hcju/isabelleps huggingface dataset:
length_alist: fixes al :: "('c \<times> 'd) list"   and al' :: "('c \<times> 'd) list" assumes "distinct (map fst al)"   and "distinct (map fst al')"   and "set al = set al'" shows "length al = length al'"
degree_zero_imp_splitted: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "domain R"   and "p \<in> carrier (poly_ring R)"   and "degree p = 0" shows "ring.splitted R p"
IH_2: fixes n :: "nat" shows "cut_bound' n F2 \<le> cbnd n (depth F2)"
flowstoE: fixes X0 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and T :: "real set"   and CX :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and X1 :: "('d \<times> 'e \<Rightarrow>\<^sub>L 'd) set"   and x0 :: "'d"   and d0 :: "'e \<Rightarrow>\<^sub>L 'd" assumes "flowsto X0 T CX X1"   and "(x0, d0) \<in> X0" obtains h :: "real" where "h \<in> T" and "h \<in> existence_ivl0 x0" and "(flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> X1" and "\<And>h'. h' \<in> {0<--<h} \<Longrightarrow> (flow0 x0 h', Dflow x0 h' o\<^sub>L d0) \<in> CX"
downset_Sup_pres_var: fixes X :: "'b set" shows "\<down> (\<Squnion> X) = (\<down> \<circ> Sup) (\<Down> X)"
perp_col2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and Y :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B X Y"   and "C \<noteq> D"   and "Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.Col Bet A B D" shows "Tarski_neutral_dimensionless.Perp Bet Cong C D X Y"
trancl_into_rtrancl: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> trancl r" shows "(a, b) \<in> r\<^sup>*"
bind_is_OK_E: fixes h :: "'x"   and f :: "('x, 'y, 'aa) prog"   and g :: "'aa \<Rightarrow> ('x, 'y, 'z) prog" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g" obtains x :: "'aa" and h' :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
I_subst_peps2: fixes \<phi> :: "atom fm"   and xs :: "'a list"   and l :: "nat"   and x :: "'a" assumes "nqfree \<phi>"   and "xs ! l < x"   and "nolb \<phi> xs (xs ! l) x"   and "x \<notin> EQ \<phi> xs"   and "\<forall>y\<in>{xs ! l<..x}. DLO.I \<phi> (y # xs)" shows "DLO.I (subst\<^sub>+ \<phi> l) xs"
ipp_cond2_def: fixes z :: "'e"   and A :: "'e set"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "ipp_cond2 z A i f \<equiv> \<forall>x\<in>addition_set f (aug_pm_set z i A). \<forall>y\<in>addition_set f (aug_pm_set z i A). \<^sub>i- (x \<^sub>f+ y) = \<^sub>i- y \<^sub>f+ \<^sub>i- x"
cases: fixes x :: "'c list" obtains "xa__ = []"   | head :: "'c" where "xa__ = [head__]"   | head :: "'c" and x :: "'c" and xs :: "'c list" where "xa__ = head # x # xs"
columns_image_basis: fixes A :: "((real, 'c) vec, 'b) vec" shows "columns A = (*v) A ` range (\<lambda>i. axis i 1)"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vint_plus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vint_plus" shows "b = c"
hom_zero: shows "to_int_mod_ring 0 = 0"
prefix_imp_sublist: fixes xs :: "'c list"   and ys :: "'c list" assumes "prefix xs ys" shows "sublist xs ys"
intT_proj: fixes \<sigma> :: "'tp"   and a :: "univ" shows "Ik_intT \<sigma> (proj \<sigma> a)"
inf_le2: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.semilattice_inf inf less_eq less" shows "less_eq (inf x y) y"
sup_aci_1: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = y \<sqinter> x"
assms_4: shows "x' \<otimes> inv x \<in> H"
rel_resource_mono: fixes A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and B' :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "A' \<le> A"   and "B \<le> B'" shows "rel_resource A B \<le> rel_resource A' B'"
allDefs_disjoint: fixes n :: "'a"   and m :: "'a" assumes "n \<in> set (gen_wf_\<alpha>n gen_cfg_wf)"   and "m \<in> set (gen_wf_\<alpha>n gen_cfg_wf)"   and "n \<noteq> m" shows "gen_wf.braun_ssa.allDefs gen_cfg_wf n \<inter> gen_wf.braun_ssa.allDefs gen_cfg_wf m = {}"
igSwapIGAbs_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme" shows "igSwapIGAbs MOD \<equiv> \<forall>zs z1 z2 xs x s X. isInBar (xs, s) \<and> igWls MOD s X \<longrightarrow> igSwapAbs MOD zs z1 z2 (igAbs MOD xs x X) = igAbs MOD xs (x @xs[z1 \<and> z2]_zs) (igSwap MOD zs z1 z2 X)"
dSk: shows "ket_plus_k (Suc k) \<in> carrier_vec (2 ^ Suc (Suc k))"
Fvars_fm_simps_6: shows "Fvars Fls = {}"
intros_2: fixes x :: "'c"   and gens :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "x \<in> gens" shows "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
trace_parallel_plan_sas_plus_head_is_initial_state: fixes I :: "'c \<Rightarrow> 'd option"   and \<psi> :: "('c, 'd) sas_plus_operator list list" shows "trace_parallel_plan_sas_plus I \<psi> ! 0 = I"
spr_simAbs_fst_snd_trc: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and as :: "'a list" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t" shows "toSet (big_union (spr_commonKnowledge envObs (fst ec) as) (toList (snd ec))) \<subseteq> toSet (fst ec)"
take_bv_prepend2: fixes n :: "nat"   and b :: "'a"   and xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "take n (bv_prepend n b xs @ ys @ zs) = bv_prepend n b []"
simps_2: fixes lc :: "'a"   and q :: "'a poly"   and r :: "'a poly"   and d :: "'a poly"   and dr :: "nat" shows "local.divide_poly_main lc q r d dr 0 = q"
le_0_eq: fixes n :: "nat" shows "(n \<le> 0) = (n = 0)"
approx_minus: fixes a :: "'a star"   and b :: "'a star" assumes "a \<approx> b" shows "- a \<approx> - b"
Ndet_is_action: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "(action.relation_of P \<or> action.relation_of Q) \<in> {p. is_CSP_process p}"
quasisym_not_less: fixes a :: "'a"   and b :: "'a" assumes "cmp a b \<noteq> Less"   and "cmp b a \<noteq> Less" shows "cmp a b = Equiv"
Liminf_le: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and l :: "'d" assumes "F \<noteq> bot"   and "\<forall>\<^sub>F x in F. f x \<le> l" shows "Liminf F f \<le> l"
Expr2Mor: fixes A :: "'t"   and e :: "('t, 'f) Expression"   and B :: "'t"   and g :: "'m" assumes "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> IMor g" shows "g maps\<^bsub>iC\<^esub> Ty\<lbrakk>A\<rbrakk> to Ty\<lbrakk>B\<rbrakk>"
coset_eq: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "H \<lhd> G" shows "\<forall>x\<in>carrier G. H #>\<^bsub>G\<^esub> x = x <#\<^bsub>G\<^esub> H"
intro: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b"   and lambda :: "nat"   and mu :: "nat" assumes "fx0.properties f x0 lambda mu" shows "properties f x0 lambda mu"
asTerm_equal_iff_alpha: fixes qX :: "('e, 'f, 'varSort, 'var, 'g) qTerm"   and qY :: "('e, 'f, 'varSort, 'var, 'g) qTerm" assumes "qGood qX \<or> qGood qY" shows "(asTerm qX = asTerm qY) = (qX #= qY)"
assms_3: fixes z :: "'a" assumes "z \<in> S" shows "f z \<noteq> 0"
altern: shows "i = pmax \<or> i \<in> kmax f PR"
coboundedI2: fixes c :: "'e"   and b :: "'e"   and a :: "'e" assumes "c \<le> b" shows "c \<le> sup a b"
uminus_carrier_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "- A \<in> carrier_mat nr nc"
pred_spmf_def: fixes P :: "'d \<Rightarrow> bool"   and p :: "'d spmf" shows "pred_spmf P p = (\<forall>x\<in>set_spmf p. P x)"
ds_ord_1: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> y) \<sqsubseteq> d x"
square_integrable_approximate_continuous_ends: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and e :: "real" assumes "f square_integrable {a..b}"   and "a < b"   and "0 < e" obtains g :: "real \<Rightarrow> real" where "continuous_on {a..b} g" and "g b = g a" and "g square_integrable {a..b}" and "l2norm {a..b} (\<lambda>x. f x - g x) < e"
pnormal_degree: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times"   and "last p \<noteq> zero" shows "semiring_0.degree zero p = length p - 1"
density_RN_deriv: fixes N :: "'a measure" assumes "absolutely_continuous M N"   and "sets N = sets M" shows "density M (RN_deriv M N) = N"
inducts: fixes p :: "'a \<times> 'b"   and x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "fstsp p x"   and "P (fst p)" shows "P x"
ChiRel_ChiPlus_trans2: fixes U :: "'d"   and S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and T :: "'d" assumes "(U, S) \<in> ChiRel A"   and "(S, T) \<in> ChiPlus A" shows "(U, T) \<in> ChiPlus A"
clauseTautologyCharacterization: fixes clause :: "Literal list" shows "clauseTautology clause = (\<exists>literal. literal el clause \<and> opposite literal el clause)"
pow_eq_id: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier (mult_of R)" shows "(x [^] n = \<one>) = (G.ord x dvd n)"
multiplicity_dvd: fixes p :: "'a"   and x :: "'a" shows "p ^ multiplicity p x dvd x"
dummyx_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and I :: "nat set"   and d :: "nat" shows "(P,C,M \<triangleright> xt /I,d) = (P,C,M \<rhd> xt / I,d)"
Transset_eclose: fixes X :: "hf" shows "Transset (eclose X)"
IQ: shows "\<forall>l s. I l s \<and> bval b s \<longrightarrow> pre C I l s \<and> 1 + preT C E s + time C s \<le> E s \<and> S s = S (postQ C s) on \<Union> (funStar SS ` (LVE \<union> LVQ))"
mojmir_accept_iff_token_set_accept: shows "accept\<^sub>M = (\<exists>i<max_rank. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i))"
ps2_P_d0: shows "ps2_P.d0 = d"
FactRing_zeroideal_1: shows "K [X] Quot {\<zero>\<^bsub>K [X]\<^esub>} \<simeq> K [X]"
ne: shows "norm (Blinfun (blinop_apply J) - 1\<^sub>L + w) = norm (J - 1 + Blinop (blinfun_apply w))"
path_via: shows "P \<turnstile> Path last Cs to C via Cs'' "
lookup_forget: fixes \<theta> :: "(name \<times> trm) list"   and x :: "name" assumes "supp (map fst \<theta>) \<sharp>* x" shows "lookup \<theta> x = Var x"
simps_9: fixes e\<^sub>1 :: "'a exp"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (e\<^sub>1\<bullet>F{D} := e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
rOutput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and C :: "'d"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K" shows "Prop C \<Psi> (M\<langle>N\<rangle>.P) K [] N P"
put_eq: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and k :: "'d"   and u :: "'d"   and \<rho> :: "'e"   and v :: "'d" assumes "vwb_lens x"   and "get\<^bsub>x\<^esub> \<sigma> = k"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "put\<^bsub>x\<^esub> \<rho> k = \<sigma>"
borel_measurable_scaleR: fixes f :: "'a \<Rightarrow> real"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
fMin_finsert: fixes x :: "'a"   and A :: "'a fset" shows "fMin (finsert x A) = (if A = {||} then x else min x (fMin A))"
simps_2: fixes get_proc :: "'h \<Rightarrow> 'g"   and Main :: "'g" shows "lift_get_proc get_proc Main NewEntry = Main"
GoodWrite_def: fixes mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs"   and v :: "Vals" shows "GoodWrite mm rs p l v \<equiv> #l \<in> #MemLoc \<and> #v \<in> #MemVal \<and> (mm!l)$ = #v \<and> (rs!p)$ = #OK"
countable_finite: fixes S :: "'d set" assumes "finite S" shows "countable S"
restrict_val_invar: fixes a :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and b :: "'a ifex"   and var :: "'a"   and val :: "bool"   and ass :: "'a \<Rightarrow> bool" assumes "\<forall>ass. a ass = val_ifex b ass" shows "bf_restrict var val a ass = val_ifex (restrict b var val) ass"
assms_1: shows "FGModuleHom G (\<star>) W smult'' S"
size_del: fixes x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "del x t = Some t'" shows "size t' = size t - 1"
to_mregex_atms: fixes c :: "Formula.formula"   and ys :: "Formula.formula list"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex" assumes "c \<in> set ys"   and "to_mregex r = (mr, ys)" shows "c \<in> atms r"
exp_add_commuting: fixes x :: "'a"   and y :: "'a" assumes "x * y = y * x" shows "exp (x + y) = exp x * exp y"
nsg_sg: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<triangleright> H" shows "G \<guillemotright> H "
discounted_AE_cong: fixes X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and n :: "nat"   and Y :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and N :: "'b measure"   and r :: "real" assumes "AEeq N (X n) (Y n)" shows "AEeq N (discounted_value r X n) (discounted_value r Y n)"
fastprod_def: fixes xs :: "int list" shows "fastprod xs = catch (work xs) (return 0)"
Framer2: fixes C :: "com"   and ps1 :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps1' :: "char list \<Rightarrow> int option"   and ps2 :: "char list \<Rightarrow> int option"   and ps :: "char list \<Rightarrow> int option"   and ps' :: "char list \<Rightarrow> int option" assumes "(C, ps1) \<Rightarrow>\<^sub>A m \<Down> ps1'"   and "ps1 ## ps2"   and "ps = ps1 + ps2"   and "ps' = ps1' + ps2" shows "(C, ps) \<Rightarrow>\<^sub>A m \<Down> ps'"
category_axioms: shows "category (\<cdot>\<^sub>B\<^sub>1\<^sub>x\<^sub>B\<^sub>2)"
pr_conv_3_to_2_def: shows "pr_conv_3_to_2 = (\<lambda>f. pr_conv_1_to_2 (pr_conv_3_to_1 f))"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a * b * c = a * (b * c)"
2: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "poly_subst (\<lambda>x. if x \<in> X then monomial (1::'a) (monomial 1 x) else 1) p \<in> P[X]"
brw_invar_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "brw_invar \<delta> \<equiv> brw_invar_add \<delta> \<inter> {s. brw_\<alpha> s \<in> br'_invar \<delta>}"
fromMaybe_def: fixes a :: "'a"   and m :: "'a option" shows "fromMaybe a m = (case m of None \<Rightarrow> a | Some a \<Rightarrow> a)"
bind_split: fixes P :: "'d option \<Rightarrow> bool"   and m :: "'e option"   and f :: "'e \<Rightarrow> 'd option" shows "P (m \<bind> f) = ((m = None \<longrightarrow> P None) \<and> (\<forall>v. m = Some v \<longrightarrow> P (f v)))"
valid_paths: shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. valid_path \<gamma>"
is_Yield_def: fixes step :: "('e, 'f) step" shows "is_Yield step = (\<exists>x31 x32. step = Yield x31 x32)"
sys_initial_state_def: fixes s :: "('a, 'b, 'c, 'd) Model.local_state" shows "sys_initial_state s = ((\<forall>m. \<not> hs_pending s m \<and> ghost_hs_in_sync s m) \<and> ghost_hs_phase s = hp_IdleMarkSweep \<and> hs_type s = ht_GetRoots \<and> obj_mark ` ran (heap s) \<subseteq> {initial_mark} \<and> fA s = initial_mark \<and> fM s = initial_mark \<and> phase s = ph_Idle \<and> ghost_honorary_grey s = {} \<and> W s = {} \<and> (\<forall>p. mem_store_buffers s p = []) \<and> mem_lock s = None)"
assms_3: fixes x :: "'s" assumes "x \<in> Dom f" shows "Fun f x = x"
vsv_vfsequence_three: fixes hgf :: "V" assumes "vsv hgf"   and "\<D>\<^sub>\<circ> hgf = 3\<^sub>\<nat>" shows "[hgf\<lparr>[]\<^sub>\<circ>\<rparr>, hgf\<lparr>1\<^sub>\<nat>\<rparr>, hgf\<lparr>2\<^sub>\<nat>\<rparr>]\<^sub>\<circ> = hgf"
le_infI2: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and b :: "'a::type"   and x :: "'a::type"   and a :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)"   and "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (b::'a::type) (x::'a::type)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) ((inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (b::'a::type)) (x::'a::type)"
supr_greater: fixes B :: "'a set"   and b :: "'a" assumes "B \<subseteq> Field r"   and "r.Above B \<noteq> {}"   and "b \<in> B" shows "(b, r.supr B) \<in> r"
normal_iff_subgroup: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" assumes "comm_group G" shows "N \<lhd> G = subgroup N G"
fsubset_eq: fixes A :: "'a fset"   and B :: "'a fset" shows "(A |\<subseteq>| B) = (\<forall>x|\<in>|A. x |\<in>| B)"
rel_refl_strong: fixes x :: "'d stream"   and Ra :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>z. z \<in> sset x \<Longrightarrow> Ra z z" shows "stream_all2 Ra x x"
rep_eq: fixes x :: "'c set"   and xa :: "'d set" shows "Rep_\<I> (\<I>_uniform x xa) = (\<lambda>x. if x \<in> x then xa else {})"
eval_list_conj_Uni_append: fixes A :: "atomUni fmUni list"   and B :: "atomUni fmUni list"   and x :: "real" shows "evalUni (list_conj_Uni (A @ B)) x = (evalUni (list_conj_Uni A) x \<and> evalUni (list_conj_Uni B) x)"
open_exceptI_3: fixes x :: "'c"   and y :: "'c" assumes "x \<noteq> y" shows "y \<notin> open_except x y"
bezout_coefficients_fst_snd: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and Gcd :: "'c set \<Rightarrow> 'c"   and Lcm :: "'c set \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "plus (times (fst (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) a) (times (snd (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b)) b) = gcd a b"
iso_iff_mono_and_retraction: fixes f :: "'a \<times> 'a" shows "CC.iso f = (CC.mono f \<and> CC.retraction f)"
lm092: fixes bids :: "'d \<times> 'e set \<Rightarrow> 'f"   and N :: "'d set"   and G :: "'e set" shows "summedBidVectorRel bids N G = graph (N \<times> (Pow G - {{}})) (summedBidSecond bids)"
fst_hd_polychain_of: fixes xs :: "'a list"   and x :: "'a" assumes "xs \<noteq> []" shows "fst (hd (polychain_of x xs)) = x"
C: fixes X :: "msg"   and K :: "nat" assumes "P X" shows "P (Crypt K X)"
vars_lp_monom: fixes r :: "rat"   and x :: "nat" shows "vars (lp_monom r x) = (if r = 0 then {} else {x})"
eval: shows "G\<turnstile>s0 \<midarrow>e-\<succ>v\<rightarrow> s1"
star_n_less: fixes X :: "nat \<Rightarrow> 'b"   and Y :: "nat \<Rightarrow> 'b" shows "(star_n X < star_n Y) = (\<forall>\<^sub>F n in \<U>. X n < Y n)"
small: shows "LLL_invariant False k fs \<or> \<bar>fs.gs.\<mu> k (k - 1)\<bar> \<le> 1 / 2"
mds_consistent_def: fixes mds :: "Mode \<Rightarrow> 'Var set"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" shows "mds_consistent mds \<Gamma> \<equiv> dom \<Gamma> = {x. dma x = Low \<and> x \<in> mds AsmNoRead \<or> dma x = High \<and> x \<in> mds AsmNoWrite}"
II: fixes it :: "'d set"   and \<sigma> :: "'c" assumes "it \<subseteq> dom m"   and "it \<noteq> dom m"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k v k' v'. k \<in> it \<and> invar v \<and> m k = Some (\<alpha> v) \<and> k' \<notin> it \<and> invar v' \<and> m k' = Some (\<alpha> v') \<longrightarrow> R (k, \<alpha> v) (k', \<alpha> v')" shows "P \<sigma>"
lhospital_at_top_at_top: fixes g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and x :: "real" assumes "filterlim g at_top at_top"   and "\<forall>\<^sub>F x in at_top. g' x \<noteq> 0"   and "\<forall>\<^sub>F x in at_top. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_top. (g has_real_derivative g' x) (at x)"   and "((\<lambda>x. f' x / g' x) \<longlongrightarrow> x) at_top" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> x) at_top"
finite_measure_Diff': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets N"   and "B \<in> sets N" shows "Sigma_Algebra.measure N (A - B) = Sigma_Algebra.measure N A - Sigma_Algebra.measure N (A \<inter> B)"
usum_def: fixes A :: "('a, 'b) node set set"   and B :: "('a, 'b) node set set" shows "usum A B \<equiv> In0 ` A \<union> In1 ` B"
x_def: shows "x = mom_select (((length xs + 4) div 5 - 1) div 2) (map slow_median (chop 5 xs))"
nonneg_le_mult_right_mono_zhmset: fixes x :: "zhmultiset"   and y :: "zhmultiset"   and z :: "zhmultiset" assumes "0 \<le> x"   and "0 < y"   and "x \<le> z" shows "x \<le> y * z"
mprojTr1: fixes M1 :: "('e, 'a, 'f) Module_scheme"   and M2 :: "('e, 'a, 'f) Module_scheme"   and M :: "('e, 'a, 'f) Module_scheme"   and x :: "'e" assumes "R module M1"   and "R module M2"   and "R module M"   and "M1 \<Oplus>\<^bsub>R,M\<^esub> M2"   and "x \<in> carrier M" shows "\<exists>!x1. x1 \<in> carrier M1 \<and> x \<plusminus>\<^bsub>M\<^esub> -\<^sub>a\<^bsub>M\<^esub> x1 \<in> carrier M2"
distinct_9: fixes x21 :: "'b aexp"   and x22 :: "'b aexp"   and x31 :: "'b aexp"   and x32 :: "'b aexp" shows "Eq x21 x22 \<noteq> Gt x31 x32"
strategyproofness_2: shows "pmf (sds R10) c + pmf (sds R10) a \<le> pmf (sds R19) c + pmf (sds R19) a"
residualInject_2: fixes x2 :: "freeRes"   and x1 :: "pi"   and y2 :: "freeRes"   and y1 :: "pi" shows "(x2 \<prec> x1 = y2 \<prec> y1) = (x2 = y2 \<and> x1 = y1)"
dom_def: fixes m :: "'e \<Rightarrow> 'f option" shows "dom m = {a. m a \<noteq> None}"
params_subterms_Union: fixes X :: "('a, 'b) Term.term list"   and f :: "'a" shows "subterms\<^sub>s\<^sub>e\<^sub>t (set X) \<subseteq> subterms (Fun f X)"
divide_nonpos_nonneg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq zero y" shows "less_eq (divide x y) zero"
bdd_belowI2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and m :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.preorder less_eq less"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq m (f x)" shows "preorder.bdd_below less_eq (f ` A)"
assms_2: shows "js \<in> set (candidate (@) (map wrap [0..<n + 1]))"