hcju/isabelleps huggingface dataset:
Node_notin_subtrees_if: fixes a :: "'c"   and t :: "'c tree"   and l :: "'c tree"   and r :: "'c tree" assumes "a \<notin> set_tree t" shows "\<langle>l, a, r\<rangle> \<notin> subtrees t"
Block_None_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and T :: "ty" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h ({V:T=None; e}, xs) ({V:T=None; e'}, xs')"
bulkload_tabulate: fixes xs :: "'b list" shows "bulkload xs = tabulate [0..<length xs] ((!) xs)"
supp_member: fixes Xs :: "'a set"   and x :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "fs TYPE('a) TYPE('b)"   and "finite Xs"   and "x \<in> supp Xs" obtains X :: "'a" where "X \<in> Xs" and "x \<in> supp X"
prod_eq_zeroE: fixes f :: "'c \<Rightarrow> 'b"   and I :: "'c set" assumes "prod f I = (0::'b)" obtains i :: "'c" where "finite I" and "i \<in> I" and "f i = (0::'b)"
name_bij: fixes pi :: "(name \<times> name) list"   and x :: "name"   and y :: "name" shows "(pi \<bullet> x = pi \<bullet> y) = (x = y)"
hseqI': fixes \<mu> :: "'a arrow_of_spans_data"   and \<nu> :: "'a arrow_of_spans_data" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<nu> = trg \<mu>" shows "hseq \<nu> \<mu>"
pm_of_poly_aux: fixes p :: "'b poly"   and x :: "'a" shows "{t. (poly.coeff p (lookup t x) when t \<in> .[{x}]) \<noteq> (0::'b)} = Poly_Mapping.single x ` {d. poly.coeff p d \<noteq> (0::'b)}"
adjoint_consequence: fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "\<And>f f'. SFunctions \<sigma> f = Some f' \<Longrightarrow> dsafe f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'"   and "Vagree \<nu> \<omega> (FVS \<sigma>)" shows "local.adjoint I \<sigma> \<nu> = local.adjoint I \<sigma> \<omega>"
finite_INF_in: fixes S :: "'c set"   and f :: "'c \<Rightarrow> ereal" assumes "finite S"   and "S \<noteq> {}" shows "Inf (f ` S) \<in> f ` S"
simple_bochner_integral_add: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f"   and "simple_bochner_integrable M g" shows "simple_bochner_integral M (\<lambda>x. f x + g x) = simple_bochner_integral M f + simple_bochner_integral M g"
sup_left_commute: fixes x :: "'g multiset"   and y :: "'g multiset"   and z :: "'g multiset" shows "x \<union># (y \<union># z) = y \<union># (x \<union># z)"
obs_consistent_refl: fixes S :: "('e, 'f) spec" shows "obs_consistent Id id S S"
set_columns_Gram_Schmidt_matrix: fixes A :: "((real, 'b) vec, 'a) vec"   and k :: "'b" shows "{column i (Gram_Schmidt_matrix A) |i. i < k} = {column i (Gram_Schmidt_upt_k A (mod_type_class.to_nat k)) |i. i < k}"
is_path_split: fixes v :: "'v"   and p1 :: "('v \<times> 'v) list"   and p2 :: "('v \<times> 'v) list"   and v' :: "'v" shows "is_path v (p1 @ p2) v' = (\<exists>u. is_path v p1 u \<and> is_path u p2 v')"
valuation_def: fixes K :: "('c, 'd) Ring_scheme"   and v :: "'c \<Rightarrow> ant" shows "valuation K v = (v \<in> extensional (carrier K) \<and> v \<in> carrier K \<rightarrow> Z\<^sub>\<infinity> \<and> v \<zero>\<^bsub>K\<^esub> = \<infinity> \<and> (\<forall>x\<in>carrier K - {\<zero>\<^bsub>K\<^esub>}. v x \<noteq> \<infinity>) \<and> (\<forall>x\<in>carrier K. \<forall>y\<in>carrier K. v (x \<cdot>\<^sub>r\<^bsub>K\<^esub> y) = v x + v y) \<and> (\<forall>x\<in>carrier K. 0 \<le> v x \<longrightarrow> 0 \<le> v (1\<^sub>r\<^bsub>K\<^esub> \<plusminus>\<^bsub>K\<^esub> x)) \<and> (\<exists>x. x \<in> carrier K \<and> v x \<noteq> \<infinity> \<and> v x \<noteq> 0))"
sum_list_append: fixes xs :: "'a list"   and ys :: "'a list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
assms: fixes x :: "'a spmf" assumes "x \<in> Y" shows "ord_spmf (=) x z"
simple_function_count_space: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "simple_function (count_space A) f = finite (f ` A)"
not_le: fixes x :: "complex"   and y :: "complex" shows "(\<not> x \<le>\<^sub>\<complex> y) = (y <\<^sub>\<complex> x)"
span_same: fixes S :: "complex vec set" assumes "S \<subseteq> mat_kernel Ag" shows "k.span S = local.span S"
strict_mono_mono: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" assumes "class.order less_eq less"   and "order.strict_mono less f" shows "order.mono less_eq f"
Min_in: fixes A :: "'c DBMEntry set" assumes "finite A"   and "A \<noteq> {}" shows "linordered_monoid.Min A \<in> A"
lve_Exit_edge: fixes e :: "'a LDCFG_node \<times> 'b edge_kind \<times> 'a LDCFG_node"   and X :: "'a"   and valid_edge :: "'c \<Rightarrow> bool"   and src :: "'c \<Rightarrow> 'a"   and trg :: "'c \<Rightarrow> 'a"   and knd :: "'c \<Rightarrow> 'b edge_kind"   and E :: "'a" assumes "e = (LDCFG_node.Node X, (\<lambda>s. True)\<^sub>\<surd>, NewExit)" shows "lift_valid_edge valid_edge src trg knd E X e"
br_run_sim: fixes \<alpha> :: "(cmd, local_state, global_state) Gen_Scheduler.global_config \<Rightarrow> (cmd, local_state, global_state) Gen_Scheduler.global_config option"   and I :: "(cmd, local_state, global_state) Gen_Scheduler.global_config \<Rightarrow> bool"   and r :: "nat \<Rightarrow> (cmd, local_state, global_state) Gen_Scheduler.global_config" assumes "br Some li'.wf_gglobal = br \<alpha> I"   and "li'.sa.is_run prog r" shows "ref_is_run prog (\<alpha> \<circ> r)"
f: shows "f \<in> borel_measurable (Pi\<^sub>M (insert i I) M)"
upt_append: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "[0..<i] @ [i..<j] = [0..<j]"
rep_eq: fixes x :: "'c \<Rightarrow> bool"   and xa :: "'c set_dlist" shows "dlist_all x xa = list_all x (list_of_dlist xa)"
IF1set: shows "IF1set \<circ> ctor1 = IF1col \<circ> F1map id IF1set IF2set"
fimage_vunion_left: fixes r :: "V"   and s :: "V"   and A :: "V" shows "(r \<union>\<^sub>\<circ> s) `\<^sub>\<bullet> A = r `\<^sub>\<bullet> A \<union>\<^sub>\<circ> s `\<^sub>\<bullet> A"
assms: shows "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2"
inv_comp: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.iso g"   and "VV.seq g f" shows "VV.inv (VV.comp g f) = VV.comp (VV.inv f) (VV.inv g)"
filter_conv_foldr: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "filter P xs = foldr (\<lambda>x xs. if P x then x # xs else xs) xs []"
HBL2_imp2: 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 num :: "'b set"   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 prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c"   and \<phi> :: "'c"   and \<chi> :: "'c" assumes "HBL1_2_3 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<chi> = {}"   and "bprv (HBL1.PP var subst P (enc (imp \<phi> \<chi>)))" shows "bprv (imp (HBL1.PP var subst P (enc \<phi>)) (HBL1.PP var subst P (enc \<chi>)))"
modular_2': fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y ; x \<cdot> z = (y \<cdot> z ; x\<^sup>T) ; x \<cdot> z"
label_wrt_adjacent_shared_facet: fixes B :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and v :: "'a"   and z :: "'a set"   and w :: "'a" assumes "label_wrt B f"   and "chamber (insert v z)"   and "chamber (insert w z)"   and "v \<notin> z"   and "w \<notin> z" shows "f v = f w"
less_eq_nmultiset_def: fixes X :: "'b nmultiset"   and Y :: "'b nmultiset" shows "(X \<le> Y) = (X < Y \<or> X = Y)"
sim_action_refl: fixes obs :: "('a, 'b) obs_event action" shows "obs \<approx> obs"
shadow_root_ptr_casts_commute3: fixes ptr :: "(_) object_ptr" assumes "is_shadow_root_ptr_kind ptr" shows "cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) = ptr"
chine_assoc_in_hom: shows "Maps.in_hom SPN_fgh.chine_assoc (Chn ((SPN f \<circ> SPN g) \<circ> SPN h)) (Chn (SPN f \<circ> SPN g \<circ> SPN h))"
take_stake: fixes n :: "nat"   and m :: "nat"   and s :: "'b stream" shows "take n (stake m s) = stake (min n m) s"
l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.type_wf get_child_nodes get_child_nodes_locs DocumentClass.known_ptrs get_parent get_parent_locs"
bin_cat_split: fixes n :: "nat"   and w :: "int"   and u :: "int"   and v :: "int" assumes "bin_split n w = (u, v)" shows "w = concat_bit n v u"
rename_free: 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 B :: "Core.term"   and x :: "variable"   and \<tau> :: "typ"   and y :: "variable" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "(x, \<tau>) \<notin> FV \<Gamma>" shows "\<Theta>,\<Gamma> \<turnstile> subst_term [((x, \<tau>), Fv y \<tau>)] B"
gen_contr_triv: fixes V :: "'a set" assumes "H.is_slim V"   and "pverts H = V" shows "gen_contr_graph (with_proj H) V = H"
abs_eq: fixes xa :: "'a set"   and x :: "'a \<Rightarrow> 'b" shows "pfun_entries xa x = Abs_pfun (\<lambda>xb. if xb \<in> xa then Some (x xb) else None)"
cring_simprules_10: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
fbox_subdist: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x] d y +  |x] d z \<le>  |x] (d y + d z)"
mat_trace_limit: fixes g :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and d :: "nat" assumes "limit_mat g A d" shows "(\<lambda>n. trace (g n)) \<longlonglongrightarrow> trace A"
assms_5: shows "range A \<subseteq> prod_algebra I M"
UnE: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
simps_1: fixes s :: "'a" shows "tLength (tInit s) = 0"
induct: fixes P :: "('a \<times> 'a) set \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "('a \<times> 'a) set"   and a1 :: "'a list" assumes "\<And>uu_. P uu_ []"   and "\<And>R x xs. \<lbrakk>P R (filter (\<lambda>y. (y, x) \<in> R) xs); P R (filter (\<lambda>y. (y, x) \<notin> R) xs)\<rbrakk> \<Longrightarrow> P R (x # xs)" shows "P a0 a1"
open_ell2_def: fixes U :: "'a ell2 set" shows "open U = (\<forall>x\<in>U. \<forall>\<^sub>F (x', y) in INF e\<in>{0<..}. principal {(x, y). norm (x - y) < e}. x' = x \<longrightarrow> y \<in> U)"
C1: fixes Q :: "('a, 'b, 'c) psi" shows "(\<Psi>, Q, Q \<parallel> \<zero>) \<in> Rel"
distinct_90: fixes pi' :: "pi"   and name :: "name"   and pi :: "pi" shows "!pi' \<noteq> <\<nu>name>pi"
thenthen: shows "eval (Cn 1 r_prod_encode [Cn 1 r_snoc [r_pdec1, Z], r_pdec2]) [arg] \<down>= prod_encode (e_snoc b 0, k)"
bounded_by_None: fixes xs :: "real list" shows "bounded_by xs (replicate (length xs) None)"
tighten_bound_root: fixes il :: "rat"   and x :: "int poly \<times> rat \<times> rat"   and ir :: "rat"   and cmpx :: "rat \<Rightarrow> order" assumes "sgn il = 1"   and "0 \<le> real_of_1 x"   and "real_of_rat il \<le> root n (real_of_1 x)"   and "root n (real_of_1 x) \<le> real_of_rat ir"   and "invariant_1 x"   and "cmpx = compare_1_rat x"   and "n \<noteq> 0" shows "converges_to (\<lambda>i. (local.tighten_bound_root cmpx ^^ i) (il, ir)) (root n (real_of_1 x))"
walk_iarray_foldl: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c iarray"   and x :: "'b" shows "walk_iarray f a x = foldl f x (IArray.list_of a)"
subst_lsst_append_inv: fixes A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and \<delta> :: "'f \<Rightarrow> ('e, 'f) Term.term"   and B1 :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list"   and B2 :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" assumes "A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = B1 @ B2" shows "\<exists>A1 A2. A = A1 @ A2 \<and> A1 \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = B1 \<and> A2 \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta> = B2"
length_selsort: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list" shows "length (ccw.selsort x xs) = length xs"
composite_cell_in_hom: fixes w :: "'b"   and u :: "'b"   and \<theta> :: "'b" assumes "D.ide w"   and "\<guillemotleft>w : src\<^sub>D u \<rightarrow>\<^sub>D src\<^sub>D (F f)\<guillemotright>"   and "\<guillemotleft>\<theta> : F f \<star>\<^sub>D w \<Rightarrow>\<^sub>D u\<guillemotright>" shows "\<guillemotleft>(F r \<star>\<^sub>D \<theta>) \<cdot>\<^sub>D \<a>\<^sub>D[F r, F f, w] \<cdot>\<^sub>D (D.inv (\<Phi> (r, f)) \<cdot>\<^sub>D F \<rho> \<star>\<^sub>D w) : F g \<star>\<^sub>D w \<Rightarrow>\<^sub>D F r \<star>\<^sub>D u\<guillemotright>"
of_nat_add: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and m :: "nat"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero (m + n) = plus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
sran_union: fixes m :: "'d \<Rightarrow> 'c set"   and n :: "'d \<Rightarrow> 'c set" shows "sran m \<union>. n = sran m \<union> sran n"
not_ephemeral_def: fixes C :: "'s set" shows "not_ephemeral C = (C \<in> UNIV // communicating \<and> (\<nexists>x. C = {x} \<and> p x x 1 = 0))"
fbox_g_ode: fixes G :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "|x\<acute>=\<lambda>t. f & G on \<lambda>s. T S @ 0] Q = (\<lambda>s. s \<in> S \<longrightarrow> (\<forall>t\<in>T. (\<forall>\<tau>\<in>down T t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s)))"
tagged_fun_relD_both: fixes f :: "'g \<Rightarrow> 'h"   and f' :: "'i \<Rightarrow> 'j"   and A :: "('g \<times> 'i) set"   and B :: "('h \<times> 'j) set"   and x :: "'g"   and x' :: "'i" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(Let x f, Let x' f') \<in> B"
b_never_blocked: fixes l :: "nat"   and s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "B.can_step l s = (l < N)"
abs_1: fixes abs :: "'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs one = one"
select_convs_6: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cAmountF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cAmountF"
tst: shows "thr s1 t = \<lfloor>(x1, ln)\<rfloor>"
up_smult_closed: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "ring R"   and "a \<in> carrier R"   and "p \<in> up R" shows "(\<lambda>i. a \<otimes>\<^bsub>R\<^esub> p i) \<in> up R"
add_nonneg_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq zero a"   and "less zero b" shows "less zero (plus a b)"
N_def: shows "N = max 2 (max (nat \<lceil>2 * (cmod z + d)\<rceil>) M)"
star_sum: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = (x\<^sup>\<star> \<squnion> y\<^sup>\<star>)\<^sup>\<star>"
sel_2: fixes x21 :: "char list"   and x22 :: "'a list" shows "get_pred (Neg x21 x22) = x21"
sel_2: fixes x2 :: "'h" shows "projr (Inr x2) = x2"
rtrancl_into_trancl3: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r\<^sup>*"   and "a \<noteq> b" shows "(a, b) \<in> r\<^sup>+"
closed_imp_pnet_trans: fixes s :: "'c net_state"   and a :: "'d node_action"   and s' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and n :: "net_tree" assumes "(s, a, s') \<in> automaton.trans (closed (pnet np n))" shows "\<exists>a'. (s, a', s') \<in> automaton.trans (pnet np n)"
assms_2: shows "B \<in> obj\<^bsub>C\<^esub>"
eq1: shows "lt (monom_mult c t f) = t \<oplus> lt f"
sift_up_step: fixes heap :: "int \<Rightarrow> int"   and l :: "int"   and r :: "int"   and x :: "int" assumes "is_heap_except_up (\<le>) heap l r x"   and "l \<le> parent l x"   and "heap x \<le> heap (parent l x)" shows "is_heap_except_up (\<le>) (swap heap (parent l x) x) l r (parent l x)"
assms_3: shows "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> (1::'a)) going_to at_top. eval_fds f s = eval_fds g s"
simps_1: fixes uu :: "nat"   and uv :: "nat" shows "pair_in_list [] uu uv = False"
prems_3: shows "P,E \<turnstile> \<langle>blocks (p # ps', Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"
execute_make: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and h :: "heap" shows "execute (Array.make n f) h = Some (Array.alloc (map f [0..<n]) h)"
min_ground_head_in_ground_heads: fixes \<zeta> :: "('s, 'v) hd" shows "min_ground_head \<zeta> \<in> local.ground_heads \<zeta>"
simps_8: fixes bs :: "'b list"   and c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "Ipoly bs (CN c n p) = Ipoly bs c + bs ! n * Ipoly bs p"
Zp_nat_inc_closed: fixes n :: "nat" shows "[n] \<cdot> \<one> \<in> carrier Zp"
intros: fixes Fstd_empty :: "'c"   and Fstd_get :: "'c \<Rightarrow> 'd \<Rightarrow> ('e, 'f, 'd, 'g) Std.instr fundef option"   and Fstd_add :: "'c \<Rightarrow> 'd \<Rightarrow> ('e, 'f, 'd, 'g) Std.instr fundef \<Rightarrow> 'c"   and Fstd_to_list :: "'c \<Rightarrow> ('d \<times> ('e, 'f, 'd, 'g) Std.instr fundef) list"   and Finca_empty :: "'h"   and Finca_get :: "'h \<Rightarrow> 'd \<Rightarrow> ('e, 'f, 'd, 'g, 'i) Inca.instr fundef option"   and Finca_add :: "'h \<Rightarrow> 'd \<Rightarrow> ('e, 'f, 'd, 'g, 'i) Inca.instr fundef \<Rightarrow> 'h"   and Finca_to_list :: "'h \<Rightarrow> ('d \<times> ('e, 'f, 'd, 'g, 'i) Inca.instr fundef) list"   and heap_empty :: "'j"   and heap_get :: "'j \<Rightarrow> 'f \<times> 'e \<Rightarrow> 'e option"   and heap_add :: "'j \<Rightarrow> 'f \<times> 'e \<Rightarrow> 'e \<Rightarrow> 'j"   and heap_to_list :: "'j \<Rightarrow> (('f \<times> 'e) \<times> 'e) list"   and is_true :: "'e \<Rightarrow> bool"   and is_false :: "'e \<Rightarrow> bool"   and \<OO>\<pp> :: "'g \<Rightarrow> 'e list \<Rightarrow> 'e"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'g \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'i \<Rightarrow> 'e list \<Rightarrow> 'e"   and \<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e list \<Rightarrow> 'i option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'i \<Rightarrow> 'e list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'i \<Rightarrow> 'g"   and F1 :: "'c"   and F2 :: "'h"   and H :: "'k"   and st :: "'l list" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>"   and "std_inca_simulation.rel_fundefs \<DD>\<ee>\<II>\<nn>\<ll> (Fstd_get F1) (Finca_get F2)" shows "std_inca_simulation.match Fstd_get Finca_get \<DD>\<ee>\<II>\<nn>\<ll> (State F1 H st) (State F2 H st)"
l11_31_2: 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 "A \<noteq> B"   and "C \<noteq> B"   and "D \<noteq> E"   and "F \<noteq> E"   and "Bet D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"
pairwise_stlD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream" assumes "PTA_Reachability.pairwise P xs" shows "PTA_Reachability.pairwise P (stl xs)"
preimages_conull_1: fixes A :: "'a set" assumes "A \<in> sets M"   and "0 < emeasure M A" shows "space M - (\<Union>n. (T ^^ n) --` A) \<in> null_sets M"
extend_indets_minus: fixes r :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and r' :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "extend_indets (r - r') = extend_indets r - extend_indets r'"
infdist_singleton: fixes x :: "'b"   and y :: "'b" shows "infdist x {y} = dist x y"
dist: shows "(if degree f = 1 then [(1, f)] else local.dist_degree_factorize_main f (monom 1 1) 0 []) = facts"
lemma_exp_total: fixes y :: "real" assumes "1 \<le> y" shows "\<exists>x\<ge>0. x \<le> y - 1 \<and> exp x = y"
jAction_def: shows "jAction \<equiv> \<lambda>M w a. concat (map (\<lambda>gc. if M, w \<Turnstile> guard gc then [action gc] else []) (jkbp a))"
ups_fv_subset: fixes S :: "stack_elem list" shows "upds S \<subseteq> fv S"
lm057: fixes a :: "('c \<times> 'd set) set"   and N :: "'c set"   and G :: "'d set" assumes "a \<in> allocationsUniverse"   and "Domain a \<subseteq> N"   and "\<Union> (Range a) = G" shows "a \<in> allAllocations N G"
da_elim_cases_57: fixes Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>[]\<rangle>\<guillemotright> A" obtains "A = \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
NonInterferenceInterGraph_axioms: shows "NonInterferenceInterGraph sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main (_Exit_) Def Use ParamDefs ParamUses H L (_High_) (_Low_)"
Diff_eq_empty_iff_mset: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A - B = {#}) = (A \<subseteq># B)"
gauss_int_norm_eq_0_iff: fixes z :: "gauss_int" shows "(gauss_int_norm z = 0) = (z = 0)"
distrib: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a + b) * c = a * c + b * c"
sup_Sup: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "sup_prod x y = Sup_prod (insert x (\<eta> y))"
image_zet_rep: fixes A :: "'a set"   and g :: "'a \<Rightarrow> ZF" assumes "A \<in> zet" shows "\<exists>z. g ` A = explode z"
ccspan_def: shows "ccspan \<equiv> map_fun id Abs_clinear_space (\<lambda>G. closure (cspan G))"
less_eq_fps_def: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "(f \<le> g) = (f + g = g)"
load_tmps_fresh_axioms: shows "load_tmps_fresh ts\<^sub>s\<^sub>b"