hcju/isabelleps huggingface dataset:
rGamma_series_nonpos_Ints_LIMSEQ: fixes z :: "'a" assumes "z \<in> \<int>\<^sub>\<le>\<^sub>0" shows "rGamma_series z \<longlonglongrightarrow> (0::'a)"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (C.L a)"
nb\<^sub>2: fixes v :: "'a" assumes "(\<Pi>\<^sub>G) v \<noteq> None" shows "\<A> (State (length \<pi>) (index (\<Pi>\<^sub>\<V>) v)) = ((\<Pi>\<^sub>G) v = Some True)"
projection_on_intersection: fixes l :: "'a list"   and X :: "'a set"   and Y :: "'a set" assumes "l \<upharpoonleft> X = []" shows "l \<upharpoonleft> (X \<inter> Y) = []"
rw: shows "\<forall>A n. A \<in> carrier_mat n n \<and> isDiagonal_mat A \<longrightarrow> (\<exists>P Q. P \<in> carrier_mat n n \<and> Q \<in> carrier_mat n n \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat (P * A * Q))"
non_volatile_reads_acquired_or_read_only_reads: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb" shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> \<O> \<union> all_acquired sb \<union> read_only_reads \<O> sb"
coerceQuant_eq_iff: fixes x :: "'d['b, 'e]" assumes "QD('b) = QD('c)" shows "QCOERCE['c] x \<cong>\<^sub>Q x"
i2: shows "\<forall>d\<in>D. bal (dblock s q) < mbal (disk s d qq)"
list_induct2: fixes xs :: "'c list"   and ys :: "'d list"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "length xs = length ys"   and "P [] []"   and "\<And>x xs y ys. \<lbrakk>length xs = length ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P xs ys"
top2I: fixes x :: "'d"   and y :: "'e" shows "\<top> x y"
base: shows "alw (\<lambda>xs. \<not> P xs) \<omega>"
Rep_range: shows "range Rep_bit0 = {0..<2 * int CARD('a)}"
borel_measurable_continuous_onI: fixes f :: "'a \<Rightarrow> 'b" assumes "continuous_on UNIV f" shows "f \<in> borel_measurable borel"
fact5: assumes "\<exists>v1 v2. fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') = 0" shows "False"
polynomial_bool_def: shows "polynomial_bool = range (\<lambda>c. (Var c)\<^sup>2 - Var c)"
has_vector_derivative_transform_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set"   and d :: "real"   and g :: "real \<Rightarrow> 'a" assumes "(f has_vector_derivative f') (at x within S)"   and "0 < d"   and "x \<in> S"   and "\<And>x'. \<lbrakk>x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g has_vector_derivative f') (at x within S)"
ENF_offending_case1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "sinvar_all_edges_normal_form P"   and "\<not> sinvar G nP" shows "{{(e1, e2). (e1, e2) \<in> edges G \<and> \<not> P (nP e1) (nP e2)}} = set_offending_flows G nP"
00: shows "rawpsubstT (rawpsubstT r (zip (map Var uus) (map snd (tx # txs)))) (zip (map fst (tx # txs)) uus) = rawpsubstT (substT (rawpsubstT r (zip (map Var us) (map snd txs) @ [(Var u, x)])) t u) (zip (map fst txs) us)"
simps_2: fixes m :: "'a"   and x :: "nat" shows "parent_node (Formal_in (m, x)) = m"
bilinear_lneg: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and x :: "'a"   and y :: "'b" assumes "bilinear h" shows "h (- x) y = - h x y"
simps_1: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list"   and accu :: "('b \<times> 'b) list" shows "TopoS_Stateful_Policy_impl.filter_compliant_stateful_ACS_accu G M accu [] = accu"
reduced_GB_pmdl_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F" shows "pmdl (reduced_GB F) = pmdl F"
ring_simprules_17: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<oplus> (\<ominus> x \<oplus> y) = y"
eq_set_preserves_inter: fixes eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and set :: "'d \<Rightarrow> 'e set"   and Ss :: "'e set set" assumes "\<And>r s. eq r s \<Longrightarrow> set r = set s"   and "Ss \<noteq> {}" shows "(\<Inter>As\<in>Ss. {(x, x'). eq x x'} `` {x. set x \<subseteq> As}) \<subseteq> {(x, x'). eq x x'} `` {x. set x \<subseteq> \<Inter> Ss}"
ssubstr_def: fixes i :: "nat"   and j :: "nat"   and xs :: "'b seq" shows "ssubstr i j xs = map (Sequence.seq.Rep_seq xs) [i..<j]"
known_ptr_impl: shows "known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = DocumentClass.known_ptr"
H2: shows "\<Gamma>\<turnstile>T2<:T11"
sel_9: fixes x21 :: "'c"   and x22 :: "'a"   and x23 :: "'c"   and x24 :: "'a" shows "pac_res (Mult x21 x22 x23 x24) = x24"
chine_simps_2: shows "C.dom \<eta>.chine = \<eta>.dom.apex"
intros: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term" assumes "t = s \<cdot> \<sigma>" shows "s \<le>\<cdot> t"
trivial_IrrFinGroupRepresentation_in_FG: assumes "of_nat (card G) \<noteq> (0::'a)" shows "IrrFinGroupRepresentation G (*) 0"
G_def: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and f :: "'d" assumes "left_adjoint_functor D C F" shows "left_adjoint_functor.G D C F f = (if partial_magma.arr C f then terminal_arrow_from_functor.the_coext D C F (left_adjoint_functor.Go D C F (partial_magma.cod C f)) (left_adjoint_functor.\<epsilon>o D C F (partial_magma.cod C f)) (left_adjoint_functor.Go D C F (partial_magma.dom C f)) (C f (left_adjoint_functor.\<epsilon>o D C F (partial_magma.dom C f))) else partial_magma.null D)"
hyps: fixes Y :: "'a multiset"   and thesis :: "bool" assumes "\<And>Y1 Y2. \<lbrakk>Y = Y1 + Y2; (X1, Y1) \<in> multpw ns; (X2, Y2) \<in> multpw ns\<rbrakk> \<Longrightarrow> thesis"   and "(X1 + X2, Y) \<in> multpw ns" shows "thesis"
cases: fixes x :: "'c constr list \<times> nat \<times> 'c TESL_atomic list \<times> 'c TESL_atomic list" obtains \<Gamma> :: "'c constr list" and n :: "nat" and \<Psi> :: "'c TESL_atomic list" and \<Phi> :: "'c TESL_atomic list" where "x = \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>"
holomorphic_hurwitz_zeta: fixes a :: "real"   and A :: "complex set" assumes "0 < a"   and "1 \<notin> A" shows "hurwitz_zeta a holomorphic_on A"
UNIV_2: fixes A :: "'c set" shows "(A \<subseteq> UNIV) = True"
f_nxt_eq_empty_iff: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b"   and x' :: "'b" shows "(f_nxt f T x x' = TTree.empty) = (f x' = TTree.empty \<or> x' = x \<and> x \<in> T)"
termdiffs_strong': fixes K :: "real"   and c :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "\<And>z. norm z < K \<Longrightarrow> summable (\<lambda>n. c n * z ^ n)"   and "norm z < K" shows "((\<lambda>z. \<Sum>n. c n * z ^ n) has_field_derivative (\<Sum>n. diffs c n * z ^ n)) (at z)"
sup_le2: fixes a :: "'a"   and b :: "'a" shows "a \<le> pseudo_hoop_algebra.sup2 (\<sqinter>) (r\<rightarrow>) (l\<rightarrow>) a b"
distinct_1: fixes x1 :: "xcpt"   and x2 :: "jump" shows "Xcpt x1 \<noteq> Jump x2"
one_in_Polys: fixes X :: "'a set" shows "1 \<in> P[X]"
minimal_antichain_nonempty: fixes A :: "'a set"   and t :: "'a" assumes "finite A"   and "t \<in> A" shows "minimal_antichain A \<noteq> {}"
mhomom3Tr2: fixes H :: "'a set"   and K :: "'a set" assumes "submodule R M H"   and "submodule R M K" shows "ker\<^bsub>mdl M H,mdl M (H \<minusplus> K) /\<^sub>m K\<^esub> compos (mdl M H) (mpj (mdl M (H \<minusplus> K)) K) (\<iota>m\<^bsub>M H,K\<^esub>) = H \<inter> K"
node_deliver_messages_Broadcast: fixes m :: "'msgid \<times> 'op" shows "node_deliver_messages [Broadcast m] = []"
hom_uminus: fixes x :: "'a poly" shows "pCons (0::'a) (- x) = - pCons (0::'a) x"
congruent_strQ1: shows "(F1map id (Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2)) (Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2)) \<circ> strT1) respects lsbis1 carT1 carT2 strT1 strT2"
map_spmf_bind_spmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'c spmf"   and g :: "'c \<Rightarrow> 'b spmf" shows "map_spmf f (bind_spmf p g) = bind_spmf p (map_spmf f \<circ> g)"
get_tag_name_reads: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads (get_tag_name_locs element_ptr) (get_tag_name element_ptr) h h'"
simps_5: fixes v :: "MIR.num" shows "dvdnumcoeff (Neg v) = (\<lambda>g. False)"
lookup_in_tree: fixes t :: "('e, 'd) RBT.rbt"   and k :: "'e"   and v :: "'d" shows "(RBT.lookup t k = Some v) = ((k, v) \<in> set (RBT.entries t))"
diagonal_imp_triangular_1x2: fixes A :: "'a mat" assumes "A \<in> carrier_mat 1 2"   and "admits_diagonal_reduction A" shows "admits_triangular_reduction A"
joinI: fixes a :: "'b"   and c :: "'b"   and A :: "('b \<times> 'b) set"   and b :: "'b" assumes "(a, c) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<down>"
Fourier_sum_limit_sine_part: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x"   and "0 < d"   and "d \<le> pi" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l = (\<lambda>x. LINT xa|lebesgue_on {0..d}. sin ((real x + 1 / 2) * xa) * ((f (t + xa) + f (t - xa) - 2 * l) / xa)) \<longlonglongrightarrow> 0"
orthogonal_matrix_transpose: fixes A :: "(('b, 'c) vec, 'c) vec" shows "orthogonal_matrix (Finite_Cartesian_Product.transpose A) = orthogonal_matrix A"
CJNB_qe: fixes qe :: "fm \<Rightarrow> fm" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bs (qe p) = Ifm bs (E p)" shows "\<forall>bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)"
merge_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" shows "mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i = {uu_. \<exists>x y n j. uu_ = (x, y) \<and> j < i \<and> (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<noteq> semi_mojmir_def.token_run \<delta> q\<^sub>0 w y n \<and> semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n \<noteq> None \<or> y = Suc n) \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc n) = semi_mojmir_def.token_run \<delta> q\<^sub>0 w y (Suc n) \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc n) \<notin> F \<and> semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some j}"
is_nff_def: fixes dflt :: "'a"   and f :: "nat \<Rightarrow> 'a"   and a :: "'a array" shows "is_nff dflt f a \<equiv> \<exists>\<^sub>Al. a \<mapsto>\<^sub>a l * \<up> (f = nff_\<alpha> dflt l)"
is_partition_alt_def: fixes cs :: "'c set list" shows "is_partition_alt cs = (\<forall>i j. i < length cs \<and> j < length cs \<and> i \<noteq> j \<longrightarrow> cs ! i \<inter> cs ! j = {})"
intuitionistic_sep_conj_sep_true: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.intuitionistic plus sep_disj (sep_algebra.sep_conj plus sep_disj (\<lambda>s. True) P)"
iso_is_section: fixes f :: "'a" assumes "C.iso f" shows "C.section f"
UN_mono: fixes A :: "'e set"   and B :: "'e set"   and f :: "'e \<Rightarrow> 'f set"   and g :: "'e \<Rightarrow> 'f set" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<subseteq> g x" shows "\<Union> (f ` A) \<subseteq> \<Union> (g ` B)"
simps_2: fixes c :: "'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b ref"   and r :: "'b \<Rightarrow> 'b ref"   and iL :: "'b \<Rightarrow> 'b ref"   and iR :: "'b \<Rightarrow> 'b ref"   and t :: "'b ref"   and p :: "'b"   and stk :: "'b list" shows "stkOk c l r iL iR t (p # stk) = (stkOk c l r iL iR (Ref p) stk \<and> iL p = (if c p then l p else t) \<and> iR p = (if c p then t else r p))"
pwd_owner_iii_msg: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "case pwd_owner_iii n of (s, S, A, B, C, D) \<Rightarrow> pwd_owner_iii_state n (S, A, B, C, D) \<subseteq> s \<and> {Key (Auth_ShaKey n)} \<in> key_sets (PriKey S) (crypts (Log -` spied s))"
the_inv_into_map_def: fixes V :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "the_inv_into_map V f x = (if x \<in> f ` V then Some (the_inv_into V f x) else None)"
beta_norm_preserves_proves: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "\<Theta>,\<Gamma> \<turnstile> t"   and "beta_norm t = Some u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> u"
bilinear_form_congruence: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "bilinear_form (M *\<^sub>m\<^sub>v v1) (M *\<^sub>m\<^sub>v v2) (congruence (mat_inv M) H) = bilinear_form v1 v2 H"
r_coset_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set"   and a :: "'b" shows "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})"
assms: fixes p :: "('a, 'b, 'c, 'd) seqp" assumes "p \<in> Range \<sigma>" shows "\<exists>pn. p \<in> subterms (\<Gamma> pn)"
e: shows "y \<sharp> (c, x, Ys, As, pi)"
1: shows "avl \<langle>l, (ab, h), r\<rangle>"
UN_binaryset_eq: fixes A :: "'c set"   and B :: "'c set" shows "\<Union> (range (binaryset A B)) = A \<union> B"
intro: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "\<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H" shows "normal_axioms H G"
real_sum: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and cnj :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and real_of :: "'a \<Rightarrow> real"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "class.real_of_extended divide inverse times one plus zero minus uminus cnj scaleR real_of" shows "real_of (comm_monoid_add.sum plus zero f A) = (\<Sum>i\<in>A. real_of (f i))"
v_def: shows "v = (\<lambda>m. ereal (u m / real m))"
subset_1: fixes x :: "nat" assumes "x \<in> A1" shows "prime x"
outputInduct: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto> a[b] \<prec> P'"   and "\<And>a b P C. F C (a{b}.P) a b P"   and "\<And>P a b P' c C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C ([c\<frown>c]P) a b P'"   and "\<And>P a b P' c d C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; \<And>C. F C P a b P'; c \<noteq> d\<rbrakk> \<Longrightarrow> F C ([c\<noteq>d]P) a b P'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b P'"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto> a[b] \<prec> Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a b Q'"   and "\<And>P a b P' Q C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P' \<parallel> Q)"   and "\<And>Q a b Q' P C. \<lbrakk>Q \<longmapsto> a[b] \<prec> Q'; \<And>C. F C Q a b Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a b (P \<parallel> Q')"   and "\<And>P a b P' x C. \<lbrakk>P \<longmapsto> a[b] \<prec> P'; x \<noteq> a; x \<noteq> b; x \<sharp> C; \<And>C. F C P a b P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a b (<\<nu>x>P')"   and "\<And>P a b P' C. \<lbrakk>P \<parallel> !P \<longmapsto> a[b] \<prec> P'; \<And>C. F C (P \<parallel> !P) a b P'\<rbrakk> \<Longrightarrow> F C (!P) a b P'" shows "F C P a b P'"
is_regular_spairI': fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and a :: "'a"   and b :: "'a" assumes "rep_list p \<noteq> 0"   and "rep_list q \<noteq> 0"   and "a + punit.lt (rep_list p) = b + punit.lt (rep_list q)"   and "a \<oplus> lt p \<noteq> b \<oplus> lt q" shows "is_regular_spair p q"
vrange_atE: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_plus" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> vreal_plus" and "vreal_plus\<lparr>a\<rparr> = b"
Proc_CFG_CondElse: fixes c\<^sub>2 :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and b :: "expr"   and c\<^sub>1 :: "cmd" assumes "c\<^sub>2 \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "n \<noteq> Entry" shows "if (b) c\<^sub>1 else c\<^sub>2 \<turnstile> n \<oplus> #:c\<^sub>1 + 1 -et\<rightarrow>\<^sub>p n' \<oplus> #:c\<^sub>1 + 1"
x0: fixes x :: "nat" assumes "x0 \<le> x" shows "b ^ x \<le> (1::'a) - b"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> R"
wf_alpha_Tree_rel_hull_rel_Tree_wf: shows "wf (FL_Validity.alpha_Tree_rel O hull_rel O Tree_wf)"
assms_1: shows "\<forall>i\<le>n. \<forall>j\<le>n. i \<noteq> j \<longrightarrow> M i j \<le> M' i j"
lift_M: fixes x :: "'a word"   and y :: "'a word" shows "uint (M x y) = L (uint x) (uint y) mod 2 ^ LENGTH('a)"
dist_triangle_half_l: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x1 :: "'c"   and y :: "'c"   and e :: "real"   and x2 :: "'c" assumes "class.metric_space dist uniformity open"   and "dist x1 y < e / 2"   and "dist x2 y < e / 2" shows "dist x1 x2 < e"
setops_list\<^sub>s\<^sub>s\<^sub>t_is_setops\<^sub>s\<^sub>s\<^sub>t: fixes S :: "('a, 'b) stateful_strand_step list" shows "setops\<^sub>s\<^sub>s\<^sub>t S = set (setops_list\<^sub>s\<^sub>s\<^sub>t S)"
rel_flip: fixes R1 :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "('d, 'f, 'h) gpv"   and b :: "('e, 'g, 'h) gpv" shows "rel_gpv R1\<inverse>\<inverse> R2\<inverse>\<inverse> a b = rel_gpv R1 R2 b a"
birkhoff_sum_cocycle: fixes f :: "'a \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat"   and x :: "'a" shows "A.birkhoff_sum f (n + m) x = A.birkhoff_sum f n x + A.birkhoff_sum f m ((induced_map A ^^ n) x)"
endpoints_in_S_are_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2" assumes "a \<noteq> b"   and "a \<in> hyp2"   and "b \<in> hyp2" shows "are_endpoints_in_S (endpoint_in_S a b) (endpoint_in_S b a) a b"
CondReds2F: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>false,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2,b\<^sub>2\<rangle>"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelRTPO Enc TRel"
Gmod2_def: shows "Gmod2 = G\<lparr>carrier := K1 <#> K \<inter> H\<rparr> Mod (K1 <#> K \<inter> H1)"
axioms_1: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subfield K R" shows "subdomain K R"
\<psi>_def: shows "\<psi> \<equiv> instInpP Pinp (Suc 0) (encF N)"
assms: fixes i :: "nat" assumes "i < dim_vec v" shows "r (v $ i) (w $ i)"
independent_same_set: fixes Us :: "'a list"   and Vs :: "'a list" assumes "set Us = set Vs"   and "length Us = length Vs"   and "local.independent K Us" shows "local.independent K Vs"
intro: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and z :: "'a" assumes "\<And>m. n < m \<Longrightarrow> f m = z" shows "bound z n f"
nhds_discrete: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.discrete_topology open" shows "topological_space.nhds open x = principal {x}"
norm_nth_le: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis" shows "norm (x \<bullet> i) \<le> norm x"
hd_sing_power: fixes k :: "nat"   and a :: "'b" assumes "k \<noteq> 0" shows "hd ([a] \<^sup>@ k) = a"
zeta_partial_sum_le_neg: fixes s :: "real" assumes "0 < s" shows "\<exists>c>0. \<forall>x\<ge>1. \<bar>sum_upto (\<lambda>n. real n powr s) x - x powr (1 + s) / (1 + s)\<bar> \<le> c * x powr s"
weak_sharing_consis_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "weak_sharing_consis ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> weak_sharing_consistent \<O> sb"
Pow_def: fixes A :: "'q set" shows "Pow A = {B. B \<subseteq> A}"
unreach_connected: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set"   and b :: "'b"   and Q\<^sub>x :: "'b"   and Q\<^sub>z :: "'b"   and Q\<^sub>y :: "'b" assumes "MinkowskiSpacetime \<E> \<P> betw"   and "Q \<in> \<P>"   and "b \<notin> Q"   and "b \<in> \<E>"   and "Q\<^sub>x \<in> MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b"   and "Q\<^sub>z \<in> MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b"   and "Q\<^sub>x \<noteq> Q\<^sub>z"   and "betw Q\<^sub>x Q\<^sub>y Q\<^sub>z" shows "Q\<^sub>y \<in> MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b"
y_1: shows "v *\<^sub>R y = cy *\<^sub>R yy"
bottom_bind: fixes P :: "'b \<Rightarrow> 'a pred" shows "bind \<bottom> P = \<bottom>"
l_cancel: fixes c :: "'a vec"   and a :: "'a vec"   and b :: "'a vec" assumes "c + a = c + b"   and "a \<in> carrier_vec n"   and "b \<in> carrier_vec n"   and "c \<in> carrier_vec n" shows "a = b"
assms_1: shows "{a, a'} \<inter> G.face_cycle_set b = {}"
s: shows "\<forall>ra. \<exists>ns. (fst ra, snd ra) \<in> set C \<longrightarrow> ns \<in> snd ` set cs \<and> is_leq_ns (s ra *R ns) \<and> s ra \<noteq> 0 \<and> subst_poly_lec \<sigma> (lec_of_nsc (fst ra *R nsc_of_atom (snd ra))) = lec_of_nsc (s ra *R ns)"
pred_sets2: fixes A :: "'b set"   and N :: "'b measure"   and f :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "A \<in> sets N"   and "f \<in> M \<rightarrow>\<^sub>M N" shows "Measurable.pred M (\<lambda>x. f x \<in> A)"
N_le_N_1: fixes n :: "int" shows "fb.N n \<le> fb.N 1"
Gauss_Jordan_upt_k_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" shows "Gauss_Jordan_upt_k A k = snd (foldl Gauss_Jordan_column_k (0, A) [0..<Suc k])"
map_to_set_map_of: fixes l :: "('e \<times> 'f) list" assumes "distinct (map fst l)" shows "map_to_set (map_of l) = set l"