hcju/isabelleps huggingface dataset:
\<tau>red1r_into_\<tau>Red1r: 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 exs :: "('addr expr1 \<times> 'addr val list) list" assumes "\<tau>red1gr uf P t h (e, xs) (e'', xs'')" shows "\<tau>Red1gr uf P t h ((e, xs), exs) ((e'', xs''), exs)"
sturm_interval: fixes a :: "real"   and b :: "real"   and p :: "real poly" assumes "a < b"   and "poly p a \<noteq> 0"   and "poly p b \<noteq> 0" shows "int (card {x. poly p x = 0 \<and> a < x \<and> x < b}) = changes_itv_smods a b p (pderiv p)"
IH: fixes B1 :: "('a strand_label \<times> ('b, 'c) stateful_strand_step) list"   and B2 :: "('a strand_label \<times> ('b, 'c) 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"
nsum_zeroA: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>n. f j = \<zero>" shows "\<Sigma>\<^sub>e M f n = \<zero>"
blue_dfs_body_def: fixes G :: "('a, 'b) b_graph_rec_scheme" shows "blue_dfs_body G \<equiv> \<lambda>D (blues, reds, onstack, s). let blues = insert s blues; onstack = insert s onstack; s_acc = s \<in> bg_F G in NDFS_SI_Statistics.vis_blue_nres \<bind> (\<lambda>_. FOREACH\<^sub>C (g_E G `` {s}) (\<lambda>(uu_, uu_, uu_, cyc). cyc = NO_CYC) (\<lambda>t (blues, reds, onstack, cyc). if t \<in> onstack \<and> (s_acc \<or> t \<in> bg_F G) then RETURN (blues, reds, onstack, init_wit_blue_early s t) else if t \<notin> blues then D (blues, reds, onstack, t) \<bind> (\<lambda>(blues, reds, onstack, cyc). RETURN (blues, reds, onstack, prep_wit_blue s cyc)) else NDFS_SI_Statistics.match_blue_nres \<bind> (\<lambda>_. RETURN (blues, reds, onstack, cyc))) (blues, reds, onstack, NO_CYC) \<bind> (\<lambda>(blues, reds, onstack, cyc). (if cyc = NO_CYC \<and> s_acc then red_dfs (g_E G) onstack reds s \<bind> (\<lambda>(reds, rcyc). RETURN (reds, init_wit_blue s rcyc)) else RETURN (reds, cyc)) \<bind> (\<lambda>(reds, cyc). let onstack = onstack - {s} in RETURN (blues, reds, onstack, cyc))))"
rangeI: fixes f :: "'h \<Rightarrow> 'g"   and x :: "'h" shows "f x \<in> range f"
aexec: shows "abc_step_l (as, lm) (Some (Inc n)) = (asa, lma)"
antiframe_def: fixes a :: "'c \<Longrightarrow> 'b"   and P :: "'b hrel" shows "{a}:\<lbrakk>P\<rbrakk> = (P \<and> $\<^bold>v\<acute> =\<^sub>u bop (\<triangleleft>\<^bsub>&a\<^esub>) $\<^bold>v\<acute> $\<^bold>v)"
compl_le_compl_iff: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (uminus x) (uminus y) = less_eq y x"
mzero_im_sub_ker: fixes R :: "('e, 'f) Ring_scheme"   and L :: "('g, 'e, 'h) Module_scheme"   and M :: "('i, 'e, 'j) Module_scheme"   and N :: "('k, 'e, 'l) Module_scheme"   and f :: "'g \<Rightarrow> 'i"   and g :: "'i \<Rightarrow> 'k" assumes "Ring R"   and "R module L"   and "R module M"   and "R module N"   and "f \<in> mHom R L M"   and "g \<in> mHom R M N"   and "compos L g f = mzeromap L N" shows "f ` carrier L \<subseteq> ker\<^bsub>M,N\<^esub> g"
ennreal_spmf_bind: fixes p :: "'e spmf"   and f :: "'e \<Rightarrow> 'd spmf"   and x :: "'d" shows "ennreal (spmf (p \<bind> f) x) = \<integral>\<^sup>+ xa. ennreal (spmf (f xa) x) \<partial>measure_spmf p"
Cart_lambda_cong: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. f x = g x" shows "vec_lambda f = vec_lambda g"
treehead_in_multiset: fixes t :: "('c, 'd) BinomialTree"   and bq :: "('c, 'd) BinomialTree list" assumes "t \<in> set bq" shows "(val t, prio t) \<in># queue_to_multiset bq"
Neg: fixes A :: "fm"   and i :: "name"   and x :: "tm" shows "(Neg A)(i::=x) = Neg (A(i::=x))"
append_interleave: fixes xs :: "'a list"   and ys :: "'a list" shows "xs @ ys \<in> xs \<otimes> ys"
max_of_antimono: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "antimono f" shows "max (f x) (f y) = f (min x y)"
simps_1: fixes n :: "'num" shows "generalize_instr (IPushUbx1 n) = IPush (box_ubx1 n)"
sets_restrict_UNIV: fixes M :: "'c measure" shows "sets (restrict_space M UNIV) = sets M"
Op_castOp: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "isOp s X" shows "Op (fst3 (castOp s X)) (snd3 (castOp s X)) (trd3 (castOp s X)) = X"
option_relE: fixes x :: "'b option"   and x' :: "'d option"   and R :: "('b \<times> 'd) set" assumes "(x, x') \<in> \<langle>R\<rangle>option_rel" obtains "x = None" and "x' = None"   | a :: "'b" and a' :: "'d" where "x = Some a" and "x' = Some a'" and "(a, a') \<in> R"
ring_distribs_2: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a + b) * c = a * c + b * c"
bind_density: fixes M :: "'c measure"   and N :: "'d measure"   and f :: "'c \<Rightarrow> 'd measure"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" assumes "sigma_finite_measure M"   and "sigma_finite_measure N"   and "space M \<noteq> {}"   and "\<And>x. x \<in> space M \<Longrightarrow> has_density (f x) N (g x)"   and "(\<lambda>(x, y). g x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "M \<bind> f = density N (\<lambda>y. \<integral>\<^sup>+ x. g x y \<partial>M)"
update_convs_2: fixes c_temp' :: "('c \<Rightarrow> 'd zmultiset) \<Rightarrow> 'c \<Rightarrow> 'd zmultiset"   and c_records :: "'d zmultiset"   and c_temp :: "'c \<Rightarrow> 'd zmultiset"   and c_msg :: "'c \<Rightarrow> 'c \<Rightarrow> 'd zmultiset list"   and c_glob :: "'c \<Rightarrow> 'd zmultiset"   and more :: "'e" shows "c_temp_update c_temp' \<lparr>c_records = c_records, c_temp = c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr> = \<lparr>c_records = c_records, c_temp = c_temp' c_temp, c_msg = c_msg, c_glob = c_glob, \<dots> = more\<rparr>"
emeasure_def: fixes M :: "'b measure" shows "emeasure M = snd (snd (Rep_measure M))"
H1_idem: shows "H1 \<circ> H1 = H1"
fac_work_def: shows "fac_work \<equiv> fix\<cdot>(unwrapB oo fac_body oo wrapB)"
lookup_alt_def: fixes x :: "'b" shows "SFI_lookup x = get_state (\<lambda>s. case apply s x of None \<Rightarrow> fail_state (fail_option return_id) | Some y \<Rightarrow> return_state (return_option return_id) y)"
assms: shows "\<forall>s. I s \<longrightarrow> (\<forall>s'. s' \<in> F s \<longrightarrow> I s')"
simps_4: fixes T :: "typ"   and t :: "Core.term"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubst (Abs T t) \<rho> = Abs (tsubstT T \<rho>) (tsubst t \<rho>)"
arr_ParI: fixes \<alpha> :: "V"   and T :: "V" assumes "\<Z> \<alpha>"   and "vfsequence T"   and "vsv (T\<lparr>ArrVal\<rparr>)"   and "vcard T = 3\<^sub>\<nat>"   and "\<D>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrDom\<rparr>"   and "\<R>\<^sub>\<circ> (T\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> T\<lparr>ArrCod\<rparr>"   and "T\<lparr>ArrDom\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"   and "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>" shows "arr_Par \<alpha> T"
adjoint_clauses_2: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and x :: "'a" assumes "linear f" shows "adjoint f y \<bullet> x = y \<bullet> f x"
LD_def: fixes L :: "'c set"   and ars :: "('d \<times> 'd) set" shows "LD L ars = (\<exists>r lrs. ars = unlabel lrs \<and> trans r \<and> wf r \<and> (\<forall>P. local_peak lrs P \<longrightarrow> (\<exists>\<sigma>' \<tau>'. DD lrs r (fst P, snd P, \<sigma>', \<tau>'))))"
topspace_product_topology: fixes T :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "topspace (product_topology T I) = (\<Pi>\<^sub>E i\<in>I. topspace (T i))"
listgcd: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "some_gcd.listgcd xs dvd x"
ae: shows "\<forall>\<^sub>F x in F. g x \<le> c"
norm_summable_moebius_over_square: shows "summable (\<lambda>n. norm (moebius_mu n / (real n)\<^sup>2))"
g_outside: fixes e :: "'v \<times> 'v" assumes "e \<notin> \<^bold>E" shows "g e = 0"
pseudoinverse': shows "\<forall>\<omega>\<in>{0<..<1}. \<forall>x. (\<omega> \<le> M.C x) = (M.I \<omega> \<le> x)"
assms_2: fixes a :: "'b" assumes "P a" shows "Q a"
f_inv_into_f: fixes y :: "'ag"   and f :: "'ah \<Rightarrow> 'ag"   and A :: "'ah set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
norm_le_componentwise_cart: fixes x :: "('b, 'a) vec"   and y :: "('c, 'a) vec" assumes "\<And>i. norm (x $ i) \<le> norm (y $ i)" shows "norm x \<le> norm y"
distinct_kv_list: fixes l :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "distinct l" shows "distinct (map (\<lambda>x. (x, f x)) l)"
top_right_normalize: fixes a :: "'a" shows "a \<^bold>* \<^bold>\<top> = normalize a"
is_Low\<^sub>C': shows "dma\<^sub>C mem\<^sub>1\<^sub>C' x\<^sub>C = Low"
bisimResComm: fixes \<Psi> :: "'b"   and x :: "name"   and y :: "name"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(\<lparr>\<nu>y\<rparr>P) \<sim> \<lparr>\<nu>y\<rparr>(\<lparr>\<nu>x\<rparr>P)"
Prot_single_not: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> bool"   and X :: "protocol"   and a :: "action"   and p :: "('b, 'c) tagged_packet_scheme" assumes "primitive_matcher_generic \<beta>" shows "matches (\<beta>, \<alpha>) (MatchNot (Match (Prot X))) a p = (\<not> match_proto X (p_proto p))"
substs_preserves_decompose_literal: fixes L :: "'c literal"   and t :: "'c trm"   and s :: "'c trm"   and polarity :: "sign"   and \<eta> :: "('c \<times> 'c trm) list" assumes "decompose_literal L t s polarity" shows "decompose_literal (subst_lit L \<eta>) (t \<lhd> \<eta>) (s \<lhd> \<eta>) polarity"
SA_States_disjunct: fixes A :: "('a, 'b, 'c) seqauto"   and A' :: "('a, 'b, 'c) seqauto" assumes "States A \<inter> States A' = {}" shows "A' \<noteq> A"
prim_spanning: fixes g :: "'a"   and r :: "'a" shows "[prim_precondition g r]   t := bot; v := r; WHILE v * - v\<^sup>T \<sqinter> g \<noteq> bot INV {prim_spanning_invariant t v g r}  VAR {\<lambda>(t, v, e). card {x. regular x \<and> x \<le> component g r \<sqinter> - v\<^sup>T}}  DO e := minarc (v * - v\<^sup>T \<sqinter> g); t := t \<squnion> e; v := v \<squnion> e\<^sup>T * top OD   [spanning_tree t g r]"
prv_\<phi>T_eqv: shows "prv (eqv \<phi>T (neg (subst T \<langle>\<phi>T\<rangle> xx)))"
comm_monoidI: fixes G :: "('a, 'b) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
floor_has_real_derivative: fixes x :: "real"   and f :: "real \<Rightarrow> 'a" assumes "isCont f x"   and "f x \<notin> \<int>" shows "((\<lambda>x. real_of_int \<lfloor>f x\<rfloor>) has_real_derivative 0) (at x)"
sublens_pres_vwb: fixes Y :: "'a \<Longrightarrow> 'b"   and X :: "'c \<Longrightarrow> 'b" assumes "vwb_lens Y"   and "X \<subseteq>\<^sub>L Y" shows "vwb_lens X"
simps_6: fixes F :: "'b formula"   and G :: "'b formula" shows "cnf (F \<^bold>\<or> G) = {C \<union> D |C D. C \<in> cnf F \<and> D \<in> cnf G}"
assms_3: shows "C\<^sub>1 \<noteq> C\<^sub>2"
rtranclpD: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
rel_of_pred_def: fixes s :: "'b \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_of_pred s \<equiv> {(a, b). s a b}"
wp_valid_def: fixes P :: "'b \<Rightarrow> real"   and prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" shows "\<lbrace>P\<rbrace> prog \<lbrace>Q\<rbrace>p \<equiv> P \<tturnstile> wp prog Q"
in_set_spmf_exec_gpv_into_results_gpv: fixes x :: "'a"   and s' :: "('b, 'c) resource"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c) resource"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> set_spmf (exec_gpv run_resource gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "lossless_resource \<I> s \<and> \<I> \<turnstile>res s \<surd>" shows "x \<in> results_gpv \<I> gpv"
assms_2: shows "(k, v) \<in> set (fst xs)"
nth_index: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "xs ! index xs x = x"
norm_le: fixes x :: "'a" shows "norm (f x) \<le> norm x * K"
val_type_eq_INTEG: fixes x :: "val" shows "(val_type x = INTEG) = (x \<in> range IntVal)"
fls_divide_base_factor: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor f / fls_base_factor g = fls_base_factor (f / g)"
pdata_def: shows "pdata = {(L, D) |L D. PData L D}"
map_PP_pps_to_list: fixes T :: "('x \<Rightarrow>\<^sub>0 nat) set" shows "map PP (pp_pm.punit.pps_to_list T) = punit.pps_to_list (PP ` T)"
ereal_inverse_le_0_iff: fixes x :: "ereal" shows "(inverse x \<le> 0) = (x < 0 \<or> x = \<infinity>)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Can (t \<^bold>\<otimes> u) = (Can t \<and> Can u)"
quicksort_by_rel_distinct: fixes l :: "('a \<times> 'b \<times> 'a) list" assumes "distinct l" shows "distinct (quicksort_by_rel edges_less_eq [] l)"
linear_on_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "linear_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 f \<equiv> vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 f"
simps_1: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "bezout_iterate A 0 i j bezout = A"
boundedByInsert: fixes N :: "nat"   and n :: "nat"   and delta :: "'b"   and B :: "(nat \<times> 'b) set" shows "boundedBy N (insert (n, delta) B) = (n < N \<and> boundedBy N B)"
simps_1: fixes t :: "'b tree" shows "merge \<langle>\<rangle> t = t"
coeff_charpoly: fixes A :: "('b, 'c) sq_matrix" shows "lead_coeff (charpoly A) = (1::'b)"
pred_numeral_simps_3: fixes k :: "num" shows "pred_numeral (num.Bit1 k) = numeral (num.Bit0 k)"
simps_1: fixes prec :: "nat"   and f :: "form"   and n :: "nat"   and ivl :: "float interval"   and bs :: "float interval option list"   and ss :: "nat list" shows "approx_form' prec f 0 n ivl bs ss = approx_form prec f (bs[n := Some ivl]) ss"
slide_eq_slide': fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree"   and as' :: "'b list" assumes "0 < fst w"   and "fst w \<le> snd w"   and "valid as t"   and "r t = length as"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length (as @ as')" shows "slide (as @ as') t w = slide' as' t w"
size_7: fixes x21 :: "'b com"   and x22 :: "'b com" shows "size (Seq x21 x22) = size x21 + size x22 + Suc 0"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<sim> P"
games: fixes ga :: "ZF set" assumes "ga = fixgames ga" shows "ga = games_lfp"
uncurry_def: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
iso_char: fixes f :: "'a1 \<times> 'a2" shows "local.iso f = (C1.iso (fst f) \<and> C2.iso (snd f))"
binop_ontoI: fixes A :: "V"   and f :: "V" assumes "nop_onto A (2\<^sub>\<nat>) f" shows "binop_onto A f"
is_hashset_def: fixes s :: "'b set"   and ht :: "('b, unit) hashtable" shows "is_hashset s ht \<equiv> is_hashmap (map_of_set s) ht"
Out_cases: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A Out B C \<or> A Out C B" shows "A Out B C"
Smith_1xn_aux_invertible_Q: fixes S' :: "'a mat"   and Q' :: "'a mat"   and i :: "nat"   and A :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat"   and n :: "nat" assumes "(S', Q') = Smith_1xn_aux i A (S, Q)"   and "A \<in> carrier_mat 1 n"   and "Q \<in> carrier_mat n n"   and "i < n"   and "invertible_mat Q" shows "invertible_mat Q'"
1: fixes a :: "real"   and c :: "'a"   and r :: "real" assumes "a \<noteq> 0" shows "(*\<^sub>R) a ` ball c r \<subseteq> ball (a *\<^sub>R c) (\<bar>a\<bar> * r)"
indicator_simps_2: fixes x :: "'c"   and S :: "'c set" assumes "x \<notin> S" shows "indicator S x = (0::'d)"
mutual_exclusion: fixes s :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and t :: "nat"   and t' :: "nat" assumes "A.reachable s"   and "t < N"   and "t' < N"   and "t \<noteq> t'"   and "is_HOLD (tts s t)"   and "is_HOLD (tts s t')" shows "False"
partial_order_on_trans: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "partial_order_on A r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
input_sizes: shows "input_sizes (insert_weights shared_weights (deep_model_l rs) wd) = input_sizes (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws)"
pos_positive: fixes x :: "'a\<^sup>+" shows "0 \<le> x"
liftPoly_def: shows "liftPoly \<equiv> map_fun id (map_fun id (map_fun mapping_of MPoly)) higherPoly\<^sub>0"
z_def: shows "z \<equiv> getFr [xx, yy] [t1, t2] []"
norm_Pinj: fixes i :: "nat"   and Q :: "pol" assumes "isnorm (Pinj i Q)" shows "isnorm Q"
simps_7: fixes x :: "'a" shows "norm_instr (Ubx.instr.IStore x) = Inca.instr.IStore x"
word_msb_sint: fixes w :: "'a Word.word" shows "msb w = (sint w < 0)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. n \<le> 3 \<Longrightarrow> P n"   and "\<And>n. \<lbrakk>3 < n; P (nat \<lfloor>real n / 2\<rfloor>); P (nat \<lceil>real n / 2\<rceil>)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
component_in_hom: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and j :: "'a" assumes "Limit.cone J C D a \<chi>"   and "partial_magma.arr J j" shows "partial_magma.in_hom C (\<chi> j) a (D (partial_magma.cod J j))"
assms_3: shows "val_Zp x \<noteq> val_Zp y"
lossless_R2_12: fixes M :: "bool \<times> bool"   and c :: "bool" shows "lossless_spmf (R2_OT12 M c)"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
d\<eta>\<epsilon>: shows "internal_equivalence e d \<eta> \<epsilon>"
pfa_shrink_def: fixes k :: "nat" shows "pfa_shrink k \<equiv> \<lambda>(a, n). return (a, k)"
bits_mod_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'c" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo zero a = zero"
bounded_translation_minus: fixes S :: "'a set"   and a :: "'a" assumes "bounded S" shows "bounded ((\<lambda>x. x - a) ` S)"
rb_aux_no_zero_red: fixes z :: "nat" assumes "hom_grading dgrad"   and "is_regular_sequence fs" shows "snd (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z)) = z"
complex_i_not_one: shows "\<i> \<noteq> 1"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "P x"
simps_3: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "upds (Upd x # S) = insert x (upds S)"
v: shows "epath (insert (Upair u v) F) w p w"
kl: fixes e :: "'c" shows "pmf (map_pmf Inr Db) (Inl e) = 0"
WTrt_env_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and E' :: "char list \<Rightarrow> ty option" assumes "P,E,h,sh \<turnstile> e : T"   and "E \<subseteq>\<^sub>m E'" shows "P,E',h,sh \<turnstile> e : T"
random_oracle_def: fixes \<sigma> :: "'b \<Rightarrow> 'a option"   and x :: "'b" shows "rf.random_oracle \<sigma> x = (case \<sigma> x of None \<Rightarrow> spmf_of_set A \<bind> (\<lambda>y. return_spmf (y, \<sigma>(x \<mapsto> y))) | Some y \<Rightarrow> return_spmf (y, \<sigma>))"
step_Normal_elim_cases_11: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Seq c1 c2 # cs, css, Normal s) \<rightarrow> u" obtains "u = (c1 # c2 # cs, css, Normal s)"
pderivs_lang_Times: fixes r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pderivs_lang UNIV1 (Times r1 r2) \<subseteq> Timess (pderivs_lang UNIV1 r1) r2 \<union> pderivs_lang UNIV1 r2"
assms: shows "\<forall>a\<in>set la. itb a = itb' a"
compat_append_left: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" assumes "lexext gt ys xs" shows "lexext gt (zs @ ys) (zs @ xs)"
frequentlyE: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "frequently P F" obtains x :: "'b" where "P x"
equivalence_in_bicategory_def: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" shows "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<equiv> adjunction_data_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon> \<and> equivalence_in_bicategory_axioms V \<eta> \<epsilon>"
linear_lower2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_real_derivative f' x) (at x within {a..b})"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> l \<le> f' x"   and "x \<in> {a..b}" shows "f a + l * (x - a) \<le> f x"
words_infI_construct: fixes w :: "'action stream"   and p :: "'state" assumes "\<And>v. v \<le>\<^sub>F\<^sub>I w \<Longrightarrow> R.path v p" shows "R.run w p"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) R local.map \<rr>.map"
X: shows "X \<in> carrier (G A_Mod a_kernel G H h)"
iso_lunit': fixes f :: "'a" assumes "ide f" shows "local.iso \<l>\<^sup>-\<^sup>1[f]"
simps_10: fixes R :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_option R None None"
Sup_ow_def: fixes U :: "'a set"   and Sup :: "'a set \<Rightarrow> 'a" shows "Sup_ow U Sup \<equiv> Sup ` Pow U \<subseteq> U"
empty_iff: fixes c :: "'k" shows "(c \<in> {}) = False"