hcju/isabelleps huggingface dataset:
nres_rel_trans1: fixes a :: "'a nres"   and b :: "'a nres"   and i :: "'b nres"   and R :: "('a \<times> 'b) set" assumes "a \<le> b"   and "(b, i) \<in> \<langle>R\<rangle>nres_rel" shows "(a, i) \<in> \<langle>R\<rangle>nres_rel"
connected_component_refl_eq: fixes S :: "'a set"   and x :: "'a" shows "connected_component S x x = (x \<in> S)"
rec_monoid_rng_of_frac_def: shows "rec_monoid_rng_of_frac \<equiv> \<lparr>carrier = set_class_of\<^bsub>rel\<^esub>, mult = mult_rng_of_frac, one = \<one> |\<^bsub>rel\<^esub> \<one>\<rparr>"
assms: shows "prefix\<cdot>xs\<cdot>ys = FF"
invpst_paint: fixes c :: "tcolor"   and t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" shows "invpst (paint c t) = invpst t"
external_from_internal_invariant: fixes S :: "('b, 'c) spec"   and P :: "'b set"   and Q :: "'c set" assumes "reach S \<subseteq> P"   and "obs S ` P \<subseteq> Q" shows "oreach S \<subseteq> Q"
prems_2: shows "P,hp s \<turnstile> lcl s (:\<le>) E"
null: shows "A - B \<union> (B - A) \<in> null_sets M"
wf2: shows "mat nr nc ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2))"
iso: shows "hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()) \<in> Group.iso (homology_group p X) (homology_group p (discrete_topology {()}))"
finite_mono_strict_prefix_implies_finite_fixpoint: fixes f :: "nat \<Rightarrow> 'a set"   and S :: "'a set" assumes "\<And>i. f i \<subseteq> S"   and "finite S"   and "\<exists>N. (\<forall>n\<le>N. \<forall>m\<le>N. m < n \<longrightarrow> f m \<subset> f n) \<and> (\<forall>n\<ge>N. f N = f n)" shows "f (card S) = \<Union> (range f)"
all_cong: fixes Q :: "'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and P' :: "'c \<Rightarrow> bool" assumes "\<And>x. Q x \<Longrightarrow> P x = P' x" shows "(\<forall>x. Q x \<longrightarrow> P x) = (\<forall>x. Q x \<longrightarrow> P' x)"
strip_while_Nil: fixes P :: "'f \<Rightarrow> bool" shows "strip_while P [] = []"
f5: shows "virt_to_phys addr (mmu (add_data_cache s1 addr data bm)) (mem (add_data_cache s1 addr data bm)) = virt_to_phys addr (mmu (add_data_cache s2 addr data bm)) (mem (add_data_cache s2 addr data bm))"
sup_right_zero_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<squnion> (- y \<squnion> - - y) = - z \<squnion> - - z"
SN_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r A" obtains "\<nexists>f. f 0 \<in> A \<and> (\<forall>i. (f i, f (Suc i)) \<in> r)"
abs_mult_self_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "times (abs a) (abs a) = times a a"
distinct_260: fixes x16 :: "Syntax.term"   and x11 :: "Syntax.term" shows "Unauth x16 \<noteq> Prj1 x11"
connected_iff_connected_component: fixes S :: "'c set" shows "connected S = (\<forall>x\<in>S. \<forall>y\<in>S. connected_component S x y)"
canonical_retraction_uniform_im: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set"   and x :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> B"   and "x \<in> B" shows "canonical_retraction A C ` x = the_apartment_iso B A ` x"
compositionality: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and multiset :: "'f multiset" shows "image_mset f (image_mset g multiset) = image_mset (f \<circ> g) multiset"
Cons_1: fixes kv :: "'a \<times> 'b" assumes "(0::'b) \<notin> snd ` set xs" shows "(0::'b) \<notin> snd ` set (update_by_pair kv xs)"
vector_smult_component: fixes c :: "'d"   and y :: "('d, 'e) Finite_Cartesian_Product.vec"   and i :: "'e" shows "(c *s y) $h i = c * y $h i"
res2: shows "wmax w1 w2 \<equiv>\<^sub>E real_of_int (sint (wmax w1 w2))"
signed_take_bit_eq_if_negative: 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "bit a n" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = or (take_bit n a) (not (mask n))"
hyps: shows "remdups l = foldr (\<lambda>i l. if i \<in> set l then l else i # l) l []"
assms: shows "as1 @ as2 \<in> valid_plans PROB"
lim_h_plus_OUT_IN: fixes x :: "'v" assumes "x \<noteq> source \<Delta>"   and "x \<noteq> sink \<Delta>" shows "KIR (\<lambda>e. \<Squnion>n. h_plus n e) x"
inputCases: fixes a :: "name"   and x :: "name"   and P :: "pi"   and b :: "subject"   and y :: "name"   and yP' :: "pi"   and Prop :: "subject \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool" assumes "a<x>.P \<longmapsto> b\<guillemotleft>y\<guillemotright> \<prec> yP'"   and "y \<noteq> a"   and "y \<noteq> x"   and "y \<sharp> P"   and "\<lbrakk>b = InputS a; yP' = [(x, y)] \<bullet> P\<rbrakk> \<Longrightarrow> Prop (InputS a) y ([(x, y)] \<bullet> P)" shows "Prop b y yP'"
ordLess_not_embed: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r <o r'" shows "\<nexists>f'. embed r' r f'"
mset_ran_subst_inside: fixes r :: "int set"   and i :: "int"   and a :: "int \<Rightarrow> 'b"   and x :: "'b" assumes "finite r"   and "i \<in> r" shows "mset_ran (a(i := x)) r = mset_ran a r + {#x#} - {#a i#}"
prems_2: shows "fmrestrict_set vs s = fmrestrict_set vs s'"
ih1: assumes "\<Gamma> \<turnstile> e\<^sub>1 : T'\<rightarrow>T" shows "\<Gamma> \<turnstile> Lam [x].e : T'\<rightarrow>T"
t0_space: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.t0_space open"   and "x \<noteq> y" shows "\<exists>U. open U \<and> (x \<in> U) \<noteq> (y \<in> U)"
subprob_not_empty: fixes M :: "'b pmf" shows "space (measure_pmf M) \<noteq> {}"
Open: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto> a[b] \<prec> P'"   and "a \<noteq> b" shows "<\<nu>b>P \<longmapsto> a<\<nu>b> \<prec> P'"
logic_actual_nec_3: fixes \<phi> :: "'a \<Rightarrow> \<o>" shows "[[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha>)]]"
dflt_image_PiE: fixes A :: "'b set"   and dflt :: "'c"   and B :: "'b \<Rightarrow> 'c set" shows "(\<lambda>h x. if x \<in> A then h x else dflt) ` Pi\<^sub>E A B = PiE_dflt A dflt B"
rqs_cost_exp_Suc: fixes n :: "nat" shows "rqs_cost_exp (Suc n) = real n + (\<Sum>i\<le>n. rqs_cost_exp i + rqs_cost_exp (n - i)) / real (Suc n)"
elims: fixes x :: "'a literal"   and y :: "'a literal" assumes "x\<^sup>c = y" obtains P :: "char list" and ts :: "'a list" where "x = Pos P__ ts__" and "y = Neg P__ ts__"   | P :: "char list" and ts :: "'a list" where "x = Neg P ts" and "y = Pos P ts"
Source_def: fixes T :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "Source T \<equiv> source ` T"
inner_scaleR_left: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and r :: "real"   and x :: "'a"   and y :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (scaleR r x) y = r * inner x y"
typeof_h_eq_NT: fixes h :: "'heap"   and v :: "'addr val" shows "(typeof\<^bsub>h\<^esub> v = \<lfloor>NT\<rfloor>) = (v = Null)"
iT_Plus_neg_cut_le2: fixes k :: "nat"   and t :: "nat"   and I :: "nat set" assumes "k \<le> t" shows "(I \<oplus>- k) \<down>\<le> (t - k) = I \<down>\<le> t \<oplus>- k"
rep_eq: fixes x :: "('a, 'b) lifted_pair" shows "Rep_lifted_pair (- x) = (\<lambda>f. triple.pairs_uminus (Rep_phi f) (Rep_lifted_pair x f))"
set_of_k_words_bound: fixes k :: "nat"   and m :: "nat" shows "real (card (set_of_k_words_length_m k m)) \<le> b ^ m"
varFIXvar: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" shows "PhiWhile b \<Phi> (s, t) = ((b, \<Phi>, s, t) \<in> var)"
continuous_of_real: fixes F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "continuous F g" shows "continuous F (\<lambda>x. of_real (g x))"
simps_18: fixes v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "'b" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> SKIP {vd}) = False"
independent_bound: fixes S :: "'b set" assumes "independent S" shows "finite S \<and> card S \<le> DIM('b)"
distinct_83: fixes x61 :: "'d set"   and x62 :: "('d, 'e, 'f) com"   and x8 :: "'d \<Rightarrow> ('d, 'e, 'f) com" shows "While x61 x62 \<noteq> DynCom x8"
fresh_5: fixes a :: "atom"   and tvar :: "Syntax.tvar"   and ty :: "ty" shows "a \<sharp> Mu tvar ty = (a \<sharp> ty \<or> a \<in> set [atom tvar])"
poly_power_card_as_sum_of_monoms: fixes h :: "'a mod_ring poly" shows "h ^ CARD('a) = (\<Sum>i\<le>degree h. monom (coeff h i) (CARD('a) * i))"
scc_rootI: fixes scc :: "'v set"   and v :: "'v"   and s :: "('v, 'a) state_scheme" assumes "is_scc E scc"   and "v \<in> dom (discovered s)"   and "v \<in> scc"   and "scc \<inter> dom (discovered s) \<subseteq> (tree_edges s)\<^sup>* `` {v}" shows "scc_root s v scc"
bind_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (rel_fset A) (rel_fun (rel_fun A (rel_fset B)) (rel_fset B)) fbind fbind"
fps_deriv_power: fixes a :: "'b fps"   and n :: "nat" shows "fps_deriv (a ^ n) = fps_const (of_nat n) * fps_deriv a * a ^ (n - 1)"
distinct_13: fixes x8 :: "'a" shows "OclAny \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
red_crit_lifting_family_empty_ord: fixes q :: "'q" assumes "q \<in> Q" shows "calculus Bot_F Inf_F (\<lambda>N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2)) (Red_I_\<G>_q q) (Red_F_\<G>_empty_q q)"
lincomb_list_Nil: fixes c :: "nat \<Rightarrow> 'a" shows "n.lincomb_list c [] = 0\<^sub>v n"
set_sel_6: fixes a :: "('a, 'b) tm"   and xk :: "'b" assumes "is_App a"   and "xk \<in> vars (arg a)" shows "xk \<in> vars a"
\<omega>_limit_set_in_compact_nonempty: fixes K :: "'a set"   and x :: "'a" assumes "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "rev.trapped_forward x K" shows "rev.\<omega>_limit_set x \<noteq> {}"
p'_continuous: fixes T :: "real set"   and a :: "real"   and v :: "real" shows "continuous_on T (movement.p' a v)"
antisym_conv2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "less_eq x y" shows "(\<not> less x y) = (x = y)"
mignotte_bound: fixes f :: "int poly"   and g :: "int poly"   and n :: "nat"   and k :: "nat" assumes "f \<noteq> 0"   and "g dvd f"   and "degree g \<le> n" shows "\<bar>poly.coeff g k\<bar> \<le> mignotte_bound f n"
map_id: fixes t :: "'a \<Rightarrow> 'b" shows "id \<cdot> t = t"
vimage_empty: fixes f :: "'b \<Rightarrow> 'c" shows "f -` {} = {}"
known_ptrs_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "object_ptr_kinds h = object_ptr_kinds h'" shows "known_ptrs h = known_ptrs h'"
fls_subdegree_leI: fixes f :: "'b fls"   and n :: "int" assumes "fls_nth f n \<noteq> (0::'b)" shows "fls_subdegree f \<le> n"
cfs_monom_mult_r: fixes p :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and k :: "nat" assumes "p \<in> carrier P"   and "a \<in> carrier R" shows "(monom P a n \<otimes>\<^bsub>P\<^esub> p) (k + n) = a \<otimes> p k"
fmdomE: fixes x :: "'a"   and m :: "'a \<rightharpoonup>\<^sub>f 'b" assumes "x |\<in>| fmdom m" obtains y :: "'b" where "fmlookup m x = Some y"
get_nth_im1_def: fixes ba :: "'a list \<times> 'a list" shows "get_nth_im1 ba = hd (fst ba)"
prime_factors_gauss_int_def: fixes z :: "gauss_int" shows "prime_factors_gauss_int z = (if z = 0 then {} else Set.filter (\<lambda>p. p dvd z) (prime_factors_gauss_int_of_nat (gauss_int_norm z)))"
exhaust: fixes y :: "D2" obtains "y = D1"   | "y = D2"
trace_parallel_plan_strips_last_cons_then: fixes s :: "'a \<Rightarrow> bool option"   and s' :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" shows "last (s # trace_parallel_plan_strips s' \<pi>) = last (trace_parallel_plan_strips s' \<pi>)"
fminus_finsert: fixes A :: "'a fset"   and a :: "'a"   and B :: "'a fset" shows "A |-| finsert a B = A |-| B |-| {|a|}"
lim_Array_set: fixes a :: "'a array"   and xs :: "'a list"   and h :: "heap" shows "lim (Array.set a xs h) = lim h"
tendsto_poly: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and p :: "'b poly" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. poly p (f x)) \<longlongrightarrow> poly p a) F"
constmap_def: fixes A :: "'e set"   and B :: "'f set" shows "constmap A B = (\<lambda>x\<in>A. SOME y. y \<in> B)"
abs_eq: fixes xa :: "ereal"   and x :: "ereal" assumes "eq_onp ((\<le>) 0) xa xa"   and "eq_onp ((\<le>) 0) x x" shows "e2ennreal xa + e2ennreal x = e2ennreal (xa + x)"
wf_prog_lift: fixes A :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and B :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool" assumes "wf_prog A P"   and "\<And>wf_md C M Ts Ca T m. \<lbrakk>wf_prog wf_md P; P \<turnstile> Ca sees M: Ts\<rightarrow>T = \<lfloor>m\<rfloor> in Ca; is_class P Ca; set Ts \<subseteq> types P; A P Ca (M, Ts, T, m)\<rbrakk> \<Longrightarrow> B P Ca (M, Ts, T, m)" shows "wf_prog B P"
not_weak_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> Collect derivable"   and "y \<in> Collect derivable" shows "(\<not> y \<sqsubseteq> x) = (x \<sqsubseteq> y \<and> x \<noteq> y)"
prm_compose_def: shows "(\<diamondop>) \<equiv> map_fun rep_prm (map_fun rep_prm abs_prm) preprm_compose"
map_replicate_const: fixes k :: "'g"   and lst :: "'h list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
norm_diff_triangle_le: fixes x :: "'a"   and y :: "'a"   and e1 :: "real"   and z :: "'a"   and e2 :: "real" assumes "norm (x - y) \<le> e1"   and "norm (y - z) \<le> e2" shows "norm (x - z) \<le> e1 + e2"
hyps: shows "\<exists>P. invertible P \<and> P ** A = echelon_form_of_upt_k A k bezout"
Usingle: fixes u :: "ZF" assumes "Elem u U" shows "Elem (Singleton u) U"
solve_rat_fps_aux: fixes cs :: "('a \<times> nat) list"   and a :: "'a poly"   and zs :: "'a list list"   and p :: "'a poly" assumes "distinct (map fst cs)"   and "(a, zs) = poly_pfd_simple p cs"   and "(0::'a) \<notin> fst ` set cs" shows "fps_of_poly p / fps_of_poly (\<Prod>(c, n)\<leftarrow>cs. [:1::'a, - c:] ^ Suc n) = Abs_fps (\<lambda>k. coeff a k + (\<Sum>i<length cs. poly (\<Sum>j\<le>snd (cs ! i). inverse_irred_power_poly (zs ! i ! j) (snd (cs ! i) + 1 - j)) (of_nat k) * fst (cs ! i) ^ k))"
is_arg_max_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" shows "ord.is_arg_max less f P x = (P x \<and> (\<nexists>y. P y \<and> less (f x) (f y)))"
assms_5: shows "\<forall>p. prop_invs (prop_config c0 p)"
less_eq_inf_bot: fixes x :: "'a"   and y :: "'a" shows "(- x \<le> - y) = (- x \<sqinter> - - y = bot)"
disc_2: fixes p :: "'b \<Rightarrow> bool"   and a :: "'b"   and g21 :: "'b \<Rightarrow> 'c"   and g22 :: "'b \<Rightarrow> 'b" assumes "\<not> p a" shows "\<not> lnull (unfold_llist p g21 g22 a)"
Gauss_Jordan_in_ij_det_P_iarrays_def: fixes A :: "'d iarray iarray"   and i :: "nat"   and j :: "nat" shows "Gauss_Jordan_in_ij_det_P_iarrays A i j = (let n = least_non_zero_position_of_vector_from_index (column_iarray j A) i in (if i = n then (1::'d) / A !! i !! j else - (1::'d) / A !! n !! j, Gauss_Jordan_in_ij_iarrays A i j))"
rank_dim_range: fixes A :: "((real, 'a) vec, 'b) vec" shows "rank A = dim (range ((*v) A))"
simps_2: fixes x :: "'b"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and uv :: "nat"   and r :: "('b \<times> nat) tree" shows "delete x \<langle>l, (a, uv), r\<rangle> = (case cmp x a of LT \<Rightarrow> balanceL (delete x l) a r | EQ \<Rightarrow> combine l r | GT \<Rightarrow> balanceR l a (delete x r))"
essentially_equal_def: fixes G :: "('c, 'd) monoid_scheme"   and fs1 :: "'c list"   and fs2 :: "'c list" shows "essentially_equal G fs1 fs2 = (\<exists>fs1'. fs1 <~~> fs1' \<and> fs1' [\<sim>]\<^bsub>G\<^esub> fs2)"
y2prop: shows "r < y2' \<and> (\<forall>x\<in>{r<..y2'}. a * x\<^sup>2 + b * x + c \<noteq> 0)"
delta_pos_neg_least_1: fixes rm :: "('i \<Rightarrow> 'd) admS"   and rp :: "('i \<Rightarrow> 'd) admS" assumes "rm \<le> F (dual rp, rm)"   and "F (dual rm, rp) \<le> rp" shows "delta_neg \<le> dual rm"
PLS_NEMPTY: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "PLS s as \<noteq> {}"
gen_dfs_outer_def: fixes E :: "('a \<times> 'a) set"   and U :: "'a set"   and V0 :: "'a set"   and it :: "'a set"   and V :: "'a set"   and brk :: "bool" shows "gen_dfs_outer E U V0 it V brk \<equiv> V0 \<subseteq> U \<and> E `` U \<subseteq> U \<and> finite U \<and> V \<subseteq> U \<and> (\<not> brk \<longrightarrow> E `` V \<subseteq> V) \<and> V0 - it \<subseteq> V"
sorted_wrt_irrefl_distinct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "irreflp r" shows "sorted_wrt r xs \<longrightarrow> distinct xs"
nprv_eql_substE012: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set"   and \<phi> :: "'fmla"   and y :: "'var"   and \<psi> :: "'fmla" assumes "eql t1 t2 \<in> F"   and "subst \<phi> t2 y \<in> F"   and "\<psi> = subst \<phi> t1 y \<or> \<psi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "\<phi> \<in> fmla"   and "y \<in> var" shows "nprv F \<psi>"
mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d" shows "f a c \<le> f b d"
Src_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "Src (Dom t) = Src t"
map_pmf_idI: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> 'c" assumes "\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = x" shows "map_pmf f p = p"
AE_ball_countable': fixes I :: "'b set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c measure" assumes "\<And>N. N \<in> I \<Longrightarrow> almost_everywhere M (P N)"   and "countable I" shows "AE x in M. \<forall>N\<in>I. P N x"
Sum_any_right_distrib: fixes g :: "'a \<Rightarrow> 'b"   and r :: "'b" assumes "finite {a. g a \<noteq> (0::'b)}" shows "r * Sum_any g = (\<Sum>n. r * g n)"
xnlrs: assumes "rs \<noteq> []" shows "x \<noteq> last rs"
StableI: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> A Co A" shows "F \<in> Stable A"
simps_1: fixes p :: "linear_poly"   and a :: "'a" shows "poly (LEQ_ns p a) = p"
list_rec_Cons: fixes a :: "'c"   and l :: "'c SList.list"   and c :: "'b"   and h :: "'c \<Rightarrow> 'c SList.list \<Rightarrow> 'b \<Rightarrow> 'b" shows "list_rec (a # l) c h = h a l (list_rec l c h)"
step: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
weak_le_iff_meet: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_lattice L"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq>\<^bsub>L\<^esub> y) = (x \<squnion>\<^bsub>L\<^esub> y .=\<^bsub>L\<^esub> y)"
type1_def: fixes init :: "'a list"   and x :: "'a"   and y :: "'a" shows "type1 init x y = bernoulli_pmf (5 / 10) \<bind> (\<lambda>a. bernoulli_pmf (5 / 10) \<bind> (\<lambda>b. return_pmf (if \<not> [a, b] ! index init x \<and> [a, b] ! index init y then ([y, x], [a, b], init) else ([x, y], [a, b], init))))"
sum_mset_constant: fixes y :: "'b"   and A :: "'c multiset" shows "(\<Sum>x\<in>#A. y) = of_nat (size A) * y"
not_le_imp_less: fixes y :: "'a list"   and x :: "'a list" assumes "\<not> y \<le>lex x" shows "x <lex y"
cases: fixes x :: "'b eclause" obtains C :: "'b literal set" and X :: "'b trm set" where "x = Ecl C X"
ac: shows "absolutely_continuous (distr M M' T) (distr N M' T)"
lens_indep_sym: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "x \<bowtie> y" shows "y \<bowtie> x"
Span_smult_closed: fixes r :: "'f"   and ms :: "'v list"   and n :: "'v" assumes "r \<in> UNIV"   and "set ms \<subseteq> V"   and "n \<in> RSpan ms" shows "r \<cdot> n \<in> RSpan ms"
that_1: fixes ti :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> ti"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, ti\<rangle> \<Rightarrow> t"   and "no_matching_Goto \<gamma> p rs\<^sub>1" shows "thesis"
language_elim: fixes w :: "'a stream"   and A :: "('a, 'b) nba" assumes "w \<in> language A" obtains r :: "'b stream" and p :: "'b" where "p \<in> initial A" and "run A (w ||| r) p" and "infs (accepting A) (p ## r)"
separatedin_subtopology: fixes X :: "'c topology"   and U :: "'c set"   and S :: "'c set"   and T :: "'c set" shows "separatedin (subtopology X U) S T = (S \<subseteq> U \<and> T \<subseteq> U \<and> separatedin X S T)"
sl_cs_1: fixes cs :: "'a list" assumes "cs \<in> set css" shows "perm_restrict f (set cs) = list_succ (some_list cs)"
mult_not_zero: fixes a :: "'c"   and b :: "'c" assumes "a * b \<noteq> bot" shows "a \<noteq> bot \<and> b \<noteq> bot"
nat_of_plus_sat: fixes x :: "'b sat"   and y :: "'b sat" shows "nat_of (x + y) = min (nat_of x + nat_of y) LENGTH('b)"
wrap_def: fixes B :: "'c set" shows "wrap B = (if B = {} then {} else {B})"
AboveS_def: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "AboveS r A \<equiv> {b \<in> Field r. \<forall>a\<in>A. b \<noteq> a \<and> (a, b) \<in> r}"
incomparable_minimal_antichain: fixes A :: "'b set" shows "incomparable (minimal_antichain A)"
simps_1: fixes M :: "('fun, 'var) Term.term set" shows "\<lbrakk>M; []\<rbrakk>\<^sub>c = (\<lambda>\<I>. True)"
mem_ass_forms: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and a :: "'b" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions" shows "(a |\<in>| Abstract_Task.ass_forms assumptions) = (a \<in> set assumptions)"
simple_rotation_bij_corollary: fixes r :: "Vertex \<Rightarrow> Vertex"   and x :: "Vertex"   and y :: "Vertex" assumes "r \<in> simple_rotations" shows "(r x \<noteq> r y) = (x \<noteq> y)"