hcju/isabelleps huggingface dataset:
Suc_diff_Suc: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "Suc (n - Suc m) = n - m"
real_of_float_power: fixes f :: "float"   and n :: "nat" shows "real_of_float (f ^ n) = real_of_float f ^ n"
shadow_root_eq_h2: fixes ptr' :: "(_) element_ptr"   and children :: "(_) shadow_root_ptr option" shows "h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children = h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r children"
less_list_def: fixes s :: "'a list"   and t :: "'a list" shows "(s < t) = (s \<le> t \<and> s \<noteq> t)"
\<psi>T_Hmap_on_indVfbasis: fixes T :: "'v \<Rightarrow> 'w"   and x :: "'g"   and f :: "('f, 'g) aezfun \<Rightarrow> 'v" assumes "T \<in> GRepHomSet (\<star>) W"   and "x \<in> H"   and "f \<in> set (concat indVfbasis)" shows "\<psi> T (x *\<currency> f) = x *\<star> \<psi> T f"
intros_2: fixes u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b"   and d :: "int" assumes "u x - u y = real_of_int d" shows "intv'_elem x y u (Const' d)"
sorted_on_insort_key_on: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" assumes "f ` set (x # xs) \<subseteq> Field r" shows "sorted_on r (map f (insort_key_on r f x xs)) = sorted_on r (map f xs)"
bdd_above_def: fixes A :: "'b multiset set" shows "subset_mset.bdd_above A = (\<exists>M. \<forall>x\<in>A. x \<subseteq># M)"
invertible: fixes u :: "'b" assumes "u \<in> J" shows "JB.additive.sub.invertible u"
assms_1: shows "theta \<subseteq> WretrT (theta \<union> WbisT)"
snd_apply_icrit: fixes crit :: "nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and data :: "nat \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "map snd (apply_icrit crit data gs bs hs ps) = ps"
Int_Collect_mono: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> P x \<longrightarrow> Q x" shows "A \<inter> Collect P \<subseteq> B \<inter> Collect Q"
quant_eq_iff_mag_eq: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" shows "(x = y) = (\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q)"
partition_ivle_def: fixes ro :: "'g"   and xse :: "('e \<times> 'f) set" shows "partition_ivle ro xse = (if op_coll_is_empty xse then RETURN (op_empty_coll ::: clw_rel elvivl_rel) else scaleR2_rep_coll xse \<bind> (\<lambda>(uu_, xs). op_image_fst_ivl_coll xs \<bind> (\<lambda>xsf. op_ivl_of_ivl_coll (xsf ::: clw_rel lvivl_rel) \<bind> (\<lambda>r. ivl_rep r \<bind> (\<lambda>(i, s). CHECK (\<lambda>_. ()) (i \<le> s) \<bind> (\<lambda>_. WHILE\<^bsup>\<lambda>(rs, ps). xse \<subseteq> rs \<times> UNIV \<union> ps\<^esup> (\<lambda>(rs, ps). \<not> op_coll_is_empty (rs ::: clw_rel lvivl_rel)) (\<lambda>(rs, ps). (split_spec_exact rs ::: \<langle>lvivl_rel \<times>\<^sub>r clw_rel lvivl_rel\<rangle>nres_rel) \<bind> (\<lambda>(r, rs'). le_post_inter_granularity_op ro r \<bind> (\<lambda>okay. if okay then op_inter_ivl_coll_scaleR2 xse r \<bind> (\<lambda>I. RETURN (rs', I \<union> ps)) else split_spec_ivl DIM('e) r \<bind> (\<lambda>(a, b). op_image_fst_ivl_coll xs \<bind> (\<lambda>fxs. op_single_inter_ivl a fxs \<bind> (\<lambda>ra'. op_single_inter_ivl b fxs \<bind> (\<lambda>rb'. RETURN (ra' \<union> rb' \<union> rs', ps)))))))) (mk_coll r ::: clw_rel lvivl_rel, op_empty_coll ::: clw_rel elvivl_rel) \<bind> (\<lambda>(rs, ps). RETURN ps)))))))"
UNION_eq: fixes B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Union> (B ` A) = {y. \<exists>x\<in>A. y \<in> B x}"
starfun_inverse: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d star" shows "inverse ((*f* f) x) = (*f* (\<lambda>x. inverse (f x))) x"
vec_nth_Basis2: fixes x :: "((real, 'a) vec, 'b) vec"   and i :: "'b"   and j :: "'a" assumes "x \<in> Basis" shows "x $ i $ j = (if x = axis i (axis j 1) then 1 else 0)"
disj: fixes i1 :: "'a set"   and i2 :: "'a set" assumes "i1 \<in> I"   and "i2 \<in> I"   and "i1 \<noteq> i2" shows "interior i1 \<inter> interior i2 = {}"
nneg_bound_nneg: fixes P :: "'c \<Rightarrow> real" assumes "Expectations.bounded P"   and "nneg P" shows "0 \<le> bound_of P"
A: shows "ipurge_tr (con_comp_pol I) (con_comp_map D E p q) u (filter (\<lambda>x. x \<in> range p \<union> range q) xs) = ipurge_tr (con_comp_pol I) (con_comp_map D E p q) u xs"
Gp_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "\<natural>\<^bsub>G\<^esub>H \<equiv> G\<lparr>carrier := H, top := (\<cdot>\<^bsub>G\<^esub>), iop := iop G, one := \<one>\<^bsub>G\<^esub>\<rparr>"
sdiv_pos_pos: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 \<le> b" shows "a sdiv b = a div b"
zero_less_iff_1_le_zhmset: fixes n :: "zhmultiset" shows "(0 < n) = (1 \<le> n)"
connected_awalkE: fixes u :: "'a"   and v :: "'a" assumes "Digraph_Component.connected (with_proj (mk_symmetric G))"   and "u \<in> pverts (mk_symmetric G)"   and "v \<in> pverts (mk_symmetric G)" obtains p :: "('a \<times> 'a) list" where "pre_digraph.awalk (with_proj (mk_symmetric (with_proj (mk_symmetric G)))) u p v"
fMax_finsert: fixes x :: "'c"   and A :: "'c fset" shows "fMax (finsert x A) = (if A = {||} then x else max x (fMax A))"
expr_sem_rf_space: shows "expr_sem_rf \<rho> e' \<in> space (stock_measure t')"
pge: fixes y :: "complex" assumes "y \<in> cball a r" shows "cmod (g y) \<le> cmod (g p)"
signzero_zero: fixes a :: "('a, 'b) IEEE.float"   and s :: "nat" assumes "is_zero a" shows "valof (zerosign s a) = 0"
unit_disc_fix_preserves_poincare_on_line: fixes M :: "moebius"   and p :: "complex_homo"   and a :: "complex_homo"   and b :: "complex_homo" assumes "unit_disc_fix M"   and "p \<in> unit_disc"   and "a \<in> unit_disc"   and "b \<in> unit_disc" shows "poincare_on_line (moebius_pt M p) (moebius_pt M a) (moebius_pt M b) = poincare_on_line p a b"
cdim: shows "d\<inverse> O m \<subseteq> ov \<union> d\<inverse> \<union> f\<inverse>"
K: shows "\<forall>x\<in>{a..b} - K. g differentiable at x within {a..b}"
insertI2: fixes a :: "'c"   and B :: "'c set"   and b :: "'c" assumes "a \<in> B" shows "a \<in> insert b B"
translation_inverse_subset: fixes a :: "'b"   and V :: "'b set"   and S :: "'b set" assumes "(+) (- a) ` V \<subseteq> S" shows "V \<subseteq> (+) a ` S"
inject_2: fixes x2 :: "'c match_expr"   and y2 :: "'c match_expr" shows "(MatchNot x2 = MatchNot y2) = (x2 = y2)"
zero_le_double_add_iff_zero_le_single_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (plus a a) = less_eq zero a"
eq_infE: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2" obtains "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2" and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>1"
vsv_ex1: fixes a :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"
last_prog_same_append: fixes p\<^sub>s\<^sub>b :: "'b"   and sb :: "'b memref list"   and xs :: "'b memref list" assumes "last_prog p\<^sub>s\<^sub>b (sb @ xs) = p\<^sub>s\<^sub>b" shows "last_prog p\<^sub>s\<^sub>b xs = p\<^sub>s\<^sub>b"
eqv_class_eqI: fixes a :: "'b"   and b :: "'b" assumes "a \<sim> b" shows "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
dg_prod_2_Cod_vsv: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>" shows "vsv ((\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Cod\<rparr>)"
finite_polyhedron_exposed_faces: fixes S :: "'b set" assumes "polyhedron S" shows "finite {F. F exposed_face_of S}"
pcr_spmf_gpv_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "spmf_to_gpv.pcr_spmf_gpv A = spmf_to_gpv.cr_spmf_gpv OO rel_gpv A (=)"
simps_5: fixes v :: "'a val" shows "call1 (Val v) = None"
\<epsilon>: shows "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C trg\<^sub>C f\<guillemotright> \<and> F \<epsilon> = unit (trg\<^sub>C f) \<cdot>\<^sub>D \<epsilon>' \<cdot>\<^sub>D D.inv (\<Phi> (f, g))"
maximalidealI: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" assumes "ideal I R"   and "carrier R \<noteq> I"   and "\<And>J. \<lbrakk>ideal J R; I \<subseteq> J; J \<subseteq> carrier R\<rbrakk> \<Longrightarrow> J = I \<or> J = carrier R" shows "maximalideal I R"
list_ex1_simps_1: fixes P :: "'a \<Rightarrow> bool" shows "list_ex1 P [] = False"
vsv_appE: fixes a :: "V"   and b :: "V" assumes "[a, b]\<^sub>\<circ>\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> [a, b]\<^sub>\<circ>" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> [a, b]\<^sub>\<circ>"
correct_aux: fixes ts :: "((nat \<times> nat) \<times> int) list"   and x :: "nat \<Rightarrow> int"   and m :: "nat \<Rightarrow> int" assumes "distinct (map fst ts)"   and "\<forall>((r, c), uu_)\<in>set ts. r < n" shows "\<forall>i. fold (\<lambda>((r, c), v) y. y(c := y c + x r * v)) ts m i = m i + local.pr (\<alpha> ts) x i"
Union_SetCompr_eq: fixes f :: "'d \<Rightarrow> 'c set"   and P :: "'d \<Rightarrow> bool" shows "\<Union> {f x |x. P x} = {a. \<exists>x. P x \<and> a \<in> f x}"
inf_double_complement_export: fixes x :: "'a"   and y :: "'a" shows "- - (- - x \<sqinter> y) = - - x \<sqinter> - - y"
is_continuous: fixes U :: "'c set" assumes "is_open' U" shows "is_open (f \<^sup>\<inverse> S U)"
try_gpv_bind_lossless: fixes gpv :: "('c, 'a, 'b) gpv"   and f :: "'c \<Rightarrow> ('d, 'a, 'b) gpv"   and gpv' :: "('d, 'a, 'b) gpv" assumes "lossless_gpv \<I>_full gpv" shows "TRY gpv \<bind> f ELSE gpv' = gpv \<bind> (\<lambda>x. TRY f x ELSE gpv')"
len_ge_0: fixes l :: "'b list" shows "0 \<le> length l"
simps: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and a1 :: "'a"   and a2 :: "'a"   and a3 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" shows "Pre_Port_Graph.path edges a1 a2 a3 = ((\<exists>v. a1 = v \<and> a2 = v \<and> a3 = []) \<or> (\<exists>e v' pth. a1 = Pre_Port_Graph.edge_begin e \<and> a2 = v' \<and> a3 = e # pth \<and> e \<in> edges \<and> Pre_Port_Graph.path edges (Pre_Port_Graph.edge_end e) v' pth))"
simps_2: fixes s :: "'sequent"   and r :: "'rule"   and uu :: "('sequent, 'rule, 'link) ctree fset" shows "seqOf (cNode (s, r) uu) = s"
e1_v1: shows "psubst \<rho>' e1 \<longrightarrow>* v1"
ln_powr_has_integral_at_top: fixes d :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "1 \<le> d * a + b"   and "c < - 1"   and "0 < d" shows "((\<lambda>x. ln (d * x + b) * (d * x + b) powr c) has_integral - ((ln (d * a + b) - 1 / (c + 1)) * (d * a + b) powr (c + 1) / (d * (c + 1)))) {a<..}"
not_nofail_inres: fixes S :: "'b nres"   and x :: "'b" assumes "\<not> nofail S" shows "inres S x"
elims: fixes x :: "('b, 'c) Combinators"   and y :: "'b set" assumes "bothNet x = y" obtains "x = DenyAll" and "y = {}"   | a :: "'b" and b :: "'b" where "x = DenyAllFromTo a__ b__" and "y = {a__, b__}"   | a :: "'b" and b :: "'b" and p :: "'c" where "x = AllowPortFromTo a b p" and "y = {a, b}"   | v :: "('b, 'c) Combinators" and va :: "('b, 'c) Combinators" where "x = v \<oplus> va" and "y = undefined"
support_preList: fixes upds :: "(char list \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and C1 :: "acom" shows "support (preList upds C1) \<subseteq> lesvars upds"
inject_4: fixes x4 :: "('a, 'b, 'c) expr"   and y4 :: "('a, 'b, 'c) expr" shows "(Ref x4 = Ref y4) = (x4 = y4)"
guest_key2_disj: fixes s :: "state"   and k\<^sub>1 :: "key"   and k :: "key"   and g\<^sub>1 :: "guest"   and k\<^sub>2 :: "key"   and g\<^sub>2 :: "guest" assumes "s \<in> reach"   and "(k\<^sub>1, k) \<in> cards s g\<^sub>1"   and "(k\<^sub>2, k) \<in> cards s g\<^sub>2" shows "g\<^sub>1 = g\<^sub>2"
seqE: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.seq g f" obtains "Maps.arr f" and "Maps.arr g" and "Maps.dom g = Maps.cod f"
2: shows "Filters.filter.Rep_filter (Rep_phi f (fst (Rep_lifted_pair (Rep_regular pf) f))) \<subseteq> {z. \<exists>qf. - Rep_regular pf \<le> Rep_dense qf \<and> z = get_dense qf f}"
Rep: fixes x :: "'b word" shows "to_bl x \<in> {bl. length bl = LENGTH('b)}"
diff_right_commute: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g"   and c :: "'g"   and b :: "'g" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (minus a c) b = minus (minus a b) c"
sets_tree_sigma_eq: fixes M :: "'d measure" shows "sets (tree_sigma M) = sigma_sets (trees (space M)) (trees_cyl ` trees (sets M))"
alt_option_def: fixes m1 :: "('a, 'm) optionT"   and m2 :: "('a, 'm) optionT" shows "local.alt_option m1 m2 = OptionT (alt (run_option m1) (run_option m2))"
reachable_While: fixes b :: "bexp"   and c :: "com" shows "reachable (WHILE b DO c) \<subseteq> {WHILE b DO c, IF b THEN c;; WHILE b DO c ELSE SKIP, SKIP} \<union> (\<lambda>c'. c';; WHILE b DO c) ` reachable c"
Cons_prefix_Cons: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(x # xs \<le>p y # ys) = (x = y \<and> xs \<le>p ys)"
Lcm_int_eq: fixes N :: "nat set" shows "Lcm (int ` N) = int (Lcm N)"
infinite_super: fixes S :: "'g set"   and T :: "'g set" assumes "S \<subseteq> T"   and "infinite S" shows "infinite T"
Red_F_Sup_subset_Red_F_Liminf: fixes Ns :: "'f set llist" assumes "chain (\<rhd>) Ns" shows "Red_F (Sup_llist Ns) \<subseteq> Red_F (Liminf_llist Ns)"
ring_isomorphism_axioms: shows "ring_isomorphism f A addA multA zeroA oneA B addB multB zeroB oneB"
le_rexp_refl: fixes r :: "'a rexp" shows "local.le_rexp r r"
distinct_9: fixes x21 :: "('a, 'b) com"   and x22 :: "('a, 'b) com"   and x31 :: "'a"   and x32 :: "('a, 'b) com"   and x33 :: "('a, 'b) com" shows "x21 ;; x22 \<noteq> if x31 then x32 else x33"
mult_scalar_monomial: fixes t :: "'a"   and c :: "'c"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monomial c t \<odot> p = monom_mult c t p"
vfind_match_elem_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and t :: "value"   and env :: "(name, value) fmap"   and pat :: "Term_Class.term"   and rhs :: "sterm" assumes "vfind_match cs t = Some (env, pat, rhs)" shows "vmatch (mk_pat pat) t = Some env"
hadd_0_left: fixes x :: "hf" shows "0 @+ x = x"
div_Suc: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "Suc m div n = (if Suc (m mod n) = n then Suc (m div n) else m div n)"
components_maximal: fixes c :: "'c set"   and s :: "'c set"   and t :: "'c set" assumes "c \<in> components s"   and "connected t"   and "t \<subseteq> s"   and "c \<inter> t \<noteq> {}" shows "t \<subseteq> c"
eq_self_neq_FF': fixes x :: "'a" shows "eq\<cdot>x\<cdot>x \<noteq> FF"
simps: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "loose \<Gamma> = ((\<forall>x. current \<Gamma> x \<longrightarrow> wave \<Gamma> x \<longrightarrow> x = (\<lambda>_. 0)) \<and> \<not> hindrance \<Gamma> (\<lambda>_. 0))"
FactGroup_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "'b set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, monoid.mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
delete_conv': fixes k :: "'d"   and al :: "('d \<times> 'e) list" shows "map_of (delete k al) = (map_of al)(k := None)"
filterlim_sequentially_iff: fixes f :: "nat \<Rightarrow> 'a"   and F1 :: "'a filter"   and k :: "nat" shows "filterlim f F1 sequentially = (LIM x sequentially. f (x + k) :> F1)"
dehomogenize_alt: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "dehomogenize x p = (\<Sum>t\<in>keys p. monomial (lookup p t) (except t {x}))"
conjr1_strict: fixes x :: "'a" shows "\<bottom> \<rhd> x = \<bottom>"
max_x_def: fixes y :: "nat list"   and i :: "nat" shows "max_x y i = (if i < m \<and> Di i y \<noteq> {} then Min (Di i y) else Max (set b))"
bitlen_Float: shows "bitlen \<bar>mantissa (Float m e)\<bar> + Float.exponent (Float m e) = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"
chainI: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b" assumes "class.po below"   and "\<And>i. below (Y i) (Y (Suc i))" shows "po.chain below Y"
preserves_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.hseq \<mu> \<nu>" shows "F (\<mu> \<star>\<^sub>C \<nu>) = \<Phi> (C.cod \<mu>, C.cod \<nu>) \<cdot>\<^sub>D (F \<mu> \<star>\<^sub>D F \<nu>) \<cdot>\<^sub>D D.inv (\<Phi> (C.dom \<mu>, C.dom \<nu>))"
iT_Minus_bound: fixes x :: "nat"   and k :: "nat"   and I :: "nat set" assumes "x \<in> k \<ominus> I" shows "x \<le> k"
mult_copy_first_matrix: fixes i :: "nat"   and nr :: "nat"   and v :: "real vec" assumes "i < nr"   and "0 < dim_vec v" shows "(copy_first_matrix nr (dim_vec v) *\<^sub>v v) $ i = v $ 0"
assms_3: shows "\<Gamma>2 = [xs [\<mapsto>] Bs]"
alw_nxt: fixes \<phi> :: "'a stream \<Rightarrow> bool" shows "alw \<phi> = (\<lambda>xs. \<phi> xs \<and> nxt (alw \<phi>) xs)"
op_list_of_eucl_image_def: fixes X :: "'c set" shows "op_list_of_eucl_image X = list_of_eucl ` X"
moebius_pt_moebius_rotation_zero_iff: fixes \<phi> :: "real"   and x :: "complex_homo" shows "(moebius_pt (moebius_rotation \<phi>) x = 0\<^sub>h) = (x = 0\<^sub>h)"
goal: shows "SecurityInvariant_withOffendingFlows.is_offending_flows_min_set sinvar_spec (set (TopoS_withOffendingFlows.minimalize_offending_overapprox (\<lambda>G. sinvar_spec G nP) (edgesL G) [] (list_graph_to_graph G))) (list_graph_to_graph G) nP"
is_scc_closed: fixes E :: "('b \<times> 'b) set"   and U :: "'b set"   and x :: "'b"   and y :: "'b" assumes "is_scc E U"   and "x \<in> U"   and "(x, y) \<in> E\<^sup>*"   and "(y, x) \<in> E\<^sup>*" shows "y \<in> U"
iso_cyclic_groups_generate: fixes a :: "'c"   and b :: "'e"   and H :: "('e, 'f) monoid_scheme" assumes "a \<in> carrier H"   and "b \<in> carrier H"   and "H.ord a = group.ord H b"   and "Group.group H" shows "{f. \<forall>k\<in>UNIV. f (a [^]\<^bsub>H\<^esub> k) = b [^]\<^bsub>H\<^esub> k} \<subseteq> Group.iso (H\<lparr>carrier := generate H {a}\<rparr>) (H\<lparr>carrier := generate H {b}\<rparr>)"
arr_cod: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.arr f" shows "S.arr (S.cod f)"
space_empty_eq_bot: fixes a :: "'a measure" shows "(space a = {}) = (a = \<bottom>)"
prod_mset_zero_iff: fixes 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 A :: "'a multiset" assumes "class.semidom plus minus zero times one" shows "(comm_monoid_mult.prod_mset times one A = zero) = (zero \<in># A)"
enum_set_get_one_None: fixes S :: "'b set"   and ss :: "'b list" assumes "S \<subseteq> set ss" shows "(enum_set_get_one ss S = None) = (S = {})"
InitSucState_StepConf: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and A :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<notin> Target TS"   and "A \<in> the (CompFun (HA ST) S)"   and "S \<notin> Conf ST"   and "S \<in> StepConf (HA ST) (Conf ST) TS" shows "InitState A \<in> StepConf (HA ST) (Conf ST) TS"
expand: fixes state :: "('a, 'b) state"   and state' :: "('a, 'b) state" assumes "runState state = runState state'" shows "state = state'"
aless_trans: fixes x :: "ant"   and y :: "ant"   and z :: "ant" assumes "x < y"   and "y < z" shows "x < z"
assms_2: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>"   and "c \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>" shows "a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c = a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> (b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> c)"
b'_nzero: shows "\<forall>x. b' < x \<and> x < b \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
eventually_at_topological: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" shows "eventually P (at a within s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
restr_ucon: fixes F :: "'a uprod set" assumes "F \<subseteq> E" shows "uconnected F = (Restr (uconnected F) verts)\<^sup>="
given_quot: fixes f :: "nat"   and l :: "nat" assumes "0 < f" shows "(f * l + (f - 1)) div f = l"
assms_1: shows "Ftor F : C \<longrightarrow> D"
update_by_fun_gr_pair_eq_update_by_fun_pair: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs" shows "comparator.update_by_fun_gr_pair comp k f xs = comparator.update_by_fun_pair comp k f xs"
l_set_shadow_root_get_tag_name_def: fixes set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_tag_name set_shadow_root_locs get_tag_name_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
IntI: fixes c :: "'h"   and A :: "'h set"   and B :: "'h set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
induct: fixes P :: "(name \<times> ty) list \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> ty \<Rightarrow> bool"   and a0 :: "(name \<times> ty) list"   and a1 :: "trm"   and a2 :: "trm"   and a3 :: "ty" assumes "\<And>\<Gamma> s t. P \<Gamma> s t TUnit"   and "\<And>\<Gamma> s t. P \<Gamma> s t TBase"   and "\<And>\<Gamma> s t T\<^sub>1 T\<^sub>2. \<lbrakk>\<And>x xa xb. \<lbrakk> \<Gamma> \<subseteq> x ; valid x\<rbrakk> \<Longrightarrow> P x xa xb T\<^sub>1; \<And>x xa xb. \<lbrakk> \<Gamma> \<subseteq> x ; valid x; x \<turnstile> xa is xb : T\<^sub>1\<rbrakk> \<Longrightarrow> P x (App s xa) (App t xb) T\<^sub>2\<rbrakk> \<Longrightarrow> P \<Gamma> s t (T\<^sub>1\<rightarrow>T\<^sub>2)" shows "P a0 a1 a2 a3"
fg: fixes x :: "'n" assumes "x \<in> cbox a b" shows "(\<lambda>k. f k x) \<longlonglongrightarrow> g x"
word_le_minus_one_leq: fixes x :: "'b word"   and y :: "'b word" assumes "x < y" shows "x \<le> y - 1"
lemma_Infinitesimal2: fixes x :: "real star" shows "(\<forall>r\<in>\<real>. 0 < r \<longrightarrow> x < r) = (\<forall>n. x < inverse (hypreal_of_nat (Suc n)))"
that: fixes point :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' x y subs = Some point"   and "fold (\<lambda>(T, U) subs. Option.bind subs (raw_match' T U)) (zip xs ys) (Some point) = Some subs'" shows "thesis"
assms: shows "H = {x \<in> A. (f x, x) \<in> r}"
cbv1: fixes v :: "lam"   and x :: "name"   and t :: "lam" assumes "val v"   and "x \<sharp> v" shows "APP LAM [x].t v \<longrightarrow>cbv t[x::=v]"