hcju/isabelleps huggingface dataset:
sigma_finite_up_in_pair_measure_generator: shows "\<exists>F. range F \<subseteq> {A \<times> B |A B. A \<in> sets M1 \<and> B \<in> sets M2} \<and> incseq F \<and> \<Union> (range F) = space M1 \<times> space M2 \<and> (\<forall>i. emeasure (M1 \<Otimes>\<^sub>M M2) (F i) \<noteq> \<infinity>)"
space_F: fixes M :: "'b measure"   and i :: "enat" shows "space (stream_filtration M i) = space (stream_space M)"
nsets_def: fixes A :: "'c set"   and n :: "nat" shows "[A]\<^bsup>n\<^esup> \<equiv> {N. N \<subseteq> A \<and> finite N \<and> card N = n}"
simps_5: fixes fv :: "'b \<Rightarrow> 'a set"   and r :: "'b Regex.regex" shows "Regex.fv_regex fv (regex.Star r) = Regex.fv_regex fv r"
inverse_def: shows "inv.inverse = (\<lambda>u\<in>inverse ` H. THE v. v \<in> inverse ` H \<and> u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
ssafe_def: fixes \<sigma> :: "('sf, 'sc, 'sz) subst" shows "ssafe \<sigma> \<equiv> (\<forall>i f'. SFunctions \<sigma> i = Some f' \<longrightarrow> dfree f') \<and> (\<forall>f f'. SPredicates \<sigma> f = Some f' \<longrightarrow> fsafe f') \<and> (\<forall>f f'. SPrograms \<sigma> f = Some f' \<longrightarrow> hpsafe f') \<and> (\<forall>f f'. SODEs \<sigma> f = Some f' \<longrightarrow> osafe f') \<and> (\<forall>C C'. SContexts \<sigma> C = Some C' \<longrightarrow> fsafe C')"
simps_2: fixes Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Length :: "'b \<Rightarrow> nat"   and satisfies0 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'b"   and a :: "'d" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> (FBase a) = satisfies0 \<AA> a"
run_step_basic_unop_testop_sound: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and b_e :: "b_e"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and t :: "t"   and iop :: "unop_i"   and fop :: "unop_f"   and testop :: "testop" assumes "run_one_step d i (s, vs, ves, $b_e) = (s', vs', RSNormal es')"   and "b_e = Unop_i t iop \<or> b_e = Unop_f t fop \<or> b_e = Testop t testop" shows "\<lparr>s;vs;vs_to_es ves @ [$b_e]\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"
maybe_or_true: fixes x :: "trilean"   and y :: "trilean" shows "(x \<or>? y = true) = ((x = true \<or> y = true) \<and> x \<noteq> invalid \<and> y \<noteq> invalid)"
elementary_cartesian_category_axioms: shows "elementary_cartesian_category (\<cdot>) pr0 pr1 \<one> trm"
induct: fixes P :: "'b hp \<Rightarrow> 'b hp \<Rightarrow> bool"   and a0 :: "'b hp"   and a1 :: "'b hp" assumes "\<And>x lx y ly. P (Hp x lx) (Hp y ly)" shows "P a0 a1"
reachable_def: fixes as :: "'a list"   and s :: "'c" shows "reachable as s = snd (the (rtrancl_while (\<lambda>_. True) (\<lambda>s. map (\<lambda>a. delta a s) as) s))"
sum_symp: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "symp R1"   and "symp R2" shows "symp (rel_sum R1 R2)"
fds_of_real_higher_deriv: fixes n :: "nat"   and f :: "real fds" shows "(fds_deriv ^^ n) (fds_of_real f) = fds_of_real ((fds_deriv ^^ n) f)"
S12l: fixes x :: "'a list set" shows "1 + x \<cdot> x\<^sup>\<star> = x\<^sup>\<star>"
cos_two_neq_zero: shows "cos 2 \<noteq> 0"
h1c: shows "((\<Sum>i = 0..<deg + 1. f i / D ^ i) \<le> 0) = ((\<Sum>i = 0..<deg + 1. f i * D ^ (deg - i)) * D ^ (deg mod 2) \<le> 0)"
l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf get_tag_name get_tag_name_locs"
wilson_theorem1: fixes p :: "nat" assumes "residues_prime p"   and "2 < p" shows "[fact (p - 1) = - 1] (mod int p)"
assms_2: shows "y \<in> set (independent_pdevs xs)"
e_cex: shows "\<not> INTERP N \<Turnstile> E"
Un_least: fixes A :: "'b set"   and C :: "'b set"   and B :: "'b set" assumes "A \<subseteq> C"   and "B \<subseteq> C" shows "A \<union> B \<subseteq> C"
LFP_greatest: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> carrier L"   and "\<And>u. \<lbrakk>u \<in> carrier L; f u \<sqsubseteq> u\<rbrakk> \<Longrightarrow> x \<sqsubseteq> u" shows "x \<sqsubseteq> LFP f"
that: fixes ys :: "'a list"   and x :: "'a"   and zs :: "'a list" assumes "xs = ys @ x # zs"   and "P x"   and "\<forall>z\<in>set zs. \<not> P z" shows "thesis"
simps_3: fixes x :: "'c"   and xs :: "'c list"   and oid :: "'c"   and ref :: "'c" shows "insert_spec (x # xs) (oid, Some ref) = (if x = ref then x # oid # xs else x # insert_spec xs (oid, Some ref))"
frange_fconst_on_ne: fixes A :: "V"   and c :: "V" assumes "A \<noteq> []\<^sub>\<circ>" shows "\<R>\<^sub>\<bullet> (fconst_on A c) = set {c}"
intros_1: fixes M :: "nat" shows "shared_weight_net (Input M)"
take_eq_map_nth: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "take n xs = map ((!) xs) [0..<n]"
sorted_mid_iff2: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "Sorted_Less.sorted (x # xs @ y # ys) = (Sorted_Less.sorted (x # xs) \<and> x < y \<and> Sorted_Less.sorted (xs @ [y]) \<and> Sorted_Less.sorted (y # ys))"
splitAt_distinct_a: fixes vs :: "'a list"   and a :: "'a list"   and b :: "'a list"   and ram :: "'a" assumes "distinct vs"   and "(a, b) = splitAt ram vs" shows "distinct a"
supp_aezfun1: shows "supp (aezfun 1) = 0"
x: shows "(\<lambda>n. exp (sum p {..<n})) \<longlonglongrightarrow> exp x"
mthd_is_static_simp: fixes m :: "'c \<times> 'b" shows "is_static (mthd m) = is_static m"
asset_i_unique: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, Crypt (Auth_ShaKey m) (PriKey A)) \<in> s"   and "(Asset n, Crypt (Auth_ShaKey n) (PriKey A)) \<in> s" shows "m = n"
stream_filtration_def: fixes M :: "'b measure"   and n :: "enat" shows "stream_filtration M n = (\<Squnion>i\<in>{x. enat x \<le> n}. vimage_algebra (streams (space M)) (\<lambda>\<omega>. \<omega> !! i) M)"
is_method_def: fixes membr :: "qtname \<times> memberdecl" shows "is_method membr = (\<exists>declC m. membr = (declC, mdecl m))"
type_less_x_Real_intro_1: fixes \<tau> :: "'a basic_type" assumes "\<tau> = UnlimitedNatural" shows "\<tau> < Real"
induct: fixes P :: "modality \<Rightarrow> safety \<Rightarrow> 'c regex \<Rightarrow> bool"   and a0 :: "modality"   and a1 :: "safety"   and a2 :: "'c regex" assumes "\<And>m uu_ n. P m uu_ (Skip n)"   and "\<And>m g \<phi>. P m g (Test \<phi>)"   and "\<And>m g r s. \<lbrakk>P m g r; P m g s\<rbrakk> \<Longrightarrow> P m g (Plus r s)"   and "\<And>g r s. \<lbrakk>P Futu g s; P Futu Lax r\<rbrakk> \<Longrightarrow> P Futu g (Times r s)"   and "\<And>g r s. \<lbrakk>P Past g r; P Past Lax s\<rbrakk> \<Longrightarrow> P Past g (Times r s)"   and "\<And>m g r. P m g r \<Longrightarrow> P m g (Star r)" shows "P a0 a1 a2"
m1_inv4_rfreshD: fixes x :: "'a m1_state_scheme"   and Rb :: "fid_t"   and A :: "agent"   and B :: "agent"   and Kab :: "key"   and Rb' :: "fid_t"   and A' :: "agent"   and B' :: "agent" assumes "runs x Rb = Some (Resp, [A, B], [aKey Kab, END])"   and "runs x Rb' = Some (Resp, [A', B'], [aKey Kab, END])"   and "B \<notin> bad"   and "A \<notin> bad"   and "Kab \<notin> Domain (m1_state.leak x)"   and "x \<in> m1_inv4_rfresh" shows "Rb = Rb'"
CAS1Null: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and v' :: "'addr val"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<bullet>compareAndSwap(D\<bullet>F, Val v, Val v'),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
assms_1: shows "h' = Ref.set r (f (get h r)) h"
norm_mult_vec: fixes x :: "'a" shows "norm (x \<bullet> x) = norm x * norm x"
C\<sigma>'_1: shows "C \<cdot> \<sigma>' = C\<sigma>'"
borel_measurable_stake: fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and w :: "bool stream"   and y :: "bool stream" assumes "f \<in> borel_measurable (nat_filtration n)"   and "stake n w = stake n y" shows "f w = f y"
simps_1: fixes f :: "('c \<Rightarrow> 'b) stream"   and x :: "'c stream" shows "shd (Applicative.ap f x) = shd f (shd x)"
maps_def: fixes f :: "'c \<Rightarrow> 'b list"   and xs :: "'c list" shows "List.maps f xs = concat (map f xs)"
wens_unfold: fixes F :: "'b program"   and act :: "('b \<times> 'b) set"   and B :: "'b set" shows "wens F act B = wp act B \<inter> awp F (B \<union> wens F act B) \<union> B"
inf_SUP: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and B :: "'b set" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf a (Sup (f ` B)) = Sup ((\<lambda>b. inf a (f b)) ` B)"
kleene_star_def: fixes f :: "'b \<Rightarrow> 'b set"   and s :: "'b" shows "(f\<^sup>*) s = \<mu> {kpower f n s |n. n \<in> UNIV}"
restrict_def: fixes f :: "'a \<Rightarrow> 'b"   and U :: "'a set" shows "f \<restriction> U = (\<lambda>n. if n \<in> U then f n else undefined)"
cm: fixes j :: "nat" shows "continuous_map (powertop_real UNIV) euclideanreal (\<lambda>x. if j = k then - x j else x j)"
Rep_update: fixes x :: "'b update" shows "Rep_update x \<in> update"
elims: fixes x :: "'b \<Rightarrow> 'a set"   and xa :: "'b option"   and y :: "'a set" assumes "\<gamma>_option x xa = y" obtains \<gamma> :: "'b \<Rightarrow> 'a set" where "x = \<gamma>'__" and "xa = None" and "y = {}"   | \<gamma> :: "'b \<Rightarrow> 'a set" and a :: "'b" where "x = \<gamma>" and "xa = Some a" and "y = \<gamma> a"
pred_BIT_simps_1: fixes x :: "int" shows "x BIT False - 1 = (x - 1) BIT True"
A: shows "TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel)"
intro: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah" assumes "category A"   and "category B"   and "constant_functor A B (partial_magma.dom B g)"   and "constant_functor A B (partial_magma.cod B g)"   and "constant_transformation_axioms B g" shows "constant_transformation A B g"
neg_def: fixes \<phi> :: "fmla" shows "Instance.neg \<phi> = \<phi> IMP fls"
vimage_algebra_vimage_algebra_eq: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and Y :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and M :: "'d measure" assumes "f \<in> X \<rightarrow> Y"   and "g \<in> Y \<rightarrow> space M" shows "vimage_algebra X f (vimage_algebra Y g M) = vimage_algebra X (\<lambda>x. g (f x)) M"
proj_finite: fixes t :: "cnf llist"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b" assumes "lfinite t" shows "lfinite (dynamic_component.proj tCMP active c t)"
ext: fixes g :: "'a"   and f :: "'a" assumes "g \<cdot>\<^sub>B f \<noteq> B.null" shows "B.seq g f"
Rep_equal: fixes x :: "'a equal" shows "Rep_equal x \<in> {(=)}"
that: shows "inj_on f (insert x (fst ` set_spmf p))"
tensor0_prod_right: fixes A :: "'b tensor"   and ds :: "nat list" shows "A \<otimes> tensor0 ds = tensor0 (dims A @ ds)"
tendsto_mult_right_iff: fixes c :: "'b"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "c \<noteq> (0::'b)" shows "((\<lambda>x. f x * c) \<longlongrightarrow> l * c) F = (f \<longlongrightarrow> l) F"
unit_factor_mult_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "times (unit_factor a) (normalize a) = a"
ord_option_simps_1: fixes ord :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "ord_option ord None x = True"
complete_empty: fixes CC :: "'b set set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "CC-complete {} r = ({} \<notin> CC)"
Im3_Hv: fixes v :: "(real, 3) vec" shows "Im3 (Hv v) = v $ 3"
get_nth_im1: fixes i :: "nat"   and ba :: "'a list \<times> 'a list"   and xs :: "'a list" assumes "list_repr i ba xs"   and "i \<noteq> 0" shows "get_nth_im1 ba = xs ! (i - 1)"
algebraic_self: fixes K :: "'a set"   and k :: "'a" assumes "subring K R"   and "k \<in> K" shows "(algebraic over K) k"
right_idem: fixes a :: "'b"   and b :: "'b" shows "inf (inf a b) b = inf a b"
pos: fixes x :: "real" assumes "x \<in> S" shows "0 < x"
assms_3: fixes X :: "'a set" assumes "X \<in> A" shows "open X \<and> connected X \<and> X \<noteq> {}"
funion_finter_assoc_eq: fixes A :: "'b fset"   and B :: "'b fset"   and C :: "'b fset" shows "(A |\<inter>| B |\<union>| C = A |\<inter>| (B |\<union>| C)) = (C |\<subseteq>| A)"
decomposeLS_basic_com: fixes c' :: "('g, 'h, 'i, 'j) com"   and ctxts :: "(('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com) \<times> (('g, 'h, 'i, 'j) com \<Rightarrow> ('g, 'h, 'i, 'j) com list)"   and s :: "('g, 'h, 'i, 'j) com list \<times> 'h option \<times> 'j" assumes "(c', ctxts) \<in> decomposeLS s" shows "basic_com c'"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
matroid_axioms_def: fixes indep :: "'b set \<Rightarrow> bool" shows "matroid_axioms indep \<equiv> \<forall>X Y. indep X \<longrightarrow> indep Y \<longrightarrow> card X = Suc (card Y) \<longrightarrow> (\<exists>x\<in>X - Y. indep (insert x Y))"
less_asym': fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g"   and P :: "bool" assumes "class.preorder less_eq less"   and "less a b"   and "less b a" shows "P"
y_le: shows "Sup (integral\<^sup>N M ` G) \<le> emeasure N (space M)"
segment_bound_lemma: fixes B :: "real"   and x :: "real"   and y :: "real"   and u :: "real" assumes "B \<le> x"   and "B \<le> y"   and "0 \<le> u"   and "u \<le> 1" shows "B \<le> (1 - u) * x + u * y"
below_Lin: fixes x :: "alt"   and y :: "alt"   and L :: "(alt \<times> alt) set" assumes "x \<noteq> y"   and "L \<in> Lin" shows "below L x y \<in> Lin"
is_lit_plus_cases: fixes xa :: "'b formula" shows "is_lit_plus xa = (case xa of Atom xda \<Rightarrow> True | \<bottom> \<Rightarrow> True | \<^bold>\<not> (Atom xea) \<Rightarrow> True | \<^bold>\<not> \<bottom> \<Rightarrow> True | \<^bold>\<not> _ \<Rightarrow> False | _ \<Rightarrow> False)"
d_OUT_def: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" shows "d_OUT g x = (\<Sum>\<^sup>+ y. g (x, y))"
module_hom_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 "module_hom_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 f \<equiv> module_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 \<and> module_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 \<and> module_hom_on_axioms U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 f"
inv_solve_left: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier C"   and "b \<in> carrier C"   and "c \<in> carrier C" shows "(a = inv\<^bsub>C\<^esub> b \<otimes>\<^bsub>C\<^esub> c) = (c = b \<otimes>\<^bsub>C\<^esub> a)"
a_3_var: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad y \<cdot> (x + y) = (zero_class.zero::'a)"
r_next_def: shows "r_next \<equiv> Cn 1 r_ifeq [Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], r_pdec1], Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], Cn 1 r_take [r_pdec2, r_pdec1]], Cn 1 r_ifeq [Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], Cn 1 r_update [r_pdec1, r_pdec2, r_const 1]], Cn 1 r_sigma [Cn 1 r_hd [r_pdec1], Cn 1 r_take [r_pdec2, r_pdec1]], Cn 1 r_prod_encode [Cn 1 r_snoc [r_pdec1, Z], r_pdec2], Cn 1 r_prod_encode [Cn 1 r_snoc [Cn 1 r_update [r_pdec1, r_pdec2, r_const 1], Z], Cn 1 r_length [r_pdec1]]], Cn 1 r_prod_encode [Cn 1 r_snoc [r_pdec1, Z], Cn 1 r_length [r_pdec1]]]"
assign_load_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and y :: "Eg1.var"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Load y) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 y)) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "y \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> \<C>"   and "y \<notin> \<C>"   and "\<forall>mds\<^sub>A'' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A'' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A'', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A'', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
funion_fempty: fixes A :: "'a fset"   and B :: "'a fset" shows "(A |\<union>| B = {||}) = (A = {||} \<and> B = {||})"
ord_less_eq_trans: fixes a :: "'t"   and b :: "'t"   and c :: "'t" assumes "a \<prec>\<^sub>t b"   and "b = c" shows "a \<prec>\<^sub>t c"
new_document_disconnected_nodes: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "h' \<turnstile> get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t new_document_ptr disconnected_nodes \<rightarrow>\<^sub>r []"
is_root_def: fixes p :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and x :: "('a list \<times> nat) multiset \<Rightarrow> 'a" shows "M.is_root p x = (x \<in> carrier M \<and> M.eval p x = \<zero>\<^bsub>M\<^esub> \<and> p \<noteq> [])"
delay_2: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "\<mu>_stable \<phi> w"   and "w \<Turnstile>\<^sub>n \<phi>" shows "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[{\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)} \<inter> restricted_subformulas \<phi>]\<^sub>\<nu>"
acc_run_reachable: fixes r :: "nat \<Rightarrow> 'Q" assumes "is_acc_run r" shows "range r \<subseteq> reachable"
Ri_limit_Sup: fixes Ns :: "'a literal multiset set llist" assumes "chain (\<triangleright>) Ns" shows "Ri (Liminf_llist Ns) = Ri (Sup_llist Ns)"
assms: shows "s' = snd (fst (load_sub3 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0"
proots_count_union_disjoint: fixes A :: "'a set"   and B :: "'a set"   and p :: "'a poly" assumes "A \<inter> B = {}"   and "p \<noteq> 0" shows "proots_count p (A \<union> B) = proots_count p A + proots_count p B"
mult_sum4sq_int: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int"   and p :: "int"   and q :: "int"   and r :: "int"   and s :: "int" shows "sum4sq_int (a, b, c, d) * sum4sq_int (p, q, r, s) = sum4sq_int (a * p + b * q + c * r + d * s, a * q - b * p - c * s + d * r, a * r + b * s - c * p - d * q, a * s - b * r + c * q - d * p)"
fwi_out_of_bounds1: fixes n :: "nat"   and i' :: "nat"   and i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and j :: "nat"   and j' :: "nat" assumes "n < i'"   and "i \<le> n" shows "fwi M n k i j i' j' = M i' j'"
trans: fixes a :: "'c \<times> 'd"   and b :: "'c \<times> 'd"   and c :: "'c \<times> 'd" assumes "less_eq_prod a b"   and "less_eq_prod b c" shows "less_eq_prod a c"
m2_2: shows "Polynomial.monom (1::'a) 1 = [:0::'a, 1::'a:]"
mdecl_visible: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list"   and S :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b" assumes "wf_prog wf_md P"   and "(C, S, fs, ms) \<in> set P"   and "(M, b, Ts, T, m) \<in> set ms" shows "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in C"
coeff_add: fixes R :: "('c, 'd) ring_scheme"   and p :: "nat \<Rightarrow> 'c"   and q :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "coeff (UP R) (p \<oplus>\<^bsub>UP R\<^esub> q) n = coeff (UP R) p n \<oplus>\<^bsub>R\<^esub> coeff (UP R) q n"
norm_pos1: fixes j :: "nat" assumes "j < m" shows "0 < \<parallel>fs.gs.gso j\<parallel>\<^sup>2"
assms_2: shows "m \<in> set (\<alpha>n g)"
integrable_on_subdivision: fixes \<D> :: "'a set set"   and i :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "\<D> division_of i"   and "f integrable_on S"   and "i \<subseteq> S" shows "f integrable_on i"
projection_split_last: fixes n :: "nat"   and \<tau> :: "'a list"   and X :: "'a set" assumes "Suc n = length (\<tau> \<upharpoonleft> X)" shows "\<exists>\<beta> x \<alpha>. x \<in> X \<and> \<tau> = \<beta> @ [x] @ \<alpha> \<and> \<alpha> \<upharpoonleft> X = [] \<and> n = length ((\<beta> @ \<alpha>) \<upharpoonleft> X)"
simple_resolution_sound: fixes F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and l\<^sub>1 :: "fterm literal"   and l\<^sub>2 :: "fterm literal" assumes "eval\<^sub>c F G C\<^sub>1"   and "eval\<^sub>c F G C\<^sub>2"   and "l\<^sub>1 \<in> C\<^sub>1"   and "l\<^sub>2 \<in> C\<^sub>2"   and "l\<^sub>1\<^sup>c = l\<^sub>2" shows "eval\<^sub>c F G (C\<^sub>1 - {l\<^sub>1} \<union> (C\<^sub>2 - {l\<^sub>2}))"
if_weak_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'c"   and y :: "'c" assumes "b = c" shows "(if b then x else y) = (if c then x else y)"
TestCommute: fixes s :: "'a" assumes "P s" shows "b s = b' (f s)"
map_closed: fixes a :: "'b" assumes "a \<in> A" shows "f a \<in> B"
intros_5: fixes iop :: "binop_i"   and c1 :: "i32"   and c2 :: "i32"   and c :: "i32" assumes "app_binop_i iop c1 c2 = Some c" shows "\<lparr>[$C ConstInt32 c1, $C ConstInt32 c2, $Binop_i T_i32 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 c]\<rparr>"
small_power: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and n :: "nat" assumes "f \<in> o[F](g)"   and "0 < n" shows "(\<lambda>x. f x ^ n) \<in> o[F](\<lambda>x. g x ^ n)"
mult_ac_1: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * b * c = a * (b * c)"