hcju/isabelleps huggingface dataset:
B_authenticates_A: fixes servK :: "nat"   and A :: "agent"   and T3 :: "nat"   and evs :: "event list"   and B :: "agent"   and Ts :: "nat" assumes "Crypt servK \<lbrace>Agent A, Number T3\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Key servK \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbIV" shows "Says A B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>, Crypt servK \<lbrace>Agent A, Number T3\<rbrace>\<rbrace> \<in> set evs"
tame10ub_def: fixes g :: "graph" shows "tame10ub g = (countVertices g \<le> 15)"
IMG_def: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "IMG f = MkIde (Map f ` Dom f)"
upd_set'_keys: fixes m :: "('a, 'b) mapping"   and d :: "'b"   and f :: "'b \<Rightarrow> 'b"   and X :: "'a set" shows "Mapping.keys (upd_set' m d f X) = Mapping.keys m \<union> X"
set_sel_2: fixes a :: "'b list"   and x :: "'b" assumes "a \<noteq> \<epsilon>"   and "x \<in> set (tl a)" shows "x \<in> set a"
tau_eqvt: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and \<tau> :: "'c"   and p :: "perm" assumes "weak_nominal_ts satisfies transition \<tau>" shows "p \<bullet> \<tau> = \<tau>"
sgdpr_def: shows "sgdpr \<equiv> {x. \<not> global_policy' x ''Eve''}"
RedSeq: fixes v :: "val"   and e\<^sub>2 :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>Val v;; e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e\<^sub>2,s,b\<rangle>"
mono_gamma_f: fixes f :: "char list \<Rightarrow> 'av"   and g :: "char list \<Rightarrow> 'av" assumes "f \<sqsubseteq> g" shows "\<gamma>\<^sub>f f \<subseteq> \<gamma>\<^sub>f g"
euclidean_product_topology: shows "product_topology (\<lambda>i. euclidean) UNIV = euclidean"
continuous_image_subset_rel_interior: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "inj_on f S"   and "f ` S \<subseteq> T"   and "aff_dim T \<le> aff_dim S" shows "f ` rel_interior S \<subseteq> rel_interior (f ` S)"
slp_of_fas_def: fixes fas :: "floatarith list" shows "slp_of_fas fas = (let (M, slp) = slp_of_fas' fas Mapping.empty []; fasi = map ((the \<circ>\<circ> Mapping.lookup) M) fas in Let (map2 (\<lambda>a b. floatarith.Var (length slp + a - Suc b)) [0..<length fasi] (rev fasi)) ((@) slp))"
sigma_sets_vimage_commute: fixes X :: "'c \<Rightarrow> 'd"   and \<Omega> :: "'c set"   and \<Omega>' :: "'d set"   and M' :: "'d set set" assumes "X \<in> \<Omega> \<rightarrow> \<Omega>'" shows "{X -` A \<inter> \<Omega> |A. A \<in> sigma_sets \<Omega>' M'} = sigma_sets \<Omega> {X -` A \<inter> \<Omega> |A. A \<in> M'}"
sel_11: fixes x21 :: "'c"   and x22 :: "'a"   and x23 :: "'c"   and x24 :: "'a" shows "pac_mult (Mult x21 x22 x23 x24) = x22"
one_QuantT_def: shows "1 \<equiv> toQ \<lparr>mag = 1::'a, dim = QD('b), unit_sys = unit\<rparr>"
cross7_eq_0: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(x \<times>\<^sub>7 y = 0) = collinear {0, x, y}"
Bind_def: fixes g :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and ab :: "bool" shows "Bind g f ab \<equiv> \<lambda>P s. let a = g s in f a ab P s"
zip_map_map: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and g :: "'d \<Rightarrow> 'b"   and ys :: "'d list" shows "zip (map f xs) (map g ys) = map2 (\<lambda>x y. (f x, g y)) xs ys"
grow1_def: fixes K :: "nat" shows "grow1 K \<equiv> \<lambda>(l, r, buf). let b = op_array_replicate (length buf + K) default in mop_list_blit buf 0 b 0 l \<bind> (\<lambda>b. mop_list_blit buf r b (r + K) (length buf - r) \<bind> (\<lambda>b. RETURN (l, r + K, b)))"
assms_2: shows "A \<subseteq> Pow \<Omega>"
real_parts_3_def: fixes p :: "int poly" shows "real_parts_3 p = roots_of_3 (root_poly_Re p)"
changes_replicate_0: fixes n :: "nat" shows "changes (replicate n (0::'b)) = 0"
poly_condD_2: fixes p :: "int poly" assumes "poly_cond p" shows "0 < lead_coeff p"
set_awalk_verts: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "set (awalk_verts u p) = {u} \<union> set (map (tail G) p) \<union> set (map (head G) p)"
remove_child_locs_impl: shows "remove_child_locs = local.a_remove_child_locs"
divide_neg_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less x zero"   and "less y zero" shows "less zero (divide x y)"
enabled_iff: fixes s :: "'s"   and \<omega> :: "'s stream" shows "K.enabled s \<omega> = (K.enabled (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
eqm: fixes n :: "nat" shows "(\<Sum>i\<le>n. Sigma_Algebra.measure lebesgue (S i)) = Sigma_Algebra.measure lebesgue (\<Union> (S ` {..n}))"
igSwapIGVar_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSwapIGVar MOD \<equiv> \<forall>zs z1 z2 xs x. igSwap MOD zs z1 z2 (igVar MOD xs x) = igVar MOD xs (x @xs[z1 \<and> z2]_zs)"
Suc_Rep_not_Zero_Rep: fixes x :: "ind" shows "Suc_Rep x \<noteq> Zero_Rep"
nonpos_Ints_cases: fixes x :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0" obtains n :: "int" where "x = of_int n" and "n \<le> 0"
narrow1: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and join :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Top :: "'a"   and widen :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and narrow :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.WN le join Top widen narrow"   and "le y x" shows "le y (narrow x y)"
brec_invar_def: fixes Qi :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" shows "brec_invar Qi \<delta> \<equiv> brec_invar_add Qi \<inter> {s. brec_\<alpha> s \<in> brw_invar \<delta>}"
m_s_def: fixes S :: "'av st"   and X :: "char list set" shows "m\<^sub>s S X = (\<Sum>x\<in>X. m (fun S x))"
ends_in_segment_2: fixes b :: "'b"   and a :: "'b" shows "b \<in> closed_segment a b"
simps_1: fixes x :: "'b" shows "replicate 0 x = []"
rbt_lookup_rbt_minus: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" assumes "class.linorder less_eq less"   and "ord.is_rbt less t1"   and "ord.is_rbt less t2" shows "ord.rbt_lookup less (ord.rbt_minus less t1 t2) = ord.rbt_lookup less t1 |` (- dom (ord.rbt_lookup less t2))"
splitAt_dist_ram: fixes vs :: "'a list"   and a :: "'a list"   and ram :: "'a"   and b :: "'a list" assumes "distinct vs"   and "vs = a @ ram # b" shows "(a, b) = splitAt ram vs"
image_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "f ` A = {y. \<exists>x\<in>A. y = f x}"
assign: fixes e :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and v :: "'val"   and x :: "'id" assumes "E e m = v" shows "SRabbr (x:=e) m [] (m(x := v))"
prod_tuple: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "span f g"   and "seq h f"   and "seq k g" shows "CCC.prod h k \<cdot> CCC.tuple f g = CCC.tuple (h \<cdot> f) (k \<cdot> g)"
dropWhile_nth: fixes j :: "nat"   and P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "j < length (dropWhile P xs)" shows "dropWhile P xs ! j = xs ! (j + length (takeWhile P xs))"
FOREACH_rule: fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>" shows "FOREACH S f \<sigma>0 \<le> SPEC P"
assms: shows "run_one_step d i (s, vs, ves, Trap) = (s', vs', res)"
hyps: fixes j :: "int" assumes "nat \<bar>j\<bar> < nat \<bar>ja__\<bar>" shows "P j"
assms_2: shows "SubRel.subpath g' v es (tgt e) subs"
subtype_irrefl: fixes \<tau> :: "'a OCL_Types.type" assumes "\<tau> < \<tau>" shows "False"
octo_mult_cnj_commute: fixes x :: "octo" shows "Octonions.cnj x * x = x * Octonions.cnj x"
P2_def: shows "\<P>\<^sup>2 \<equiv> {\<lambda>x y. eval r [x, y] |r. recfn 2 r}"
fls_lr_inverse_one_1: fixes x :: "'a" shows "fls_left_inverse 1 x = fls_const x"
simps_2: fixes v :: "'b"   and va :: "'b list" shows "inter_sorted_rev (v # va) [] = []"
arcsin_lbound: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "- (pi / 2) \<le> arcsin y"
simple_function_indicator: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "simple_function M (indicator A)"
errMOD_igSubstIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igSubstIPresIGWls MOD"   and "igWlsDisj MOD" shows "igSubstIPresIGWlsSTR (errMOD MOD)"
disjSD_2_def: fixes x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'd) Combinators" shows "disjSD_2 x y = (\<forall>a b c d. (a, b) \<in> sdnets x \<and> (c, d) \<in> sdnets y \<longrightarrow> twoNetsDistinct a b c d \<and> twoNetsDistinct a b d c)"
fVectorSpace: shows "VectorSpace (\<sharp>\<cdot>) V"
dma_consistent: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and mem\<^sub>C :: "'e \<Rightarrow> 'd"   and x\<^sub>A :: "'c" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of" shows "dma\<^sub>A (\<lambda>x\<^sub>A. mem\<^sub>C (var\<^sub>C_of x\<^sub>A)) x\<^sub>A = dma\<^sub>C mem\<^sub>C (var\<^sub>C_of x\<^sub>A)"
shift_operator_ergodic_def: fixes M :: "'b measure" shows "shift_operator_ergodic M \<equiv> prob_space M"
set_sum_mem: fixes a :: "'a"   and I :: "'a set"   and b :: "'a"   and J :: "'a set" assumes "a \<in> I"   and "b \<in> J"   and "I \<subseteq> carrier R"   and "J \<subseteq> carrier R" shows "a \<plusminus> b \<in> I \<minusplus> J"
zip_rev: fixes xs :: "'a list"   and ys :: "'b list" assumes "length xs = length ys" shows "zip (rev xs) (rev ys) = rev (zip xs ys)"
card_map: fixes x :: "'a set" assumes "x \<in> X" shows "card (f ` x) = card x"
abs_summable_on_cmult_right: assumes "inverse c \<noteq> (0::'b) \<Longrightarrow> (\<lambda>x. c * f x) abs_summable_on A" shows "(\<lambda>x. inverse c * (c * f x)) abs_summable_on A"
\<alpha>'_ide: fixes f :: "'a arrow_of_spans_data"   and g :: "'a arrow_of_spans_data"   and h :: "'a arrow_of_spans_data" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<alpha>' (f, g, h) = \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc' (\<cdot>) prj0 prj1 f g h, Dom = Cod (f \<star> g \<star> h), Cod = Dom ((f \<star> g) \<star> h)\<rparr>"
cases: fixes a1 :: "'c DBMEntry"   and a2 :: "'c DBMEntry" assumes "a1 \<prec> a2" obtains uu_ :: "'c" where "a1 = Lt uu_" and "a2 = \<infinity>"   | uv_ :: "'c" where "a1 = Le uv_" and "a2 = \<infinity>"   | a :: "'c" and b :: "'c" where "a1 = Le a__" and "a2 = Le b__" and "a__ < b__"   | a :: "'c" and b :: "'c" where "a1 = Le aa__" and "a2 = Lt ba__" and "aa__ < ba__"   | a :: "'c" and b :: "'c" where "a1 = Lt ab__" and "a2 = Le bb__" and "ab__ \<le> bb__"   | a :: "'c" and b :: "'c" where "a1 = Lt a" and "a2 = Lt b" and "a < b"
simps_4: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "negation_free (And \<phi>\<^sub>1 \<phi>\<^sub>2) = (negation_free \<phi>\<^sub>1 \<and> negation_free \<phi>\<^sub>2)"
add_idem': fixes x :: "'a ppath set" shows "x \<union> x = x"
makeCArray_def: fixes arraySize :: "nat"   and fillValue :: "'a"   and index :: "nat" shows "makeCArray arraySize fillValue index \<equiv> if index < arraySize then fillValue else undefined"
not_eq_extremum: fixes a :: "'a list" shows "(a \<noteq> []) = strict_prefix [] a"
bl2nat_simps_1: fixes x :: "nat" shows "bl2nat (Oc # Oc \<up> x) 0 = 2 * 2 ^ x - Suc 0"
aux2: fixes M :: "trm"   and M' :: "trm"   and M'' :: "trm" assumes "M \<longrightarrow>\<^sub>l M'"   and "M' = M''" shows "M \<longrightarrow>\<^sub>l M''"
lepoll_cardinal_le: fixes A :: "V"   and i :: "V" assumes "elts A \<lesssim> elts i"   and "Ord i" shows "vcard A \<le> i"
thmD14: fixes k :: "nat"   and u :: "nat" assumes "k \<le> length Doc" shows "items_le k (\<J> k u) = Gen (paths_le k (\<P> k u)) \<and> \<T> k u = \<Z> k u \<and> items_le k (\<I> k) = Gen (paths_le k (\<Q> k))"
const_lam: fixes e :: "'e" shows "const (\<lambda>_. e)"
obj_simps_3: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.obj a" shows "Span.trg a = a"
gamma_smooth: shows "\<forall>x\<in>{0..1}. \<gamma> differentiable at x"
total: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "le x y \<or> le y x"
map_last_def: fixes f :: "'a \<Rightarrow> 'a"   and xs :: "'a list" shows "map_last f xs = (if xs = [] then [] else butlast xs @ [f (last xs)])"
exM0_def: shows "exM0 = tensor_P (1\<^sub>m N) M0"
Int_Un_eq_2: fixes S :: "'a set"   and T :: "'a set" shows "S \<inter> T \<union> T = T"
med1a1_def: fixes s :: "m1_state" shows "med1a1 s \<equiv> \<lparr>runs = runs1a1 (runs s), m1x_state.leak = Domain (m1_state.leak s)\<rparr>"
Posix1_1: fixes s :: "'b list"   and r :: "'b rexp"   and v :: "'b val" assumes "s \<in> r \<rightarrow> v" shows "s \<in> lang r"
undefined: fixes a :: "'a"   and b :: "'a" assumes "h a \<noteq> h b"   and "a \<in> A" shows "m a b = None"
clock_mkSuccs_def: fixes envObs :: "'b \<Rightarrow> 'c"   and obs :: "'c"   and Y' :: "'b odlist" shows "clock_mkSuccs envObs obs Y' \<equiv> (Y', ODList.filter (\<lambda>s. envObs s = obs) Y')"
simps_1: fixes R :: "('b \<times> 'b) set" shows "R ^^ 0 = Id"
geometric_sums: assumes "norm c < 1" shows "(^) c sums ((1::'a) / ((1::'a) - c))"
tautology_ptautology: fixes p :: "'a fm" assumes "tautology p" shows "ptautology (lift p)"
eq_TT_dest: fixes x :: "'d"   and y :: "'d" assumes "eq\<cdot>x\<cdot>y = TT" shows "x = y"
that: fixes s :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(red F)\<^sup>*\<^sup>* q1 s"   and "(red F)\<^sup>*\<^sup>* q2 s" shows "thesis"
part_fn_def: fixes r :: "nat"   and s :: "nat"   and Y :: "'b set"   and f :: "'b set \<Rightarrow> nat" shows "part_fn r s Y f = (f \<in> [Y]\<^bsup>r\<^esup> \<rightarrow> {..<s})"
DESTRimpl_rule2: fixes s :: "'s"   and ni :: "'ni" assumes "I s"   and "(ni, Falseif) \<in> R s" shows "ospec (DESTRimpl ni s) (\<lambda>r. r = FD)"
shift_append: fixes xs :: "'c list"   and ys :: "'c list"   and s :: "'c stream" shows "(xs @ ys) @- s = xs @- ys @- s"
simps_2: fixes y :: "name"   and T :: "trm"   and \<theta> :: "(name \<times> trm) list"   and x :: "name" shows "lookup ((y, T) # \<theta>) x = (if x = y then T else lookup \<theta> x)"
match_tcp_flags_conjunct_option_Some: fixes f1 :: "ipt_tcp_flags"   and f2 :: "ipt_tcp_flags"   and f3 :: "ipt_tcp_flags"   and pkt :: "tcp_flag set" assumes "match_tcp_flags_conjunct_option f1 f2 = Some f3" shows "(match_tcp_flags f1 pkt \<and> match_tcp_flags f2 pkt) = match_tcp_flags f3 pkt"
scaleR_eq_0_iff: fixes a :: "real"   and x :: "'d" shows "(a *\<^sub>R x = (0::'d)) = (a = 0 \<or> x = (0::'d))"
simps_12: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and y21 :: "'f"   and y22 :: "'f list" shows "\<not> list_all2 R [] (y21 # y22)"
Si_def: fixes t :: "real" shows "Si t \<equiv> LBINT x=0..ereal t. sin x / x"
p_max_eq: fixes a :: "real"   and v :: "real"   and s0 :: "real" shows "movement.p_max a v s0 = s0 - v\<^sup>2 / a / 2"
INF_less_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and 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 bot :: "'b"   and top :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set"   and a :: "'b" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf (f ` A)) a = (\<exists>x\<in>A. less (f x) a)"
prems_1: fixes c :: "'b" assumes "c \<in> ideal B"   and "(\<Prod>i\<in>insert j I. (f i + g i) ^ m i) = (\<Prod>i\<in>insert j I. f i ^ m i) + c" shows "thesis"
no_overlaps_defeq: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and fe :: "('a, 'c) flow_entry_match list"   and p :: "'b" assumes "no_overlaps \<gamma> fe" shows "OF_same_priority_match2 \<gamma> fe p = OF_priority_match \<gamma> fe p"
cond_spmf_fst_eq_return_None: fixes p :: "('b \<times> 'a) spmf"   and x :: "'b" shows "(cond_spmf_fst p x = return_pmf None) = (x \<notin> fst ` set_spmf p)"
has_contour_integral_trivial: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex" shows "(f has_contour_integral 0) (linepath a a)"
p12_eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> T"   and "y \<in> T'" shows "p1 (x + y) = x \<and> p2 (x + y) = y"
add_right_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "(y + x = z + x) = (y = z)"
IH: fixes y :: "V" assumes "y \<in> elts \<alpha>" obtains \<alpha>s :: "V list" and ms :: "nat list" where "list.set \<alpha>s \<subseteq> ON" and "list.set ms \<subseteq> {0<..}" and "length \<alpha>s = length ms" and "sorted_wrt (\<lambda>x y. y < x) \<alpha>s" and "y = Cantor_sum \<alpha>s ms"
simplicial_chain_sum: fixes I :: "'c set"   and p :: "nat"   and S :: "('d \<Rightarrow> real) set"   and f :: "'c \<Rightarrow> ((nat \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real) \<Rightarrow>\<^sub>0 'e" assumes "\<And>i. i \<in> I \<Longrightarrow> simplicial_chain p S (f i)" shows "simplicial_chain p S (sum f I)"
0: shows "f2 * (real l0 * min e c) \<le> e * (f2 * real l0)"
slide_inj: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "inj_on (slide i) {k. k \<le> j - i}"
point_rep_number_alt_def: fixes x :: "'a" shows "\<B> rep x = int (size (filter_mset ((\<in>) x) \<B>))"
word_bw_comms_1: fixes x :: "'a word"   and y :: "'a word" shows "x AND y = y AND x"
density_context_axioms: shows "density_context V V' \<Gamma> (\<lambda>\<sigma>. \<delta> \<sigma> * (if P' \<sigma> then 1 else 0))"
Rel: fixes P :: "pi"   and Q :: "pi"   and x :: "name" assumes "(P, Q) \<in> Rel" shows "(<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"
intersect'_nodes_card: fixes A :: "('d, 'c) dba"   and B :: "('d, 'e) dba" assumes "finite (DBA.nodes A)"   and "finite (DBA.nodes B)" shows "card (DGBA.nodes (intersect' A B)) \<le> card (DBA.nodes A) * card (DBA.nodes B)"
sel_2: fixes x2 :: "'b" shows "atm_of (Neg x2) = x2"
DERIV_add: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'b"   and E :: "'b" assumes "(f has_field_derivative D) (at x within s)"   and "(g has_field_derivative E) (at x within s)" shows "((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
sum_list_simps_2: fixes x :: "'c"   and xs :: "'c list" shows "sum_list (x # xs) = x + sum_list xs"
cmod_Re_le_iff: fixes x :: "complex"   and y :: "complex" assumes "Im x = Im y" shows "(cmod x \<le> cmod y) = (\<bar>Re x\<bar> \<le> \<bar>Re y\<bar>)"
SHOmsgVectors_HO: fixes A :: "('a, 'c, 'b, 'd) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'a"   and cfg :: "'a \<Rightarrow> 'c"   and HO :: "'a set" shows "SHOmsgVectors A r p cfg HO HO = {HOrcvdMsgs A r p HO cfg}"
cases: fixes x :: "('e, 'b, 'c, 'f) SLin_state_scheme \<times> ('a, 'b, 'c, 'g) SLin_state_scheme" obtains s1 :: "('e, 'b, 'c, 'f) SLin_state_scheme" and s2 :: "('a, 'b, 'c, 'g) SLin_state_scheme" where "x = (s1, s2)"
entropy_simple_distributed: fixes X :: "'key \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real" assumes "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) X f" shows "K.entropy_Pow X = - (\<Sum>x\<in>X ` space (point_measure keys (\<lambda>x. ennreal (K x))). f x * log b (f x))"
inject: fixes x1 :: "symbol \<times> symbol list"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat"   and y1 :: "symbol \<times> symbol list"   and y2 :: "nat"   and y3 :: "nat"   and y4 :: "nat" shows "(Item x1 x2 x3 x4 = Item y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
compile_heads: shows "fst |`| fset_of_list (Rewriting_Sterm.compile rs) = fst |`| rs"
UseCertVerify_lemma: fixes A :: "agent"   and nb :: "msg"   and B :: "agent"   and PMS :: "nat"   and evs :: "event list" assumes "Crypt (priEK A) (Hash \<lbrace>nb, Agent B, Nonce PMS\<rbrace>) \<in> parts (knows Spy evs)"   and "evs \<in> tls"   and "A \<notin> bad" shows "Notes A \<lbrace>Agent B, Nonce PMS\<rbrace> \<in> set evs"
inv_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "C.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>C.inv f : b \<Rightarrow>\<^sub>C a\<guillemotright>"
term_path_stable: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<pi> i = IFC_def.return prob"   and "i \<le> j" shows "\<pi> j = IFC_def.return prob"
npesub_def: fixes x :: "pexpr"   and y :: "pexpr" shows "npesub x y = (case y of PExpr1 (PCnst d) \<Rightarrow> if d = 0 then x else case x of PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (c - d)) | PExpr1 _ \<Rightarrow> PExpr1 (PSub x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PSub x y) | PExpr1 _ \<Rightarrow> case x of PExpr1 (PCnst c) \<Rightarrow> if c = 0 then PExpr1 (PNeg y) else PExpr1 (PSub x y) | PExpr1 _ \<Rightarrow> PExpr1 (PSub x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PSub x y) | PExpr2 pexpr2 \<Rightarrow> case x of PExpr1 (PCnst c) \<Rightarrow> if c = 0 then PExpr1 (PNeg y) else PExpr1 (PSub x y) | PExpr1 _ \<Rightarrow> PExpr1 (PSub x y) | PExpr2 pexpr2 \<Rightarrow> PExpr1 (PSub x y))"
assms: shows "run_one_step d i (s, vs, ves, Label ln les es) = (s', vs', RSBreak n res)"
if_split: fixes P :: "'d \<Rightarrow> bool"   and Q :: "bool"   and x :: "'d"   and y :: "'d" shows "P (if Q then x else y) = ((Q \<longrightarrow> P x) \<and> (\<not> Q \<longrightarrow> P y))"
reads_consistent_flush_other: fixes sb :: "'b memref list"   and xs :: "'c memref list"   and pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "outstanding_refs (Not \<circ> is_volatile_Read\<^sub>s\<^sub>b) xs \<inter> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "reads_consistent pending_write \<O> m xs" shows "reads_consistent pending_write \<O> (flush sb m) xs"
image_Nset_Suc: fixes f :: "nat \<Rightarrow> 'h"   and n :: "nat" shows "f ` {j. j \<le> Suc n} = insert (f (Suc n)) (f ` {j. j \<le> n})"