hcju/isabelleps huggingface dataset:
vec_to_lpoly_from_lpoly_coeff_dual2: fixes i :: "nat"   and v :: "rat vec" assumes "i < dim_vec (lpoly_to_vec (vec_to_lpoly v))" shows "lpoly_to_vec (vec_to_lpoly v) $ i = v $ i"
intros: fixes ve :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and ve' :: "(nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set"   and c :: "call"   and \<beta> :: "nat \<Rightarrow> 'a option"   and b :: "'a" assumes "ve \<lessapprox> ve'" shows "(c, \<beta>, ve, b) \<lessapprox> (c, \<beta>, ve', b)"
sort_by_pri_edges': fixes edges :: "'a edge_scheme list"   and A :: "'a edge_scheme set"   and min_pri :: "nat" assumes "set edges \<subseteq> A" shows "set (sort_by_pri min_pri edges) \<subseteq> {xs. set xs \<subseteq> A}"
hyps_1: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma>1"
inf_WhileTrue: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and cs :: "('d, 'e, 'f) com list"   and css :: "(('d, 'e, 'f) com list \<times> ('d, 'e, 'f) com list) list" assumes "s \<in> b" shows "AlternativeSmallStep.inf \<Gamma> (While b c # cs) css (Normal s) = AlternativeSmallStep.inf \<Gamma> (c # While b c # cs) css (Normal s)"
linear_span_image: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f" shows "span (f ` S) = f ` span S"
inv_solve_right: fixes a :: "'a set"   and b :: "'a set"   and c :: "'a set" assumes "a \<in> carrier (R Quot J)"   and "b \<in> carrier (R Quot J)"   and "c \<in> carrier (R Quot J)" shows "(a = b \<oplus>\<^bsub>R Quot J\<^esub> \<ominus>\<^bsub>R Quot J\<^esub> c) = (b = a \<oplus>\<^bsub>R Quot J\<^esub> c)"
step: fixes s :: "'a" assumes "guard s" shows "p \<le> spmf (map_spmf (\<lambda>s'. f s' < f s) (body s)) True"
f: shows "\<forall>x. f x (s(fld := val)) = f x s"
lappend_lnil_id_left: shows "lappend\<cdot>lnil = ID"
P5: shows "B A C CongA D A E' \<and> D A OS E' B"
matches_alist_and: fixes \<gamma> :: "('d \<Rightarrow> 'e \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'e \<Rightarrow> bool)"   and l :: "'d negation_type list"   and a :: "action"   and p :: "'e" shows "matches \<gamma> (alist_and l) a p = ((\<forall>m\<in>set (getPos l). matches \<gamma> (Match m) a p) \<and> (\<forall>m\<in>set (getNeg l). matches \<gamma> (MatchNot (Match m)) a p))"
hyps: assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "trace (matrix_sum d f n) = (\<Sum>k = 0..<n. trace (f k))"
elims_3: fixes x :: "'d ifex"   and xa :: "('d, bool) mapping" assumes "\<not> taut_test_rec x xa" obtains env :: "('d, bool) mapping" where "x = Falseif" and "xa = env__"   | xb :: "'d" and t1 :: "'d ifex" and t2 :: "'d ifex" and env :: "('d, bool) mapping" where "x = IF xb t1 t2" and "xa = env" and "\<not> (case Mapping.lookup env xb of None \<Rightarrow> taut_test_rec t1 (Mapping.update xb True env) \<and> taut_test_rec t2 (Mapping.update xb False env) | Some b \<Rightarrow> taut_test_rec (if b then t1 else t2) env)"
mono_funpow: fixes Q :: "'a \<Rightarrow> 'a" assumes "mono Q" shows "incseq (\<lambda>i. (Q ^^ i) bot)"
tendsto_ln_powr_over_ln_powr: fixes a :: "real"   and c :: "real"   and d :: "real" assumes "0 < a"   and "0 < c" shows "((\<lambda>x. ln (a * x) powr d / ln (c * x) powr d) \<longlongrightarrow> 1) at_top"
ge_ubound_def: fixes c :: "'a"   and b :: "'a option" shows "c \<ge>\<^sub>u\<^sub>b b = \<unrhd>\<^sub>u\<^sub>b (<) c b"
eval_wellf_Mn: fixes n :: "nat"   and f :: "recf"   and xs :: "nat list" shows "eval_wellf (Mn n f) xs = (if \<exists>z. eval_wellf f (z # xs) \<down>= 0 \<and> (\<forall>y<z. eval_wellf f (y # xs) \<down>) then Some (LEAST z. eval_wellf f (z # xs) \<down>= 0) else None)"
mask_eq_exp_minus_1: fixes n :: "nat" shows "mask n = (2::'a) ^ n - (1::'a)"
outstanding_volatile_writes_unowned_by_others_instructions_read_value_independent: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "outstanding_volatile_writes_unowned_by_others (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb, \<D>', \<O>, \<R>')])"
CondFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c\<^sub>1 :: "('a, 'b, 'c) com"   and c\<^sub>2 :: "('a, 'b, 'c) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> (Cond b c\<^sub>1 c\<^sub>2, Normal s) \<rightarrow> (c\<^sub>2, Normal s)"
assms: shows "p \<in> ctermsl (\<Gamma>\<^sub>T\<^sub>O\<^sub>Y pn)"
fmlookup_restrict_fset: fixes A :: "'f fset"   and m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" shows "fmlookup (fmrestrict_fset A m) x = (if x |\<in>| A then fmlookup m x else None)"
NN: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "NN t1 t2 \<in> fmla"
lnth_ltl: fixes xs :: "'b llist"   and n :: "nat" assumes "\<not> lnull xs" shows "ltl xs ?! n = xs ?! Suc n"
dims1_def: shows "stm2.dims1 = nths dims {0..<Suc m}"
path_compose_join: fixes f :: "'b \<Rightarrow> 'a"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "f \<circ> p +++ q = (f \<circ> p) +++ (f \<circ> q)"
L_in_interval: fixes a :: "'a"   and b :: "'a"   and S :: "'a set"   and L :: "'a" assumes "a \<in> A"   and "b \<in> A"   and "S \<subseteq> interval r a b"   and "S \<noteq> {}"   and "isLub S cl L"   and "interval r a b \<noteq> {}" shows "L \<in> interval r a b"
term_to_nterm_nterm_to_term: fixes t :: "Term_Class.term"   and S :: "name fset"   and Q :: "name fset" assumes "wellformed t"   and "frees t |\<subseteq>| S" shows "nterm_to_term' (frun_fresh (term_to_nterm [] t) (S |\<union>| Q)) = t"
prob_space_measure_pmf: fixes p :: "'c pmf" shows "prob_space (measure_pmf p)"
H\<^sub>L_def: fixes g :: "'a" shows "H\<^sub>L g \<equiv> (\<star>) g"
goedel_numbering_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option" shows "goedel_numbering \<psi> \<equiv> \<psi> \<in> \<P>\<^sup>2 \<and> (\<forall>\<chi>\<in>\<P>\<^sup>2. \<exists>c\<in>\<R>. \<forall>i. \<chi> i = \<psi> (the (c i)))"
rbl_sizes_4: fixes bl :: "bool list"   and cl :: "bool list" shows "length (rbl_mult bl cl) = length bl"
lastBlockInvariant_def: fixes state :: "CFile" shows "lastBlockInvariant state \<equiv> \<forall>index. \<not> index < fileSize state \<and> nextFreeBlock state = index div blockSize + 1 \<longrightarrow> data state (index div blockSize) (index mod blockSize) = fillByte"
nrows_less: shows "nrows A - Suc 0 < nrows (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n))"
unat_one_word_le: fixes a :: "1 word" shows "unat a \<le> Suc 0"
implements_ind_jkbpC: fixes jp :: "'a \<Rightarrow> ('obs, 'aAct, 'b) Protocol" assumes "\<And>a n t. \<lbrakk>{t \<in> jpTraces jp. tLength t = n} = jkbpC\<^bsub>n\<^esub>; t \<in> jkbpC\<^bsub>n\<^esub>\<rbrakk> \<Longrightarrow> actJP jp t a = jAction MC t a" shows "implements jp"
dM1: shows "det (mat n n (\<lambda>(i, j). if i = j then 1 else if j < dfj then 0 else if i < j then [:b (int db - int i + int (j - dfj)):] else 0)) = 1"
case_6: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x61 :: "'b ltln"   and x62 :: "'b ltln" shows "(case x61 or\<^sub>n x62 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f6 x61 x62"
hyps_4: shows "lookup (tabulate (fromList xs) f) = (Some \<circ> f) |` toSet (fromList xs)"
t: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "finite {w. P w}" shows "finite {w. Q w \<and> P w}"
IH: shows "(ref \<in> set (deleted_ids ops)) = (\<exists>i. (i, Delete ref) \<in> set ops)"
simps_2: fixes P :: "'a"   and uu :: "'b"   and uv :: "('a, 'b) chain" shows "pre (\<lbrace> P \<rbrace> \<cdot> uu \<cdot> uv) = P"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "min_dnf (\<phi> and\<^sub>n \<psi>) = min_dnf \<phi> \<otimes>\<^sub>m min_dnf \<psi>"
strict_mono_imp_inj_on: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "inj_on f A"
wadjust_backto_standard_pos_B_Bk_Oc_via_O: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_backto_standard_pos_O m rs (c, Oc # list)"   and "c \<noteq> []"   and "hd c = Bk" shows "wadjust_backto_standard_pos_B m rs (tl c, Bk # Oc # list)"
hyps: assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = Zero" shows "last (fin_make_untimed (inf_truncate z t)) = Zero"
that: fixes t :: "dbtm" assumes "wf_dbtm t"   and "x = \<lbrakk>quot_dbtm t\<rbrakk>e" shows "thesis"
bet__suppa: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "B \<noteq> A'"   and "Bet A B A'" shows "A B C SuppA C B A' "
simps_1: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
fls_divide_fls_base_factor_to_fps': fixes f :: "'b fls"   and g :: "'b fls" shows "fls_base_factor_to_fps f / fls_base_factor_to_fps g = fls_regpart (fls_shift (fls_subdegree f - fls_subdegree g) (f / g))"
Inj: fixes X :: "msg"   and H :: "msg set" assumes "X \<in> H" shows "X \<in> synth H"
simps_11: fixes e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" shows "fv (e\<^sub>1;; e\<^sub>2) = fv e\<^sub>1 \<union> fv e\<^sub>2"
not_lfinite_llength: fixes xs :: "'c llist" assumes "\<not> lfinite xs" shows "llength xs = \<infinity>"
that: fixes v :: "'a \<Rightarrow> nat"   and n :: "nat" assumes "bij_betw v A {1..n}"   and "\<forall>c\<in>A. 0 < v c"   and "\<forall>c. c \<notin> A \<longrightarrow> n < v c" shows "thesis"
assms_3: shows "y \<in> cl (insert x X) - cl X"
less_or_eq_imp_le: fixes m :: "nat"   and n :: "nat" assumes "order.greater n m \<or> m = n" shows "order.greater_eq n m"
functor_axioms: shows "functor DopxD.comp S.comp HomD.map"
ref_loc: fixes r :: "Value"   and f :: "AttId" assumes "isObjV r"   and "\<tau> r \<le> dtype f" shows "ref (r..f) = r"
exec1E_1: fixes s :: "state"   and cs' :: "com list"   and s' :: "state"   and P :: "bool" assumes "([], s) \<rightarrow> (cs', s')" shows "P"
Class_widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and D :: "char list" shows "P \<turnstile> Class C \<le> Class D = P \<turnstile> C \<preceq>\<^sup>* D"
doctor_optimal_match_optimal: fixes ds :: "'d set"   and Y :: "'x set"   and x :: "'x"   and X :: "'x set" assumes "doctor_optimal_match ds Y"   and "x \<in> X"   and "stable_on ds X" shows "\<exists>y\<in>Y. (x, y) \<in> Pd (Xd x)"
lambda_system_positive: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal"   and \<Omega> :: "'b set" assumes "positive M f" shows "positive (lambda_system \<Omega> M f) f"
row_col: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat"   and j :: "nat" assumes "mat nr nc m"   and "i < nr"   and "j < nc" shows "row m i ! j = col m j ! i"
coboundedI2: fixes b :: "'b::semilattice_inf"   and c :: "'b::semilattice_inf"   and a :: "'b::semilattice_inf" assumes "(b::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<le> (c::'b::semilattice_inf)"
elims_1: fixes x :: "'a set list"   and y :: "bool" assumes "min_gallery x = y" obtains "x = []" and "y"   | xa :: "'a set" where "x = [xa__]" and "y = folding.chamber xa__"   | xa :: "'a set" and xs :: "'a set list" and ya :: "'a set" where "x = xa # xs @ [ya]" and "y = (xa \<noteq> ya \<and> is_arg_min length (\<lambda>zs. folding.gallery (xa # zs @ [ya])) xs)"
arr_inv_set_locals: fixes l :: "lname \<Rightarrow> val option"   and s :: "st" shows "arr_inv (set_locals l s) = arr_inv s"
gcdof_greatestLower: fixes a :: "'c"   and G :: "('c, 'd) monoid_scheme"   and b :: "'c"   and x :: "'c" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "(x \<in> carrier G \<and> x gcdof\<^bsub>G\<^esub> a b) = is_glb (division_rel G) x {a, b}"
inj_on_image_set_diff: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "A - B \<subseteq> C"   and "B \<subseteq> C" shows "f ` (A - B) = f ` A - f ` B"
universal_lookup_sem: fixes fml :: "nat fml"   and qs :: "rat poly list"   and signs :: "rat list list" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"   and "set signs = consistent_sign_vectors qs UNIV" shows "(\<forall>x. lookup_sem fml (map (\<lambda>p. rpoly p x) qs)) = list_all (lookup_sem fml) signs"
is_non_volatile_Write\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_non_volatile_Write\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = False"
range_rcut_subset_Pow_rat: shows "range rcut \<subseteq> Pow \<rat>"
eqn_hyp: shows "matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) *\<^sub>v (M_mat_R signs subsets *\<^sub>v w_vec_R p qs signs) = matr_option (dim_row (M_mat_R signs subsets)) (mat_inverse (M_mat_R signs subsets)) *\<^sub>v v_vec_R p qs subsets"
assms: fixes x' :: "Vars.var"   and e :: "exp"   and a :: "'a" assumes "(x', e) \<in> set \<Gamma>" shows "(t a e)[x::=y] = t a e[x::=y]"
closed_injective_linear_image: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "closed S"   and "linear f"   and "inj f" shows "closed (f ` S)"
has_field_derivative_Polygamma: fixes z :: "'a"   and n :: "nat"   and A :: "'a set" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(Polygamma n has_field_derivative Polygamma (Suc n) z) (at z within A)"
plus_poly_mapping_def: shows "(+) \<equiv> map_fun lookup (map_fun lookup Abs_poly_mapping) (\<lambda>f1 f2 k. f1 k + f2 k)"
max_in_chainD: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "nat"   and Y :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "class.po below"   and "po.max_in_chain i Y"   and "i \<le> j" shows "Y i = Y j"
c: 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> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>s t"
floor_le_ceiling: fixes x :: "'b" shows "\<lfloor>x\<rfloor> \<le> \<lceil>x\<rceil>"
hyps_3: fixes xs :: "'a memref list" shows "(m'', sb'' @ xs, \<O>'', \<R>'', \<S>'') \<rightarrow>\<^sub>f\<^sup>* (m', sb' @ xs, \<O>', \<R>', \<S>')"
prime_s_reverse: fixes n :: "nat" assumes "primerec s n" obtains "n = Suc 0"
seqr_left_one_point_false: fixes x :: "bool \<Longrightarrow> 'b"   and P :: "('c, 'b) urel"   and Q :: "('b, 'd) urel" assumes "vwb_lens x" shows "(P \<and> \<not> $x\<acute>) ;; Q = P\<lbrakk>false/$x\<acute>\<rbrakk> ;; Q\<lbrakk>false/$x\<rbrakk>"
ring_simprules_15: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
simple_match_src_alt: fixes r :: "('b, 'c) simple_match_scheme"   and p :: "'b word" assumes "simple_match_valid r" shows "simple_match_ip (src r) p = prefix_match_semantics (PrefixMatch (fst (src r)) (snd (src r))) p"
le_imp_erf_real_le: fixes a :: "real"   and b :: "real" assumes "a \<le> b" shows "erf a \<le> erf b"
apath_ends: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "G.apath u p v"   and "G.apath u' p v'" shows "p \<noteq> [] \<and> u \<noteq> v \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
cases: fixes s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('x \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and a1 :: "'b + 'c + 'c"   and a2 :: "'c set" assumes "must_wait s t a1 a2" obtains l :: "'b" and t' :: "'c" and Ts :: "'c set" where "a1 = Inl l" and "a2 = Ts__" and "has_lock (locks s $ l) t'__" and "t'__ \<noteq> t" and "t'__ \<in> Ts__"   | t' :: "'c" and Ts :: "'c set" where "a1 = Inr (Inl t'a__)" and "a2 = Tsa__" and "not_final_thread s t'a__" and "t'a__ \<in> Tsa__"   | Ts :: "'c set" and t' :: "'c" where "a1 = Inr (Inr t')" and "a2 = Ts" and "all_final_except s Ts" and "t' \<notin> interrupts s"
le0: fixes n :: "nat" shows "0 \<le> (n::nat)"
simps_2: fixes final :: "'c \<Rightarrow> bool"   and s :: "('d, 'e) locks \<times> (('e \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option) \<times> 'f) \<times> ('e \<Rightarrow> 'g wait_set_status option) \<times> 'e set"   and t :: "'e" shows "final_thread.cond_action_ok final s t Yield = True"
fset_simps_2: fixes x :: "'b"   and S :: "'b fset" shows "fset (insert_fset x S) = insert x (fset S)"
simps_3: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Trg (t \<^bold>\<star> u) = Trg t"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c tree \<Rightarrow> 'c \<Rightarrow> 'c tree \<Rightarrow> 'b"   and tree :: "'c tree" shows "P (case tree of \<langle>\<rangle> \<Rightarrow> f1 | \<langle>x, xa, xb\<rangle> \<Rightarrow> f2 x xa xb) = ((tree = \<langle>\<rangle> \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23. tree = \<langle>x21, x22, x23\<rangle> \<longrightarrow> P (f2 x21 x22 x23)))"
uint_sshiftr_def: fixes x :: "uint"   and n :: "integer" shows "uint_sshiftr x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined (>>>) x n else x >>> nat_of_integer n)"
dist_0_norm: fixes x :: "'e" shows "dist (0::'e) x = norm x"
ceiling_mono: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "\<lceil>y\<rceil> \<le> \<lceil>x\<rceil>"
tranclp_into_rtranclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
mod_eq_imp_diff_mod_eq2: fixes x :: "nat"   and m :: "nat"   and y :: "nat"   and t :: "nat" assumes "x mod m = y mod m"   and "t \<le> x"   and "t \<le> y" shows "(x - t) mod m = (y - t) mod m"
llist_all2_lhdD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys"   and "\<not> lnull xs" shows "P (lhd xs) (lhd ys)"
T: shows "T = (\<lambda>l. {ps. length ps = l + 1 \<and> adj_path (hd ps) (tl ps)})"
inject: fixes x21 :: "'b"   and x22 :: "'b"   and x23 :: "'c"   and y21 :: "'b"   and y22 :: "'b"   and y23 :: "'c" shows "(MkArr x21 x22 x23 = MkArr y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
dim_psubset: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "vec.span S \<subset> vec.span T" shows "vec.dim S < vec.dim T"
purgeIdle_Cons_iff: fixes sl :: "('St, 'U, 'C) state list"   and s :: "('St, 'U, 'C) state"   and sll :: "('St, 'U, 'C) state list" shows "(purgeIdle sl = s # sll) = (\<exists>sl1 sl2. sl = sl1 @ s # sl2 \<and> (\<forall>s1\<in>set sl1. \<not> isState s1) \<and> isState s \<and> purgeIdle sl2 = sll)"
distinct_mset_def: fixes S :: "'b multiset" shows "distinct_mset S = (\<forall>a. a \<in># S \<longrightarrow> count S a = 1)"
nat_pow_consistent: fixes x :: "'a"   and n :: "nat"   and H :: "'a set" shows "x [^]\<^bsub>G\<lparr>carrier := H__\<rparr>\<^esub> n = x [^]\<^bsub>G\<lparr>carrier := H__, carrier := H\<rparr>\<^esub> n"
finite_measure_axioms: fixes M :: "'c pmf" shows "finite_measure (measure_pmf M)"
suppD_contra: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "f x = (0::'a)" shows "x \<notin> supp f"
borel_measurable_min: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. min (g x) (f x)) \<in> borel_measurable M"
"1_2": shows "\<forall>x\<in>greachable A w (clean A w V) u. \<not> gaccepting A x"
scale_left_diff_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "('a, 'b) vec" shows "(a - b) *s x = a *s x - b *s x"
respectfulness_8: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" assumes "\<phi> \<equiv>\<^sub>P \<psi>" shows "Unf \<phi> \<equiv>\<^sub>P Unf \<psi>"
ag_pOp_commute: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier M"   and "y \<in> carrier M" shows "x \<plusminus> y = y \<plusminus> x"
mr_in_dom: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x p = \<lfloor>a\<rfloor>" shows "x \<in> dom (Cp a)"
rewriteR_comp_comp: fixes g :: "'f \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'f"   and r :: "'d \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'g" assumes "g \<circ> h = r" shows "f \<circ> g \<circ> h = f \<circ> r"
sup_compl_top: fixes x :: "'a upred" shows "(x \<or> \<not> x) = true"
fds_abs_converges_cong: fixes f :: "'b fds"   and g :: "'b fds"   and s :: "'b"   and s' :: "'b" assumes "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n"   and "s = s'" shows "fds_abs_converges f s = fds_abs_converges g s'"
GatewaySystemReq_def: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and vc :: "nat \<Rightarrow> vcType list" shows "GatewaySystemReq req dt stop lose d ack vc \<equiv> msg 1 req \<and> msg 1 stop \<and> ts lose \<longrightarrow> (\<forall>t k. ack t = [init_state] \<and> req (Suc t) = [init] \<and> (\<forall>t1\<le>t. req t1 = []) \<and> req (t + 2) = [] \<and> (\<forall>m<k + 3. req (t + m) \<noteq> [send]) \<and> req (t + 3 + k) = [send] \<and> inf_last_ti dt (t + 2) \<noteq> [] \<and> (\<forall>j\<le>4 + k + d + d. lose (t + j) = [False]) \<longrightarrow> vc (t + 4 + k + d + d) = [vc_com])"
coeff_int_def: fixes p :: "'b poly"   and i :: "int" shows "coeff_int p i = (if i < 0 then 0::'b else coeff p (nat i))"
amin_mem_i: fixes x :: "ant"   and y :: "ant" assumes "x \<in> Z\<^sub>\<infinity>"   and "y \<in> Z\<^sub>\<infinity>" shows "amin x y \<in> Z\<^sub>\<infinity>"
subst_mset_Var: fixes M :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" shows "subst_mset Var M = M"
bind_returns_heap_E: fixes h :: "'m"   and f :: "('m, 'n, 'p) prog"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and h'' :: "'m" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h''" obtains x :: "'p" and h' :: "'m" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>h h''"