hcju/isabelleps huggingface dataset:
distinct_63: fixes x61 :: "'b ltlr"   and x62 :: "'b ltlr"   and x81 :: "'b ltlr"   and x82 :: "'b ltlr" shows "x61 or\<^sub>r x62 \<noteq> x81 U\<^sub>r x82"
k_2: fixes n :: "nat" shows "norm (ln (ln (real n + 1))) \<le> k"
substNext_code_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val" shows "uninst_code.substNext_code phis g next \<equiv> \<lambda>v. if v = snd next then uninst_code.substitution_code phis g next else v"
eq: shows "null_sets lborel = null_sets (distr lebesgue lborel T)"
poly_f_is: shows "poly_f qs = [:0, 1:]"
sel_10: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "rchild (Node x21 x22 x23 x24 x25) = x25"
simps_8: fixes x2 :: "'c" shows "\<infinity> \<noteq> Lt x2"
card2_eI: fixes S :: "'c set" assumes "2 \<le> card S" shows "\<exists>x y. x \<noteq> y \<and> x \<in> S \<and> y \<in> S"
unique: fixes q :: "'a list" assumes "q \<in> carrier (K [X]) \<and> pirreducible K q \<and> lead_coeff q = \<one> \<and> a_kernel (K [X]) R (\<lambda>p. local.eval p x) = PIdl\<^bsub>K [X]\<^esub> q" shows "q = p"
2: shows "\<forall>(i, p)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. \<forall>(j, q)\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A \<union> set D. p = q \<longrightarrow> i = j"
Abs_freeword_singleton_conv_apply_sign_freeletter: fixes x :: "'a \<times> bool" shows "Abs_freeword [x] = apply_sign Abs_freeletter x"
left_adjoint_prod: fixes b :: "'a" assumes "ide b" shows "left_adjoint_functor C C (\<lambda>x. local.prod x b)"
s2'_def: shows "s2' \<equiv> activate_cond_action2 (ls, (ts1, m1), ws, is) (ls, (ts2, m2), ws, is) ct"
eq_card_imp_inj_on: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite A"   and "card (f ` A) = card A" shows "inj_on f A"
ordinal_of_nat_le: fixes x :: "nat"   and y :: "nat" shows "(ordinal_of_nat x \<le> ordinal_of_nat y) = (x \<le> y)"
select_convs_4: fixes iiface :: "iface"   and oiface :: "iface"   and src :: "'a word \<times> nat"   and dst :: "'a word \<times> nat"   and proto :: "protocol"   and sports :: "16 word \<times> 16 word"   and dports :: "16 word \<times> 16 word"   and more :: "'b" shows "dst \<lparr>iiface = iiface, oiface = oiface, src = src, dst = dst, proto = proto, sports = sports, dports = dports, \<dots> = more\<rparr> = dst"
free_gen_in: fixes w :: "'b list"   and G :: "'b list set" assumes "w \<in> G" shows "w \<in> \<langle>G\<rangle>\<^sub>F"
update_correct_2: fixes m :: "'s"   and k :: "'k"   and v :: "'v" assumes "invar m" shows "invar (update k v m)"
arr_comps_def: fixes T :: "'a"   and k :: "int" shows "arr_comps T k = (\<lambda>i. if i in_bounds k then Some T else None)"
cnj_coeff: fixes x :: "nat" shows "cnj (coeff p x) = coeff p x"
sspace_DAG_def: fixes PROB :: "(('b, 'c) fmap \<times> ('b, 'c) fmap) set"   and lss :: "('b, 'c) fmap list" shows "sspace_DAG PROB lss \<equiv> set lss = valid_states PROB \<and> top_sorted_abs (\<lambda>x y. y \<in> state_successors PROB x) lss"
induct: fixes P :: "'c rexp \<Rightarrow> bool"   and a0 :: "'c rexp" assumes "P Zero"   and "P Full"   and "P One"   and "\<And>r. P (Star r)"   and "\<And>v. P (Atom v)"   and "\<And>v va. P (Plus v va)"   and "\<And>v va. P (Times v va)"   and "\<And>v. P (rexp.Not v)"   and "\<And>v va. P (rexp.Inter v va)"   and "\<And>v. P (Pr v)" shows "P a0"
of_nat_0: fixes n :: "nat" assumes "word_of_nat n = 0"   and "n < 2 ^ LENGTH('b)" shows "n = 0"
excess'_if: fixes x :: "nat" shows "excess f' x = (if x = u then excess f u - \<Delta> else if x = v then excess f v + \<Delta> else excess f x)"
has_derivative_sequence_Lipschitz: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and f' :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and g' :: "'a \<Rightarrow> 'a \<Rightarrow> 'b"   and e :: "real" assumes "convex S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_derivative f' n x) (at x within S)"   and "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"   and "0 < e" shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
least_least: fixes k :: "'d"   and A :: "'d set" assumes "k \<in> A" shows "least A \<le> k"
DERIV_NS_iff: fixes f :: "'b \<Rightarrow> 'b"   and D :: "'b"   and x :: "'b" shows "(f has_field_derivative D) (at x) = (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0::'b\<rightarrow>\<^sub>N\<^sub>S D"
assms: shows "c \<in>\<^sub>\<circ> Vset \<alpha>"
carrier_local_ring_at_def: shows "pi.carrier_local_ring_at \<equiv> qr.carrier_quotient_ring"
cases_scheme: fixes r :: "('d, 'e, 'f, 'g) pre_RedBlack_scheme" obtains red :: "('d \<times> nat) rgraph" and black :: "('d, 'e, 'f) lts" and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" and init_conf :: "('e, 'f) conf" and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf" and marked :: "'d \<times> nat \<Rightarrow> bool" and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool" and more :: "'g" where "r = \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr>"
allD: fixes P :: "'d \<Rightarrow> bool"   and ll :: "'d" assumes "\<forall>ll. P ll" shows "P ll"
pmf_eq_0_set_pmf: fixes M :: "'a pmf"   and p :: "'a" shows "(pmf M p = 0) = (p \<notin> set_pmf M)"
pconj_nneg: fixes a :: "real"   and b :: "real" shows "0 \<le> a .& b"
measurable_count_space_eq2: fixes A :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and M :: "'e measure" assumes "finite A" shows "(f \<in> M \<rightarrow>\<^sub>M count_space A) = (f \<in> space M \<rightarrow> A \<and> (\<forall>a\<in>A. f -` {a} \<inter> space M \<in> sets M))"
divide_out_primepow: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and n :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "n \<noteq> zero"   and "\<not> dvd.dvd times n one" obtains p :: "'b" and k :: "nat" and n' :: "'b" where "normalization_semidom.prime zero times one normalize p" and "dvd.dvd times p n" and "\<not> dvd.dvd times p n'" and "0 < k" and "n = times (power.power one times p k) n'"
da_elim_cases_60: fixes Env :: "env"   and B :: "lname set"   and e :: "expr"   and es :: "expr list"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>In3 (e # es)\<guillemotright> A" obtains E :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e\<rangle>\<guillemotright> E" and "Env\<turnstile> nrm E \<guillemotright>\<langle>es\<rangle>\<guillemotright> A"
assms: shows "z \<in> at_least_as_good y B Pr"
n_kat_3_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(t x \<cdot> y \<cdot> !z = (0::'a)) = (y \<cdot> !z \<le> !x \<cdot> y)"
disc_2: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "\<not> lnull (lalternate xs)"
neq_if_length_neq: fixes xs :: "'b list"   and ys :: "'b list" assumes "length xs \<noteq> length ys" shows "xs = ys \<equiv> False"
confluent_distinct_aux: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q1 :: "'t \<Rightarrow>\<^sub>0 'c"   and f1 :: "'t \<Rightarrow>\<^sub>0 'c"   and t1 :: "'a"   and q2 :: "'t \<Rightarrow>\<^sub>0 'c"   and f2 :: "'t \<Rightarrow>\<^sub>0 'c"   and t2 :: "'a"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "red_single p q1 f1 t1"   and "red_single p q2 f2 t2"   and "t1 \<oplus> lt f1 \<prec>\<^sub>t t2 \<oplus> lt f2"   and "f1 \<in> F"   and "f2 \<in> F" obtains s :: "'t \<Rightarrow>\<^sub>0 'c" where "(red F)\<^sup>*\<^sup>* q1 s" and "(red F)\<^sup>*\<^sup>* q2 s"
Spy_spies_bad_shrK: fixes A :: "agent"   and evs :: "event list" assumes "A \<in> bad" shows "Key (RPwd A) \<in> knows Spy evs"
in_idem: fixes A :: "'b sat set"   and x :: "'b sat" assumes "finite A"   and "x \<in> A" shows "min x (Inf A) = Inf A"
times_dense: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and top :: "'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linorder_stone_relation_algebra_expansion one times uminus less_eq less bot top inf sup conv"   and "x \<noteq> bot"   and "y \<noteq> bot" shows "times x y \<noteq> bot"
permute_plus: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "perm"   and q :: "perm"   and x :: "'b" assumes "class.pt permute" shows "permute (p + q) x = permute p (permute q x)"
of_real_eq_1_iff: fixes x :: "real" shows "(of_real x = (1::'b)) = (x = 1)"
cpt_def: fixes P :: "'a set"   and env :: "'a \<Rightarrow> nat"   and \<pi>l :: "'b list" shows "cpt P env \<pi>l \<equiv> \<forall>p\<in>P. env p < length \<pi>l"
igWlsBinp_def: fixes MOD :: "('a, 'b, 'varSort, 'sort, 'c, 'd, 'e, 'f, 'g) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'f option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
bounded_clinear_apply_cblinfun: fixes b :: "'a" shows "bounded_clinear (\<lambda>a. a *\<^sub>V b)"
classes_changed_class_set: fixes S :: "char list set"   and P :: "'b prog"   and P' :: "'b prog" assumes "S \<inter> classes_changed P P' = {}" shows "\<forall>C\<in>S. class P C = class P' C"
"case": fixes f :: "('b \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> 'a"   and x1 :: "('b \<times> real) msllist"   and x2 :: "real \<Rightarrow> real" shows "(case MS x1 x2 of MS x xa \<Rightarrow> f x xa) = f x1 x2"
vars_term_supteq: fixes x :: "'d"   and t :: "('e, 'd) Term.term" assumes "x \<in> vars_term t" shows "t \<unrhd> Var x"
ba_top: shows "top = - (THE x. \<forall>z. x = - (z \<squnion> - z))"
convergentI: fixes X :: "nat \<Rightarrow> 'e"   and L :: "'e" assumes "X \<longlonglongrightarrow> L" shows "convergent X"
bind_fv2_subst_bv1_cancel: fixes lev :: "nat"   and t :: "Core.term"   and v :: "variable \<times> typ" assumes "\<And>x. lev < x \<Longrightarrow> \<not> loose_bvar1 t x"   and "\<not> occs (case v of (x, xa) \<Rightarrow> Fv x xa) t" shows "bind_fv2 v lev (subst_bv1 t lev (case v of (x, xa) \<Rightarrow> Fv x xa)) = t"
N1_live: shows "\<turnstile> \<box>($PsiInv \<and> [(N1 \<or> N2) \<and> \<not> beta1]_(x, y, sem, pc1, pc2)) \<and> SF(N1)_(x, y, sem, pc1, pc2) \<and> \<box>SF(N2)_(x, y, sem, pc1, pc2) \<longrightarrow> \<diamond>pc1 = #b"
rel_mono: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and R3 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R3a :: "'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a"   and "R3 \<le> R3a" shows "rel_pre_G\<^sub>M R1 R2 R3 \<le> rel_pre_G\<^sub>M R1a R2a R3a"
cod_char: shows "V.cod = (\<lambda>\<mu>. if V.arr \<mu> then \<lparr>Chn = span_in_category.apex (\<cdot>) (Cod \<mu>), Dom = Cod \<mu>, Cod = Cod \<mu>\<rparr> else V.null)"
Cod_Diagonalize_rep: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f" shows "monoidal_language.Cod C (monoidal_language.Diagonalize C (free_monoidal_category.rep C f)) = monoidal_language.Diagonalize C (free_monoidal_category.rep C (partial_magma.cod (free_monoidal_category.comp C) f))"
is_lub_ttree: fixes S :: "'b ttree set" shows "S <<| Either S"
monoid_homomorphism_def: fixes \<eta> :: "'b \<Rightarrow> 'c"   and M :: "'b set"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit :: "'b"   and M' :: "'c set"   and composition' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit' :: "'c" shows "monoid_homomorphism \<eta> M composition unit M' composition' unit' \<equiv> (map \<eta> M M' \<and> Group_Theory.monoid M composition unit) \<and> Group_Theory.monoid M' composition' unit' \<and> monoid_homomorphism_axioms \<eta> M composition unit composition' unit'"
map_cong0: fixes x :: "'a multiset"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in># x \<Longrightarrow> f z = g z" shows "image_mset f x = image_mset g x"
psubst_cong: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "good X"   and "goodEnv rho"   and "goodEnv rho'"   and "\<And>ys y. fresh ys y X \<or> rho ys y = rho' ys y" shows "X #[rho] = X #[rho']"
simps: fixes x :: "tm" shows "HF_axP x = x EQ \<guillemotleft>HF1\<guillemotright> OR x EQ \<guillemotleft>HF2\<guillemotright>"
ih_5: shows "InvariantWatchListsUniq (getWatchList state')"
rel_mono_strong: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c list"   and y :: "'d list"   and Ra :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "list_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> set x; yb \<in> set y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "list_all2 Ra x y"
Im_complex_div_lemma: fixes z :: "complex" shows "(Im ((1 - \<i> * z) / (1 + \<i> * z)) = 0) = (Re z = 0)"
worker_wrapper_fusion: fixes wrap :: "'c \<rightarrow> 'b"   and unwrap :: "'b \<rightarrow> 'c"   and work :: "'c"   and body :: "'b \<rightarrow> 'b" assumes "wrap oo unwrap = ID"   and "work = fix\<cdot>(unwrap oo body oo wrap)" shows "(unwrap oo wrap)\<cdot>work = work"
invar_rel_def: fixes a :: "'h \<Rightarrow> 'g set"   and X :: "('f \<times> 'h) set"   and S :: "('e \<times> 'g set) set" shows "\<langle>X, S\<rangle>invar_rel a = {((x, s'), y). \<exists>s. (s', s) \<in> X \<and> (x, y) \<in> S \<and> y \<subseteq> a s}"
intT_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "intT \<sigma> a \<equiv> if unprot \<sigma> then eintT \<sigma> a else if protFw \<sigma> then eintT \<sigma> a \<and> eintF (Tag \<sigma>) [a] = a else eintT \<sigma> a \<and> a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}"
run_ask_env: fixes f :: "'b \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (ask_env f) r = run_env (f r) r"
finite_config_BIT: fixes init :: "'c list"   and qs :: "'c list" assumes "distinct init" shows "finite (set_pmf (config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs))"
VDM_Complete: fixes c :: "OBJ"   and A :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" assumes " \<Turnstile> c : A " shows "{} \<rhd> c : A"
intro: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure" assumes "Filtration.filtration M F" shows "filtrated_prob_space_axioms M F"
adv: assumes "exception_\<I> (\<I>_middle \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C)) \<surd>" shows "advantage (absorb \<A> (obsf_converter (sim_outer |\<^sub>= 1\<^sub>C))) (obsf_resource (sim_inner |\<^sub>= 1\<^sub>C \<rhd> ideal)) (obsf_resource middle) \<le> adv1"
oalist_inv_raw_def: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" shows "oalist_inv_raw x xs = ((0::'c) \<notin> snd ` set xs \<and> sorted_wrt (lt x) (map fst xs))"
transpose_empty: fixes xs :: "'a list list" shows "(transpose xs = []) = (\<forall>x\<in>set xs. x = [])"
unsigned_eq_0_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and w :: "'b Word.word" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.unsigned one plus zero w = zero) = (w = 0)"
arr_dom_iff_arr: fixes f :: "('a, 'a set) Maps.arr" shows "Maps.arr (Maps.dom f) = Maps.arr f"
mset_subst: fixes C' :: "'a literal set"   and D :: "'a literal set"   and \<theta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list" assumes "C' = subst_cl D \<theta>"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"   and "finite D" shows "mset_cl (C', \<eta>) = mset_cl (D, \<sigma>) \<or> (mset_cl (C', \<eta>), mset_cl (D, \<sigma>)) \<in> mult (mult trm_ord)"
transpose_transpose: fixes xs :: "'a list list" assumes "sorted (rev (map length xs))" shows "List.transpose (List.transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs"
convert_term_frees: fixes t :: "'b" assumes "no_abs t" shows "frees (convert_term t) = frees t"
wf_projectQuery: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and I :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and QQ :: "(nat set \<times> 'b option list set) set" assumes "getIJ getIJ"   and "1 \<le> card I"   and "wf_query n I Q Qn"   and "non_empty_query Q"   and "covering I Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> I)"   and "QQ = New_max.projectQuery I Q"   and "included I Qn"   and "non_empty_query Qn" shows "rwf_query n I QQ Qn"
char_matrix_jordan_block: fixes n :: "nat"   and a :: "'b"   and b :: "'b" shows "char_matrix (jordan_block n a) b = jordan_block n (a - b)"
eklr: shows "length (as @ [x] @ bs @ [x]) = Suc (length (as @ [x] @ bs))"
min_less_iff_disj: fixes x :: "'a Word.word"   and y :: "'a Word.word"   and z :: "'a Word.word" shows "(signed.min x y <s z) = (x <s z \<or> y <s z)"
class_of_def: fixes U :: "'a set set"   and s :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "st.class_of U s \<equiv> equivalence.Class (Sigma {U. is_zariski_open U \<and> \<pp> \<in> U} sheaf_spec) {(x, y). st.rel x y} (U, s)"
inv_unique: fixes G :: "('c, 'd) monoid_scheme"   and y :: "'c"   and x :: "'c"   and y' :: "'c" assumes "Group.monoid G"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"   and "x \<otimes>\<^bsub>G\<^esub> y' = \<one>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "y' \<in> carrier G" shows "y = y'"
indep_imp_subset_basis: fixes X :: "'a set" assumes "indep X" shows "\<exists>B. basis B \<and> X \<subseteq> B"
assms_1: shows "x \<cdot> y = u \<cdot> v"
injective_sup: fixes t :: "'a"   and e :: "'a" assumes "is_inj t"   and "e ; t\<^sup>T \<le> 1'"   and "is_inj e" shows "is_inj (t + e)"
respect_2: shows "\<forall>v\<in>C. \<phi> (g v) = \<phi> v"
ginjec_def: fixes F :: "('c, 'd) Group_scheme"   and G :: "('e, 'f) Group_scheme"   and f :: "'c \<Rightarrow> 'e" shows "ginj\<^bsub>F,G\<^esub> f = (f \<in> gHom F G \<and> inj_on f (carrier F))"
24: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<squnion> z) \<squnion> (x \<sqinter> y \<squnion> x \<sqinter> z) = x \<sqinter> (y \<squnion> z)"
simps_1: fixes x :: "'b" shows "subterms (Var x) = {Var x}"
assms_2: shows "\<forall>\<^sub>F x in at_top. \<forall>y z. x \<le> y \<and> y \<le> z \<longrightarrow> f z \<le> f y"
unit_factor_mult_normalize: fixes a :: "'b poly" shows "[:lead_coeff a:] * smult (inverse (lead_coeff a)) a = a"
EF_lem2a: fixes x :: "'a"   and f :: "'a set" assumes "x \<in> f" shows "x \<in> EF f"
gf: fixes y :: "'a" assumes "y \<in> S" shows "g (f y) = y"
pcr_cr_eq: shows "pcr_hmultiset = cr_hmultiset"
add_le_same_cancel2: fixes a :: "'a"   and b :: "'a" shows "(a + b \<le> b) = (a \<le> (0::'a))"
Sup_distl: fixes x :: "'b \<times> 'c"   and Y :: "('b \<times> 'c) set" shows "times_prod' x (Sup_prod Y) = Sup_prod (times_prod' x ` Y)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s1)) \<and> t2 = snd (fst (sub_instr_sub1 instr_name result rs1_val operand2 s2))"
cases: fixes a :: "'a set" assumes "finite a" obtains "aa__ = {}"   | A :: "'a set" and a :: "'a" where "aa__ = insert a A" and "finite A"
uminus_vec_def: fixes v :: "'a vec" shows "- v \<equiv> vec (dim_vec v) (\<lambda>i. - v $ i)"
fperms: fixes init :: "'c list" shows "finite {x. length x = length init \<and> distinct x \<and> set x = set init}"
spr_simTrans_def: fixes agents :: "'b odlist"   and jkbp :: "'b \<Rightarrow> ('b, 'e, 'f) GC list"   and envAction :: "('b, 'c, 'd) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'f) \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> ('b, 'c, 'd) BEState"   and envVal :: "('b, 'c, 'd) BEState \<Rightarrow> 'e \<Rightarrow> bool"   and envObsC :: "'c \<Rightarrow> 'h"   and envObs :: "'b \<Rightarrow> ('b, 'c, 'd) BEState \<Rightarrow> 'h \<times> 'd option" shows "spr_simTrans agents jkbp envAction envTrans envVal envObsC envObs \<equiv> \<lambda>a ec. let aSuccs = spr_trans agents jkbp envAction envTrans envVal envObs (fst ec) (snd ec); cec' = fromList (spr_trans agents jkbp envAction envTrans envVal envObs (fst ec) (fst ec)) in map (\<lambda>aec'. (ODList.filter (\<lambda>s. envObsC (es (snd s)) = spr_simObsC envObsC aec') cec', aec')) (map fromList (List_local.partition (envObs_rel (envObs a)) aSuccs))"
test_num': fixes i :: "int"   and a :: "'av" shows "test_num' i a = (i \<in> \<gamma> a)"
Ints_diff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "minus a b \<in> ring_1.Ints one plus zero minus"
add_block_design: fixes bl :: "'a set" assumes "finite bl"   and "bl \<noteq> {}" shows "design (\<V> \<union> bl) (add_block bl)"
assms: shows "\<forall>\<^sub>F n in \<U>. norm (X n) < u"
leadsETo_Un_distrib: fixes F :: "'d program"   and A :: "'d set"   and B :: "'d set"   and CC :: "'d set set"   and C :: "'d set" shows "(F \<in> (A \<union> B) leadsTo[CC] C) = (F \<in> A leadsTo[CC] C \<and> F \<in> B leadsTo[CC] C)"
ident_rel_trans: fixes idt :: "'c"   and G :: "('c, 'b) labeled_graph" assumes "ident_rel idt G" shows "maintained (transitive_rule idt) G"
assms_2: shows "squarefree_part m = squarefree_part n"
Z_def: shows "Z \<equiv> mat 2 2 (\<lambda>(i, j). if i \<noteq> j then 0 else if i = 0 then 1 else - 1)"
hyps_3: shows "mset xs \<noteq> Ys \<and> (\<forall>x. count Ys x < count (mset xs) x \<longrightarrow> (\<exists>y. gt y x \<and> count (mset xs) y < count Ys y))"
nn_integral_count_space_eq: fixes A :: "'c set"   and B :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = 0"   and "\<And>x. x \<in> B - A \<Longrightarrow> f x = 0" shows "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) f"
spr_simVal_def: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and agent :: "'b"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "FiniteSingleAgentEnvironment jkbp agent" shows "FiniteSingleAgentEnvironment.spr_simVal envVal \<equiv> envVal \<circ> snd"
right_translations_of_subgroup_are_transformation_group: shows "transformation_group (translation ` H) G"
lt: fixes i :: "nat" assumes "i < n" shows "p i < n"
strict_sorted_concat_I: fixes xs :: "'a list list" assumes "\<And>x. x \<in> list.set xs \<Longrightarrow> strict_sorted x"   and "\<And>n. Suc n < length xs \<Longrightarrow> xs ! n < xs ! Suc n"   and "xs \<in> lists (- {[]})" shows "strict_sorted (concat xs)"
preserves_reflects_arr: fixes f :: "'j" shows "A_B.arr (\<chi> f) = J.arr f"