hcju/isabelleps huggingface dataset:
tiny_dg_Cod_vdomain: shows "\<D>\<^sub>\<circ> (\<CC>\<lparr>Cod\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
jordan_nf_matrix_poly_bound: fixes A :: "'b mat"   and n :: "nat"   and n_as :: "(nat \<times> 'b) list"   and N :: "nat" assumes "A \<in> carrier_mat n n"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; 0 < n\<rbrakk> \<Longrightarrow> norm a \<le> 1"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; norm a = 1\<rbrakk> \<Longrightarrow> n \<le> N"   and "jordan_nf A n_as" shows "\<exists>c1 c2. \<forall>k. norm_bound (A ^\<^sub>m k) (c1 + c2 * real k ^ (N - 1))"
swap_cs_Recv_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
n_val_n_val1: fixes P :: "('a \<Rightarrow> ant) set" assumes "P \<in> Pds" shows "n_val K (\<nu>\<^bsub>K P\<^esub>) = \<nu>\<^bsub>K P\<^esub>"
smult_monom_mult: fixes a :: "'a"   and b :: "'a"   and n :: "nat"   and f :: "'a poly" shows "smult a (monom b n * f) = monom (a * b) n * f"
bernoulli_inequality: fixes x :: "'b"   and n :: "nat" assumes "- (1::'b) \<le> x" shows "(1::'b) + of_nat n * x \<le> ((1::'b) + x) ^ n"
axioms_1: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set" assumes "rational_preference carrier relation" shows "preference carrier relation"
ps\<^sub>L'_def: shows "ps\<^sub>L' = {p \<in> ps\<^sub>L. dist (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, snd (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y))) of (x, y) \<Rightarrow> (real_of_int x, y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))}"
HomSet_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" shows "Hom\<^bsub>C\<^esub> X Y \<equiv> implode (HOMCollection C X Y)"
language_state_def: fixes M :: "('e, 'f, 'g) FSM"   and q :: "'g" shows "LS M q \<equiv> {map fst r |r. path M r q}"
assms_8: shows "aux = compl gs bs (ps -- sps) sps (snd data)"
l_one: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<one>\<^bsub>H\<^esub> \<otimes>\<^bsub>H\<^esub> x = x"
iterative_HSem': fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and e :: "exp" assumes "x \<notin> domA \<Gamma>" shows "(\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<N>\<lbrace>\<Gamma>\<rbrace>xa))(x := \<N>\<lbrakk> e \<rbrakk>\<^bsub>xa\<^esub>)) = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> (\<N>\<lbrace>\<Gamma>\<rbrace>xa))(x := \<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>xa\<^esub>))"
x1: shows "Abs_clinear_space (closure ((*\<^sub>V) (\<alpha> *\<^sub>C A) ` space_as_set S)) = \<alpha> *\<^sub>C Abs_clinear_space (closure ((*\<^sub>V) A ` space_as_set S))"
ord_resolve_sound: fixes M :: "'a literal multiset set"   and CAs :: "'a literal multiset list"   and DA :: "'a literal multiset"   and AAs :: "'a multiset list"   and As :: "'a list"   and E :: "'a literal multiset"   and I :: "'a set" assumes "gr.ord_resolve M CAs DA AAs As E"   and "I \<TTurnstile>m mset CAs"   and "I \<TTurnstile> DA" shows "I \<TTurnstile> E"
id_to_mor_locally_ringed_spaces: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "morphism_locally_ringed_spaces S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S is_open \<FF> \<rho> b add_str mult_str zero_str one_str (identity S) (\<lambda>U. identity (\<FF> U))"
mevalp_meval_eq: shows "mevalp = (\<lambda>x xa xb xc xd. (x, xa) \<leadsto>\<^bsub>xb\<^esub> (xc, xd))"
natUnion_subset: fixes f :: "nat \<Rightarrow> 'b set"   and g :: "nat \<Rightarrow> 'b set" assumes "\<And>n. \<exists>m. f n \<subseteq> g m" shows "natUnion f \<subseteq> natUnion g"
reflexive: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto>\<^sup>^<Rel> P"
rep_eq: shows "generator rbt_keys_generator = (rbt_has_next, rbt_keys_next)"
coprime_linear_poly: fixes c :: "'a"   and c' :: "'a" assumes "c \<noteq> c'" shows "coprime [:c, 1::'a:] [:c', 1::'a:]"
check_in_heap_returns_result: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and f :: "unit \<Rightarrow> ((_) heap, exception, 'n) prog"   and x :: "'n" assumes "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> Heap_Error_Monad.bind (check_in_heap ptr) f \<rightarrow>\<^sub>r x = h \<turnstile> f () \<rightarrow>\<^sub>r x"
lower_asymptotic_density_one_intersection: fixes A :: "nat set"   and B :: "nat set" assumes "lower_asymptotic_density A = 1"   and "lower_asymptotic_density B = 1" shows "lower_asymptotic_density (A \<inter> B) = 1"
is_auto_reducedD: fixes B :: "('t \<Rightarrow>\<^sub>0 'd) set"   and b :: "'t \<Rightarrow>\<^sub>0 'd" assumes "is_auto_reduced B"   and "b \<in> B" shows "\<not> is_red (B - {b}) b"
red_rtrancl_\<tau>_heapD_inv: fixes t :: "'t"   and s :: "'x1 \<times> 'm1"   and s' :: "'x1 \<times> 'm1"   and wfs :: "'t \<Rightarrow> 'x1 \<times> 'm1 \<Rightarrow> bool" assumes "r1.silent_moves t s s'"   and "wfs t s" shows "snd s' = snd s"
DBImplBD: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> \<diamond>\<box>F \<longrightarrow> \<box>\<diamond>F"
SE3: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<and> \<diamond>G \<longrightarrow> \<diamond>(G \<and> F)"
mult_matrix_n_def: fixes n :: "nat"   and fmul :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and fadd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix" shows "mult_matrix_n n fmul fadd A B \<equiv> Abs_matrix (\<lambda>j i. foldseq fadd (\<lambda>k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
degh2: shows "MPoly_Type.degree (Const (real (i + 1)) * Var x ^ i) x = i"
flowD_finite_OUT: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "flow (residual_network f) f"   and "x \<noteq> source (residual_network f)"   and "x \<noteq> sink (residual_network f)" shows "d_OUT f x \<noteq> \<top>"
strand_substI_1: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and S :: "('d, 'c) strand_step list" assumes "subst_domain \<theta> \<inter> fv\<^sub>s\<^sub>t S = {}" shows "S \<cdot>\<^sub>s\<^sub>t \<theta> = S"
bind_spmf_assoc: fixes x :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf"   and g :: "'d \<Rightarrow> 'b spmf" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
subseq_filter_left: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "subseq (filter P xs) xs"
extreme_bound_singleton_refl: fixes A :: "'c set"   and x :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "extreme_bound A less_eq {x} x = (x \<in> A \<and> less_eq x x)"
sumlist_dim: fixes xs :: "'a vec list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> carrier_vec n" shows "dim_vec (M.sumlist xs) = n"
funionM_comp_fun_idem: shows "comp_fun_idem funionM"
fmdiff_fmupd: fixes m :: "('d, 'c) fmap"   and k :: "'d"   and v :: "'c" assumes "m $$ k = None" shows "fmdom' {k $$:= v} \<lhd>/ m(k $$:= v) = m"
lhs: shows "filter_mset P ({#a#} \<times># add_mset x B) = filter_mset P ({#a#} \<times># B) + filter_mset P {#(a, x)#}"
order_iff_strict: fixes a :: "'a"   and b :: "'a" shows "cless_eq a b = (cless a b \<or> a = b)"
simps_6: fixes b :: "nat"   and \<phi> :: "formula" shows "fvi b (Neg \<phi>) = fvi b \<phi>"
RDirProd_hom2_2: fixes f :: "'a \<Rightarrow> 'b"   and S :: "('a, 'c) ring_scheme"   and T :: "('b, 'd) ring_scheme"   and R :: "('e, 'f) ring_scheme" assumes "f \<in> ring_hom S T" shows "(\<lambda>(x, y). (f x, y)) \<in> ring_hom (RDirProd S R) (RDirProd T R)"
Infinitesimal_approx_minus: fixes x :: "'c star"   and y :: "'c star" shows "(x - y \<in> Infinitesimal) = (x \<approx> y)"
sample_commute: fixes p :: "'p pmf"   and q :: "'p pmf"   and f :: "'p \<Rightarrow> 'p \<Rightarrow> 'm" shows "sample p (\<lambda>x. sample q (f x)) = sample q (\<lambda>y. sample p (\<lambda>x. f x y))"
assms: shows "t \<in> space (tree_sigma (count_space A \<Otimes>\<^sub>M lborel))"
phl_cons2: fixes w :: "'a"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "w \<le> x"   and "\<lbrace>z\<rbrace>y\<lbrace>w\<rbrace>" shows "\<lbrace>z\<rbrace>y\<lbrace>x\<rbrace>"
preserves_right_adjoint: fixes g :: "'b" assumes "D.is_right_adjoint g" shows "C.is_right_adjoint (F.right_map g)"
succ_drest_eq_drest_succ: fixes vs :: "'d set"   and s :: "('d, 'e) fmap"   and a :: "('d, 'e) fmap \<times> ('d, 'e) fmap" shows "state_succ (fmrestrict_set vs s) (action_proj a vs) = fmrestrict_set vs (state_succ s (action_proj a vs))"
measure_pair_spmf_times: fixes p :: "'a spmf"   and q :: "'b spmf"   and A :: "'a set"   and B :: "'b set" shows "Sigma_Algebra.measure (measure_spmf (pair_spmf p q)) (A \<times> B) = Sigma_Algebra.measure (measure_spmf p) A * Sigma_Algebra.measure (measure_spmf q) B"
inf_sup_aci_4: fixes 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 x :: "'b"   and y :: "'b" assumes "class.lattice inf less_eq less sup" shows "inf x (inf x y) = inf x y"
iptables_goto_bigstep_Undecided_Undecided_deterministic: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t" shows "t = Undecided"
Eq_IC_cfn: shows "\<bar>IC p (i + 1) (te p (i + 1)) - IC q (i + 1) (te p (i + 1))\<bar> = \<bar>cfn q (\<lambda>n. \<theta> q (i + 1) n + (PC q (te p (i + 1)) - PC q (te q (i + 1)))) - cfn p (\<theta> p (i + 1))\<bar>"
l1_iagreement_Resp_def: shows "l1_iagreement_Resp \<equiv> {s. \<forall>A B N. signalsResp s (Commit A B N) \<le> signalsResp s (Running A B N)}"
g1: shows "poly_of_vec g = (q' + smult k q'') * u + smult k r''"
iterate_add_eq_add_iff1: fixes i :: "nat"   and j :: "nat"   and u :: "'k"   and m :: "'k"   and n :: "'k" assumes "i \<le> j" shows "(iterate_add j u + m = iterate_add i u + n) = (iterate_add (j - i) u + m = n)"
psiFreshVec_7: fixes xvec :: "name list"   and P :: "('d, 'e, 'f) psi" shows "xvec \<sharp>* !P = xvec \<sharp>* P"
that: shows "e \<in> UNIV \<rightarrow> {- 1..1}"
hoare: shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
diff_gt_0_iff_gt: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (minus a b) = less b a"
Inf_insert: fixes a :: "'a"   and A :: "'a set" shows "Inf (insert a A) = inf a (Inf A)"
identity\<^sub>E_infix_def: fixes x :: "\<kappa>"   and y :: "\<kappa>" shows "x \<^bold>=\<^sub>E y \<equiv> \<lparr>basic_identity\<^sub>E,x,y\<rparr>"
list_map_to_fun_def: fixes m :: "('a, ('a \<times> 'b) list) mapping"   and i :: "'a" shows "list_map_to_fun m i = (case Mapping.lookup m i of None \<Rightarrow> [] | Some ias \<Rightarrow> ias)"
ssap: 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 fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and F :: "('b \<Rightarrow> 'b) set"   and encF :: "('b \<Rightarrow> 'b) \<Rightarrow> 'c"   and N :: "'b \<Rightarrow> 'b"   and ssap :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'c" assumes "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {Generic_Syntax.Variable var (Suc 0)}" shows "ssap \<phi> \<in> F"
prv_eql_subst_trm_id: fixes y :: "'var"   and \<phi> :: "'fmla"   and n :: "'trm" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "n \<in> num" shows "bprv (imp (eql (Var y) n) (imp \<phi> (subst \<phi> n y)))"
establish_rwof_invar: assumes "m0 \<le>\<^sub>n SPEC I"   and "\<And>s. \<lbrakk>rwof m0 cond step s; I s; cond s\<rbrakk> \<Longrightarrow> step s \<le>\<^sub>n SPEC I"   and "rwof m0 cond step s" shows "I s"
a_Roll_inv_I: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e' :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<alpha> :: "Syntax.tvar"   and \<tau> :: "ty" assumes "\<Gamma> \<turnstile> Roll e', eP, eV : Mu \<alpha> \<tau>" obtains eP' :: "Syntax.term" and eV' :: "Syntax.term" where "eP = Roll eP'" and "eV = Roll eV'" and "\<Gamma> \<turnstile> e', eP', eV' : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>"
simps_4: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_aexp_inputs (Plus v va) = enumerate_aexp_inputs v \<union> enumerate_aexp_inputs va"
gen_model_def: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g" shows "gen_model M w \<equiv> let ws' = worlds M \<inter> (\<Union> (range (relations M)))\<^sup>* `` {w} in \<lparr>worlds = ws', relations = \<lambda>a. Restr (relations M a) ws', valuation = valuation M\<rparr>"
f4_red_aux_not_zero: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" shows "0 \<notin> set (f4_red_aux bs ps)"
connected_component_of_eq: fixes X :: "'b topology"   and x :: "'b"   and y :: "'b" shows "(connected_component_of X x = connected_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> x \<in> topspace X \<and> y \<in> topspace X \<and> connected_component_of X x y)"
set_mult_eq_generate_subgroup: fixes H :: "'a set"   and J :: "'a set" assumes "subgroup H G"   and "subgroup J G" shows "generate G (H \<union> J) = H <#> J"
split_sel_asm: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (\<not> (bool = True \<and> \<not> P f1 \<or> bool = False \<and> \<not> P f2))"
assms_2: shows "m\<^sub>2 $$ k = Some v'"
simps_2: fixes \<S> :: "nat \<Rightarrow> bool option"   and x :: "'b memref"   and xs :: "'b memref list" shows "no_write_to_read_only_memory \<S> (x # xs) = (case x of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> a \<notin> read_only \<S> \<and> (if volatile then no_write_to_read_only_memory (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) xs else no_write_to_read_only_memory \<S> xs) | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> no_write_to_read_only_memory (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) xs | _ \<Rightarrow> no_write_to_read_only_memory \<S> xs)"
times0_right: fixes A :: "(('a, 'd) vec, 'c) vec" shows "A ** 0 = 0"
simps_5: fixes t :: "'c local.term" shows "Can \<^bold>\<l>\<^bold>[t\<^bold>] = Can t"
surjective_top_closed: shows "matrix_bounded_idempotent_semiring.surjective mtop"
cong_r: fixes H :: "'a set" assumes "H \<in> normalfilters" shows "cong H = cong_r H"
yoneda_lemma_fixed_e_axioms: shows "yoneda_lemma_fixed_e (\<cdot>) (\<cdot>\<^sub>S) \<phi> F a e"
inverse_arrows_can: fixes f :: "'c local.term set" assumes "can f" shows "inverse_arrows f (mkarr (Inv (DOM f\<^bold>\<down>) \<^bold>\<cdot> \<^bold>\<lfloor>rep f\<^bold>\<rfloor> \<^bold>\<cdot> COD f\<^bold>\<down>))"
simps_1: fixes x :: "char"   and \<theta> :: "trm" shows "ODEo x (Aterm \<theta>) = Agame (ODE x \<theta>)"
inf_compl_bot: fixes x :: "'a" shows "x \<cdot> - x = 0"
i_expand_i_take_Suc: fixes k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "0 < k" shows "f \<odot> k \<Down> Suc n = f \<Down> (n div k) \<odot> k @ f (n div k) # NoMsg\<^bsup>n mod k\<^esup>"
get_tag_name_new_document: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr"   and "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
inv_comp_right_1: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.iso (VV.comp g f)"   and "VV.iso f" shows "VV.inv (VV.comp g f) = VV.comp (VV.inv f) (VV.inv g)"
mat_kernelI: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v :: "'b vec" assumes "A \<in> carrier_mat nr nc"   and "v \<in> carrier_vec nc"   and "A *\<^sub>v v = 0\<^sub>v nr" shows "v \<in> mat_kernel A"
closure_scaleR: fixes c :: "real"   and S :: "'b set" shows "(*\<^sub>R) c ` closure S = closure ((*\<^sub>R) c ` S)"
assms_3: shows "s1 O s1 \<subseteq> s1"
last_index_eq_index_conv: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b" assumes "x \<in> set xs \<or> y \<in> set xs" shows "(last_index xs x = last_index xs y) = (x = y)"
zero_right: fixes a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a" shows "cblinfun_apply a (0::'b) = (0::'a)"
load_tmps_fresh: shows "load_tmps_fresh ts\<^sub>s\<^sub>b"
assms_2: shows "tt p \<prec>\<^sub>t tt q"
npibnd: shows "\<forall>x. \<not> PP \<and> P x \<longrightarrow> (\<exists>u\<in>U. x \<sqsubseteq> u)"
tan_sec: fixes x :: "'d" assumes "cos x \<noteq> (0::'d)" shows "(1::'d) + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
assms_2: shows "isRecv ev \<longrightarrow> getId ev \<noteq> i"
preserves_ide: fixes a :: "'a \<times> 'a" assumes "VoV.ide a" shows "VoV.ide (VoV.map a)"
ifex_ite_opt_rel_bf: fixes fi :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and i :: "'a ifex"   and ft :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and t :: "'a ifex"   and fe :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and e :: "'a ifex" assumes "(fi, i) \<in> bf_ifex_rel"   and "(ft, t) \<in> bf_ifex_rel"   and "(fe, e) \<in> bf_ifex_rel" shows "(bf_ite fi ft fe, ifex_ite_opt i t e) \<in> bf_ifex_rel"
iMODb_subset_imp_divisor_mod_0: fixes c' :: "nat"   and r' :: "nat"   and m' :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "0 < c'"   and "[ r', mod m', c' ] \<subseteq> [ r, mod m, c ]" shows "m' mod m = 0"
a_elemrcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "a \<in> carrier R"   and "a' \<in> I +> a" shows "a' \<in> carrier R"
measure_eqI_countable_AE: fixes M :: "'a measure"   and N :: "'a measure"   and \<Omega> :: "'a set" assumes "sets M = UNIV"   and "sets N = UNIV"   and "AE x in M. x \<in> \<Omega>"   and "AE x in N. x \<in> \<Omega>"   and "countable \<Omega>"   and "\<And>x. x \<in> \<Omega> \<Longrightarrow> emeasure M {x} = emeasure N {x}" shows "M = N"
Red_I_of_subset: fixes N :: "'g set"   and N' :: "'g set" assumes "N \<subseteq> N'" shows "Red_I_G N \<subseteq> Red_I_G N'"
vec_map: fixes nr :: "nat"   and u :: "'b list"   and f :: "'b \<Rightarrow> 'b" assumes "vec nr u" shows "vec nr (vec_map f u)"
star_induct: fixes w :: "'c list"   and A :: "'c list set"   and P :: "'c list \<Rightarrow> bool" assumes "w \<in> star A"   and "P []"   and "\<And>u v. \<lbrakk>u \<in> A; v \<in> star A; P v\<rbrakk> \<Longrightarrow> P (u @ v)" shows "P w"
tendsto_ereal: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
fold_def: shows "RBT_Set2.fold \<equiv> map_fun id (map_fun mapping_rbt.impl_of id) (\<lambda>f. RBT_Impl.fold (\<lambda>a _. f a))"
iT_cut_le1_5: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "t \<in> [ r, mod m, c ]" shows "[ r, mod m, c ] \<down>\<le> t = [ r, mod m, (t - r) div m ]"
lprefix_antisym: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lprefix xs ys"   and "lprefix ys xs" shows "xs = ys"
neq_if_length_neq: fixes xs :: "'g list"   and ys :: "'g list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
frac_def: fixes r :: "'a"   and s :: "'a" shows "r / s \<equiv> rel.Class (r, s)"
less_imp_not_eq2: fixes x :: "'k multiset"   and y :: "'k multiset" assumes "x \<subset># y" shows "(y = x) = False"
set_iterator_abs_simple_rule_P: assumes "set_iterator_abs \<alpha> invar it S0"   and "\<And>xc. invar xc \<Longrightarrow> f xc = f' (\<alpha> xc)"   and "I S0 \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; x \<in> S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (S - {x}) (f' x \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> {}; \<not> True; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
map_add_restrict: fixes m\<^sub>0 :: "'c \<Rightarrow> 'd option"   and m\<^sub>1 :: "'c \<Rightarrow> 'd option"   and S :: "'c set" shows "(m\<^sub>0 ++ m\<^sub>1) |` S = m\<^sub>0 |` S ++ m\<^sub>1 |` S"
induces_monoidal_category: shows "monoidal_category (\<cdot>\<^sub>S) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C"
P4_cong: fixes tms :: "tm list"   and P :: "tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> fm"   and H :: "fm set"   and x1 :: "tm"   and x1' :: "tm"   and x2 :: "tm"   and x2' :: "tm"   and x3 :: "tm"   and x3' :: "tm"   and x4 :: "tm"   and x4' :: "tm" assumes "\<And>i t1 t2 t3 t4 x. atom i \<sharp> tms \<Longrightarrow> (P t1 t2 t3 t4)(i::=x) = P (subst i x t1) (subst i x t2) (subst i x t3) (subst i x t4)"   and "H \<turnstile> x1 EQ x1'"   and "H \<turnstile> x2 EQ x2'"   and "H \<turnstile> x3 EQ x3'"   and "H \<turnstile> x4 EQ x4'" shows "H \<turnstile> P x1 x2 x3 x4 IFF P x1' x2' x3' x4'"
pseudo_proj_True_Suc_False_proj: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True (Suc n) (pseudo_proj_False n w) = pseudo_proj_False n w"
iter_narrow_def: fixes f :: "'a acom \<Rightarrow> 'a acom" shows "iter_narrow f = while_option (\<lambda>c. \<not> c \<sqsubseteq> c \<triangle>\<^sub>c f c) (\<lambda>c. c \<triangle>\<^sub>c f c)"
pfp_wn_def: fixes f :: "'b option acom \<Rightarrow> 'b option acom"   and c :: "com" shows "pfp_wn f c = (case iter_widen f (\<bottom>\<^sub>c c) of None \<Rightarrow> None | Some c' \<Rightarrow> iter_narrow f c')"
assms_1: shows "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"
discr_While: fixes c :: "(test, atom) com"   and tst :: "test" assumes "discr c" shows "discr (while tst do c)"