hcju/isabelleps huggingface dataset:
assms_1: fixes s :: "'a"   and t :: "'a" shows "d (s + t) = max (d s) (d t)"
keys_mult_monomial: fixes k :: "nat \<Rightarrow>\<^sub>0 nat"   and n :: "int"   and a :: "int mpoly" shows "keys (monomial n k * mapping_of a) = (if n = 0 then {} else (+) k ` keys (mapping_of a))"
comp_fun_commute: shows "Suc \<circ> Suc = Suc \<circ> Suc"
zero_less_mult_iff: fixes x :: "V"   and y :: "V" assumes "Ord x" shows "(0 < x * y) = (0 < x \<and> 0 < y)"
invariant_1D_6: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "0 < degree (poly_real_alg_1 plr)"
pick_drop_zero: fixes xs :: "(natural \<times> 'b) list" shows "pick (filter (\<lambda>(k, uu_). 0 < k) xs) = pick xs"
inner_same_Basis: 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 Basis :: "'a set"   and u :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "inner u u = 1"
greek_less_app_mono1: fixes r :: "('a \<times> 'a) set"   and s :: "(accent \<times> 'a) list"   and t :: "(accent \<times> 'a) list"   and p :: "(accent \<times> 'a) list" assumes "trans r"   and "(s, t) \<in> greek_less r" shows "(s @ p, t @ p) \<in> greek_less r"
redT_upd_inv_ext: fixes final :: "'g \<Rightarrow> bool"   and r :: "('j, 'h, 'g, 'i, 'k, 'l) semantics"   and convert_RA :: "'j \<Rightarrow>f nat \<Rightarrow> 'l list"   and s :: "('j, 'h) locks \<times> (('h \<Rightarrow> ('g \<times> 'j \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'k wait_set_status option) \<times> 'h set"   and t :: "'h"   and ta :: "('j, 'h, 'g, 'i, 'k, 'l) thread_action"   and s' :: "('j, 'h) locks \<times> (('h \<Rightarrow> ('g \<times> 'j \<Rightarrow>f nat) option) \<times> 'i) \<times> ('h \<Rightarrow> 'k wait_set_status option) \<times> 'h set"   and I :: "'h \<Rightarrow> 'm option"   and P :: "'m \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> bool" assumes "multithreaded_base.redT final r convert_RA s (t, ta) s'"   and "ts_inv_ok (thr s) I" shows "I \<subseteq>\<^sub>m upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
card_face_cycle_sets_da: shows "card G.face_cycle_sets = Suc (card H.face_cycle_sets)"
fbox_arden_while_iff: fixes t :: "'a"   and x :: "'a"   and p :: "'a"   and q :: "'a" assumes "\<nabla>(d t \<cdot> x) = (zero_class.zero::'a)" shows "(d p = (d t + d q) \<cdot>  |d t \<cdot> x] p) = (d p =  |while t do x od] q)"
trans: fixes A :: "wall"   and B :: "wall"   and C :: "wall" assumes "A ~ B"   and "B ~ C" shows "A ~ C"
set_dropWhileD: fixes x :: "'f"   and P :: "'f \<Rightarrow> bool"   and xs :: "'f list" assumes "x \<in> set (dropWhile P xs)" shows "x \<in> set xs"
max_atm_imp_Interp: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and A :: "'a" assumes "C \<in> N"   and "Pos A \<in># C"   and "A = Max (atms_of C)"   and "S C = {#}" shows "Interp N C \<Turnstile> C"
b3: fixes s1 :: "lam"   and s2 :: "lam"   and a :: "name" assumes " s1 \<longrightarrow>\<^sub>\<beta> s2" shows " Lam [a].s1 \<longrightarrow>\<^sub>\<beta> Lam [a].s2"
hcomp_assoc_isomorphic: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "(f \<star>\<^sub>D g) \<star>\<^sub>D h \<cong>\<^sub>D f \<star>\<^sub>D g \<star>\<^sub>D h"
matrix_of_matrix_vector_mul: fixes A :: "(('a, 'b) vec, 'c) vec" shows "matrix ((*v) A) = A"
simps_15: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (Plus e1 e2) (Times v va) = True"
scalars_zero_if_independent_euclidean: fixes A :: "('a, 'b) vec set"   and f :: "('a, 'b) vec \<Rightarrow> 'a" assumes "vec.independent A"   and "(\<Sum>x\<in>A. f x *s x) = 0" shows "\<forall>x\<in>A. f x = (0::'a)"
IH: assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>2"   and "subst_domain \<sigma>1 \<inter> subst_domain \<theta>2 = {}"   and "subst_domain \<sigma>1 \<inter> range_vars \<theta>2 = {}"   and "range_vars \<sigma>1 \<inter> subst_domain \<theta>2 = {}" shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t (\<lambda>v. if \<sigma>1 v = Var v then \<theta>2 v else \<sigma>1 v)"
matrix_mult_0_left: fixes A :: "('c, 'f, 'e) matrix" shows "0 *\<^sub>M A = 0"
ChurchRosser_conversion: shows "is_ChurchRosser = ({(x, y). x \<rightarrow> y}\<^sup>\<leftrightarrow>\<^sup>* \<subseteq> {(x, y). x \<rightarrow> y}\<^sup>\<down>)"
bounded_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set" assumes "MinkowskiChain \<E> \<P> betw" shows "MinkowskiChain.bounded betw Q \<equiv> \<exists>Q\<^sub>b. MinkowskiChain.is_bound betw Q\<^sub>b Q"
transitionE_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P'"   and "x \<sharp> P" shows "\<exists>P'' P'''. P \<Longrightarrow>\<^sub>\<tau> P''' \<and> P''' \<longmapsto> a<\<nu>x> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P'"
has_densityD_1: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "f \<in> borel_measurable N"
hsubset_imp_le: fixes x :: "hf"   and y :: "hf" assumes "x \<le> y" shows "Rep_hf x \<le> Rep_hf y"
l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf set_disconnected_nodes set_disconnected_nodes_locs"
a_ge_0: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and a :: "real" assumes "akra_bazzi_params_nonzero k as bs"   and "a \<in> set as" shows "0 \<le> a"
fU0: fixes z :: "complex" assumes "z \<in> U" shows "f z = g z"
diff_single_trivial: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<notin># M" shows "remove1_mset x M = M"
subtermeq_Var_const_1: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term"   and v :: "'b" assumes "s \<sqsubseteq> t"   and "t = Var v" shows "s = Var v"
Thin_lemma: fixes A' :: "triple set"   and C :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "vassn" shows "(A' |\<turnstile> C \<longrightarrow> (\<forall>A. A' \<subseteq> A \<longrightarrow> A |\<turnstile> C)) \<and> (A' \<turnstile>\<^sub>e {P} e {Q} \<longrightarrow> (\<forall>A. A' \<subseteq> A \<longrightarrow> A \<turnstile>\<^sub>e {P} e {Q}))"
hyps_1: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
lt_sum_distinct_in_lt_set: fixes I :: "'c set"   and p :: "'c \<Rightarrow> 't \<Rightarrow>\<^sub>0 'd" assumes "finite I"   and "sum p I \<noteq> 0"   and "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; p i1 \<noteq> 0; p i2 \<noteq> 0; lt (p i1) = lt (p i2)\<rbrakk> \<Longrightarrow> i1 = i2" shows "lt (sum p I) \<in> lt_set (p ` I)"
sKs_def: fixes p :: "PROB" shows "sKs p \<equiv> SOME kid. p = prob kid"
generate_pow_on_finite_carrier: fixes a :: "'a" assumes "finite (carrier G)"   and "a \<in> carrier G" shows "generate G {a} = {a [^] k |k. k \<in> UNIV}"
reflp_on_reflclp_simp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "reflp_on P A"   and "a \<in> A"   and "b \<in> A" shows "P\<^sup>=\<^sup>= a b = P a b"
Buchberger_criterion_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "F \<subseteq> dgrad_p_set d m"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0; p \<noteq> q; component_of_term (lt p) = component_of_term (lt q)\<rbrakk> \<Longrightarrow> (red F)\<^sup>*\<^sup>* (spoly p q) 0" shows "is_Groebner_basis F"
merge_height: fixes l :: "'b tree"   and r :: "'b tree" shows "height (merge l r) \<le> Suc (max (height l) (height r))"
States_HARoot_HAStates: fixes HA :: "('a, 'b, 'c) hierauto" shows "States (HARoot HA) \<subseteq> HAStates HA"
cm: shows "c = a - m * x \<and> 2 * \<bar>c\<bar> < x"
GreatestI: fixes P :: "'d \<Rightarrow> bool"   and k :: "'d" assumes "P k" shows "P (GREATEST x. P x)"
rb_aux_inv_preserved_3_2: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and p :: "('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv (bs, ss, p # ps)"   and "\<not> sig_crit bs (new_syz_sigs ss bs p) p"   and "rep_list (sig_trd bs (poly_of_pair p)) \<noteq> 0" shows "lt (sig_trd bs (poly_of_pair p)) \<notin> lt ` set bs"
length_Cons: fixes x :: "'d"   and xs :: "'d list" shows "length (x # xs) = Suc (length xs)"
Class_def: fixes S :: "'c set"   and E :: "('c \<times> 'c) set" assumes "equivalence S E" shows "equivalence.Class S E = (\<lambda>a\<in>S. {b \<in> S. (b, a) \<in> E})"
filter_filter_mset_cond_simp: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and A :: "'a multiset" assumes "\<And>a. P a \<Longrightarrow> Q a" shows "filter_mset P A = filter_mset P (filter_mset Q A)"
R1: shows "(tsl s, l') \<in> \<langle>Rk \<times>\<^sub>r Rv\<rangle>list_rel"
LHS: shows "to_fun p a = a [^] Suc n \<ominus> b [^] Suc n"
1: fixes t :: "'c local.term" assumes "Can t"   and "coherent t" shows "coherent (Inv t)"
finite_cspan_closed: fixes S :: "'d set" assumes "finite S" shows "closed (cspan S)"
K_def: shows "K = 92 * lambda\<^sup>2 * (C + deltaG TYPE('b))"
typeof_notclass_heap: fixes T :: "ty"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "\<forall>C. T \<noteq> Class C" shows "(P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>) = (P \<turnstile> typeof\<^bsub>h'\<^esub> v = \<lfloor>T\<rfloor>)"
dualization_4: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> \<not> \<diamond>\<langle>A\<rangle>_v = \<box>[\<not> A]_v"
universal: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and h :: "('a, 'a set) Maps.arr"   and k :: "('a, 'a set) Maps.arr" assumes "Maps.commutative_square f g h k" shows "\<exists>!l. Maps.PRJ\<^sub>1 f g \<odot> l = h \<and> Maps.PRJ\<^sub>0 f g \<odot> l = k"
act_in_UNION_preserves_fst: fixes act :: "(('c \<times> 'd) \<times> 'c \<times> 'd) set" assumes "act \<subseteq> {(x, x'). fst x = fst x'}" shows "act \<in> \<Union> (Acts ` preserves fst)"
simps_2: fixes n :: "nat"   and nd :: "('c, 'd) Node" shows "is_leveln_ftree n (Single nd) = is_leveln_node n nd"
assms: shows "proj_stoch_proc geom_proc m x = proj_stoch_proc geom_proc m y"
rank_transpose: fixes A :: "((real, 'b) vec, 'a) vec" shows "rank (Finite_Cartesian_Product.transpose A) = rank A"
pointwise_convergent_to_def: fixes x :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and l :: "'c \<Rightarrow> 'd" shows "x \<midarrow>pointwise\<rightarrow> l = (\<forall>t. (\<lambda>n. x n t) \<longlonglongrightarrow> l t)"
red2_in_Hom_2: fixes u :: "'c local.term"   and v :: "'c local.term" assumes "Ide u"   and "Nml u"   and "Ide v"   and "Nml v"   and "Src u = Trg v" shows "u \<^bold>\<Down> v \<in> VHom (u \<^bold>\<star> v) \<^bold>\<lfloor>u \<^bold>\<star> v\<^bold>\<rfloor>"
leg1_commutes: shows "\<mu>.cod.leg1 \<cdot> \<mu>.chine = \<mu>.dom.leg1"
rep_eq: fixes x :: "'a regular" shows "Rep_regular (- x) = - Rep_regular x"
abs_def: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a set \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a set"   and x :: "'a" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
elts_ord_of_nat: fixes k :: "nat" shows "elts (k\<^sub>\<nat>) = ord_of_nat ` {..<k}"
some_ap_option: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "Applicative.ap (pure f) x = map_option f x"
induct: fixes P :: "('c, 'd) FingerTreeStruc \<Rightarrow> ('c, 'd) Node list \<Rightarrow> ('c, 'd) FingerTreeStruc \<Rightarrow> bool"   and a0 :: "('c, 'd) FingerTreeStruc"   and a1 :: "('c, 'd) Node list"   and a2 :: "('c, 'd) FingerTreeStruc" assumes "\<And>xs t. P Empty xs t"   and "\<And>v xs. P (Single v) xs Empty"   and "\<And>v va vb vc xs. P (Deep v va vb vc) xs Empty"   and "\<And>x xs v. P (Single x) xs (Single v)"   and "\<And>x xs v va vb vc. P (Single x) xs (Deep v va vb vc)"   and "\<And>v va vb vc xs x. P (Deep v va vb vc) xs (Single x)"   and "\<And>uu_ pr1 m1 sf1 ts uv_ pr2 m2 sf2. P m1 (nodes (digitToNlist sf1 @ ts @ digitToNlist pr2)) m2 \<Longrightarrow> P (Deep uu_ pr1 m1 sf1) ts (Deep uv_ pr2 m2 sf2)" shows "P a0 a1 a2"
qfN_primedivisor_not: fixes P :: "int"   and Q :: "int"   and N :: "int" assumes "prime P \<and> 0 < Q \<and> is_qfN (P * Q) N \<and> \<not> is_qfN P N" shows "\<exists>R. prime R \<and> R dvd Q \<and> \<not> is_qfN R N"
leftderivations_subset_derivations: shows "leftderivations \<subseteq> derivations"
hd_in: fixes xs :: "('b, 'c) oalist_ntm" assumes "fst (list_of_oalist_ntm xs) \<noteq> []" shows "OAlist_hd_ntm xs \<in> set (fst (list_of_oalist_ntm xs))"
mset_heap_del_min: fixes ts :: "'b tree list" assumes "ts \<noteq> []" shows "mset_heap ts = mset_heap (del_min ts) + {#get_min ts#}"
minus_scalar_prod_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and v\<^sub>3 :: "'b vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 - v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 - v\<^sub>2 \<bullet> v\<^sub>3"
step: fixes x :: "'a vec"   and y :: "'a vec"   and a :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "(0::'a) \<le> a"   and "a \<le> (1::'a)" shows "a \<cdot>\<^sub>v x + ((1::'a) - a) \<cdot>\<^sub>v y \<in> S"
zero: fixes K :: "'a set"   and x :: "'a" shows "\<zero>\<^bsub>R\<lparr>carrier := K\<rparr>\<^esub> \<in> K.simple_extension K x"
assms_2: shows "P (\<lambda>v c. return_pmf None)"
index_conv_size_if_notin: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "index xs x = length xs"
trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubseteq> x"   and "z \<sqsubseteq> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
greaterThanLessThan_upt: fixes n :: "nat"   and m :: "nat" shows "{n<..<m} = list.set [Suc n..<m]"
relabel': fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "pr_algo_lts'p f l RELABEL' f (relabel_effect f l u)"
set_child_nodes_def: shows "set_child_nodes = invoke a_set_child_nodes_tups"
intros: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp" shows "\<Gamma> \<turnstile>\<^sub>a e \<in> \<Union> (to_total \<Gamma> ` aexp_vars e)"
app_args_spp_of_vec_of: fixes args :: "(('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list \<times> 't list \<times> (('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat) list) \<times> nat" assumes "rb_aux_inv (fst (app_args vec_of args))" shows "app_args spp_of (app_args vec_of args) = args"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool" assumes "\<And>h. type_wf h \<Longrightarrow> ShadowRootClass.type_wf h" shows "l_type_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t type_wf"
rhs_def: shows "rhs \<equiv> nth_cons (flatten (\<lambda>s. (s, 1)) (fix_gen (\<lambda>x. upto_prod (id x))) (upto_prod z)) (2, None) 8"
disjnt_def: fixes A :: "'a set"   and B :: "'a set" shows "disjnt A B = (A \<inter> B = {})"
strictI: fixes f :: "'c \<rightarrow> 'b"   and x :: "'c" assumes "f\<cdot>x = \<bottom>" shows "f\<cdot>\<bottom> = \<bottom>"
Q_closed_1: shows "supp Q\<^sub>1 = {}"
le_diff: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A \<noteq> B"   and "A B Le C D" shows "C \<noteq> D"
isTrue_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {x}"   and "isTrue (exi x \<phi>)" shows "\<exists>n\<in>num. isTrue (subst \<phi> n x)"
codomains_simp: fixes f :: "'a" assumes "arr f" shows "codomains f = {cod f}"
polypoly'_poly: fixes p :: "poly"   and n0 :: "nat"   and x :: "'a"   and bs :: "'a list" assumes "isnpolyh p n0" shows "\<lparr>p\<rparr>\<^sub>p\<^bsup>x # bs\<^esup> = poly (polypoly' bs p) x"
fbrelation: fixes r :: "V" assumes "fbrelation r" shows "fpairs r = r"
Rf_mono_on: fixes X :: "'a set set" shows "Basis.mono_on X (Rf linear_cf)"
simps_1: fixes f :: "'a \<Rightarrow> 'b" shows "groupF f [] = []"
Join_absorb: fixes F :: "'b program" shows "F \<squnion> F = F"
particular_affirmation_def: fixes A :: "'b set"   and B :: "'b set" shows "A I B \<equiv> \<exists>b\<in>B. b \<in> A"
left_idem: fixes a :: "'b"   and b :: "'b" shows "a \<squnion> (a \<squnion> b) = a \<squnion> b"
Units_r_inv: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
no_steps_final: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and v :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and w :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate" assumes "\<Gamma> \<turnstile> v \<rightarrow>\<^sup>* w"   and "final v" shows "w = v"
map_indets_inverseE: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" obtains g :: "'d \<Rightarrow> 'c" where "g = the_inv f" and "g \<circ> f = id" and "map_indets g \<circ> map_indets f = id"
one_dim_iso_idem: fixes x :: "'d" shows "one_dim_iso (one_dim_iso x) = one_dim_iso x"
abs_poly_def: fixes x :: "'b poly" shows "\<bar>x\<bar> = (if x < 0 then - x else x)"
lprefix_infinite: fixes u :: "'b llist"   and v :: "'b llist" assumes "u \<le> v"   and "linfinite u" shows "u = v"
Lyndon_sing: fixes a :: "'a" shows "Lyndon [a]"
poly_add_append_zero: fixes p :: "'a list"   and q :: "'a list" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R" shows "poly_add (p @ [\<zero>]) (q @ [\<zero>]) = local.normalize (poly_add p q @ [\<zero>])"
mrbst_join_correct: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x < y" shows "random_bst A \<bind> (\<lambda>t1. random_bst B \<bind> mrbst_join t1) = random_bst (A \<union> B)"
set_take_subset: fixes n :: "nat"   and xs :: "'a list" shows "list.set (take n xs) \<subseteq> list.set xs"
vpathE: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "vpath p G" obtains "vwalk p G" and "distinct p"
toFull: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "bij_betw (InfModel.toFull intT \<sigma>) {a. intT \<sigma> a} UNIV"
lcm_self: fixes a :: "'a" shows "lcm a a = normalize a"
perp_not_col2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Perp C D" shows "\<not> Col A B C \<or> \<not> Col A B D"
fst_def: fixes prod :: "'b \<times> 'c" shows "fst prod = (case prod of (x1, x2) \<Rightarrow> x1)"
span_span: fixes W :: "'b vec set"   and n :: "nat" assumes "W \<subseteq> carrier_vec n" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W) = LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W"
param_map_option: fixes A :: "('a \<times> 'd) set"   and B :: "('c \<times> 'e) set" shows "(map_option, map_option) \<in> (A \<rightarrow> B) \<rightarrow> \<langle>A\<rangle>option_rel \<rightarrow> \<langle>B\<rangle>option_rel"
prems_1: shows "[:- a, 1:] * p \<noteq> 0"
cases: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and \<psi> :: "'b stream \<Rightarrow> bool"   and a :: "'b stream" assumes "(\<phi> until \<psi>) a" obtains xs :: "'b stream" where "a = xs__" and "\<psi> xs__"   | xs :: "'b stream" where "a = xs" and "\<phi> xs" and "(\<phi> until \<psi>) (stl xs)"
aff_dim_dim_affine_diffs: fixes S :: "'a set"   and a :: "'a" assumes "affine S"   and "a \<in> S" shows "aff_dim S = int (dim ((\<lambda>x. x - a) ` S))"
range_reducerE: fixes p :: "nat"   and l :: "float"   and ivl :: "float interval" assumes "range_reducer p l = Some ivl" obtains n :: "int" where "real_of_int n * (2 * pi) \<in>\<^sub>r ivl"
set_sorted_list_of_set: fixes A :: "'c set" assumes "finite A" shows "set (compare_linorder.sorted_list_of_set A) = A"
greatest_less_zero_row: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "'f" assumes "echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "\<not> (\<forall>a. is_zero_row_upt_k a k A)" shows "(GREATEST m. \<not> is_zero_row_upt_k m k A) < i"
prime_power_inj: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize a"   and "power.power one times a m = power.power one times a n" shows "m = n"
orientable_8_implies_5: fixes x :: "'a" assumes "orientable_8 (x \<sqinter> - (1::'a))" shows "orientable_5 x"
ltakeWhile_eq_LNil_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a llist" shows "(ltakeWhile P xs = LNil) = (xs \<noteq> LNil \<longrightarrow> \<not> P (lhd xs))"
indexlt_simp: fixes k' :: "nat"   and u' :: "nat"   and k :: "nat"   and u :: "nat" shows "indexlt k' u' k u = (k' < k \<or> k' = k \<and> u' < u)"
ncols_singleton: fixes j :: "nat"   and i :: "nat"   and e :: "'a" shows "ncols (singleton_matrix j i e) = (if e = (0::'a) then 0 else Suc i)"
subst_support_comp: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<theta> supports \<I>"   and "\<delta> supports \<I>" shows "\<theta> \<circ>\<^sub>s \<delta> supports \<I>"
list_all_list: fixes A :: "'a set"   and \<phi> :: "'a \<Rightarrow> bool" assumes "finite A" shows "list_all \<phi> (list A) = (\<forall>a\<in>A. \<phi> a)"
bounded_linear_axioms: shows "bounded_linear (if x \<in> S then f' x else (\<lambda>x. 0::'b))"
elims_3: fixes x :: "'b set"   and xa :: "'b interrupt_action list" assumes "\<not> interrupt_actions_ok' x xa" obtains "is" :: "'b set" and ia :: "'b interrupt_action" and ias :: "'b interrupt_action list" where "x = is" and "xa = ia # ias" and "\<not> (interrupt_action_ok' is ia \<and> interrupt_actions_ok' (redT_updI is ia) ias)"
gaugeI: fixes \<gamma> :: "'a \<Rightarrow> 'a set" assumes "\<And>x. x \<in> \<gamma> x"   and "\<And>x. open (\<gamma> x)" shows "gauge \<gamma>"