noteq> z \<and> (if x = z then x \<noteq> y else x <\<^bsub>L\<^esub> y))"
simps: fixes a :: "(('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent" shows "(a \<in> edges) = ((\<exists>c is. a = edge_at c is \<and> c \<in> set conclusions \<and> is \<in> it_paths (it' c)) \<or> (\<exists>c is n s ants. a = hyp_edge_at c is n s \<and> c \<in> set conclusions \<and> is \<in> it_paths (it' c) \<and> tree_at (it' c) is = HNode n s ants))"
divmod_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "divmod m n = (divide (numeral.numeral one plus m) (numeral.numeral one plus n), modulo (numeral.numeral one plus m) (numeral.numeral one plus n))"
lexordp_eq_simps_3: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "ord.lexordp_eq less (x # xs) [] = False"
iT_Minus_Minus_eq: fixes I :: "nat set"   and k :: "nat" assumes "finite I"   and "Max I \<le> k" shows "k \<ominus> (k \<ominus> I) = I"
set_less_eq_aux''_total: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> infinite_complement_partition"   and "B \<in> infinite_complement_partition" shows "A \<sqsubseteq>''' B \<or> B \<sqsubseteq>''' A"
conjugate_isotone: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugate f g" shows "isotone f"
not_transC_transT: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and s' :: "'state"   and phi :: "bool" assumes "cf \<rightarrow>c cf'"   and "cf \<rightarrow>t s'" shows "phi"
a1: shows "get_S (cpu_reg_val PSR s1) = 0 \<and> good_context s1 \<and> get_delayed_pool s1 = [] \<and> get_trap_set s1 = {} \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> get_delayed_pool s2 = [] \<and> get_trap_set s2 = {} \<and> good_context s2 \<and> low_equal s1 s2"
assms: shows "C1.seq g1 f1 \<and> C2.seq g2 f2"
type_weaken: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and B :: "binding" assumes "\<Delta> @ \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f\<^sub>B B" shows "\<up>\<^sub>e 1 0 \<Delta> @ B \<Colon> \<Gamma> \<turnstile> \<up> 1 \<parallel>\<Delta>\<parallel> t : \<up>\<^sub>\<tau> 1 \<parallel>\<Delta>\<parallel> T"
sq_norm_poly_pos: fixes p :: "'a poly" shows "((0::'a) < \<parallel>p\<parallel>\<^sup>2) = (p \<noteq> 0)"
SUP_mult_right_ennreal: fixes f :: "'b \<Rightarrow> ennreal"   and I :: "'b set"   and c :: "ennreal" shows "Sup (f ` I) * c = (SUP i\<in>I. f i * c)"
ex_un_lam_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b" assumes "cartesian_closed_category C"   and "partial_magma.ide C b"   and "partial_magma.ide C c" shows "partial_magma.in_hom C (cartesian_closed_category.eval C b c) (elementary_category_with_binary_products.prod C (category_with_binary_products.pr0 C) (category_with_binary_products.pr1 C) (cartesian_closed_category.exp C b c) b) c"
invariant_eq_rev: fixes M :: "'a set" shows "rev.invariant M = invariant M"
exists_wt_sym: fixes \<zeta> :: "('s, 'v) hd" shows "\<exists>f\<in>local.ground_heads \<zeta>. wt (Hd \<zeta>) = wt_sym f + the_enat (enat \<delta> * arity_sym f)"
bigo_plus_eq: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. (0::'d) \<le> f x"   and "\<forall>x. (0::'d) \<le> g x" shows "O(f + g) = O(f) + O(g)"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"
cring_simprules_14: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b"   and y :: "'b" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"
r_4: fixes w :: "complex" assumes "w \<in> cball z0 r" shows "f w = zor_poly f z0 w * (w - z0) ^ nat n"
trcl_def: shows "trcl \<equiv> \<lambda>t. {(xb, xa, x). trclp (\<lambda>x xa xb. (x, xa, xb) \<in> t) xb xa x}"
mem_CD_on_Cd: fixes x :: "X4"   and ds :: "D2 set"   and X :: "X4 set" shows "(x \<in> StableNoDecomp.CD_on ds X) = (x \<in> StableNoDecomp.Cd (X4d x) X \<and> X4d x \<in> ds)"
assms: shows "\<forall>t\<in>M. \<exists>s \<delta>. s \<in> N \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t = s \<cdot> \<delta>"
only_fork_introduces_rids': fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r" assumes "revision_step r s s'"   and "\<And>\<sigma> \<tau> \<E> e. s r \<noteq> Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "r' \<notin> RID\<^sub>G s" shows "r' \<notin> RID\<^sub>G s'"
cases: fixes x :: "'c list" obtains xa :: "'c" and xs :: "'c list" where "x = xa # xs"   | "x = []"
extensionalityI: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and g :: "'g \<Rightarrow> 'h" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
that_8: fixes CTa :: "nat \<Rightarrow> classDef option"   and Ca :: "nat"   and e0 :: "exp"   and e0' :: "exp" assumes "x = CTa"   and "xa = Cast Ca e0"   and "xb = Cast Ca e0'"   and "CTa \<turnstile> e0 \<rightarrow> e0'" shows "thesis"
LLq_num2: fixes p :: "'trm" assumes "p \<in> num" shows "\<exists>P\<subseteq>num. finite P \<and> prv (dsj (sdsj {eql (Var yy) r |r. r \<in> P}) (LLq p (Var yy)))"
less_eq_Assertion_def: fixes p :: "'a Assertion"   and q :: "'a Assertion" shows "(p \<le> q) = ({\<cdot> p } \<le> {\<cdot> q })"
InvariantUniqQ_def: fixes Q :: "Literal list" shows "InvariantUniqQ Q = uniq Q"
soundness: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_hoare P c c' Q" shows "ir_valid P c c' Q"
map_replicate_trivial: fixes x :: "'m"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
order: fixes p :: "'a poly"   and a :: "'a" assumes "p \<noteq> 0" shows "[:- a, 1::'a:] ^ Polynomial.order a p dvd p \<and> \<not> [:- a, 1::'a:] ^ Suc (Polynomial.order a p) dvd p"
disj_cancel_left: fixes x :: "'a" shows "NOT x OR x = - (1::'a)"
circle_def: fixes \<mu> :: "complex"   and r :: "real" shows "circle \<mu> r = {z. cmod (z - \<mu>) = r}"
ZFImageRangeExplode: fixes x :: "ZF set"   and f :: "ZF \<Rightarrow> ZF" assumes "x \<in> range explode" shows "f ` x \<in> range explode"
insert_first: fixes l :: "'c list"   and b :: "'c" shows "list_insert l 0 b = b # l"
span_superset: fixes Sa :: "'a set" assumes "Sa \<subseteq> S" shows "Sa \<subseteq> span Sa"
simps_2: fixes n :: "nat"   and e :: "exp" shows "eta_expand (Suc n) e = Lam [fresh_var e]. eta_expand n (App e (fresh_var e))"
comp_in_homI': fixes f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and c :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.arr f"   and "S.VV.arr g"   and "S.VV.dom f = a"   and "S.VV.cod g = c"   and "S.VV.dom g = S.VV.cod f" shows "S.VV.in_hom (S.VV.comp g f) a c"
subcls_ObjectI: fixes G :: "prog"   and C :: "qtname" assumes "is_class G C"   and "ws_prog G" shows "G\<turnstile>C\<preceq>\<^sub>C Object"
B0b: fixes x :: "i" shows "E x \<^bold>\<leftarrow> E (dom x)"
choiceT2: fixes \<xi> :: "'e"   and q :: "('e, 'f, 'g, 'h) seqp"   and a :: "'f seq_action"   and \<xi>' :: "'e"   and q' :: "('e, 'f, 'g, 'h) seqp"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and p :: "('e, 'f, 'g, 'h) seqp" assumes "((\<xi>, q), a, \<xi>', q') \<in> seqp_sos \<Gamma>" shows "((\<xi>, p \<oplus> q), a, \<xi>', q') \<in> seqp_sos \<Gamma>"
tendsto_dist: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and m :: "'b" assumes "(f \<longlongrightarrow> l) F"   and "(g \<longlongrightarrow> m) F" shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
Abs_JF2_inverse: fixes y :: "((bd_type_F + bd_type_F) list set \<times> ((bd_type_F + bd_type_F) list \<Rightarrow> ('d, bd_type_F, bd_type_F) F1 + ('d, bd_type_F, bd_type_F) F2)) set" assumes "y \<in> car_final2" shows "Rep_JF2 (Abs_JF2 y) = y"
hd_in_set: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "hd xs \<in> set xs"
hypnat_add_one_self_less: fixes x :: "nat star" shows "x < x + 1"
extremum: fixes a :: "'a multiset" shows "\<Empt> \<subseteq># a"
finite_measure_mono: fixes A :: "'s stream set"   and B :: "'s stream set"   and s :: "'s" assumes "A \<subseteq> B"   and "B \<in> T.events s" shows "T.prob s A \<le> T.prob s B"
intros_7: fixes v :: "'b val"   and r :: "'b rexp"   and vs :: "'b val list" assumes "\<turnstile> v : r"   and "\<turnstile> Stars vs : Star r" shows "\<turnstile> Stars (v # vs) : Star r"
add_edge_def: fixes v :: "'c"   and v' :: "'c"   and G :: "('c, 'd) graph_scheme" shows "add_edge v v' G = \<lparr>nodes = nodes G \<union> {v, v'}, edges = {(v, v')} \<union> edges G\<rparr>"
assms_1: shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>h h'"
in_arcsD2: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs (subdivide G e w)" shows "v \<in> pverts (subdivide G e w)"
in_set_spmf_iff_spmf: fixes x :: "'a"   and p :: "'a spmf" shows "(x \<in> set_spmf p) = (spmf p x \<noteq> 0)"
mult_vec_minus_mult_right: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "p ** - q = - (p ** q)"
factor_idom: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.idom times one plus zero minus uminus" shows "x divides\<^bsub>\<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>\<^esub> y = (if y = zero then x = zero else dvd.dvd times x y)"
compiler_env_config_rec_def: fixes f :: "bool \<Rightarrow> (string \<times> string list \<times> string) option \<Rightarrow> internal_oids \<Rightarrow> nat \<times> nat \<Rightarrow> generation_semantics_toy \<Rightarrow> toy_class option \<Rightarrow> all_meta_embedding list \<Rightarrow> (string\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<times> toy_instance_single \<times> internal_oids) list \<Rightarrow> (string\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<times> (internal_oids \<times> (string \<times> toy_instance_single) toy_def_state_core) list) list \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<times> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<Rightarrow> string\<^sub>b\<^sub>a\<^sub>s\<^sub>e list \<Rightarrow> generation_lemma_mode option \<times> bool \<Rightarrow> 'b \<Rightarrow> 'a"   and env :: "'b compiler_env_config_scheme" shows "compiler_env_config_rec f env = compiler_env_config_rec0 f env (compiler_env_config.more env)"
eq_iff: fixes x :: "'b list"   and y :: "'b list" shows "(x = y) = (prefix x y \<and> prefix y x)"
nat_to_string___is_strong_nat_string: fixes n :: "nat" shows "is_strong_nat_string (nat_to_string n)"
unx_none_simps_1: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (const name) = None"
eucl_rel_poly: fixes x :: "'a poly"   and y :: "'a poly" shows "eucl_rel_poly x y (x div y, x mod y)"
simps: fixes fs :: "'b list"   and rs :: "'b list" shows "norm (fs, rs) = (if fs = [] then (itrev rs [], []) else (fs, rs))"
cond_spmf_def: fixes p :: "'d spmf"   and A :: "'d set" shows "cond_spmf p A = (if set_spmf p \<inter> A = {} then return_pmf None else cond_pmf p (Some ` A))"
ennreal_top_divide: fixes x :: "ennreal" shows "top / x = (if x = top then 0 else top)"
star_insert_eps: fixes A :: "'b list set" shows "star (insert [] A) = star A"
assms_2: fixes c :: "nat"   and n :: "nat" assumes "0 < c" shows "\<exists>m>n. c * f m < g m"
csrp_fn_def: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set" shows "csrp_fn R I = (\<lambda>x\<in>carrier (R /\<^sub>r I). if x = I then \<zero>\<^bsub>R\<^esub> else SOME y. y \<in> x)"
bbox_def: fixes x :: "'b ppath set"   and y :: "'b ppath set" shows "wp x y = pp_a (x ; pp_a y)"
R_def: fixes A :: "'a set"   and u :: "nat"   and l :: "nat" shows "R A u l = map_pmf (\<lambda>f. steps A f 0 u l) (SL A)"
d: shows "CARD('pq) = CARD('q) * CARD('p)"
TAbs: fixes T :: "POPLmarkRecord.type"   and t :: "trm" shows "(\<lambda><:T. t) \<in> value"
length_ci_goto: fixes ly :: "nat list"   and ss :: "nat"   and n :: "nat" shows "length (ci ly ss (Goto n)) = 2"
proj_tuple_alt: fixes bs :: "bool list"   and as :: "'b option list" shows "proj_tuple bs as = map2 (\<lambda>x y. if x then y else None) bs as"
eq_fold': fixes A :: "'a set" shows "Max A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> max x z)) None A)"
assms_4: shows "\<^bold>g [^] fst z' \<otimes> g' [^] snd z' = a \<otimes> h [^] e'"
lex_conv: fixes r :: "('g \<times> 'g) set" shows "lex r = {(xs, ys). length xs = length ys \<and> (\<exists>xys x y xs' ys'. xs = xys @ x # xs' \<and> ys = xys @ y # ys' \<and> (x, y) \<in> r)}"
sup_left_commute: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool)" shows "(sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) (sup (y::'b) (z::'b)) = sup y (sup x z)"
unit_factor_is_unit: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "is_unit (unit_factor a)"
range_vars_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "range_vars \<sigma> = \<Union> (vars_term ` subst_range \<sigma>)"
quat_norm_pow_2_alt: fixes q :: "quat" shows "(quat_of_real (norm q))\<^sup>2 = Quaternions.cnj q * q"
2320: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> (y \<squnion> top)) = - x"
bigo_imp_conv_radius_ge: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> O(g)" shows "conv_radius g \<le> conv_radius f"
b_def: shows "b \<equiv> \<lambda>n. q ^ fact (Suc n)"
cr_lifted_distrib_lattice_top_def: shows "cr_lifted_distrib_lattice_top \<equiv> \<lambda>x y. x = Rep_lifted_distrib_lattice_top y"
exhaust_sel: fixes state :: "('b, 'c) state" obtains "state = State (run_state state)"
Sup_Sec_def: fixes S :: "Sec set" shows "\<Squnion> S = (if High \<in> S then High else Low)"
ib: shows "i + 1 \<le> j \<and> \<not> ps (S t (i + 1)) p \<noteq> None \<and> (\<forall>k. i + 1 \<le> k \<and> k < j \<longrightarrow> occurs_on (t ! k) \<noteq> p) \<and> j - (i + 1) = n"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set" assumes "K_\<Lambda>_GDD point_set block_collection sizes groups 1" shows "K_GDD point_set block_collection sizes groups"
bounded_clinear_mult_right: fixes x :: "'b" shows "bounded_clinear ((*) x)"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)"   and "P []" shows "P a0"
u'_def: fixes t :: "real" shows "u' t = (if t \<le> \<delta> then ego2.q' t else ego2.s' (t - \<delta>))"