hcju/isabelleps huggingface dataset:
induct: fixes P :: "('a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> ('b, 'c) result list \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> ('b, 'c) result \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "('b, 'c) result list" assumes "\<And>Pa x xs y ys. (\<And>x1. y = Rval x1 \<Longrightarrow> P Pa xs ys) \<Longrightarrow> P Pa (x # xs) (y # ys)"   and "\<And>Pa. P Pa [] []"   and "\<And>Pa v va. P Pa [] (v # va)"   and "\<And>Pa v va. P Pa (v # va) []" shows "P a0 a1 a2"
LLL_measure_def: fixes n :: "nat"   and m :: "nat"   and \<alpha> :: "rat"   and i :: "nat"   and fs :: "int vec list" shows "LLL.LLL_measure n m \<alpha> i fs = 2 * LLL.logD n m \<alpha> fs + m - i"
is_lb_downward: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and l :: "'c"   and y :: "'c" assumes "class.po below"   and "po.is_lb below S l"   and "below y l" shows "po.is_lb below S y"
rev_less: fixes xs :: "nat list"   and ys :: "nat list" shows "(rev xs <\<^sub>v rev ys) = (xs <\<^sub>v ys)"
zero: fixes w :: "complex" assumes "w \<notin> U" shows "winding_number \<gamma> w = 0"
Entry_no_phis: fixes g :: "'g"   and v :: "'val" shows "phis g (Entry g, v) = None"
findim: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "FinDimVectorSpace smult V" shows "\<exists>ms. set ms \<subseteq> V \<and> R_scalar_mult.RSpan UNIV smult ms = V"
cblinfun_compose_image: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and S :: "'b ccsubspace" shows "(A o\<^sub>C\<^sub>L B) *\<^sub>S S = A *\<^sub>S B *\<^sub>S S"
mk_inter_coll_def: shows "mk_inter_coll \<equiv> (\<inter>)"
H_1: shows "\<not> (a = 0 \<and> b = 0 \<and> c = 0 \<and> m \<le> M)"
ilist_Previous_nth: fixes f :: "nat \<Rightarrow> 'd"   and init :: "'d"   and n :: "nat" shows "f\<^bsup>\<leftarrow> init\<^esup> n = (case n of 0 \<Rightarrow> init | Suc n' \<Rightarrow> f n')"
ide_compE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "CC.ide (CC.comp g f)" obtains "CC.seq g f" and "CC.seq f g" and "CC.comp g f = CC.dom f" and "CC.comp g f = CC.cod g"
less: fixes p :: "'a poly" assumes "degree p < degree pa__" shows "P p"
disj_zero_left: fixes x :: "'a" shows "(0::'a) OR x = x"
one_rat_def: shows "1 \<equiv> Abs_Rat (1, 1)"
trees_arity: fixes t :: "'a" assumes "t \<in> trees" shows "length (args t) = snd (root t)"
WV: shows "wv = mpath_weight' (snd \<sigma> v)"
curry_preserves_functors: fixes F :: "'a2 \<times> 'a1 \<Rightarrow> 'b" assumes "functor A2xA1.comp (\<cdot>\<^sub>B) F" shows "functor (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>[\<^sub>A\<^sub>1,\<^sub>B\<^sub>]) (local.curry F F F)"
i_2: shows "denormal_exponent TYPE(('e, 'f) IEEE.float) = Float.exponent f - int i"
atLeast0_atMost_Suc: fixes n :: "nat" shows "{0..Suc n} = insert (Suc n) {0..n}"
AB_3: fixes i :: "'a"   and n :: "nat" assumes "i \<in> Basis" shows "B (Suc n) \<bullet> i \<le> B n \<bullet> i"
hball_succ: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" shows "HBall (succ k) P = (P k \<and> HBall k P)"
IH_2: fixes ps :: "(real, 'a) vec list" assumes "invar r"   and "sorted_wrt_dist p ps"   and "set ps \<inter> set_kdt r = {}"   and "distinct ps"   and "0 < n" shows "\<forall>q\<in>set_kdt r \<union> set ps - set (nearest_nbors n ps p r). dist (last (nearest_nbors n ps p r)) p \<le> dist q p"
dependent_live_vars_lastnode: fixes V :: "'var"   and as' :: "'edge list"   and as :: "'edge list"   and n' :: "'node" assumes "(V, as', as) \<in> dependent_live_vars n'"   and "as \<noteq> []" shows "n' = last (targetnodes as)"
step_3_c_dims_1: fixes x :: "'b"   and l :: "nat"   and k :: "nat"   and i :: "(nat \<times> nat) list"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.step_3_c x l k i A) = dim_row A"
simps: fixes p :: "(action \<times> nat) list"   and off :: "nat" shows "tm_wf (p, off) = (2 \<le> length p \<and> is_even (length p) \<and> (\<forall>(a, s)\<in>set p. s \<le> length p div 2 + off \<and> off \<le> s))"
syzygy_module_iff: fixes s :: "'b \<Rightarrow>\<^sub>0 'a"   and B :: "'b set" shows "(s \<in> syzygy_module B) = represents B s (0::'b)"
le_diff_conv2: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "(c \<subseteq># b - a) = (c + a \<subseteq># b)"
crel_vsI: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "['b]" assumes "(state_dp_consistency.crel_vs R OO local.rel_state (=)) a b" shows "crel_vs R a b"
assms_3: shows "\<forall>S'\<in>C. \<forall>S\<subseteq>S'. S \<in> C"
w2_def: shows "w2 = (\<lambda>r. Re (winding_number (poly p \<circ> part_circlepath 0 r 0 pi) 0))"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "\<And>C. Prop C \<Psi> P M xvec N P'"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop C \<Psi> (Cases Cs) M xvec N P'"
char_eq_0: fixes a :: "'a" assumes "a \<notin> carrier G" shows "\<chi> a = 0"
rpr_less_trans: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and y :: "'a"   and z :: "'a"   and A :: "'a set" assumes "x \<^bsub>r\<^esub>\<prec> y"   and "y \<^bsub>r\<^esub>\<prec> z"   and "rpr A r" shows "x \<^bsub>r\<^esub>\<prec> z"
set_sort_oalist: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw ox xs" shows "set (sort_oalist ko xs) = set xs"
group_card_non_zero_groups_of_size_min: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and g :: "'b set"   and a :: "nat" assumes "group_design point_set block_collection groups"   and "g \<in> groups"   and "card g = a" shows "1 \<le> group_design.groups_of_size groups a"
tendsto_diff_smallo: fixes f1 :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and f2 :: "'b \<Rightarrow> 'c" assumes "(f1 \<longlongrightarrow> a) F"   and "f2 \<in> o[F](f1)" shows "((\<lambda>x. f1 x - f2 x) \<longlongrightarrow> a) F"
sup_ge1: fixes x :: "'e multiset"   and y :: "'e multiset" shows "x \<subseteq># x \<union># y"
klift_def: fixes f :: "'c \<Rightarrow> 'd set" shows "f\<^sup>\<dagger> = \<mu> \<circ> \<P> f"
s0_secrecy_def: shows "l1_secrecy \<equiv> {s. synth (analz (ik s)) \<inter> secret s = {}}"
ordinal_exp_monoL: fixes x :: "ordinal"   and x' :: "ordinal"   and y :: "ordinal" assumes "x \<le> x'" shows "x ** y \<le> x' ** y"
scaleRat_leq2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleRat :: "rat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and k :: "rat" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat"   and "less_eq a b"   and "k < 0" shows "less_eq (scaleRat k b) (scaleRat k a)"
bres_canc1: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> proto_near_quantale.bres Sup (\<le>) (\<sqinter>) x y \<le> y"
the_inv_into_G_binop_G: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and g :: "'a"   and x :: "'a" assumes "BinOpSetGroup G binop e"   and "g \<in> G"   and "x \<in> G" shows "the_inv_into G (binop g) x = binop (BinOpSetGroup.the_inverse G binop e g) x"
aux: fixes x :: "'a"   and l :: "'a list"   and y :: "'b"   and m :: "'b list" shows "hr_comp (list_assn A) (\<langle>B\<rangle>list_rel) (x # l) (y # m) = hr_comp A B x y * hr_comp (list_assn A) (\<langle>B\<rangle>list_rel) l m"
ACI_norm_rderivs: fixes w :: "'a list"   and r :: "'b rexp" shows "\<guillemotleft>rderivs w \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>rderivs w r\<guillemotright>"
image_mset_add_mset: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and M :: "'b multiset" shows "image_mset f (a, M) = f a, image_mset f M"
assoc: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" shows "signed.min (signed.min a b) c = signed.min a (signed.min b c)"
append_assoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs \<cdot> ys) \<cdot> zs = xs \<cdot> ys \<cdot> zs"
inv_greek_def: fixes s :: "(accent \<times> 'a) list" shows "inv_greek s = rev (map inv_letter s)"
finprod_insert: fixes F :: "'d set"   and a :: "'d"   and f :: "'d \<Rightarrow> 'a" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier R"   and "f a \<in> carrier R" shows "finsum R f (insert a F) = f a \<oplus> finsum R f F"
WbisT_trace2: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and tr :: "(('test, 'atom) com \<times> 'state) list \<times> 'state" assumes "c \<approx>wT d"   and "s \<approx> t"   and "finTrace tr"   and "iconfig tr = (c, s)" shows "\<exists>tr'. finTrace tr' \<and> iconfig tr' = (d, t) \<and> fstate tr \<approx> fstate tr'"
equivalence_in_bicategory_axioms: shows "D.internal_equivalence (\<tau>\<^sub>0 a) (map\<^sub>0 a) (unit a) (counit a)"
JVM_heap_axioms: shows "JVM_heap addr2thread_id thread_id2addr allocate typeof_addr heap_write P"
read_only_unowned_nth_update: fixes i :: "nat"   and \<O> :: "nat set"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "unit"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "\<O> \<inter> read_only \<S> = {}" shows "read_only_unowned \<S> (ts[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
interval_doublesplit: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real"   and e :: "real" assumes "k \<in> Basis" shows "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> e} = cbox (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) (c - e) else a \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) (c + e) else b \<bullet> i) *\<^sub>R i)"
local_flow_tank: fixes k :: "real" shows "local_flow (f k) UNIV UNIV (\<phi> k)"
rel_carrier_Eps_in_2: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and X :: "('c set \<times> 'd) set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "X \<in> dlim \<FF> \<rho> I" shows "(SOME x. x \<in> X) \<in> Sigma I \<FF>"
nFOR_Cons: fixes x :: "('a, 'b) aformula"   and xs :: "('a, 'b) aformula list" shows "nFOR (x # xs) = (if xs = [] then x else FOr x (nFOR xs))"
thetaSSeqI_Sbis: shows "thetaSSeqI \<subseteq> Sbis"
du: shows "dist u \<le> ereal (awalk_cost c p')"
double_dual_iso_eq_iff: fixes x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "x \<in> carrier (Characters G)"   and "y \<in> carrier (Characters G)" shows "(G'.double_dual_iso x = G'.double_dual_iso y) = (x = y)"
q3: fixes f :: "'a" assumes "\<forall>f. f \<in> M \<longrightarrow> (\<forall>c>0. 2 * Re \<langle>h - k, f\<rangle> \<le> c * \<parallel>f\<parallel>\<^sup>2)"   and "f \<in> M"   and "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) k" shows "\<forall>c>0. 2 * Re \<langle>h - k, f\<rangle> \<le> 0"
ABinds_restr_fresh: fixes S :: "Vars.var set"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "atom ` S \<sharp>* \<Gamma>" shows "ABinds \<Gamma>\<cdot>ae f|` (- S) = ABinds \<Gamma>\<cdot>(ae f|` (- S)) f|` (- S)"
1: shows "NT YMapInv C X F (YMap C X \<eta>) : YFtor C @@ X \<Longrightarrow> F"
ccomp_exits: fixes c :: "com" shows "exits (ccomp c) \<subseteq> {size (ccomp c)}"
elims: fixes x :: "nat"   and xa :: "'b list"   and y :: "'b list" assumes "delete_index x xa = y" obtains uu_ :: "nat" where "x = uu_" and "xa = []" and "y = []"   | xb :: "'b" and xs :: "'b list" where "x = 0" and "xa = xb__ # xsb__" and "y = xsb__"   | n :: "nat" and xb :: "'b" and xs :: "'b list" where "x = Suc n" and "xa = xb # xs" and "y = xb # delete_index n xs"
extremum_unique: fixes a :: "'k multiset" shows "(a \<subseteq># {#}) = (a = {#})"
sm: fixes bl :: "'a set" assumes "p \<notin> bl" shows "bl - {p} = bl"
SS1: shows "PRED \<subseteq>\<^sub>m PRED'"
mplus_assoc: fixes a :: "'a\<cdot>'b"   and b :: "'a\<cdot>'b"   and c :: "'a\<cdot>'b" shows "mplus\<cdot>(mplus\<cdot>a\<cdot>b)\<cdot>c = mplus\<cdot>a\<cdot>(mplus\<cdot>b\<cdot>c)"
paths_tri2: fixes ab :: "'a set"   and a :: "'a"   and b :: "'a"   and bc :: "'a set"   and c :: "'a"   and ca :: "'a set" assumes "path ab a b"   and "path bc b c"   and "path ca c a"   and "ab \<noteq> bc" shows "\<triangle> a b c"
psubstBinp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and binp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qAbs set option" shows "binp %%[rho] \<equiv> lift (psubstAbs rho) binp"
that: fixes g :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>n. g n \<in> borel_measurable (lebesgue_on S)"   and "\<And>n. finite (range (g n))"   and "\<And>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x" shows "thesis"
NSLIM_def: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and L :: "'b" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L = (\<forall>x. x \<noteq> star_of a \<and> x \<approx> star_of a \<longrightarrow> (*f* f) x \<approx> star_of L)"
g_def: shows "g \<equiv> smult (sgn (lead_coeff f)) f"
of_nat_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
inter_aform_plane_ortho_def: fixes p :: "nat"   and Z :: "'a \<times> 'a pdevs"   and n :: "'a"   and g :: "real" shows "inter_aform_plane_ortho p Z n g = those (map (\<lambda>b. bound_intersect_2d_ud p (inner2_aform Z n b) g) Basis_list) \<bind> (\<lambda>mMs. let l = sum_list (map2 (\<lambda>b m. m *\<^sub>R b) Basis_list (map fst mMs)); u = sum_list (map2 (\<lambda>b M. M *\<^sub>R b) Basis_list (map snd mMs)) in Some (aform_of_ivl l u))"
simps_3: fixes t :: "('a, 'b) Term.term"   and s :: "('a, 'b) Term.term"   and A :: "('a, 'b) stateful_strand_step list"   and I :: "'b \<Rightarrow> ('a, 'b) Term.term"   and D :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) list" shows "db'\<^sub>s\<^sub>s\<^sub>t (delete\<langle>t,s\<rangle> # A) I D = db'\<^sub>s\<^sub>s\<^sub>t A I (removeAll ((t, s) \<cdot>\<^sub>p I) D)"
\<sigma>: shows "\<theta> \<circ>\<^sub>s \<delta> \<circ>\<^sub>s \<sigma> = \<I>"
map_poly_hom_smult: fixes c :: "'b"   and p :: "'b poly" shows "map_poly to_fract (Polynomial.smult c p) = Polynomial.smult (to_fract c) (map_poly to_fract p)"
b_1: shows "exec_plan s as1 = exec_plan s as''"
M_poly_def: fixes p :: "int poly"   and x :: "int" shows "M_poly p x \<equiv> M (poly p x)"
max_bot: fixes x :: "'d upred" shows "ord.max (\<le>) false x = x"
in_restricted_global_in_unrestricted_global_1: fixes r' :: "'d"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and r :: "'d" assumes "r' \<in> RID\<^sub>G (s(r := None))" shows "r' \<in> RID\<^sub>G s"
IS_BELOW_ID_fun_rel_aux: assumes "Id \<subseteq> Id"   and "IS_BELOW_ID R2" shows "IS_BELOW_ID (Id \<rightarrow> R2)"
append_eq_symm: fixes t2 :: "(val \<times> val) list"   and t1 :: "(val \<times> val) list" shows "t2 @ t1 \<sim> t1 @ t2"
is_closed_eq: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and x :: "'a"   and x' :: "'a" assumes "equivalence S"   and "is_closed\<^bsub>S\<^esub> A"   and "x \<in> A"   and "x .=\<^bsub>S\<^esub> x'"   and "x' \<in> carrier S" shows "x' \<in> A"
frees_list_comb: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c"   and xs :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "frees (term.list_comb app t xs) = frees t |\<union>| pre_term.freess frees xs"
nodes_ins_nodes: fixes v :: "'b"   and g :: "'b ugraph" shows "nodes (ins_node v g) = insert v (nodes g)"
ind_hyp: shows "(k + l + 1) \<otimes> x = k \<otimes> x \<squnion> l \<otimes> x"
Scoped_Graph_axioms: shows "Scoped_Graph nodes inPorts outPorts vertices nodeOf edges hyps"
weakenQ: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and m :: "'b" assumes "Q \<sigma> m" shows "Q' \<sigma> m"
image_sub: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set"   and A1 :: "'h set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A" shows "f ` A1 \<subseteq> B"
bup: shows "eval (Cn 1 r_update [r_pdec1, r_pdec2, r_const 1]) [arg] \<down>= e_update b k 1"
6: shows "selected_edge h j g * forest_components f * selected_edge h j g = bot"
res: shows "LLL_Impl.basis_reduction_step \<alpha> n m upw i state = (upw', i', state')"
multiset_inter_single: fixes a :: "'f"   and b :: "'f" assumes "a \<noteq> b" shows "{#a#} \<inter># {#b#} = {#}"
det_upperdiagonal: fixes A :: "(('c, 'b) vec, 'b) vec" assumes "\<And>i j. j < i \<Longrightarrow> A $ i $ j = (0::'c)" shows "det A = (\<Prod>i\<in>UNIV. A $ i $ i)"
emeasure_completion_Un: fixes S :: "'b set"   and M :: "'b measure"   and T :: "'b set" assumes "S \<in> sets (completion M)"   and "T \<in> sets (completion M)" shows "emeasure (completion M) (S \<union> T) = emeasure M (main_part M S \<union> main_part M T)"
\<omega>_limit_set_in_compact_existence: fixes y :: "'a" assumes "y \<in> \<omega>_limit_set x" shows "existence_ivl0 y = UNIV"
bind_left_unit: fixes c :: "'f"   and P :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E (unit\<^sub>S\<^sub>E c) P = P c"
elims_1: fixes x :: "('c, 'd) strand_step"   and y :: "bool" assumes "simple\<^sub>s\<^sub>t\<^sub>p x = y" obtains t :: "('c, 'd) Term.term" where "x = receive\<langle>t\<rangle>\<^sub>s\<^sub>t" and "y"   | v :: "'d" where "x = send\<langle>Var v__\<rangle>\<^sub>s\<^sub>t" and "y"   | X :: "'d list" and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" where "x = \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t" and "y = (\<exists>\<I>. ineq_model \<I> X F)"   | va :: "'c" and vb :: "('c, 'd) Term.term list" where "x = send\<langle>Fun va__ vb__\<rangle>\<^sub>s\<^sub>t" and "\<not> y"   | v :: "poscheckvariant" and va :: "('c, 'd) Term.term" and vb :: "('c, 'd) Term.term" where "x = \<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t" and "\<not> y"
minimal_inD2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set"   and x :: "'b"   and z :: "'b" assumes "ordering less_eq less"   and "ordering.minimal_in less P x"   and "z \<in> P" shows "\<not> less z x"
hyps: fixes x :: "real" shows "sin (x + real n * (2 * pi)) = sin x"
one_dim_iso_adjoint: fixes A :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "one_dim_iso (A*) = one_dim_iso A*"
param_pattern: fixes f :: "recf"   and xs :: "nat list"   and p :: "abc_inst list"   and arity :: "nat"   and fp :: "nat" assumes "terminate f xs"   and "rec_ci f = (p, arity, fp)" shows "length xs = arity"
P_all: fixes x :: "'a"   and y :: "'b" assumes "G x y" shows "P x y"
relative_equality: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<squnion> z = y \<squnion> z"   and "x \<sqinter> z = y \<sqinter> z" shows "x = y"
Cauchy_derivative_integral_circlepath_2: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "w \<in> ball z r" shows "(f has_field_derivative 1 / (2 * complex_of_real pi * \<i>) * contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2)) (at w)"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<^bold>\<squnion> a \<^bold>\<squnion> b = a \<^bold>\<squnion> b"
UN: fixes K :: "'c set set"   and S :: "'c set set" assumes "\<And>k. k \<in> K \<Longrightarrow> generate_topology S k" shows "generate_topology S (\<Union> K)"
in_path_image_part_circlepath: fixes w :: "complex"   and z :: "complex"   and r :: "real"   and st :: "real"   and tt :: "real" assumes "w \<in> path_image (part_circlepath z r st tt)"   and "0 \<le> r" shows "cmod (w - z) = r"
subseqs_not_empty: fixes xs :: "'b list" shows "subseqs xs \<noteq> []"
in_set_gb_red_auxE: fixes h :: "'t \<Rightarrow>\<^sub>0 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) \<times> ('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e) list" assumes "h \<in> set (gb_red_aux bs ps)" obtains p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" where "(p, q) \<in> set ps" and "h = trdsp (map fst bs) (p, q)"
set_plus_rearrange3: fixes a :: "'c"   and B :: "'c set"   and C :: "'c set" shows "a +o B + C = a +o (B + C)"
mat_row_first_carrier: fixes A :: "'b mat"   and n :: "nat" shows "mat_row_first A n \<in> carrier_mat n (dim_col A)"
norm: shows "dg - J + (df - J) = df - J + (dg - J)"
h3: fixes G :: "'a \<Rightarrow> atom list \<times> atom fm list \<times> nat \<Rightarrow> bool" shows "(\<exists>x. \<exists>f\<in>set F. G x f) = (\<exists>f\<in>set F. \<exists>x. G x f)"
matrix_inv_Gauss: fixes A :: "(('a, 'b) vec, 'b) vec"   and P :: "(('a, 'b) vec, 'b) vec" assumes "invertible A"   and "Gauss_Jordan A = P ** A" shows "matrix_inv A = P"
complex_homo_coords_eq_transp: shows "transp (\<approx>)"
collapse_1: fixes code_status :: "'a code_status" assumes "is_cfailed code_status" shows "CFAILED (the_error code_status) = code_status"
setops\<^sub>s\<^sub>s\<^sub>t_pair_image_cons_6: fixes t :: "('fun, 'a) Term.term"   and s :: "('fun, 'a) Term.term"   and S :: "('fun, 'a) stateful_strand_step list" shows "pair ` setops\<^sub>s\<^sub>s\<^sub>t (delete\<langle>t,s\<rangle> # S) = {pair (t, s)} \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
3: shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G \<^bold>\<rightarrow> H"
IH_1: fixes i :: "nat"   and v :: "'a list" assumes "i < Monitor.progress \<sigma> \<phi>1 (plen \<pi>)" shows "MFOTL.sat \<sigma> v i \<phi>1 = MFOTL.sat \<sigma>' v i \<phi>1"
z: shows "(u \<cdot> v) \<^sup>@ k \<cdot> u = z"