hcju/isabelleps huggingface dataset:
card: fixes Rep :: "'b \<Rightarrow> 'c"   and Abs :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "type_definition Rep Abs A" shows "CARD('b) = card A"
res_6: shows "E = (\<Sum>\<^sub># (mset Cs) + DA) \<cdot> \<sigma>''"
tag_eq_def: fixes tag :: "'a list \<Rightarrow> 'b" shows "=tag= \<equiv> {(x, y). tag x = tag y}"
lcm_gcd_prod: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times (lcm a b) (gcd a b) = normalize (times a b)"
real_singleton_in_borel: fixes a :: "real" shows "{a} \<in> sets borel"
subformulas\<^sub>\<nu>_semantics: fixes \<phi> :: "'b ltln" shows "subformulas\<^sub>\<nu> \<phi> = {\<psi> \<in> subfrmlsn \<phi>. \<exists>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi> = \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<or> \<psi> = \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2}"
roofed_circ_def: fixes \<Gamma> :: "('k, 'l) web_scheme"   and S :: "'k set" shows "RF\<^sup>\<circ>\<^bsub>\<Gamma>\<^esub> S = RF\<^bsub>\<Gamma>\<^esub> S - \<E>\<^bsub>\<Gamma>\<^esub> S"
rBang: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and C :: "'d" assumes "\<Psi> \<rhd> P \<parallel> !P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "guarded P"   and "\<And>C. Prop C \<Psi> (P \<parallel> !P) M N P'" shows "Prop C \<Psi> (!P) M N P'"
simps_6: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x2 :: "'c" shows "(case Wit x2 of Oldf x \<Rightarrow> f1 x | Wit x \<Rightarrow> f2 x) = f2 x2"
lin_dep_cols_imp_det_0': fixes ws :: "'a vec list" assumes "\<forall>w\<in>set ws. dim_vec w = n"   and "mat_of_cols n ws \<in> carrier_mat n n"   and "lin_dep (set (cols (mat_of_cols n ws)))" shows "det (mat_of_cols n ws) = (0::'a)"
poly: fixes X :: "'a set" assumes "X \<in> \<G>" shows "polytope X"
abs_eq_iff: fixes x :: "('b, 'c, 'd, 'e) Tree"   and y :: "('b, 'c, 'd, 'e) Tree" shows "(abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> y) = (x =\<^sub>\<alpha> y)"
mult_1_right: fixes a :: "'a" shows "a \<parallel> 1\<^sub>\<pi> = a"
globally_release: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n G\<^sub>n (\<phi> R\<^sub>n \<psi>) = w \<Turnstile>\<^sub>n G\<^sub>n \<psi>"
left_exact_injec: fixes M :: "('f, 'a) Module"   and N :: "('g, 'a) Module"   and z :: "'h \<Rightarrow> 'f"   and e :: "'h"   and f :: "'f \<Rightarrow> 'g" assumes "R module M"   and "R module N"   and "z \<in> mHom R (Zm R e) M"   and "f \<in> mHom R M N"   and "exact3 R (Zm R e) z M f N" shows "injec\<^bsub>M,N\<^esub> f"
PiE_mem2: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and x :: "'a" assumes "f \<in> S \<rightarrow>\<^sub>E T"   and "x \<in> S" shows "f x \<in> T"
neutral_const: fixes A :: "'c set" shows "F.F (\<lambda>_. \<^bold>1) A = \<^bold>1"
assms_2: shows "e = Unop_i t iop \<or> e = Unop_f t fop \<or> e = Testop t testop"
dep_vars_Use: fixes V :: "'var"   and n' :: "'node" assumes "V \<in> Use n'" shows "(V, [], []) \<in> dependent_live_vars n'"
Class_closed: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Class b"   and "b \<in> S" shows "a \<in> S"
infsetsum_conv_nn_integral: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space A \<noteq> \<infinity>"   and "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "infsetsum f A = enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space A)"
mom_select_correct: fixes k :: "nat"   and xs :: "'a list" assumes "k < length xs" shows "mom_select k xs = select k xs"
le_divide_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq a (divide b c) = (if less zero c then less_eq (times a c) b else if less c zero then less_eq b (times a c) else less_eq a zero)"
rech: shows "sorted_list_of_set {x. poly [:0, 1:] x = 0} = [0]"
subadditive_imp_eventually_subadditive: fixes u :: "nat \<Rightarrow> real" assumes "Fekete.subadditive u" shows "eventually_subadditive u 0"
prog_step: shows "\<theta>\<turnstile> p \<rightarrow>\<^sub>s (p', is')"
rev_cases: fixes xs :: "'f list" obtains "xs = []"   | ys :: "'f list" and y :: "'f" where "xs = ys @ [y]"
mini_maxi_keys_eq: fixes key :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and x :: "'d" assumes "key (xs ! mini xs key) = key (xs ! maxi xs key)"   and "x \<in> set xs" shows "key x = key (xs ! maxi xs key)"
cod_cod: fixes f :: "'c" assumes "C.arr f" shows "C.cod (C.cod f) = C.cod f"
simps_3: fixes v :: "'b"   and xs :: "'b negation_type list" shows "getNeg (Pos v # xs) = getNeg xs"
contains_initialEVs: fixes gamma :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula" assumes "branch subs (pseq gamma) f"   and "A \<in> set gamma" shows "EV (contains f (0, A))"
Dup_touches: fixes p :: "('c, 'd) fm"   and i :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and xs :: "(nat \<times> nat) set" assumes "Dup p i branch xs" shows "only_touches i branch xs"
Gp_mult_induced1: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a"   and b :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "a \<in> H \<inter> K"   and "b \<in> H \<inter> K" shows "a \<cdot>\<^bsub>\<natural>H \<inter> K\<^esub> b = a \<cdot>\<^bsub>\<natural>H\<^esub> b"
nths_nil: fixes A :: "nat set" shows "nths [] A = []"
assms_7: shows "h \<turnstile> get_dom_component ptr' \<rightarrow>\<^sub>r c'"
norm_blinfun_eqI: fixes n :: "real"   and f :: "'c \<Rightarrow>\<^sub>L 'b"   and x :: "'c" assumes "n \<le> norm (blinfun_apply f x) / norm x"   and "\<And>x. norm (blinfun_apply f x) \<le> n * norm x"   and "0 \<le> n" shows "norm f = n"
Aheap_subst: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and y :: "Vars.var"   and e :: "exp" assumes "x \<notin> domA \<Gamma>"   and "y \<notin> domA \<Gamma>" shows "Aheap \<Gamma>[x::h=y] e[x::=y] = Aheap \<Gamma> e"
seqE: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" obtains "D.arr f" and "D.arr g" and "D.dom g = D.cod f"
Seq_bconf_preserve_aux: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,sh \<turnstile>\<^sub>b (e;; e\<^sub>2,b) \<surd>"   and "P,sh \<turnstile>\<^sub>b (e,b) \<surd> \<longrightarrow> P,sh' \<turnstile>\<^sub>b (e',b') \<surd>" shows "P,sh' \<turnstile>\<^sub>b (e';; e\<^sub>2,b') \<surd>"
le_infE: fixes x :: "'c multiset"   and a :: "'c multiset"   and b :: "'c multiset" assumes "x \<subseteq># a \<inter># b" obtains "x \<subseteq># a" and "x \<subseteq># b"
homologous_rel_refl: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "homologous_rel p X S c c"
history_consistent_appendD: fixes xs :: "'a memref list"   and ys :: "'a memref list"   and \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a" assumes "\<forall>sop\<in>write_sops xs. valid_sop sop"   and "read_tmps xs \<inter> read_tmps ys = {}"   and "history_consistent \<theta> p (xs @ ys)" shows "history_consistent (\<theta> |` (dom \<theta> - read_tmps ys)) p xs \<and> history_consistent \<theta> (last_prog p xs) ys \<and> read_tmps ys \<inter> \<Union> (fst ` write_sops xs) = {}"
\<epsilon>_def: shows "\<epsilon> \<equiv> Min {r, e'} / 2"
stable_on_def: fixes CX :: "'c set"   and trap :: "'c set" shows "stable_on CX trap = (\<forall>t x0. flow0 x0 t \<in> trap \<longrightarrow> t \<in> existence_ivl0 x0 \<longrightarrow> 0 < t \<longrightarrow> (\<forall>s\<in>{0<..t}. flow0 x0 s \<in> CX) \<longrightarrow> x0 \<in> trap)"
internal_iface_name_wildcard_longest_correct: fixes i1 :: "char list"   and i2 :: "char list"   and p_i :: "char list" assumes "iface_name_is_wildcard i1"   and "iface_name_is_wildcard i2" shows "(match_iface (Iface i1) p_i \<and> match_iface (Iface i2) p_i) = (case internal_iface_name_wildcard_longest i1 i2 of None \<Rightarrow> False | Some x \<Rightarrow> match_iface (Iface x) p_i)"
emeasure_lborel_box_eq: fixes l :: "'a"   and u :: "'a" shows "emeasure lborel (box l u) = ennreal (if \<forall>b\<in>Basis. l \<bullet> b \<le> u \<bullet> b then prod ((\<bullet>) (u - l)) Basis else 0)"
antisym_conv: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "(y \<le> x) = (x = y)"
sig_red_zero_def: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and r :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_red_zero sing_reg F r = (\<exists>s. (sig_red sing_reg (\<preceq>) F)\<^sup>*\<^sup>* r s \<and> rep_list s = 0)"
emeasure_real: fixes M :: "'a pmf"   and A :: "'a set" shows "\<exists>r\<ge>0. emeasure (measure_pmf M) A = ennreal r"
fresh_vrs_of: fixes x :: "vrs"   and b :: "binding" shows "x \<sharp> vrs_of b = x \<sharp> b"
d_glb_iff: fixes z :: "'a"   and x :: "'a"   and y :: "'a" shows "(d z \<le> d x \<and> d z \<le> d y) = (d z \<le> d x \<cdot> d y)"
adjacentchain_Cons_reduce: fixes x :: "'b set"   and xs :: "'b set list" assumes "adjacentchain (x # xs)" shows "adjacentchain xs"
domain_subset_valid_pub: shows "Domain Rel0 \<subseteq> valid_pub0"
ab_mod_f0: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly" assumes "a * b mod f = 0"   and "degree a < degree f"   and "degree b < degree f" shows "a = 0 \<or> b = 0"
exactly_length: fixes x :: "char list"   and s :: "char list"   and y :: "char list"   and r :: "char list" assumes "exactly x s = Inr (y, r)" shows "length x + length r \<le> length s"
standard_redundancy_criterion_counterex_reducing_def: fixes \<Gamma> :: "'b Inference_System.inference set"   and I_of :: "'b literal multiset set \<Rightarrow> 'b set" shows "standard_redundancy_criterion_counterex_reducing \<Gamma> I_of \<equiv> Inference_System.counterex_reducing_inference_system \<Gamma> I_of"
stut_Enabled: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "STUTINV Enabled \<langle>F\<rangle>_v"
foo_cost_nonneg: fixes n :: "nat" shows "0 \<le> foo_cost n"
lb_impl_job_sum: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "lb T A j" shows "sum T {1..m} = sum t {1..j}"
IH: fixes y :: "nat"   and t :: "tm"   and u :: "tm" assumes "y < n"   and "size t + size u < y"   and "ground t"   and "ground u" shows "(\<lbrakk>t\<rbrakk>e \<le> \<lbrakk>u\<rbrakk>e \<longrightarrow> {} \<turnstile> t SUBS u) \<and> (\<lbrakk>t\<rbrakk>e \<^bold>\<in> \<lbrakk>u\<rbrakk>e \<longrightarrow> {} \<turnstile> t IN u)"
inside_same_component: fixes S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected_component (- S) x y"   and "x \<in> inside S" shows "y \<in> inside S"
img_is_cring: assumes "cring S" shows "cring (S\<lparr>carrier := h ` carrier R\<rparr>)"
m_comm: fixes x :: "'a list list"   and y :: "'a list list" assumes "x \<in> carrier (mat_monoid nr nc b)"   and "y \<in> carrier (mat_monoid nr nc b)" shows "x \<otimes>\<^bsub>mat_monoid nr nc b\<^esub> y = y \<otimes>\<^bsub>mat_monoid nr nc b\<^esub> x"
fcs_a: shows "face_cycle_set a = orbit face_cycle_succ a0"
l_no_fwd: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j < length l" shows "l ! j \<times> l ! i \<inter> E\<^sup>* = {}"
snth_Stream: fixes x :: "'d"   and s :: "'d stream"   and i :: "nat" shows "(x ## s) !! Suc i = s !! i"
chine_in_homs_4: shows "\<guillemotleft>chine_hcomp \<lparr>Chn = \<mu>\<nu>\<pi>.chine_assoc, Dom = Dom ((\<mu> \<star> \<nu>) \<star> \<pi>), Cod = Cod (\<mu> \<star> \<nu> \<star> \<pi>)\<rparr> \<rho> : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C HH\<mu>H\<nu>\<pi>\<rho>.chine\<guillemotright>"
length_hd_merge2: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs" shows "length (hd bs) \<le> length (hd vs)"
fls_base_factor_to_fps_nonzero: fixes f :: "'a fls" assumes "f \<noteq> 0" shows "fls_base_factor_to_fps f \<noteq> 0"
At: fixes n :: "nat" shows "(case three.f n of (x, y, z) \<Rightarrow> At x y z) n"
N_def: shows "N = (SOME N. \<forall>ij. \<exists>n\<le>N. ij \<in> G ^^ n)"
spr_repRels_def: shows "spr_repRels \<equiv> \<lambda>a. rel_ext (\<lambda>(s, s'). envObs a s' = envObs a s)"
AxT: fixes Ax :: "('a, 'b) Axioms" assumes "Axioms Ax" shows "aAxioms Ax \<subseteq> {Vx : A \<turnstile> e1\<equiv>e2 : B |A B e1 e2. Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e1\<equiv>e2 : B}"
inverse_Infinitesimal_iff_HInfinite: fixes x :: "'a star" assumes "x \<noteq> 0" shows "(inverse x \<in> Infinitesimal) = (x \<in> HInfinite)"
rl: fixes u :: "'t" assumes "is_pred_syz (new_syz_sigs ss bs p) u"   and "dgrad (pp_of_term u) \<le> dgrad_max dgrad" shows "is_syz_sig dgrad u"
singleton_pow: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "{a} \<in> Pow A"
permute_mode_def: fixes \<pi> :: "perm"   and h :: "mode" shows "\<pi> \<bullet> h = h"
sum: shows "unity_root_sum k n = (unity_root k n ^ k - 1) / (unity_root k n - 1)"
dom_restrict_def: fixes S :: "'d set"   and p :: "'d \<Rightarrow> 'e decision option" shows "S \<triangleleft> p \<equiv> \<lambda>x. if x \<in> S then p x else \<bottom>"
SAT_Sup_upper: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "f \<in> Y" shows "SAT \<Gamma> f \<subseteq> SAT \<Gamma> (\<Squnion> Y)"
call: fixes i :: "nat"   and \<C> :: "t_context"   and tf :: "tf" assumes "i < length (func_t \<C>)"   and "func_t \<C> ! i = tf" shows "\<C> \<turnstile> [Call i] : tf"
poly_sgn_eventually_at_top: fixes p :: "real poly" shows "\<forall>\<^sub>F x in at_top. sgn (poly p x) = sgn_pos_inf p"
arr_char: fixes f :: "'a" shows "arr f = (f \<in> Obj)"
abs_eq: fixes xa :: "'b ltl"   and x :: "'b set list" shows "af_G_abs_opt.f_foldl_abs (Abs xa) x = Abs (af\<^sub>G\<^sub>\<UU> xa x)"
assms_1: fixes n :: "nat" assumes "k < n" shows "p n = \<zero>"
IH: fixes gpv :: "('a, 'call, 'ret) gpv"   and s :: "'s" assumes "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "expectation_gpv' gpv \<le> expectation_gpv2 (\<lambda>a. case a of (x, s) \<Rightarrow> f x) (inline callee gpv s)"
keys_monom_mult: fixes c :: "'a"   and t :: "('x, nat) pp"   and p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'a" assumes "c \<noteq> (0::'a)" shows "keys (local.punit.monom_mult c t p) = local.punit.splus t ` keys p"
hseq_char: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "C.hseq \<nu> \<mu> = (C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<nu> = trg\<^sub>C \<mu>)"
log_ceiling_sound_1: fixes b :: "int"   and x :: "int"   and y :: "nat" assumes "1 < b"   and "0 < x"   and "log_ceiling b x = y" shows "x \<le> b ^ y"
set_induct: fixes x :: "'b"   and a :: "'b llist"   and P :: "'b \<Rightarrow> 'b llist \<Rightarrow> bool" assumes "x \<in> lset a"   and "\<And>z1 z2. P z1 (LCons z1 z2)"   and "\<And>z1 z2 xa. \<lbrakk>xa \<in> lset z2; P xa z2\<rbrakk> \<Longrightarrow> P xa (LCons z1 z2)" shows "P x a"
prod_cases7: fixes y :: "'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g \<times> 'h" obtains a :: "'b" and b :: "'c" and c :: "'d" and d :: "'e" and e :: "'f" and f :: "'g" and g :: "'h" where "y = (a, b, c, d, e, f, g)"
refl_imp_locally_refl: fixes A :: "'a multiset" shows "locally_refl NS A"
inf_dres_addsimps_4: fixes v :: "'b"   and x :: "'b dres" shows "inf (dRETURN v) x \<noteq> dFAILi"
theorem3_part2: assumes "All admits_triangular_reduction" shows "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'b}"
transPD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "transp r"   and "r x y"   and "r y z" shows "r x z"
eval_Neg_Neg: fixes \<phi> :: "atom fm"   and L :: "real list" shows "eval (Neg (Neg \<phi>)) L = eval \<phi> L"
simplical_face_in_standard_simplex: fixes p :: "nat"   and k :: "nat"   and x :: "nat \<Rightarrow> real" assumes "1 \<le> p"   and "k \<le> p"   and "x \<in> standard_simplex (p - Suc 0)" shows "simplical_face k x \<in> standard_simplex p"
poly_add_length: fixes p1 :: "'a list"   and p2 :: "'a list" shows "length (p1 +++ p2) = max (length p1) (length p2)"
subspace_affine: fixes S :: "'a set" shows "subspace S = (affine S \<and> (0::'a) \<in> S)"
I: shows "loop \<in> brec_invar (hs.\<alpha> Qi) (ls.\<alpha> \<delta>)"
lnth_prefix: fixes i :: "nat" assumes "i < length xss'" shows "lnth xss i = LNil"
nn_integral_count_space_nat: fixes f :: "nat \<Rightarrow> ennreal" shows "integral\<^sup>N (count_space UNIV) f = (\<Sum>i. f i)"
connected_ivt_component: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and k :: "'b"   and a :: "real" assumes "connected S"   and "x \<in> S"   and "y \<in> S"   and "x \<bullet> k \<le> a"   and "a \<le> y \<bullet> k" shows "\<exists>z\<in>S. z \<bullet> k = a"
A: fixes as :: "('a poly \<times> nat) set"   and p :: "'a poly"   and n :: "nat" assumes "monic_factorization as p" shows "monic_factorization.A p n = (\<Prod>(a, i)\<in>as \<inter> UNIV \<times> {n}. a)"
map_cong0: fixes x :: "('l, 'k, 'm, 'n) F"   and f1 :: "'k \<Rightarrow> 'o"   and g1 :: "'k \<Rightarrow> 'o"   and f2 :: "'m \<Rightarrow> 'p"   and g2 :: "'m \<Rightarrow> 'p"   and f3 :: "'n \<Rightarrow> 'q"   and g3 :: "'n \<Rightarrow> 'q" assumes "\<And>z1. z1 \<in> F'set3 x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> F'set1 x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> F'set2 x \<Longrightarrow> f3 z3 = g3 z3" shows "F'map f2 f3 f1 x = F'map g2 g3 g1 x"
not_le_imp_less: fixes y :: "'b word"   and x :: "'b word" assumes "\<not> y \<le>s x" shows "x <s y"
nat_term_compare_POT: fixes to :: "'b nat_term_order" shows "nat_term_compare (POT to) = pot_comp (nat_term_compare to)"
topological_space_quotient: fixes f :: "'c \<Rightarrow> 'b" assumes "surj f" shows "class.topological_space (openin (map_topology f euclidean))"
add_less_cancel_left: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" shows "(c + a \<subset># c + b) = (a \<subset># b)"
l2_inv6I: fixes x :: "l2_state" assumes "\<And>R. in_progress (progress x R) xsk \<Longrightarrow> \<exists>X N. guessed_frame R xsk = Some (Exp X (NonceF (R $ N)))" shows "x \<in> l2_inv6"
analz_LtKeys: fixes H :: "msg set" assumes "H \<subseteq> payload \<union> sym.valid \<union> range LtK \<union> Tags" shows "analz H \<inter> range LtK \<subseteq> H"
smallI: fixes g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<And>c. 0 < c \<Longrightarrow> \<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)" shows "f \<in> \<omega>[F](g)"
fst_vimage_eq_Times: fixes S :: "'c set" shows "fst -` S = S \<times> UNIV"
unity_root_sum_1: fixes k :: "nat"   and n :: "int" assumes "1 \<le> k"   and "int k dvd n" shows "unity_root_sum k n = of_nat k"
intros_4: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and b :: "'BExp" assumes "no_await c" shows "no_await (While b c)"
hom_induced_trivial: fixes p :: "int" assumes "p < 0" shows "hom_induced p = (\<lambda>X S Y T f c. undefined)"
vsv_valeq_v11I: fixes a :: "V"   and b :: "V" assumes "\<And>x y. \<lbrakk>x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}); y \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}); set {\<langle>a, b\<rangle>}\<lparr>x\<rparr> = set {\<langle>a, b\<rangle>}\<lparr>y\<rparr>\<rbrakk> \<Longrightarrow> x = y" shows "v11 (set {\<langle>a, b\<rangle>})"
iASSERT_def: fixes return :: "unit \<Rightarrow> 'a"   and \<Phi> :: "bool" shows "iASSERT return \<Phi> \<equiv> if \<Phi> then return () else top"
\<tau>red1t_inj_\<tau>reds1t: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and es :: "'addr expr1 list" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>reds1gt uf P t h (e # es, xs) (e' # es, xs')"
Psi_diff_Ins: fixes i :: "real"   and d :: "real"   and b :: "bool"   and x\<^sub>1 :: "real"   and x\<^sub>2 :: "real"   and n :: "nat" assumes "0 < i"   and "0 < d" shows "\<Psi> b i d x\<^sub>1 x\<^sub>2 (Suc n) - \<Psi> b i d x\<^sub>1 x\<^sub>2 n \<le> i"
T1_2: fixes w :: "i"   and F :: "\<Pi>\<^sub>2"   and x :: "\<kappa>"   and y :: "\<kappa>" shows "[\<lparr>F,x,y\<rparr> in w] = (\<exists>r o\<^sub>1 o\<^sub>2. Some r = d\<^sub>2 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> (o\<^sub>1, o\<^sub>2) \<in> ex2 r w)"
replace_def: fixes k :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" shows "Mapping.replace k v m = (if k \<in> Mapping.keys m then Mapping.update k v m else m)"
a: shows "{0..<m} - (X \<union> Y) \<union> X \<union> Y = {0..<m}"
has_fps_expansion_tan: fixes c :: "'a" shows "(\<lambda>x. tan (c * x)) has_fps_expansion fps_tan c"
not_None_eq: fixes x :: "'a option" shows "(x \<noteq> \<bottom>) = (\<exists>y. x = Some y)"
free_means_no_cars: shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>x. \<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True))"
l2norm_neg: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "l2norm S (\<lambda>x. - f x) = l2norm S f"
type_soundness: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<turnstile> \<Gamma> {c} \<Gamma>'"   and "mds_consistent mds \<Gamma>"   and "\<forall>x. to_total \<Gamma> x = Low \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x" shows "\<langle>c, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c, mds, mem\<^sub>2\<rangle>"
walk_array_Cons: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c"   and xs :: "'c list"   and b :: "'b" shows "walk_array f (Array (a # xs)) b = walk_array f (Array xs) (f b a)"