hcju/isabelleps huggingface dataset:
ringed_space_axioms: shows "ringed_space Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
Ord_ordertype: fixes A :: "V set" shows "Ord (tp A)"
exiists: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla"   and i :: "name" assumes "H \<turnstile> A IMP B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> exi i A IMP B"
starset_Compl: fixes A :: "'b set" shows "*s* - A = - (*s* A)"
bounded_suc_ind: fixes m :: "nat"   and P :: "nat \<Rightarrow> 'b"   and z :: "nat"   and k :: "nat" assumes "\<And>k. k < m \<Longrightarrow> P (Suc (z + k)) = P (z + k)"   and "k < m" shows "P (Suc (z + k)) = P z"
\<E>\<tau>_in_Fa: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and a :: "'a"   and \<tau> :: "'a \<Rightarrow> 'b" assumes "yoneda_lemma C S \<phi> F a"   and "natural_transformation (dual_category.comp C) S (yoneda_functor.Y C S \<phi> a) F \<tau>" shows "yoneda_lemma.\<E> S \<phi> a \<tau> \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)"
fact2: shows "msb (~~ w) = (\<not> msb w)"
surjective: fixes r :: "('a, 'b) partial_object_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = partial_object.more r\<rparr>"
sndE: fixes x :: "'a \<times> 'b"   and a :: "'a"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "x = (a, b)"   and "P (snd x)" shows "P b"
list_nonMt_append: fixes s :: "'c list" assumes "s \<noteq> []" shows "\<exists>a t. s = t @ [a]"
fact_nonzero: fixes n :: "nat" shows "fact n \<noteq> (0::'a)"
msg_fresh_3: fixes \<sigma> :: "nat \<Rightarrow> state"   and dests :: "nat \<Rightarrow> nat option"   and sip :: "nat" shows "msg_fresh \<sigma> (Rerr dests sip) = (\<forall>ripc\<in>dom dests. ripc \<in> kD (rt (\<sigma> sip)) \<and> the (dests ripc) - 1 \<le> nsqn (rt (\<sigma> sip)) ripc)"
M_dim_3: shows "gs1.M (length (map of_int_hom.vec_hom fs)) \<in> carrier_mat (length (map of_int_hom.vec_hom fs)) (length (map of_int_hom.vec_hom fs))"
substL_o: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm"   and l :: "('c, 'd) lit" shows "substL (subst \<pi>1 \<circ> \<pi>2) l = substL \<pi>1 (substL \<pi>2 l)"
anonymous: fixes \<pi> :: "'agent \<Rightarrow> 'agent"   and R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<pi> permutes agents"   and "is_pref_profile R" shows "scf (R \<circ> \<pi>) = scf R"
abs_eq: fixes y :: "int" assumes "eq_onp (\<lambda>x. x \<in> {0..<int CARD('b)}) y y" shows "to_int_mod_ring (Abs_mod_ring y) = y"
select_convs_2: fixes init :: "'a set"   and trans :: "('a \<times> 'b \<times> 'a) set"   and more :: "'c" shows "automaton.trans \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = trans"
sel_3: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (x21 ## x22) = x22"
derg: fixes x :: "complex" assumes "x \<in> ball z r" shows "deriv g x = g'__ x"
INF_inf_distrib: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a" shows "Inf (f ` A) \<sqinter> Inf (g ` A) = (INF a\<in>A. f a \<sqinter> g a)"
antisym_conv3: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> y \<prec> x" shows "(\<not> x \<prec> y) = (x = y)"
tan_arctan: fixes y :: "real" shows "tan (arctan y) = y"
bij_betw_nth: fixes xs :: "'b list"   and A :: "nat set"   and B :: "'b set" assumes "distinct xs"   and "A = {..<length xs}"   and "B = set xs" shows "bij_betw ((!) xs) A B"
PChain: shows "\<Psi>\<^sub>G \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"
fin: fixes i :: "nat" assumes "i \<le> n" shows "finite (range (X i))"
simps_4: fixes x11 :: "int"   and x12 :: "int list"   and x21 :: "int"   and x22 :: "int"   and x23 :: "int list" shows "Le x11 x12 \<noteq> Dvd x21 x22 x23"
counit_in_hom_2: shows "\<guillemotleft>\<tau> : h \<star> k \<Rightarrow> trg h\<guillemotright>"
finite_boundaries: shows "\<forall>B\<in>boundary ` two_chain. finite B"
5: shows "(supp \<alpha>1 - bn \<alpha>1) \<sharp>* p"
induct: fixes P :: "'a withERR \<Rightarrow> bool"   and withERR :: "'a withERR" assumes "P ERR"   and "\<And>x. P (OK x)" shows "P withERR"
mset_zero_iff: fixes x :: "'b list" shows "(mset x = {#}) = (x = [])"
T_opt_def: fixes s :: "'c list"   and rs :: "'c list" shows "T\<^sub>p_opt s rs = \<Sqinter> {T\<^sub>p s rs as |as. length as = length rs}"
ltake_lappend: fixes n :: "enat"   and xs :: "'a llist"   and ys :: "'a llist" shows "ltake n (lappend xs ys) = lappend (ltake n xs) (ltake (n - llength xs) ys)"
eval_monom_list_mono_1: fixes g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and m :: "('c \<times> nat) list" assumes "\<And>x. g x \<le> f x"   and "\<And>x. (0::'d) \<le> g x" shows "eval_monom_list g m \<le> eval_monom_list f m"
disj_eq_classes: fixes P :: "'b set set"   and X :: "'b set" assumes "is_non_overlapping P"   and "X \<in> P" shows "X \<inter> \<Union> (P - {X}) = {}"
getlocs_def2: fixes g :: "glb \<Rightarrow> nat"   and l :: "loc \<Rightarrow> nat" shows "loc (st g l) = l"
norm_beta_sound: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^sub>\<N> \<langle>l', D'\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "valid_dbm D" shows "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l', [D']\<^bsub>v,n\<^esub>\<rangle>"
countable_Un: fixes A :: "'d set"   and B :: "'d set" assumes "countable A"   and "countable B" shows "countable (A \<union> B)"
VectorSpaceEndI: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceHom smult V smult T"   and "T ` V \<subseteq> V" shows "VectorSpaceEnd smult V T"
ordinal_plus_mono: fixes x :: "ordinal"   and x' :: "ordinal"   and y :: "ordinal"   and y' :: "ordinal" assumes "x \<le> x'"   and "y \<le> y'" shows "x + y \<le> x' + y'"
roots_of_rat_poly: fixes p :: "rat poly" assumes "p \<noteq> 0" shows "set (roots_of_rat_poly p) = {x. poly p x = 0}"
simps_2: fixes f :: "recf"   and n :: "nat" shows "wrap_Id f (Suc n) = Cn (arity f) (recf.Id 1 0) [wrap_Id f n]"
Rep_dense_filter_type_inject: fixes x :: "'c dense_filter_type"   and y :: "'c dense_filter_type" shows "(Rep_dense_filter_type x = Rep_dense_filter_type y) = (x = y)"
field_differentiable_imp_continuous_at: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set" assumes "f field_differentiable at x within S" shows "continuous (at x within S) f"
map_nth: fixes xs :: "'b list" shows "list.map ((!) xs) [0..<length xs] = xs"
rel_FGl_def: fixes L1 :: "'m \<Rightarrow> 'z \<Rightarrow> bool"   and L2 :: "'n \<Rightarrow> 'aa \<Rightarrow> bool"   and L3 :: "'t \<Rightarrow> 'af \<Rightarrow> bool"   and Co1 :: "'o \<Rightarrow> 'ab \<Rightarrow> bool"   and Co2 :: "'p \<Rightarrow> 'ac \<Rightarrow> bool"   and Co3 :: "'u \<Rightarrow> 'ag \<Rightarrow> bool"   and Co4 :: "'v \<Rightarrow> 'ah \<Rightarrow> bool"   and Contra1 :: "'q \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra2 :: "'r \<Rightarrow> 'ae \<Rightarrow> bool"   and Contra3 :: "'w \<Rightarrow> 'ai \<Rightarrow> bool"   and Contra4 :: "'x \<Rightarrow> 'aj \<Rightarrow> bool" shows "rel_FGl L1 L2 L3 Co1 Co2 Co3 Co4 Contra1 Contra2 Contra3 Contra4 = rel_F (rel_G L1 L2 Co1 Co2 Contra1 Contra2) L1 L3 Co1 Co3 Co4 Contra1 Contra3 Contra4"
frange_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "\<R>\<^sub>\<bullet> (vinsert [a, b]\<^sub>\<circ> r) = vinsert b (\<R>\<^sub>\<bullet> r)"
fls_trivial_delta_eq_zero: fixes b :: "'b"   and a :: "int" assumes "b = (0::'b)" shows "Abs_fls (\<lambda>n. if n = a then b else (0::'b)) = 0"
fun_dual3: fixes f :: "'b set \<Rightarrow> 'c"   and g :: "'b set \<Rightarrow> 'c" shows "(f = g \<circ> (`) \<partial>) = (f \<circ> (`) \<partial> = g)"
sum_singleton: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "sum f {x} = f x"
simps_4: fixes \<phi> :: "('e, 'f + 'h, 'g) formula"   and \<psi> :: "('e, 'f + 'h, 'g) formula"   and \<sigma> :: "'h \<Rightarrow> ('e, 'f, 'g) formula" shows "PFsubst (\<phi> && \<psi>) \<sigma> = (PFsubst \<phi> \<sigma> && PFsubst \<psi> \<sigma>)"
lfp_lowerbound: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq (f A) A" shows "less_eq (complete_lattice.lfp Inf less_eq f) A"
elims_2: fixes x :: "'c arctic_delta" assumes "pos_arctic_delta x" obtains n :: "'c" where "x = Num_arc_delta n" and "(0::'c) \<le> n"
subgraph_def: fixes G :: "('a, 'b) graph"   and H :: "('a, 'b) graph" shows "subgraph G H \<equiv> nodes G = nodes H \<and> edges G \<subseteq> edges H"
monic_monom_smult: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "a \<odot>\<^bsub>P\<^esub> monom P \<one> n = monom P a n"
Lim_in_closed_set: fixes S :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter"   and l :: "'b" assumes "closed S"   and "\<forall>\<^sub>F x in F. f x \<in> S"   and "F \<noteq> bot"   and "(f \<longlongrightarrow> l) F" shows "l \<in> S"
euler_lipschitz: fixes t :: "real"   and t0 :: "real"   and T :: "real"   and L :: "real"   and D' :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "real" assumes "t \<in> {t0..T}"   and "\<forall>t\<in>{t0..T}. L-lipschitz_on D' (f t)" shows "L-lipschitz_on D' (euler_increment f h t)"
rank_add_distrib: fixes x :: "V"   and y :: "V" shows "rank (x + y) = rank x + rank y"
order_trans: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
linorder_cases: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" obtains "x \<prec> y"   | "x = y"   | "y \<prec> x"
a: shows "\<forall>k<i. t ! k = swap_events i j t ! k"
compute_far_float_plus_down: fixes m2 :: "int"   and e1 :: "int"   and e2 :: "int"   and p :: "nat"   and m1 :: "int" assumes "bitlen \<bar>m2\<bar> \<le> e1 - e2 - int (Suc p - nat (bitlen \<bar>m1\<bar>)) - 2"   and "m1 \<noteq> 0"   and "m2 \<noteq> 0"   and "e2 \<le> e1" shows "float_plus_down p (Float m1 e1) (Float m2 e2) = float_round_down p (Float (m1 * 2 ^ Suc (Suc (Suc p - nat (bitlen \<bar>m1\<bar>))) + sgn m2) (e1 - int (Suc p - nat (bitlen \<bar>m1\<bar>)) - 2))"
total_lenlex: fixes r :: "('a \<times> 'a) set" assumes "total r" shows "total (lenlex r)"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x .= y"   and "y .= z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x .= z"
degree_pCons_eq_if: fixes a :: "'a"   and p :: "'a poly" shows "Polynomial.degree (pCons a p) = (if p = 0 then 0 else Suc (Polynomial.degree p))"
vunion_vempty_left: fixes a :: "V" shows "[]\<^sub>\<circ> \<union>\<^sub>\<circ> a = a"
value_mprod_exp: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v"   and "\<forall>j\<le>n. e j \<in> Zset"   and "\<forall>j\<le>n. f j \<in> carrier K - {\<zero>}" shows "v (mprod_exp K e f n) = ASum (\<lambda>j. e j *\<^sub>a v (f j)) n"
exhaust_sel: fixes inference :: "'b inference" obtains "inference = Infer (prems_of inference) (concl_of inference)"
disj_commute: fixes P :: "bool"   and Q :: "bool" shows "(P \<or> Q) = (Q \<or> P)"
infdist_triangle_abs: fixes x :: "'c"   and A :: "'c set"   and y :: "'c" shows "\<bar>infdist x A - infdist y A\<bar> \<le> dist x y"
lens_indep_left_comp: fixes z :: "'a \<Longrightarrow> 'b"   and x :: "'c \<Longrightarrow> 'a"   and y :: "'d \<Longrightarrow> 'a" assumes "mwb_lens z"   and "x \<bowtie> y" shows "x ;\<^sub>L z \<bowtie> y ;\<^sub>L z"
homotopic_paths_join: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and p' :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b"   and q' :: "real \<Rightarrow> 'b" assumes "homotopic_paths s p p'"   and "homotopic_paths s q q'"   and "pathfinish p = pathstart q" shows "homotopic_paths s (p +++ q) (p' +++ q')"
drop_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'a" shows "drop i (replicate k x) = replicate (k - i) x"
ccINF_const: fixes A :: "'c set"   and f :: "'a" assumes "A \<noteq> {}" shows "(INF i\<in>A. f) = f"
stream_numeral: fixes n :: "num" shows "stream (numeral n) = numeral n"
summable_fps: fixes z :: "'a"   and f :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f" shows "summable (\<lambda>n. fps_nth f n * z ^ n)"
inject_2: fixes x31 :: "expr"   and x32 :: "expr"   and x33 :: "expr"   and x34 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x35 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x36 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and x37 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y31 :: "expr"   and y32 :: "expr"   and y33 :: "expr"   and y34 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y35 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y36 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and y37 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set" shows "(CAS x31 x32 x33 x34 x35 x36 x37 = CAS y31 y32 y33 y34 y35 y36 y37) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33 \<and> x34 = y34 \<and> x35 = y35 \<and> x36 = y36 \<and> x37 = y37)"
abs_prod_def: fixes x :: "'a \<times> 'b" shows "\<bar>x\<bar> = (\<bar>fst x\<bar>, \<bar>snd x\<bar>)"
intro: fixes final :: "'c \<Rightarrow> bool"   and r :: "('f, 'd, 'c, 'e, 'g, 'h) semantics"   and P :: "'i \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool" assumes "if_multithreaded final r"   and "lifting_inv final r P" shows "if_lifting_inv final r P"
simps_6: fixes h :: "'a \<Rightarrow> 'b list \<Rightarrow> bool"   and \<phi> :: "'a fm"   and xs :: "'b list" shows "interpret h (Neg \<phi>) xs = (\<not> interpret h \<phi> xs)"
root_int_ceiling_pos: fixes x :: "int"   and p :: "nat" assumes "0 \<le> x" shows "root_int_ceiling_pos p x = \<lceil>root p (real_of_int x)\<rceil>"
conv_path: fixes x :: "'a" shows "path x = path (x\<^sup>T)"
prems_4: assumes "snd b \<noteq> 0"   and "\<not> punit.lt (snd b) adds t" shows "thesis"
parallel_converter2_comp1_out: fixes conv :: "('i, 'k, 'q, 'r) converter"   and conv' :: "('q, 'r, 'm, 'o) converter"   and conv'' :: "('j, 'l, 'n, 'p) converter" shows "conv \<odot> conv' |\<^sub>= conv'' = (conv |\<^sub>= 1\<^sub>C) \<odot> (conv' |\<^sub>= conv'')"
path_through_components: fixes H :: "'a"   and e :: "'a"   and d :: "'a" assumes "equivalence H"   and "bijective (e * top) \<and> bijective (e[[top]])" shows "(H * (d \<squnion> e))\<^sup>\<star> = (H * d)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star> * H * e * (H * d)\<^sup>\<star>"
Throw: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and a' :: "val"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>a'\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Throw e\<rightarrow> abupd (throw a') s1"
harm_altdef: fixes n :: "nat" shows "harm n = (\<Sum>k<n. inverse (of_nat (Suc k)))"
prems_2: fixes dx :: "'a"   and dt :: "'b" shows "higher_differentiable_on (S \<times> cbox a b) (\<lambda>x. D x (dx, dt)) k"
similar_mat_wit_def: fixes A :: "'c mat"   and B :: "'c mat"   and P :: "'c mat"   and Q :: "'c mat" shows "similar_mat_wit A B P Q = (let n = dim_row A in {A, B, P, Q} \<subseteq> carrier_mat n n \<and> P * Q = 1\<^sub>m n \<and> Q * P = 1\<^sub>m n \<and> A = P * B * Q)"
simps_1: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and pf :: "'b" assumes "Solved_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "Solved_Task.nodeOf freshenLC renameLCs lconsts subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions (pf, []) = Conclusion pf"
deep_def: fixes pr :: "('c, 'd) Digit"   and m :: "('c, 'd) FingerTreeStruc"   and sf :: "('c, 'd) Digit" shows "deep pr m sf = Deep (gmd pr + gmft m + gmd sf) pr m sf"
neg_cong: fixes x :: "int" shows "\<ominus>\<^bsub>R\<^esub> (x mod int p) = - x mod int p"
assms_2: shows "D \<in> g \<turnstile> folding_g.\<C>"
1: shows "\<guillemotleft>\<r>[f] : R f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> R \<r>[f] = (f \<star> \<iota>) \<cdot>\<^sub>S \<a>[f, a, a]"
cod_closed: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>" shows "\<guillemotleft>cod f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"
mult_right_le_imp_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq (times a c) (times b c)"   and "less zero c" shows "less_eq a b"
marked_insertions_dequeue_Mutate: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and r :: "'g"   and f :: "'e"   and r' :: "'g option"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers p s = mw_Mutate r f r' # ws"   and "mut_m.marked_insertions m s" shows "mut_m.marked_insertions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (\<lambda>obj. obj\<lparr>obj_fields := (obj_fields obj)(f := r')\<rparr>) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))"
independent_finite: fixes B :: "'c set" assumes "independent B" shows "finite B"
progress: fixes xp :: "nat option"   and frs :: "frame list"   and h :: "heap"   and sh :: "sheap"   and P :: "jvm_prog" assumes "xp = None"   and "frs \<noteq> []" shows "\<exists>\<sigma>'. P \<turnstile> (xp, h, frs, sh) -jvm\<rightarrow>\<^sub>1 \<sigma>'"
sigma_sets_subset: fixes a :: "'a set set" assumes "a \<subseteq> {{}, X, S - X, S}" shows "sigma_sets S a \<subseteq> {{}, X, S - X, S}"
Key_synth_eq: fixes K :: "key"   and H :: "msg set" shows "(Key K \<in> synth H) = (Key K \<in> H)"
Beta_2: shows "(r[u/0] \<degree>\<degree> ss) \<degree> s \<rightarrow>\<^sub>s t \<degree> s'"
in_set_replicate: fixes x :: "'a"   and n :: "nat"   and y :: "'a" shows "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)"
row_space_row_echelon_eq_phull: fixes ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "Keys (set ps) \<subseteq> set ts" shows "row_to_poly ts ` row_space (row_echelon (polys_to_mat ts ps)) = phull (set ps)"
range_disjointed_sets: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> M" shows "range (disjointed A) \<subseteq> M"
ssubst_insert: fixes V :: "name set"   and i :: "name"   and t :: "tm" assumes "finite V"   and "i \<notin> V" shows "ssubst t (insert i V) F = subst i (F i) (ssubst t V F)"
rep_eq: fixes x :: "'c \<Rightarrow> bool"   and xa :: "('c, 'd) fmap" shows "($$) (fmfilter x xa) = map_filter x (($$) xa)"
orderE: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" obtains "a = a + b"
E_App2: fixes v :: "trm"   and t :: "trm"   and t' :: "trm" assumes "val v"   and "t \<longmapsto> t'" shows "v \<cdot> t \<longmapsto> v \<cdot> t'"
semigroup_pair_def: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "semigroup_pair f\<^sub>a f\<^sub>b \<equiv> Type_Semigroups.semigroup f\<^sub>a \<and> Type_Semigroups.semigroup f\<^sub>b"
length_tl: fixes xs :: "'d list" shows "length (tl xs) = length xs - 1"
declclass_qtname_ext_def: fixes q :: "'a qtname_scheme" shows "declclass q = \<lparr>pid = pid q, tid = tid q\<rparr>"
gram_schmidt0_result_2: fixes ws :: "complex vec list" assumes "set ws \<subseteq> carrier_vec d" shows "distinct (gram_schmidt0 d ws)"
Cdom: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e" assumes "Category C"   and "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> f \<in> obj\<^bsub>C\<^esub>"
abs_sum_list_coll: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "list_all (coll 0 x) xs"   and "x \<noteq> 0"   and "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs" shows "\<bar>sum_list xs\<bar> = sum_list (map abs xs)"
Affine_def: fixes X :: "'b \<times> 'b pdevs" shows "Affine X = valuate (\<lambda>e. aform_val e X)"
sets_into_space: fixes x :: "'a set"   and M :: "'a measure" assumes "x \<in> sets M" shows "x \<subseteq> space M"
set_in_finite_cone: fixes Vs :: "'a vec set" assumes "Vs \<subseteq> carrier_vec n"   and "finite Vs" shows "Vs \<subseteq> finite_cone Vs"
that: fixes \<phi> :: "'i \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> diff_fun k charts (\<phi> i)"   and "\<And>i x. \<lbrakk>i \<in> A; x \<in> carrier\<rbrakk> \<Longrightarrow> 0 \<le> \<phi> i x"   and "\<And>i x. \<lbrakk>i \<in> A; x \<in> carrier\<rbrakk> \<Longrightarrow> \<phi> i x \<le> 1"   and "\<And>x. x \<in> carrier \<Longrightarrow> (\<Sum>i | i \<in> A \<and> \<phi> i x \<noteq> 0. \<phi> i x) = 1"   and "\<And>i. i \<in> A \<Longrightarrow> csupport_on carrier (\<phi> i) \<inter> carrier \<subseteq> X i"   and "locally_finite_on carrier A (\<lambda>i. csupport_on carrier (\<phi> i))" shows "thesis"
boffa_aux2: fixes x :: "('b, 'c) boffa_mon" assumes "conway_assm1 x" shows "\<Sum> {x\<^bsub>i\<^esub> \<cdot> x\<^bsub>j\<^esub> |i j. i \<in> UNIV \<and> j \<in> UNIV} \<le> \<Sum> {x\<^bsub>i \<cdot> j\<^esub> |i j. i \<in> UNIV \<and> j \<in> UNIV}"
isl_def: fixes sum :: "'b + 'c" shows "isl sum = (\<exists>x1. sum = Inl x1)"
succ_def: fixes x :: "'a" shows "succ x = val (pos x + 1)"
cong2_lt__lt: 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"   and D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Lt Bet Cong A B C D"   and "Cong A B A' B'"   and "Cong C D C' D'" shows "Tarski_neutral_dimensionless.Lt Bet Cong A' B' C' D'"
pivot_positions_def: fixes zer :: "'b"   and A :: "'b mat" shows "pivot_positions_gen zer A \<equiv> pivot_positions_main_gen zer A (dim_row A) (dim_col A) 0 0"
vars_polynomial_p: shows "vars polynomial_p \<subseteq> {..<weight_space_dim}"