hcju/isabelleps huggingface dataset:
distrib_2: fixes y :: "'b"   and z :: "'b"   and x :: "'b" shows "y \<sqinter> z \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
a_inv_closed: fixes G :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "abelian_group G"   and "x \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x \<in> carrier G"
assms_2: shows "Group.group (DirProds g (f ` A))"
ordinal_minus_less_eq: fixes y :: "ordinal"   and x :: "ordinal"   and z :: "ordinal" assumes "y \<le> x" shows "(x - y < z) = (x < y + z)"
mult_right_le_imp_le: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" 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"
bounded_closed_chain: fixes B :: "'a set"   and \<F> :: "'a set set" assumes "B \<in> \<F>"   and "bounded B"   and "\<And>S. S \<in> \<F> \<Longrightarrow> closed S"   and "{} \<notin> \<F>"   and "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S" shows "\<Inter> \<F> \<noteq> {}"
Rep_N: fixes x :: "'b N" shows "Rep_N x \<in> N"
units_comm_group: shows "comm_group (units_of (add_monoid R))"
hyps: fixes s :: "'a" assumes "s t\<Longrightarrow>\<^bsub>SES\<^esub> s'"   and "s' e\<longrightarrow>\<^bsub>SES\<^esub> sn" shows "s (t @ [e])\<Longrightarrow>\<^bsub>SES\<^esub> sn"
IH: shows "dnf y = Finite_Set.fold (\<lambda>\<phi>. (\<union>) (dnf \<phi>)) {} A"
C_case_Cinf: fixes f :: "C \<rightarrow> 'a" shows "C_case\<cdot>f\<cdot>C\<^sup>\<infinity> = f\<cdot>C\<^sup>\<infinity>"
univ_poly_carrier_subfield_of_consts: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "subfield (ring.poly_of_const R ` carrier R) (carrier R [X]\<^bsub>R\<^esub>)"
cardinal_eqpoll: fixes a :: "V" shows "elts (vcard a) \<approx> elts a"
assms_1: fixes u' :: "'t"   and v' :: "'t" assumes "u' \<in> set ss"   and "v' \<in> set ss"   and "u' adds\<^sub>t v'" shows "u' = v'"
degree_poly_mult_rat: fixes r :: "rat"   and p :: "int poly" assumes "r \<noteq> 0" shows "degree (poly_mult_rat r p) = degree p"
simps_3: fixes c1 :: "'c acom"   and c2 :: "'c acom" shows "strip (c1;; c2) = strip c1;; strip c2"
pred_def: fixes P :: "bexp set" shows "pred P \<equiv> \<lambda>mem. \<forall>p\<in>P. ev\<^sub>B mem p"
simps_26: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'c"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'c"   and x11 :: "nat"   and x12 :: "nat"   and x13 :: "nat"   and x14 :: "nat"   and x15 :: "k"   and x16 :: "nat"   and x17 :: "nat"   and x18 :: "nat" shows "(case Rreq x11 x12 x13 x14 x15 x16 x17 x18 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f1 x11 x12 x13 x14 x15 x16 x17 x18"
zmset_subset_eq_zmultiset_union_diff_commute: fixes A :: "'b zmultiset"   and B :: "'b zmultiset"   and C :: "'b zmultiset" shows "A - B + C = A + C - B"
sum_list_closed: fixes f :: "'a \<Rightarrow> 'r"   and as :: "'a list" assumes "set (map f as) \<subseteq> R" shows "sum_list (map f as) \<in> R"
sams_distincts: 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 E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F" shows "A \<noteq> B \<and> B \<noteq> C \<and> D \<noteq> E \<and> E \<noteq> F"
a_type_wf_def: fixes h :: "(_) heap" shows "l_type_wf_def\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_type_wf h = (DocumentClass.type_wf h \<and> (\<forall>shadow_root_ptr\<in>fset (shadow_root_ptr_kinds h). get shadow_root_ptr h \<noteq> None))"
pochhammer_poly_0: shows "pochhammer_poly 0 = 1"
heap_upds_ok_append: fixes \<Delta> :: "(Vars.var \<times> 'a) list"   and S :: "stack_elem list"   and \<Gamma> :: "(Vars.var \<times> 'a) list" assumes "domA \<Delta> \<inter> upds S = {}"   and "heap_upds_ok (\<Gamma>, S)" shows "heap_upds_ok (\<Delta> @ \<Gamma>, S)"
mcalc2_iff: fixes \<sigma> :: "'tp"   and c :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" shows "mcalc2 \<sigma> c = (infTp \<sigma> \<or> (\<forall>l x. l \<in> set c \<and> x \<in> nv2L l \<and> GE.tpOfV x = \<sigma> \<longrightarrow> isGuard x (grdOf c l x)))"
isTrivialPhi_def: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" shows "uninst_code.ssa.isTrivialPhi phis g v v' = (v' \<noteq> v \<and> (case uninst_code.ssa.phi phis g v of None \<Rightarrow> False | Some vs \<Rightarrow> set vs = {v, v'} \<or> set vs = {v'}))"
face_cycle_succ_inD: fixes x :: "'b"   and y :: "'b" assumes "H.face_cycle_succ x \<in> H.face_cycle_set y" shows "x \<in> H.face_cycle_set y"
assms_3: assumes "\<Phi>" shows "mi \<le> \<Down>\<^sub>E E R m"
HH1: shows "H1 = (A1, B1, C1, D1)"
used_Says_rev: fixes evs :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" shows "used (evs @ [Says A B X]) = parts {X} \<union> used evs"
simps_14: fixes v :: "nat" shows "eta_reducible (Bv v) = False"
1: shows "rev v @ [a] @ [b] @ rev u =\<^sub>S rev v @ [b] @ [a] @ rev u"
3: shows "\<forall>x\<in>subst_domain \<theta>. \<theta> x \<notin> Var ` X"
assms_4: shows "alternating_list n s t @ cs = xs @ [x] @ ys"
ord_less_eq_trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a <s b"   and "b = c" shows "a <s c"
le_infI2: fixes b :: "'e multiset"   and x :: "'e multiset"   and a :: "'e multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
simps_3: fixes shared :: "bool"   and m1 :: "(nat \<times> nat) convnet"   and m2 :: "(nat \<times> nat) convnet" shows "count_weights shared (Pool m1 m2) = (if shared then max (count_weights shared m1) (count_weights shared m2) else count_weights shared m1 + count_weights shared m2)"
eq_effect_eq_vset: fixes x :: "('d, bool) fmap \<times> ('d, bool) fmap"   and y :: "('d, bool) fmap \<times> ('d, bool) fmap"   and vs :: "'d set" assumes "snd x = snd y" shows "varset_action x vs = varset_action y vs"
wt_App_\<delta>: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm"   and t :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (App s t) = kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym t" shows "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s = \<delta>"
\<F>\<G>_elim: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> \<F>\<G> \<phi> w" shows "w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)"
distinct_3: fixes x1 :: "Vars.var"   and x31 :: "assn"   and x32 :: "exp" shows "Var x1 \<noteq> LetA x31 x32"
eventually_at_ball': fixes d :: "real"   and z :: "'b"   and A :: "'b set" assumes "0 < d" shows "\<forall>\<^sub>F t in at z within A. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A"
retractionE: fixes g :: "'b" assumes "D.retraction g" obtains f :: "'b" where "D.ide (g \<cdot>\<^sub>D f)"
card_face_cycle_sets_da: shows "card G.face_cycle_sets \<le> Suc (card H.face_cycle_sets)"
finprod_insert: fixes F :: "'e set"   and a :: "'e"   and f :: "'e \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier G"   and "f a \<in> carrier G" shows "finprod G f (insert a F) = f a \<otimes> finprod G f F"
trivial_limit_eq: fixes net :: "'c filter" shows "(net = bot) = (\<forall>P. eventually P net)"
Cons: fixes v :: "'a"   and w :: "'a"   and vs :: "'a list" assumes "v \<rightarrow> w"   and "walk (w # vs)" shows "walk (v # w # vs)"
ex_n: shows "\<exists>n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n"
assms: shows "llist_all2 (\<lambda>xs ys. llength xs = llength ys) xss yss"
set_less_eq_aux_Compl_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and ao :: "'a option"   and s1 :: "'d"   and s2 :: "'e" shows "set_less_eq_aux_Compl_fusion g1 g2 ao s1 s2 = set_less_eq_aux_Compl ao (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
m_def: shows "m = Mn 3 (Cn 4 r_le [Cn 4 r_pdec1 [r_both], r_constn 3 1])"
has_derivative_bounded_linear: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "D f \<mapsto> f' F" shows "bounded_linear f'"
simps_19: fixes sh :: "sheap"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "'a exp" shows "iconf sh (INIT C (Cs,b) \<leftarrow> e) = ((case Cs of [] \<Rightarrow> initPD sh C | C' # Cs' \<Rightarrow> last Cs = C) \<and> \<not> sub_RI e)"
enabled_Stream: fixes x :: "'a cfg"   and y :: "'a cfg"   and \<omega> :: "'a cfg stream" shows "MC.enabled x (y ## \<omega>) = (y \<in> set_pmf (K_cfg x) \<and> MC.enabled y \<omega>)"
vics: shows "P,h,sh \<turnstile>\<^sub>i (C, M, pc, ics)"
incidence_system_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset" shows "incidence_system point_set block_collection \<equiv> \<forall>b. b \<in># block_collection \<longrightarrow> b \<subseteq> point_set"
has_integral_implies_lebesgue_measurable: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'d"   and \<Omega> :: "'c set" assumes "(f has_integral I) \<Omega>" shows "(\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable lebesgue"
enat_less_enat_plusI: fixes x :: "nat"   and y :: "nat"   and z :: "enat" assumes "x < y" shows "enat x < enat y + z"
bigo_measure_trans: fixes A :: "'a set"   and t :: "'a \<Rightarrow> real"   and t' :: "nat \<Rightarrow> real"   and m :: "'a \<Rightarrow> nat"   and f :: "nat \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> t x \<le> (t' \<circ> m) x"   and "t' \<in> O(f)"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> t x" shows "t \<in> O[m going_to sequentially within A](f \<circ> m)"
castVar_def: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" shows "castVar s X \<equiv> SOME xs_x. s = asSort (fst xs_x) \<and> X = Var (fst xs_x) (snd xs_x)"
prime_factorization_gcd_factorial: fixes a :: "'a"   and b :: "'a" assumes "a \<noteq> (0::'a)"   and "b \<noteq> (0::'a)" shows "prime_factorization (gcd_factorial a b) = prime_factorization a \<inter># prime_factorization b"
False: shows "\<not> (x = 0 \<or> sin (x / 2) = 0)"
discI_2: fixes llist :: "'a llist"   and x21 :: "'a"   and x22 :: "'a llist" assumes "llist = x21 #\<^sub>l x22" shows "\<not> lnull llist"
mult_mat_vec_scalar: fixes A :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec"   and v :: "('d, 'f) Finite_Cartesian_Product.vec" shows "A *v v = local.mk_vec (\<lambda>i. scalar_product (row i A) v)"
MP_subset_SMP_1: fixes S :: "('fun, 'var) strand_step list" shows "\<Union> (trms\<^sub>s\<^sub>t\<^sub>p ` set S) \<subseteq> SMP (trms\<^sub>s\<^sub>t S)"
finite_Collect_subsets: fixes A :: "'d set" assumes "finite A" shows "finite {B. B \<subseteq> A}"
resChainI: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and xs :: "name list" assumes "P \<leadsto>[Rel] Q"   and "eqvt Rel"   and "\<And>P Q x. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel" shows "resChain xs P \<leadsto>[Rel] resChain xs Q"
CALL_cor: fixes q :: "pname"   and pre :: "state" shows "(\<Union>p. {(\<lambda>z s. (z = s \<and> body p \<down> s) \<and> ((p, s), q, pre) \<in> termi_call_steps, CALL p, \<lambda>z. exec z (body p))}) \<turnstile>\<^sub>t {\<lambda>z s. (z = s \<and> body q \<down> s) \<and> s = pre} body q {\<lambda>z. exec z (body q)}"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<omega>.chine\<guillemotright>"
Gauss_Jordan_upt_k_det_P_iarrays_def: fixes A :: "'b iarray iarray"   and k :: "nat" shows "Gauss_Jordan_upt_k_det_P_iarrays A k = (let foldl = foldl Gauss_Jordan_column_k_det_P_iarrays (1::'b, 0, A) [0..<Suc k] in (fst foldl, snd (snd foldl)))"
to_precondition_def: shows "SAS_Plus_Representation.to_precondition \<equiv> sas_plus_operator.precondition_of"
Fvars_all: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'a"   and \<phi> :: "'c" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "x \<in> var"   and "\<phi> \<in> fmla" shows "Fvars (all x \<phi>) = Fvars \<phi> - {x}"
measure\<^sub>s\<^sub>t_alt_def: fixes s :: "('a, 'b) strand_step list"   and x :: "'b \<Rightarrow> ('a, 'b) Term.term"   and t :: "('a, 'b) strand_step list"   and y :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "(((s, x), t, y) \<in> measure\<^sub>s\<^sub>t) = (card (fv\<^sub>s\<^sub>t s) < card (fv\<^sub>s\<^sub>t t) \<or> card (fv\<^sub>s\<^sub>t s) = card (fv\<^sub>s\<^sub>t t) \<and> size\<^sub>s\<^sub>t s < size\<^sub>s\<^sub>t t)"
f_bounded: fixes T :: "real set"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "real \<times> 'b \<Rightarrow> real \<times> 'b \<Rightarrow> 'b"   and B :: "real"   and B' :: "real"   and t :: "real"   and x :: "'b" assumes "derivative_norm_bounded T X f f' B B'"   and "t \<in> T"   and "x \<in> X" shows "norm (f t x) \<le> B"
set_of_sub_inc: fixes A :: "'a interval"   and A' :: "'a interval"   and B :: "'a interval"   and B' :: "'a interval" assumes "set_of A \<subseteq> set_of A'"   and "set_of B \<subseteq> set_of B'" shows "set_of (A - B) \<subseteq> set_of (A' - B')"
IH: fixes T :: "secLevel" assumes "\<Gamma>,T \<turnstile> c'" shows "nonInterference \<Gamma> c'"
hyps: shows "\<Gamma>\<turnstile> (cs @ xs, css, Fault f) \<rightarrow>\<^sup>* (xs, css, Fault f)"
Bleast_def: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Bleast S P = (LEAST x. x \<in> S \<and> P x)"
Least_equality: fixes P :: "'c list \<Rightarrow> bool"   and x :: "'c list" assumes "P x"   and "\<And>y. P y \<Longrightarrow> prefix x y" shows "prefix_order.Least P = x"
ereal_uminus_eq_iff: fixes a :: "ereal"   and b :: "ereal" shows "(- a = - b) = (a = b)"
AE_lim_acc: fixes x :: "'a"   and t :: "real" shows "AE \<omega> in K.lim_stream (t, x). \<forall>t z. trace_in {z} t x \<omega> \<longrightarrow> (x, z) \<in> (SIGMA x:UNIV. set_pmf (J x))\<^sup>*"
any_\<Sigma>: shows "any \<in> set \<Sigma>"
analz_image_freshK_lemma: fixes K :: "nat"   and nE :: "nat set"   and H :: "msg set" assumes "Key K \<in> analz (Key ` nE \<union> H) \<longrightarrow> K \<in> nE \<or> Key K \<in> analz H" shows "(Key K \<in> analz (Key ` nE \<union> H)) = (K \<in> nE \<or> Key K \<in> analz H)"
P2: shows "Bet D E D0 \<and> D' E' F' CongA F E D0"
agents_eq: shows "agents' = {i. R i \<noteq> (\<lambda>_ _. False)}"
him: fixes x :: "'b" assumes "x \<in> S'" shows "h x \<in> X"
cube_line_integ: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> one_chain1" shows "line_integral_exists F basis \<gamma> \<and> one_chain_line_integral F basis (f (k, \<gamma>)) = real_of_int k * line_integral F basis \<gamma>"
surjI: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. g (f x) = x" shows "surj g"
that: fixes s' :: "'s" assumes "removel s = (hd (\<alpha> s), s')"   and "invar s'"   and "\<alpha> s' = tl (\<alpha> s)" shows "thesis"
sat_push_alt: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and e :: "nat \<times> nat" assumes "sat_push_precond f l e" shows "push_effect f e = augment_edge f e (cf_of f e)"
is_weak_composition: shows "weak_composition (\<cdot>) (\<star>)"
fib_Suc_mono: fixes m :: "nat" shows "fib m \<le> fib (Suc m)"
Compl_step2: fixes I :: "'a set"   and s :: "'a set" assumes "\<forall>x\<in>I. \<exists>y\<in>s. x \<rightarrow>\<^sub>i* y" shows "\<forall>x\<in>I. x \<in> s \<or> (\<exists>sl. sl \<noteq> [] \<and> tl sl \<noteq> [] \<and> (sl ! 0, sl ! (length sl - 1)) = ({x}, s) \<and> (\<forall>i<length sl - 1. \<turnstile>\<N>\<^bsub>(sl ! i, sl ! (i + 1))\<^esub>))"
gcd_0_right: fixes a :: "'a" shows "gcd a (0::'a) = normalize a"
prj_tuple_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b"   and k :: "'b" assumes "elementary_category_with_pullbacks C prj0 prj1"   and "category.commutative_square C f g h k" shows "C (prj0 f g) (elementary_category_with_pullbacks.tuple C prj0 prj1 h f g k) = k"
vec_le_sq_norm: fixes v :: "'a vec"   and n :: "nat"   and i :: "nat" assumes "v \<in> carrier_vec n"   and "i < n" shows "\<bar>v $ i\<bar>\<^sup>2 \<le> \<parallel>v\<parallel>\<^sup>2"
zdvd_not_zless: fixes m :: "int"   and n :: "int" assumes "0 < m"   and "m < n" shows "\<not> n dvd m"
vcType_empty: fixes a :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> [vc_com]" shows "a t = []"
simps: fixes os :: "'t iarray"   and s :: "'s"   and t_end :: "'s"   and n :: "nat" shows "forward_ix_rec os s t_end n = (if IArray.length os \<le> n then indicator {t_end} s else \<Sum>t\<leftarrow>state_list. ennreal (pmf (\<O> t) (os !! n)) * ennreal (pmf (\<K> s) t) * forward_ix_rec os t t_end (n + 1))"
enum_idx_def: fixes x :: "'b" shows "enum_idx x = index_of x enum_class.enum"
exhaust: fixes y :: "('a, 'b, 'c) state" obtains x1 :: "'a" and x2 :: "'b" and x3 :: "'c list" where "y = State x1 x2 x3"
real_power_nonzero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and real_power :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and d :: "real"   and a :: "'a" assumes "class.nat_power_normed_field one times plus zero nat_power divide inverse minus uminus dist norm scaleR sgn uniformity open inner real_power"   and "0 < d" shows "real_power d a \<noteq> zero"
simps_1: fixes L :: "('c \<times> nat) option"   and t :: "'c" shows "acquire_locks L t 0 = L"
FG_Alm_all: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<phi>) = (\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n \<phi>)"
arg_min_list_in: fixes xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "xs \<noteq> []" shows "arg_min_list f xs \<in> set xs"
Abs_hyp2_inject: fixes x :: "proj2"   and y :: "proj2" assumes "x \<in> hyp2"   and "y \<in> hyp2" shows "(Abs_hyp2 x = Abs_hyp2 y) = (x = y)"
nonzero_mult_closed: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp"   and "b \<in> nonzero Zp" shows "a \<otimes> b \<in> nonzero Zp"
Un_Image: fixes R :: "('f \<times> 'e) set"   and S :: "('f \<times> 'e) set"   and A :: "'f set" shows "(R \<union> S) `` A = R `` A \<union> S `` A"
open_core_eq: fixes a :: "'a llist set" assumes "a open" shows "core a = a"
PC_monotone: shows "\<forall>p s t. correct p t \<and> s \<le> t \<longrightarrow> PC p s \<le> PC p t"
isomorphic_transitive: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr" assumes "Maps.isomorphic f g"   and "Maps.isomorphic g h" shows "Maps.isomorphic f h"
remove_def: fixes x :: "'c"   and A :: "'c set" shows "Set.remove x A = A - {x}"
simps: fixes a1 :: "'b"   and a2 :: "'c"   and r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" shows "((a1, a2) \<in> r O s) = (\<exists>a b c. a1 = a \<and> a2 = c \<and> (a, b) \<in> r \<and> (b, c) \<in> s)"
integrable_scaleR_left: fixes c :: "'d"   and M :: "'e measure"   and f :: "'e \<Rightarrow> real" assumes "c \<noteq> (0::'d) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x *\<^sub>R c)"
ball_trivial: fixes x :: "'a" shows "ball x 0 = {}"
monic_monom: fixes n :: "nat" shows "Q (monom P \<one> n)"
frameResComm': fixes x :: "name"   and xvec :: "name list"   and F :: "'b frame" shows "\<lparr>\<nu>x\<rparr>\<lparr>\<nu>*xvec\<rparr>F \<simeq>\<^sub>F \<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>F"
bres_galois: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b"   and z :: "'a \<times> 'b" shows "less_eq_prod (times_prod' x y) z = less_eq_prod y (prod.bres x z)"
one_neq_neg_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus (numeral.numeral one plus n)"
id_nec2_2_Aux: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<diamond>\<phi> \<^bold>\<equiv> \<psi> in v]" shows "[\<^bold>\<not>\<psi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<not>\<phi> in v]"
l_get_parent_get_host_wf_is_get_parent_get_host_wf: shows "l_get_parent_get_host_wf Shadow_DOM.heap_is_wellformed Shadow_DOM.get_parent get_shadow_root get_host Shadow_DOM.get_child_nodes get_disconnected_document get_disconnected_nodes"
fls_shift_fls_shift: fixes m :: "int"   and k :: "int"   and f :: "'b fls" shows "fls_shift m (fls_shift k f) = fls_shift (k + m) f"
case_2: fixes f1 :: "'c"   and f2 :: "nat \<Rightarrow> 'c"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
bounded_derivative_imp_lipschitz: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and C :: "real" assumes "\<And>x. x \<in> X \<Longrightarrow> (f has_derivative f' x) (at x within X)"   and "convex X"   and "\<And>x. x \<in> X \<Longrightarrow> onorm (f' x) \<le> C"   and "0 \<le> C" shows "C-lipschitz_on X f"
extend_def: fixes forms :: "('a, 'b) sequent"   and seq :: "('a, 'b) sequent" shows "extend forms seq \<equiv>  antec forms + antec seq \<Rightarrow>* succ forms + succ seq"
le_imp_less_or_eq: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<preceq> y" shows "x \<prec> y \<or> x = y"
sdsj_def: fixes F :: "'fmla set" shows "sdsj F = ldsj (asList F)"