hcju/isabelleps huggingface dataset:
closest_pair_rec_simps: fixes n :: "nat"   and xs :: "(int \<times> int) list" assumes "n = length xs"   and "\<not> n \<le> 3" shows "closest_pair_rec xs = (let (xs\<^sub>L, xs\<^sub>R) = split_at (n div 2) xs; (ys\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) = closest_pair_rec xs\<^sub>L; (ys\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) = closest_pair_rec xs\<^sub>R; ys = Common.merge snd ys\<^sub>L ys\<^sub>R in (ys, combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) (fst (hd xs\<^sub>R)) ys))"
transT_MtransT: fixes cf :: "(test, atom) com \<times> (Concrete.var \<Rightarrow> nat)"   and s' :: "Concrete.var \<Rightarrow> nat" assumes "Example_PL.transT cf s'" shows "Example_PL.MtransT cf s'"
div_abs_eq: fixes a :: "int"   and b :: "int" assumes "a mod b = 0"   and "0 < b" shows "\<bar>a div b\<bar> = \<bar>a\<bar> div b"
cproper_interval_set_Some_Some: fixes A :: "'b set"   and B :: "'b set" shows "cproper_interval (Some A) (Some B) = (finite UNIV \<and> (\<exists>C. cless A C \<and> cless C B))"
poly_mult: fixes p1 :: "'a list"   and p2 :: "'a list"   and x :: "'a" shows "poly (p1 *** p2) x = poly p1 x * poly p2 x"
convol_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "BNF_Def.convol f g \<equiv> \<lambda>a. (f a, g a)"
unlex_induct: fixes y :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "y \<in> {xs. True}"   and "\<And>x. P (unlex x)" shows "P y"
spl: shows "split_block B k k = (B1, B2, B3, B4)"
implmt_methd_access: fixes G :: "prog"   and I :: "qtname"   and sig :: "sig"   and dynC :: "qtname"   and accC :: "qtname" assumes "DeclConcepts.imethds G I sig \<noteq> {}"   and "G\<turnstile>dynC\<leadsto>I"   and "is_iface G I"   and "wf_prog G" shows "\<exists>dynM. methd G dynC sig = Some dynM \<and> G \<turnstile>Methd sig dynM in dynC dyn_accessible_from accC"
with_infos_def: fixes h :: "'f"   and x :: "'e set" shows "with_infos h x = x"
Abs3: fixes x :: "'e"   and y :: "'f" assumes "T3 x y" shows "Abs3 x = y"
suppI: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d" assumes "f x \<noteq> (0::'c)" shows "x \<in> supp f"
word_log_esimps_1: fixes x :: "'b word" shows "x AND 0 = 0"
normal_of_no_zero_root: fixes f :: "real poly"   and i :: "nat" assumes "poly f 0 \<noteq> 0"   and "i \<le> degree f"   and "normal_poly f" shows "0 < coeff f i"
Less_1: shows "s \<in> \<gamma>\<^sub>o S"
elims_2: fixes x :: "'a ext"   and xa :: "'a ext" assumes "x \<le> xa" obtains uu_ :: "'a ext" where "xa__ = Bot" and "xa = uu_"   | x :: "'a" and y :: "'a" where "xa__ = Val x" and "xa = Val y" and "x \<le> y"   | uw_ :: "'a" where "xa__ = Val uw_" and "xa = Top"   | "xa__ = Top" and "xa = Top"
induct: fixes x1 :: "prog"   and x2 :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and x3 :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and P :: "prog \<Rightarrow> (fin_string + fin_string \<Rightarrow> 32 Word.word) \<Rightarrow> (fin_string + fin_string \<Rightarrow> 32 Word.word) \<Rightarrow> bool" assumes "[[x1]]x2 \<down> x3"   and "\<And>\<phi> \<nu> \<omega>. \<lbrakk>[[\<phi>]]\<nu> \<down> True; \<nu> = \<omega>\<rbrakk> \<Longrightarrow> P (? \<phi>) \<nu> \<omega>"   and "\<And>\<alpha> \<nu> \<mu> \<beta> \<omega>. \<lbrakk>[[\<alpha>]]\<nu> \<down> \<mu>; P \<alpha> \<nu> \<mu>; [[\<beta>]]\<mu> \<down> \<omega>; P \<beta> \<mu> \<omega>\<rbrakk> \<Longrightarrow> P (\<alpha> ;; \<beta>) \<nu> \<omega>"   and "\<And>\<omega> \<nu> x \<theta>. \<omega> = \<nu>(Inr x := snd ([\<theta>]<>\<nu>), Inl x := fst ([\<theta>]<>\<nu>)) \<Longrightarrow> P (x := \<theta>) \<nu> \<omega>"   and "\<And>\<alpha> \<nu> \<omega> \<beta>. \<lbrakk>[[\<alpha>]]\<nu> \<down> \<omega>; P \<alpha> \<nu> \<omega>\<rbrakk> \<Longrightarrow> P (\<alpha> \<union>\<union> \<beta>) \<nu> \<omega>"   and "\<And>\<beta> \<nu> \<omega> \<alpha>. \<lbrakk>[[\<beta>]]\<nu> \<down> \<omega>; P \<beta> \<nu> \<omega>\<rbrakk> \<Longrightarrow> P (\<alpha> \<union>\<union> \<beta>) \<nu> \<omega>" shows "P x1 x2 x3"
elementLevel_def: fixes e :: "'a"   and t :: "('a \<times> bool) list" shows "elementLevel e t = length (markedElementsTo e t)"
closest_is_closest: fixes s :: "('a, 'b) IEEE.float set"   and v :: "('a, 'b) IEEE.float \<Rightarrow> real"   and x :: "real"   and p :: "('a, 'b) IEEE.float \<Rightarrow> bool" assumes "s \<noteq> {}" shows "is_closest v s x (closest v p s x)"
wf_trm_arity: fixes f :: "'fun"   and T :: "('fun, 'a) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)" shows "length T = arity f"
coll_eq: shows "collision {0..} = collision {0<..<min ego.t_stop other.t_stop}"
Field_insert: fixes a :: "'d"   and b :: "'d"   and r :: "('d \<times> 'd) set" shows "Field (insert (a, b) r) = {a, b} \<union> Field r"
mono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'g" shows "mono f = (\<forall>n. f n \<le> f (Suc n))"
B_def: shows "B = {c * arccos (- b / a), - c * arccos (- b / a)}"
SComplex_iff: fixes x :: "complex star" shows "(x \<in> SComplex) = (\<exists>y. x = hcomplex_of_complex y)"
ax4: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[P \<longrightarrow> Q]_v \<longrightarrow> \<box>[P]_v \<longrightarrow> \<box>[Q]_v"
PsiInv_beta1: shows "\<turnstile> beta1 \<and> $PsiInv \<longrightarrow> PsiInv$"
eq_eqvt: fixes p :: "perm"   and x :: "'e"   and y :: "'e" shows "p \<bullet> (x = y) = (p \<bullet> x = p \<bullet> y)"
eWls_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and s :: "'sort" shows "eWls MOD s ERR = False"
ab_obey_1: fixes v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>\<rightarrow> ((\<^bold>\<forall>F. \<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<equiv> \<lbrace>y\<^sup>P,F\<rbrace>) \<^bold>\<rightarrow> x\<^sup>P \<^bold>= y\<^sup>P) in v]"
admI: fixes P :: "'c \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i)\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)" shows "adm P"
is_trail_intro: fixes v' :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v" assumes "is_trail v' ps v" shows "is_path v' ps v"
proots_half_empty: fixes a :: "complex"   and b :: "complex"   and p :: "complex poly" assumes "a = b" shows "proots_half p a b = 0"
p_def: shows "p = (\<lambda>a. if a \<in> A then b a else a)"
simps_10: fixes V :: "'a set"   and va :: "'a list"   and vb :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and vc :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and S :: "('b, 'a) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (\<forall>va\<langle>\<or>\<noteq>: vb \<or>\<notin>: vc\<rangle> # S) = wf'\<^sub>s\<^sub>s\<^sub>t V S"
succ_to_list_def: fixes g :: "('d, 'c) graph"   and v :: "'d" shows "succ_to_list g v \<equiv> it_to_sorted_list (\<lambda>_ _. True) (succ g v)"
normalize_mod_ring_def: fixes x :: "'a mod_ring" shows "normalize x = (if x = 0 then 0 else 1)"
simps_1: fixes v :: "'a val" shows "fin1 (Val v) = Val v"
db_defs_5: fixes inp :: "state \<Rightarrow> 'b"   and mid :: "state \<Rightarrow> 'b"   and out :: "state \<Rightarrow> 'b"   and q1 :: "state \<Rightarrow> 'b list"   and q2 :: "state \<Rightarrow> 'b list"   and ic :: "state \<Rightarrow> 'c"   and q :: "state \<Rightarrow> 'c list"   and oc :: "state \<Rightarrow> 'c" assumes "DBuffer inp mid out q1 q2" shows "ISpec ic q oc \<equiv> BInit ic q oc \<and> \<box>[Nxt ic q oc]_(ic, q, oc) \<and> WF(Deq ic q oc)_(ic, q, oc)"
flatten_raise: fixes f :: "'d \<Rightarrow> 'd" shows "flatten (raise f) = [Basic f, Throw]"
B: fixes D' :: "'a set set" assumes "D' \<subseteq> \<D>"   and "finite D'" shows "Sigma_Algebra.measure lebesgue (\<Union> D') \<le> B"
S1Env: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> ENext p \<and> $S1 rmhist p \<and> unchanged (c p, r p, m p, rmhist!p) \<longrightarrow> S2 rmhist p$"
dyadics_iff: shows "dyadics = \<nat> \<union> (\<Union>k m. {of_nat (4 * m + 1) / (2::'b) ^ Suc k}) \<union> (\<Union>k m. {of_nat (4 * m + 3) / (2::'b) ^ Suc k})"
linear: fixes x :: "'a"   and y :: "'a" shows "le x y \<or> le y x"
simps_1: fixes x :: "'b" shows "lists_succ [] x = x"
prj_chine_composite_2: shows "cods.Prj\<^sub>0\<^sub>1 \<cdot> \<mu>\<nu>_\<pi>.chine = \<nu>.chine \<cdot> doms.Prj\<^sub>0\<^sub>1"
allAllocationsBridgingLemma: fixes N :: "'c set"   and G :: "'d list" assumes "0 < card N"   and "distinct G" shows "allAllocations N (set G) = set (allAllocationsAlg N G)"
filter_single_mset: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k" shows "filter_mset P {#x#} = (if P x then {#x#} else {#})"
not_lnull_conv: fixes xs :: "'c llist" shows "(\<not> lnull xs) = (\<exists>x xs'. xs = LCons x xs')"
val_tick: fixes x :: "'b" shows "Time_Monad.val (tick x) = x"
inf_chain_ltl_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" assumes "chain R xs"   and "\<not> lfinite xs" shows "chain R (ltl xs)"
inv_comp_right: fixes f :: "'d \<Rightarrow> 'd" assumes "bijection f" shows "f \<circ> inv f = id"
j_of_min: fixes k :: "nat" shows "\<forall>j>k. (f (Suc j), f j) \<in> R \<longrightarrow> j_of k \<le> j"
leg_simps_3: shows "C.arr doms.\<nu>.leg1"
map_def: fixes f :: "'c local.term set" shows "local.map f = (if arr f then f else null)"
assms_2: shows "\<forall>x\<in>U\<^sub>M\<^sub>_\<^sub>1. f x \<in> U\<^sub>M\<^sub>_\<^sub>2"
res_fold_A_chamber_images_2: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z" shows "restrict1 (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) (\<Union> A) ` D = C"
Coplanar_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Coplanar Bet A B C D \<equiv> \<exists>X. Tarski_neutral_dimensionless.Col Bet A B X \<and> Tarski_neutral_dimensionless.Col Bet C D X \<or> Tarski_neutral_dimensionless.Col Bet A C X \<and> Tarski_neutral_dimensionless.Col Bet B D X \<or> Tarski_neutral_dimensionless.Col Bet A D X \<and> Tarski_neutral_dimensionless.Col Bet B C X"
norm_eq_zero: 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 x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x = 0) = (x = zero)"
mod_eq_mod_0_imp_mod_eq: fixes a :: "nat"   and m' :: "nat"   and b :: "nat"   and m :: "nat" assumes "a mod m' = b mod m'"   and "m' mod m = 0" shows "a mod m = b mod m"
nat_neq_iff: fixes m :: "nat"   and n :: "nat" shows "(m \<noteq> n) = (m < n \<or> n < m)"
poly_mapping_sum_monomials: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "(\<Sum>a\<in>keys p. monomial (lookup p a) a) = p"
s: shows "learn_lim \<phi> V\<^bsub>BC-LIM\<^esub> s"
assms: shows "s' = snd (fst (div_write_new_val i result temp_V s)) \<and> get_S (cpu_reg_val PSR s) = 0"
p51: shows "\<forall>br\<in>allBlocksRead s p. mbal br < mbal (dblock s' p) \<and> bal br \<le> bal (dblock s' p)"
alpha_trans_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "perm"   and x :: "'a"   and y :: "'a"   and q :: "perm"   and z :: "'a"   and bs :: "atom set"   and f :: "'a \<Rightarrow> atom set"   and cs :: "atom set"   and ds :: "atom set" assumes "\<lbrakk>R (p \<bullet> x) y; R (q \<bullet> y) z\<rbrakk> \<Longrightarrow> R ((q + p) \<bullet> x) z"   and "(bs, x) \<approx>set R f p (cs, y)"   and "(cs, y) \<approx>set R f q (ds, z)" shows "(bs, x) \<approx>set R f (q + p) (ds, z)"
intersection_calculus_axioms: shows "intersection_calculus Bot_F Inf_F Q (\<lambda>q N1 N2. entails_q q (\<G>_Fset_q q N1) (\<G>_Fset_q q N2)) Red_I_\<G>_q Red_F_\<G>_q"
rel_flip: fixes R :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'g \<rightharpoonup>\<^sub>f 'e"   and b :: "'g \<rightharpoonup>\<^sub>f 'f" shows "fmrel R\<inverse>\<inverse> a b = fmrel R b a"
measure_while_option_Some: fixes P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> nat"   and s :: "'b" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s) \<and> f (c s) < f s"   and "P s" shows "\<exists>t. while_option b c s = Some t"
finite_doms_1: fixes \<Gamma> :: "binding list" shows "finite (ty_dom \<Gamma>)"
Gramian_matrix_alt_alt_def: fixes k :: "nat" assumes "k \<le> length fs" shows "Gramian_matrix fs k = mat k k (\<lambda>(i, j). fs ! i \<bullet> fs ! j)"
seq_app_first_tail: fixes s :: "nat \<Rightarrow> 'b" shows "first s ## tail s = s"
distinct_9: fixes x21 :: "'d rexp"   and x22 :: "'d rexp"   and x4 :: "'d rexp" shows "Alt x21 x22 \<noteq> Star x4"
subtype_x_OrderedSet: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<sqsubset> OrderedSet \<sigma>" obtains \<tau>' :: "'a OCL_Types.type" where "\<tau> = OrderedSet \<tau>'" and "\<tau>' \<sqsubset> \<sigma>"
nres_more_simps_4: fixes X :: "'a set"   and Y :: "'a set" shows "(RES X = RES Y) = (X = Y)"
lm073: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set" assumes "a \<in> allAllocations N G" shows "pseudoAllocation a \<subseteq> N \<times> finestpart G"
termFSwImorph_wlsBinp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" assumes "wlsBinp delta binp"   and "termFSwImorph h hA MOD" shows "igWlsBinp MOD delta (lift hA binp)"
pdevs_apply_msum_pdevs: fixes n :: "nat"   and f :: "'b pdevs"   and g :: "'b pdevs"   and i :: "nat" shows "pdevs_apply (msum_pdevs n f g) i = (if i < n then pdevs_apply f i else pdevs_apply g (i - n))"
final_finished: fixes s :: "('fenv, 'henv, ('fun, ('dyn, 'num, 'bool) unboxed) frame) state" assumes "final s" shows "finished (\<rightarrow>) s"
arr_char: fixes f :: "'c \<times> 'c" shows "C.VV.arr f = (C.arr (fst f) \<and> C.arr (snd f) \<and> src\<^sub>C (fst f) = trg\<^sub>C (snd f))"
adopt_node_writes: fixes h :: "(_) heap"   and node :: "(_) node_ptr"   and document_ptr :: "(_) document_ptr"   and h' :: "(_) heap" shows "writes (Shadow_DOM.adopt_node_locs |h \<turnstile> Shadow_DOM.get_parent node|\<^sub>r |h \<turnstile> Shadow_DOM.get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node)|\<^sub>r document_ptr) (Shadow_DOM.adopt_node document_ptr node) h h'"
assms_2: fixes child :: "(_) object_ptr" assumes "\<And>parent child_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node; h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent\<rbrakk> \<Longrightarrow> P parent"   and "\<And>host shadow_root. \<lbrakk>child = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root; h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host\<rbrakk> \<Longrightarrow> P (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host)"   and "\<And>disc_doc disc_node. \<lbrakk>child = cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_node; h \<turnstile> get_disconnected_document disc_node \<rightarrow>\<^sub>r disc_doc\<rbrakk> \<Longrightarrow> P (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r disc_doc)" shows "P child"
SUP_ereal_add_pos: fixes f :: "nat \<Rightarrow> ereal"   and g :: "nat \<Rightarrow> ereal" assumes "incseq f"   and "incseq g"   and "\<And>i. 0 \<le> f i"   and "\<And>i. 0 \<le> g i" shows "(SUP i. f i + g i) = Sup (range f) + Sup (range g)"
f_map: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and r :: "'a"   and m :: "'b" assumes "VectorSpaceEnd smult V T"   and "r \<in> UNIV"   and "m \<in> V" shows "T (smult r m) = smult r (T m)"
finite_Range: fixes r :: "('d \<times> 'e) set" assumes "finite r" shows "finite (Range r)"
hyps_2: shows "P,E \<turnstile> \<langle>e'',s''\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
nonnegative_eigenvector_has_ev_sr: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec"   and v :: "(real, 'a) Finite_Cartesian_Product.vec"   and mu :: "real" assumes "perron_frobenius A"   and "eigen_vector A v mu"   and "\<forall>i. 0 $h i \<le> v $h i" shows "mu = perron_frobenius.r A (perron_frobenius.z A)"
div_by_1: fixes a :: "'a" shows "a / (1::'a) = a"
llist_all2_lfilterI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist"   and Q1 :: "'a \<Rightarrow> bool"   and Q2 :: "'b \<Rightarrow> bool" assumes "llist_all2 P xs ys"   and "\<And>x y. P x y \<Longrightarrow> Q1 x = Q2 y" shows "llist_all2 P (lfilter Q1 xs) (lfilter Q2 ys)"
sets_measure_pmf: fixes p :: "'c pmf" shows "measure_pmf.events p = UNIV"
hom_0_iff: fixes x :: "'a poly" shows "(poly_x_mult_y x = 0) = (x = 0)"
UP_one_closed: shows "\<one>\<^bsub>P\<^esub> \<in> carrier P"
f': shows "(\<lambda>x. if x \<in> A then f x else null_measure (f (SOME x. x \<in> A \<and> x \<in> space M))) \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
induct: fixes P :: "('a, 'b) Combinators \<Rightarrow> bool"   and a0 :: "('a, 'b) Combinators" assumes "P DenyAll"   and "\<And>a b. P (DenyAllFromTo a b)"   and "\<And>a b p. P (AllowPortFromTo a b p)"   and "\<And>a b. \<lbrakk>P b; P a\<rbrakk> \<Longrightarrow> P (a \<oplus> b)" shows "P a0"
seqI: fixes f :: "'j"   and g :: "'j" assumes "J.arr f"   and "J.arr g"   and "J.dom g = J.cod f" shows "J.seq g f"
refl_on_domain: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "refl_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
elims_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq x" obtains "x = []"   | xa :: "'b" and ys :: "'b list" where "x = xa # ys" and "Ball (set ys) (less_eq xa) \<and> linorder.sorted less_eq ys"
prod_list_pow_suc: fixes i :: "nat"   and bs :: "'c list" shows "(\<Prod>x\<leftarrow>bs. x * x ^ i) = prod_list bs * prod_list bs ^ i"
in_Gr: fixes x :: "'e"   and y :: "'f"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" shows "((x, y) \<in> BNF_Def.Gr A f) = (x \<in> A \<and> f x = y)"
size_Diff1_less: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "size (remove1_mset x M) < size M"
LN_5: shows "\<S>\<bullet>\<C> \<turnstile> vs @ [Label n es' lfilledk] @ es'' : tsa__ _> ts'"
Suc_2: fixes i :: "nat" assumes "i \<le> Suc (Suc n)" shows "f i \<le> Suc n"
borel_measurable_cond_exp: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "real_cond_exp M F f \<in> borel_measurable F"
intros_9: fixes \<Gamma> :: "'b formula set"   and G :: "'b formula"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<or> G"
set: fixes x1 :: "nat"   and x2 :: "(nat \<times> linear_poly) list"   and x3 :: "('a \<times> QDelta atom) list"   and x4 :: "linear_poly \<Rightarrow> nat option"   and x5 :: "'a list" shows "set_istate (IState x1 x2 x3 x4 x5) = \<Union> (Basic_BNFs.fsts ` set x3) \<union> set x5"
unique_cross_ratio: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo"   and z :: "complex_homo"   and z' :: "complex_homo" assumes "u \<noteq> v"   and "v \<noteq> w"   and "u \<noteq> w"   and "cross_ratio z u v w = cross_ratio z' u v w" shows "z = z'"
solves_ode_cong: fixes T :: "real set"   and x :: "real \<Rightarrow> 'd"   and y :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "real set"   and X :: "'d set"   and Y :: "'d set" assumes "\<And>t. t \<in> T \<Longrightarrow> x t = y t"   and "\<And>t. t \<in> T \<Longrightarrow> f t (x t) = g t (x t)"   and "T = S"   and "X = Y" shows "(x solves_ode f) T X = (y solves_ode g) S Y"
that: fixes b :: "'a" assumes "b \<in> set bs" shows "a dvd b"
validD_L: fixes \<phi> :: "'a"   and A :: "'a set" assumes "\<tturnstile> \<phi>" shows "\<phi> \<in> Cn A"
prems: shows "\<exists>nid'. (n', nid') \<in> his t n nid"
coboundedI2: fixes c :: "'d"   and b :: "'d"   and a :: "'d" assumes "c \<le> b" shows "c \<le> max a b"
moebius_mu_conv_moebius_mu_aux: fixes qs :: "unit \<Rightarrow> nat list"   and n :: "nat" assumes "mset (qs ()) = prime_factorization n" shows "moebius_mu n = of_int (moebius_mu_aux n qs)"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_loop_goon_on_rightmost m lm (l, r) = (\<exists>rn. l = Bk # <rev lm> @ Bk # Bk # Oc \<up> Suc m \<and> r = Bk \<up> rn)"
double_complement_dist_sup: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) = - - x \<squnion> - - y"
corec_2: fixes p :: "'a \<Rightarrow> bool"   and a :: "'a"   and g1 :: "'a \<Rightarrow> 'c"   and g21 :: "'a \<Rightarrow> 'b"   and q22 :: "'a \<Rightarrow> bool"   and g221 :: "'a \<Rightarrow> ('b, 'c) tllist"   and g222 :: "'a \<Rightarrow> 'a" assumes "\<not> p a" shows "corec_tllist p g1 g21 q22 g221 g222 a = TCons (g21 a) (if q22 a then g221 a else corec_tllist p g1 g21 q22 g221 g222 (g222 a))"
2: fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'c" assumes "\<tau> \<Turnstile> \<upsilon> X" shows "\<tau> \<Turnstile> Pair{X,Y} .Second() \<triangleq> Y"
cases: fixes a :: "'c"   and g :: "('c \<Rightarrow> bool) \<times> ('c \<Rightarrow> 'd \<times> 'c)" assumes "a \<in> terminates_on g" obtains s :: "'c" where "a = s__" and "\<not> fst g s__"   | s :: "'c" where "a = s" and "fst g s" and "snd (snd g s) \<in> terminates_on g"
invar_update: fixes m :: "'a"   and a :: "'v"   and b :: "nat" assumes "M_invar m" shows "M_invar (M_update a b m)"
pre_pure_rule: fixes P :: "assn"   and b :: "bool"   and f :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" shows "(\<not> <P * \<up> b> f <Q>) = (\<not> <P> f <Q> \<and> b)"