hcju/isabelleps huggingface dataset:
simps: fixes C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and proto :: "8 word"   and pts :: "(16 word \<times> 16 word) list" shows "l4_ports_negate_one C (L4Ports proto pts) = MatchOr (MatchNot (Match (Prot (Proto proto)))) (Match (C (L4Ports proto (raw_ports_invert pts))))"
take_update_swap: fixes m :: "nat"   and xs :: "'c list"   and n :: "nat"   and x :: "'c" shows "take m (xs[n := x]) = (take m xs)[n := x]"
insert_def: shows "Challenge1_short.insert \<equiv> \<lambda>uu uua. rec_textbuffer (\<lambda>p t x. BUF (p + 1) (take p t @ x # drop p t)) uua uu"
filterlim_INF: fixes f :: "'c \<Rightarrow> 'd"   and G :: "'e \<Rightarrow> 'd filter"   and B :: "'e set"   and F :: "'c filter" shows "filterlim f (Inf (G ` B)) F = (\<forall>b\<in>B. filterlim f (G b) F)"
simps_2: fixes E :: "'a uprod set"   and u :: "'a"   and x :: "'a uprod"   and xs :: "'a uprod list"   and v :: "'a" shows "epath E u (x # xs) v = ((\<exists>w. u \<noteq> w \<and> Upair u w = x \<and> epath E w xs v) \<and> x \<in> E)"
assms: shows "g1 +++ g2 piecewise_differentiable_on {0..1}"
join: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "rha a = rha b"   and "a \<in> A" shows "\<exists>ab. ma a b = Some ab \<and> boa a ab \<and> boa b ab \<and> (\<forall>u. boa a u \<longrightarrow> boa b u \<longrightarrow> boa ab u)"
lconst_unfold: fixes a :: "'b" shows "lconst a = a ## lconst a"
that: fixes val :: "'val" assumes "var g val = v"   and "phis g (n, val) \<noteq> None" shows "thesis"
sem: shows "\<forall>I \<nu>. is_interp I \<longrightarrow> dterm_sem I \<theta> \<nu> = dterm_sem I \<theta>' \<nu>"
leg_simps_1: shows "C.arr \<epsilon>.dom.leg0"
similar_mat_wit_pow: fixes A :: "'a mat"   and B :: "'a mat"   and P :: "'a mat"   and Q :: "'a mat"   and k :: "nat" assumes "similar_mat_wit A B P Q" shows "similar_mat_wit (A ^\<^sub>m k) (B ^\<^sub>m k) P Q"
t_new: fixes CT :: "nat \<Rightarrow> classDef option"   and C :: "nat"   and Df :: "varDef list"   and es :: "exp list"   and Ds :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and Cs :: "nat list" assumes "fields(CT,C) = Df"   and "length es = length Df"   and "varDefs_types Df = Ds"   and "CT;\<Gamma> \<turnstile>+ es : Cs"   and "CT \<turnstile>+ Cs <: Ds" shows "CT;\<Gamma> \<turnstile> New C es : C"
un_ui_le: fixes a :: "'b word"   and b :: "'b word" shows "(unat a \<le> unat b) = (uint a \<le> uint b)"
some_geodesic_commute: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "{x--S--y} = {y--S--x}"
Borsuk_homotopy_extension_homotopic: fixes T :: "'a set"   and S :: "'a set"   and U :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "closedin (top_of_set T) S"   and "ANR S \<and> ANR T \<or> ANR U"   and "continuous_on T f"   and "f ` T \<subseteq> U"   and "homotopic_with_canon (\<lambda>x. True) S U f g" obtains g' :: "'a \<Rightarrow> 'b" where "homotopic_with_canon (\<lambda>x. True) T U f g'" and "continuous_on T g'" and "g' ` T \<subseteq> U" and "\<And>x. x \<in> S \<Longrightarrow> g' x = g x"
t_mtf_MTF: fixes n :: "nat"   and s :: "'a list" assumes "n < length rs" shows "t_mtf n = int (t (s_mtf n) (rs ! n) (off MTF s rs ! n))"
P_def: shows "P = (\<lambda>zs a. \<not> (\<exists>z\<in>set zs. rel z a))"
hom_prod: fixes f :: "'b \<Rightarrow> rat poly"   and X :: "'b set" shows "map_poly of_rat (prod f X) = (\<Prod>x\<in>X. map_poly of_rat (f x))"
VDec: shows "initReachable c \<and> <\<bottom>, outM v> \<in># msgs c"
img: shows "g ` ({x. b \<bullet> x = 0} \<inter> T) = {x. norm x = 1 \<and> b \<bullet> x \<noteq> 1} \<inter> T"
Int_Un_distrib: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<inter> (B \<union> C) = A \<inter> B \<union> A \<inter> C"
arctan_eq_zero_iff: fixes x :: "real" shows "(arctan x = 0) = (x = 0)"
d_def_split: fixes x :: "'a" shows "d (x \<sqinter> nc) + d (x \<cdot> (0::'a)) = d x"
ln2: shows "ln 2 \<in> {2 / 3..25 / 36}"
chine_eq_apex: shows "HH\<mu>\<nu>\<pi>.chine = HH\<mu>\<nu>\<pi>.apex"
Liminf_grounding_of_state_ground: fixes C :: "'a literal multiset" assumes "C \<in> Liminf_llist (lmap (\<lambda>St. grounding_of_clss (N_of_state St \<union> P_of_state St \<union> Q_of_state St)) Sts)" shows "is_ground_cls C"
simple_function_finite: fixes f :: "'key \<times> 'message list \<Rightarrow> 'c" shows "simple_function \<mu> f"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" obtains "B.arr f" and "B.dom f = a" and "B.cod f = b"
cfun_cong: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "f = g"   and "x = y" shows "f\<cdot>x = g\<cdot>y"
simps_8: fixes v1 :: "'b"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (V v1) (Plus v va) = True"
sum_distrib_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and r :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "class.semiring_0 plus zero times" shows "times r (comm_monoid_add.sum plus zero f A) = comm_monoid_add.sum plus zero (\<lambda>n. times r (f n)) A"
collapse: fixes list :: "'f list" assumes "list \<noteq> []" shows "hd list # tl list = list"
code: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "ltakeWhile P xs = (if lnull xs \<or> \<not> P (lhd xs) then LNil else LCons (lhd xs) (ltakeWhile P (ltl xs)))"
coerce_cfun: fixes f :: "'e \<rightarrow> 'f" shows "COERCE('e \<rightarrow> 'f, 'c \<rightarrow> 'd)\<cdot>f = COERCE('f, 'd) oo f oo COERCE('c, 'e)"
subst_cls_lists_append: fixes Cs :: "'a literal multiset list"   and \<sigma>s :: "'s list"   and Cs' :: "'a literal multiset list"   and \<sigma>s' :: "'s list" assumes "length Cs = length \<sigma>s"   and "length Cs' = length \<sigma>s'" shows "(Cs @ Cs') \<cdot>\<cdot>cl (\<sigma>s @ \<sigma>s') = Cs \<cdot>\<cdot>cl \<sigma>s @ Cs' \<cdot>\<cdot>cl \<sigma>s'"
LeadsETo_subset_LeadsTo: fixes A :: "'d set"   and CC :: "'d set set"   and B :: "'d set" shows "A LeadsTo[CC] B \<subseteq> A \<longmapsto>w B"
awalk_verts_non_Nil: fixes u :: "'a"   and p :: "('a \<times> 'a) list" shows "pawalk_verts u p \<noteq> []"
0: fixes u :: "'a local.term" assumes "Diag t"   and "Diag u" shows "Diag (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<and> Dom (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Dom u \<and> Cod (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> Cod u"
hyp2: shows "matrix_to_iarray (snd (Gauss_Jordan_upt_k_det_P A k)) = snd (Gauss_Jordan_upt_k_det_P_iarrays (matrix_to_iarray A) k)"
lunit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<l>[a] : \<I> \<otimes> a \<rightarrow> a\<guillemotright>"
Bv: fixes x :: "'a" shows "(\<lambda>n. cblinfun_apply (X n) x) \<longlonglongrightarrow> cblinfun_apply (CBlinfun v) x"
unity_def: shows "M.unity \<equiv> S.cod \<i>[a]"
integral_indicator: fixes M :: "'b measure"   and A :: "'b set" shows "integral\<^sup>L M (indicat_real A) = Sigma_Algebra.measure M (A \<inter> space M)"
continuous_at_ennreal: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> real" assumes "continuous F f" shows "continuous F (\<lambda>x. ennreal (f x))"
bounded_clinear_intro: fixes f :: "'d \<Rightarrow> 'e"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>C x) = r *\<^sub>C f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_clinear f"
apply_ocrit_connectible: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and m :: "nat"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and ps' :: "(bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'e"   and ps :: "((('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list" assumes "ocrit_spec crit"   and "dickson_grading d"   and "set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "unique_idx (p # q # hs @ map (fst \<circ> snd) ps' @ map (snd \<circ> snd) ps') data"   and "\<And>p' q'. \<lbrakk>(p', q') \<in> snd ` set ps'; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "p \<in> B"   and "q \<in> B"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "(p, q) \<in> set ps"   and "(p, q) \<notin> set (apply_ocrit crit data hs ps' ps)" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
1: shows "take_cols A (map snd (pivot_positions A)) = 1\<^sub>m (length (pivot_positions A)) @\<^sub>r 0\<^sub>m (dim_row A - length (pivot_positions A)) (length (pivot_positions A))"
comparables_empty: fixes M :: "nat set" shows "comparables {} M = Fpow M"
cross3_simps_9: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b + (a + c) = a + (b + c)"
lm069: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G"   and "finite G" shows "sum (sum bids) (omega ` a) = sum bids (\<Union> (omega ` a))"
prod_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq zero (f i) \<and> less_eq (f i) (g i)" shows "less_eq (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
map_add_empty: fixes m :: "'c \<Rightarrow> 'd option" shows "m ++ Map.empty = m"
behaves_def: shows "L1.behaves = load1 OO L1.sem_behaves"
left_def: shows "left = (\<lambda>f. \<Sum>x | jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b. jumpF f (at_left x))"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "ext_irrefl ext"   and "ext_irrefl_before_trans_axioms ext" shows "ext_irrefl_before_trans ext"
not_Cons_self: fixes xs :: "'g list"   and x :: "'g" shows "xs \<noteq> x # xs"
rewrite_X_enat_correct: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> = w \<Turnstile>\<^sub>n to_ltln (rewrite_X_enat \<phi>)"
le_add_diff_inverse2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq b a" shows "plus (minus a b) b = a"
simps_1: fixes at :: "('b, 'c) atm" shows "nvL (Pos at) = nvA at"
mult_strict_right_mono_neg: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b < a"   and "c < (0::'a)" shows "a * c < b * c"
galois_connection_def: fixes \<alpha> :: "'b \<Rightarrow> 'c"   and \<gamma> :: "'c \<Rightarrow> 'b" shows "galois_connection \<alpha> \<gamma> \<equiv> \<forall>c a. (c \<le> \<gamma> a) = (\<alpha> c \<le> a)"
VEnd_end_smult_VEnd: fixes a :: "'f" shows "VEnd (a \<cdot>\<cdot> T)"
sorted_list_of_set_numbers_of_marks: fixes n :: "nat"   and bs :: "bool list" shows "sorted_list_of_set (numbers_of_marks n bs) = map fst (filter snd (enumerate n bs))"
valid_path_cons_simp: fixes G :: "('c, 'd) Graph_scheme"   and x :: "'c"   and xs :: "'c llist" assumes "Digraph G" shows "Digraph.valid_path G (LCons x xs) = (xs = LNil \<and> x \<in> V\<^bsub>G\<^esub> \<or> x \<in> V\<^bsub>G\<^esub> \<and> lhd xs \<in> V\<^bsub>G\<^esub> \<and> x \<rightarrow>\<^bsub>G\<^esub> lhd xs \<and> Digraph.valid_path G xs \<and> \<not> lnull xs)"
get_attribute_pure: fixes ptr :: "(_) element_ptr"   and k :: "char list"   and h :: "(_) heap" shows "pure (get_attribute ptr k) h"
mini_stable: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd"   and A :: "nat set"   and k :: "'d" assumes "xs \<noteq> []"   and "mini xs key \<in> A" shows "filter (\<lambda>x. key x = k) ([xs ! mini xs key] @ nths xs (A - {mini xs key})) = filter (\<lambda>x. key x = k) (nths xs A)"
mem_agree_def: fixes bs :: "mem"   and m :: "nat" shows "mem_agree bs m = (m \<le> mem_size bs)"
edge_det: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and a :: "'f"   and a' :: "'f" assumes "CFG sourcenode targetnode valid_edge Entry"   and "valid_edge a"   and "valid_edge a'"   and "sourcenode a = sourcenode a'"   and "targetnode a = targetnode a'" shows "a = a'"
sum_list_mult_const_prod: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and xys :: "('b \<times> 'c) list"   and r :: "'a" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) * r = (\<Sum>(x, y)\<leftarrow>xys. f x y * r)"
representsE_full: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" obtains "p \<noteq> 0" and "ipoly p x = (0::'b)" and "Polynomial.degree p \<noteq> 0"
preferred_alts_subset: fixes x :: "'a" shows "preferred_alts le x \<subseteq> carrier"
rep_eq: shows "blinfun_apply embed1_blinfun = (\<lambda>x. (x, 0::'c))"
termi: shows "\<Gamma>\<turnstile>Catch c1' c2' \<down> Normal s"
exists_least_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "(\<exists>n. P n) = (\<exists>n. P n \<and> (\<forall>m. less m n \<longrightarrow> \<not> P m))"
assms_1: shows "[a] \<preceq>\<^sub>F u @ v"
inverse_mod_def: fixes x :: "int"   and m :: "int" shows "inverse_mod x m = fst (bezout_coefficients x m)"
dual_weak_lattice: shows "weak_lattice (inv_gorder (inv_gorder L))"
is_gcdE: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "is_gcd x a b" obtains "x dvd a" and "x dvd b" and "\<And>y. \<lbrakk>y dvd a; y dvd b\<rbrakk> \<Longrightarrow> y dvd x"
split_bst'_altdef: fixes x :: "'a"   and t :: "'a tree" shows "split_bst' x t = (isin t x, split_bst x t)"
Enc_synth_eq2: fixes X :: "msg"   and H :: "msg set"   and K :: "msg" assumes "X \<notin> synth H" shows "(Enc X K \<in> synth H) = (Enc X K \<in> H)"
simps_2: fixes start :: "nat"   and i :: "nat" shows "Goodstein_Sequence.goodstein start (Suc i) = decode (i + 3) 0 (encode (i + 2) 0 (Goodstein_Sequence.goodstein start i)) - 1"
derivable_clauses_are_well_constrained: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and P :: "'b eclause set"   and S :: "'b eclause set"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "basic_superposition.derivable trm_ord sel pos_ord filter_trms C P S \<sigma> k C'" shows "basic_superposition.well_constrained trm_ord C"
frequently_imp_iff: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b filter" shows "(\<exists>\<^sub>F x in F. P x \<longrightarrow> Q x) = (eventually P F \<longrightarrow> frequently Q F)"
irrefl: fixes xs :: "'a list"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> ext gt xs xs"
valid2: shows "valid_net' (insert_weights shared_weights (shallow_model (rs ! 0) Z (last rs) (2 * N_half - 1)) ws)"
finite_gpv_rel''D1: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'i, 'k) gpv"   and gpv' :: "('h, 'j, 'l) gpv"   and \<I> :: "('i, 'k) \<I>"   and \<I>' :: "('j, 'l) \<I>" assumes "rel_gpv'' A C R gpv gpv'"   and "finite_gpv \<I> gpv"   and "rel_\<I> C R \<I> \<I>'" shows "finite_gpv \<I>' gpv'"
fresh_star_bij_2: fixes pi :: "(name \<times> name) list"   and b :: "name list"   and x :: "'e" shows "(pi \<bullet> b) \<sharp>* (pi \<bullet> x) = b \<sharp>* x"
RES_leof_RES_iff: fixes X :: "'b set"   and Y :: "'b set" shows "(RES X \<le>\<^sub>n RES Y) = (X \<subseteq> Y)"
subgame_VV_subset: fixes p :: "Player"   and V' :: "'a set" shows "(if p = Even then V0\<^bsub>G'.subgame V'\<^esub> else V\<^bsub>G'.subgame V'\<^esub> - V0\<^bsub>G'.subgame V'\<^esub>) \<subseteq> G'.VV p"
max_fun_diff_def: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "wo_rel r" shows "wo_rel.max_fun_diff r f g \<equiv> wo_rel.maxim r {a \<in> Field r. f a \<noteq> g a}"
3: shows "d \<sqinter> H * a * top \<le> top * a"
wt1_nneg: fixes I :: "nat set"   and i :: "nat" assumes "I \<in> P"   and "i \<in> I" shows "0 \<le> wt (fst cf1) (snd cf1) i"
assms_3: shows "domA \<Gamma> \<subseteq> S"
leading_coeff_def: shows "leading_coeff = Polynomial.lead_coeff"
ge_iff_diff_ge_0: fixes b :: "'b"   and a :: "'b" shows "b \<le> a \<equiv> (0::'b) \<le> a - b"
lr_3: shows "(x \<circ> r) \<longlonglongrightarrow> l"
map_of_Cons_code_2: fixes l :: "'c"   and v :: "'b"   and ps :: "('c \<times> 'b) list"   and k :: "'c" shows "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
sets_measure_of_eq: fixes \<mu> :: "'a set \<Rightarrow> ennreal" shows "sets (measure_of \<Omega> (sigma_sets \<Omega> A) \<mu>) = sigma_sets \<Omega> A"
geodesic_segment_param_in_geodesic_spaces_5: fixes x :: "'a"   and y :: "'a" shows "geodesic_segment_param {x--y} x ` {0..dist x y} = {x--y}"
map_of_map_entry: fixes k :: "'b"   and f :: "'c \<Rightarrow> 'c"   and xs :: "('b \<times> 'c) list" shows "map_of (AList.map_entry k f xs) = (map_of xs)(k := case map_of xs k of None \<Rightarrow> None | Some v' \<Rightarrow> Some (f v'))"
simps_2: fixes fn :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and e :: "'b" shows "fold_impl fn e [] = e"
prv_all_gen: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "bprv \<phi>" shows "bprv (all x \<phi>)"
elims: fixes x :: "'b tree"   and y :: "'b list" assumes "list x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "y = xa # splice (list l) (list r)"
zero_less_iff_neq_zero: fixes n :: "'b" shows "((0::'b) < n) = (n \<noteq> (0::'b))"
norm_bound_Basis_lt: fixes b :: "'b"   and x :: "'b"   and e :: "real" assumes "b \<in> Basis"   and "norm x < e" shows "\<bar>x \<bullet> b\<bar> < e"
forest_components_wcc: fixes x :: "'a" assumes "x \<sqinter> id x \<le> top" shows "top = top \<sqinter> top"
phi_bot: shows "phi bot = Filters.filter.Abs_filter {top}"
object_at_def: fixes P :: "cdl_object \<Rightarrow> bool"   and p :: "32 word"   and s :: "32 word \<Rightarrow> cdl_object option" shows "object_at P p s \<equiv> \<exists>object. s p = Some object \<and> P object"
emeasure_le_1: fixes S :: "'a set" shows "emeasure M S \<le> 1"
constrains_Un: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> A co A'"   and "F \<in> B co B'" shows "F \<in> A \<union> B co A' \<union> B'"
elims: fixes x :: "'fsym trm"   and y :: "'tp" assumes "Ik.tpOf x = y" obtains xa :: "Preliminaries.var" where "x = Var xa" and "y = GE.tpOfV xa"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "y = resOf f"
sup_idem: shows "\<rho> \<squnion> \<rho> = \<rho>"
assms_2: shows "f' \<in> A \<rightarrow>\<^sub>E B"
total_finite_trans_max: fixes X :: "'d set"   and r :: "('d \<times> 'd) set" assumes "X \<noteq> {}"   and "finite X"   and "total_on X r"   and "trans r" shows "\<exists>x\<in>X. \<forall>y\<in>X. x \<noteq> y \<longrightarrow> (y, x) \<in> r"
0: shows "bprv (imp (subst \<chi> \<langle>neg \<phi>\<rangle> xx') (imp (NN \<langle>\<phi>\<rangle> (Var xx')) \<chi>))"
word_adjacent_union: fixes e :: "'b word"   and s' :: "'b word"   and s :: "'b word"   and e' :: "'b word" assumes "word_next e = s'"   and "s \<le> e"   and "s' \<le> e'" shows "{s..e} \<union> {s'..e'} = {s..e'}"
maximally_connected_def: fixes H :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "maximally_connected H G \<equiv> \<forall>v\<in>nodes G. \<forall>v'\<in>nodes G. nodes_connected G v v' \<longrightarrow> nodes_connected H v v'"
stirling_integral_holomorphic: fixes m :: "nat"   and A :: "complex set" assumes "0 < m"   and "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}" shows "stirling_integral m holomorphic_on A"
of_nat_take_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.of_nat one plus zero (take_bit n m) = take_bit n (semiring_1.of_nat one plus zero m)"
element_ptr_kinds_eq: shows "element_ptr_kinds h = element_ptr_kinds h'"
inject_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(Le x1 = Le y1) = (x1 = y1)"
eval_fds_zeroD: fixes h :: "'a fds" assumes "conv_abscissa h < \<infinity>"   and "\<exists>\<^sub>F s in (\<lambda>s. s \<bullet> (1::'a)) going_to at_top. eval_fds h s = (0::'a)" shows "h = 0"
II: fixes it :: "'u set"   and \<sigma> :: "'\<sigma>" assumes "it \<subseteq> \<alpha> m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>"   and "\<forall>k\<in>it. \<forall>j\<in>\<alpha> m - it. j \<le> k" shows "P \<sigma>"
bind_advantage_def: fixes key_gen :: "('b \<times> 'c) spmf"   and verify :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and valid_msg :: "'d \<Rightarrow> bool"   and \<A> :: "'b \<Rightarrow> ('e \<times> 'd \<times> 'f \<times> 'd \<times> 'f) spmf" shows "abstract_commitment.bind_advantage key_gen verify valid_msg \<A> \<equiv> spmf (abstract_commitment.bind_game key_gen verify valid_msg \<A>) True"
simps_2: fixes empty :: "'a"   and get :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and add :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and to_list :: "'a \<Rightarrow> ('b \<times> 'c) list"   and e :: "'a"   and x :: "'b \<times> 'c"   and xs :: "('b \<times> 'c) list" assumes "env empty get add to_list" shows "env.add_list add e (x # xs) = add (env.add_list add e xs) (fst x) (snd x)"
UNIV_rw: shows "UNIV = insert j (UNIV - {j})"
length_remdups_card: fixes l :: "'a list" shows "length (remdups l) = card (set l)"