hcju/isabelleps huggingface dataset:
pelims: fixes x :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and xa :: "nat"   and xb :: "int vec"   and xc :: "int iarray"   and y :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" assumes "upd_fi_mu_state x xa xb xc = y"   and "upd_fi_mu_state_dom (x, xa, xb, xc)" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and d :: "int iarray" and i :: "nat" and fi :: "int vec" and mu_i :: "int iarray" where "x = (f, mu, d)" and "xa = i" and "xb = fi" and "xc = mu_i" and "y = (update_i f fi, iarray_update mu i mu_i, d)" and "upd_fi_mu_state_dom ((f, mu, d), i, fi, mu_i)"
factors_of_int_poly_const: fixes p :: "int poly" assumes "degree p = 0" shows "factors_of_int_poly p = []"
ennreal_minus_mono: fixes a :: "ennreal"   and c :: "ennreal"   and d :: "ennreal"   and b :: "ennreal" assumes "a \<le> c"   and "d \<le> b" shows "a - b \<le> c - d"
rep_Tree\<^sub>\<alpha>_def: shows "FL_Formula.rep_Tree\<^sub>\<alpha> \<equiv> quot_type.rep FL_Formula.Tree\<^sub>\<alpha>.Rep_Tree\<^sub>\<alpha>"
UN_least: fixes A :: "'a set"   and B :: "'a \<Rightarrow> 'b set"   and C :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> order.greater_eq C (B x)" shows "order.greater_eq C (\<Union> (B ` A))"
memberid_method_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "memberid (method sig m) = mid sig"
scalar_prod_smult_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and a :: "'a" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (a \<cdot>\<^sub>v v\<^sub>2) = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
csum_cong: fixes p1 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set"   and p2 :: "('f \<times> 'f) set"   and r2 :: "('g \<times> 'g) set" assumes "p1 =o r1"   and "p2 =o r2" shows "p1 +c p2 =o r1 +c r2"
a3: shows "butlast i1 = take (length i1 - Suc 0) x"
tendsto_proj: fixes a :: "'d \<Rightarrow>\<^sub>F 'e"   and F :: "('d \<Rightarrow>\<^sub>F 'e) filter"   and i :: "'d" assumes "((\<lambda>x. x) \<longlongrightarrow> a) F" shows "((\<lambda>x. (x)\<^sub>F i) \<longlongrightarrow> (a)\<^sub>F i) F"
mult_zero_ow_axioms: shows "mult_zero_ow U (*\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w"
to_nat_from_nat_id: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and x :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "x < CARD('a)" shows "mod_type.to_nat Rep (mod_type.from_nat Abs x) = x"
lex_def: fixes r :: "('g \<times> 'g) set" shows "lex r = \<Union> (range (lexn r))"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "M.satL \<xi> (Pos a) = M.satA \<xi> a"
not_image: shows "(z - b) / c \<notin> path_image \<gamma>"
bind_sample1: fixes return :: "'d \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and sample :: "'f pmf \<Rightarrow> ('f \<Rightarrow> 'e) \<Rightarrow> 'e"   and p :: "'f pmf"   and f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "monad_prob return bind sample" shows "bind (sample p f) g = sample p (\<lambda>x. bind (f x) g)"
min_distrib: fixes c :: "real"   and a :: "real"   and b :: "real" assumes "0 \<le> c" shows "c * min a b = min (c * a) (c * b)"
positively_invariant_trapG: shows "g.positively_invariant trapG"
RCALL_def: fixes D :: "'a" shows "RCALL D \<equiv> D"
conf_widen: fixes P :: "'b prog"   and h :: "'heap"   and v :: "'addr val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "P \<turnstile> T \<le> T'" shows "P,h \<turnstile> v :\<le> T'"
integral_cmul: fixes S :: "'c set"   and c :: "real"   and f :: "'c \<Rightarrow> 'b" shows "integral S (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral S f"
empty_interpretation_validate: fixes L :: "'at Literal" shows "{} \<Turnstile> L = (\<exists>A. L = Neg A)"
lookup_trie_update: fixes ks :: "'c list"   and v :: "'b"   and t :: "('c, 'b) trie"   and ks' :: "'c list" shows "lookup_trie (trie_update ks v t) ks' = (if ks = ks' then Some v else lookup_trie t ks')"
assms_2: shows "\<forall>x y. x \<in> s \<and> y \<in> s \<longrightarrow> (f x = f y) = (x = y)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>G f : G a \<rightarrow>\<^sub>D G b\<guillemotright>"
elims_2: fixes x :: "'a ext"   and xa :: "'a ext" assumes "x < xa" obtains x :: "'a ext" and y :: "'a ext" where "xa__ = x" and "xa = y" and "x \<le> y \<and> \<not> y \<le> x"
prime_elem_const_poly_iff: fixes c :: "'a" shows "prime_elem [:c:] = prime_elem c"
assms_6: shows "X' \<equiv> (\<lambda>\<sigma>. <|\<sigma> x, \<sigma> y|>) -` X \<inter> space (state_measure V \<Gamma>)"
factorcount_unique: fixes as :: "'a list"   and a :: "'a" assumes "wfactors G as a"   and "a \<in> carrier G"   and "set as \<subseteq> carrier G" shows "factorcount G a = length as"
induct: fixes x1 :: "cmd"   and x2 :: "cmd"   and x3 :: "char list \<Rightarrow> val option"   and x4 :: "nat"   and x5 :: "cmd"   and x6 :: "char list \<Rightarrow> val option"   and x7 :: "nat"   and P :: "cmd \<Rightarrow> cmd \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> nat \<Rightarrow> cmd \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> nat \<Rightarrow> bool" assumes "x1 \<turnstile> \<langle>x2,x3,x4\<rangle> \<leadsto> \<langle>x5,x6,x7\<rangle>"   and "\<And>V e s. P (V:=e) (V:=e) s 0 Skip (s(V := interpret e s)) 1"   and "\<And>c\<^sub>1 c\<^sub>2 l s. \<lbrakk>labels (c\<^sub>1;; c\<^sub>2) l (Skip;; c\<^sub>2); labels (c\<^sub>1;; c\<^sub>2) #:c\<^sub>1 c\<^sub>2; l < #:c\<^sub>1\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (Skip;; c\<^sub>2) s l c\<^sub>2 s #:c\<^sub>1"   and "\<And>b c' l s. labels (while (b) c') l (Skip;; while (b) c') \<Longrightarrow> P (while (b) c') (Skip;; while (b) c') s l (while (b) c') s 0"   and "\<And>b s c\<^sub>1 c\<^sub>2. interpret b s = Some true \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (if (b) c\<^sub>1 else c\<^sub>2) s 0 c\<^sub>1 s 1"   and "\<And>b s c\<^sub>1 c\<^sub>2. interpret b s = Some false \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (if (b) c\<^sub>1 else c\<^sub>2) s 0 c\<^sub>2 s (#:c\<^sub>1 + 1)"   and "\<And>b s c. interpret b s = Some true \<Longrightarrow> P (while (b) c) (while (b) c) s 0 (c;; while (b) c) s 2"   and "\<And>b s c. interpret b s = Some false \<Longrightarrow> P (while (b) c) (while (b) c) s 0 Skip s 1"   and "\<And>prog c s l c' s' l' c\<^sub>2. \<lbrakk>prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>; P prog c s l c' s' l'\<rbrakk> \<Longrightarrow> P (prog;; c\<^sub>2) (c;; c\<^sub>2) s l (c';; c\<^sub>2) s' l'"   and "\<And>prog c s l c' s' l' c\<^sub>1. \<lbrakk>prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>; P prog c s l c' s' l'\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; prog) c s (l + #:c\<^sub>1) c' s' (l' + #:c\<^sub>1)"   and "\<And>prog c s l c' s' l' b c\<^sub>2. \<lbrakk>prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>; P prog c s l c' s' l'\<rbrakk> \<Longrightarrow> P (if (b) prog else c\<^sub>2) c s (l + 1) c' s' (l' + 1)"   and "\<And>prog c s l c' s' l' b c\<^sub>1. \<lbrakk>prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>; P prog c s l c' s' l'\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else prog) c s (l + #:c\<^sub>1 + 1) c' s' (l' + #:c\<^sub>1 + 1)"   and "\<And>cx c s l c' s' l' b. \<lbrakk>cx \<turnstile> \<langle>c,s,l\<rangle> \<leadsto> \<langle>c',s',l'\<rangle>; P cx c s l c' s' l'\<rbrakk> \<Longrightarrow> P (while (b) cx) (c;; while (b) cx) s (l + 2) (c';; while (b) cx) s' (l' + 2)" shows "P x1 x2 x3 x4 x5 x6 x7"
sym_sum_mset_inter_sets_size: fixes blv :: "'a set" assumes "blv \<in># \<B>" shows "int (size (\<Sum>bl\<in>#remove1_mset blv \<B>. mset_set (bl \<inter> blv))) = \<k> * (\<Lambda> * (\<v> - 1) div (\<k> - 1) - 1)"
is_empty_lookup_empty: fixes t :: "('c, 'd) trie" assumes "invar_trie t" shows "is_empty_trie t = (lookup_trie t = Map.empty)"
exhaust_disc: fixes vertex :: "'a vertex" obtains "vertex = SOURCE"   | "vertex = vertex.SINK"   | "is_Inner vertex"
distinct_3: fixes x1 :: "'d" shows "mutator x1 \<noteq> sys"
meval_def: shows "meval \<equiv> {((xa, x), xc, xb). mevalp xa x xc xb}"
in_lset_ltlD: fixes x :: "'a"   and xs :: "'a llist" assumes "x \<in> lset (ltl xs)" shows "x \<in> lset xs"
ring_of_sets_Pow: fixes sp :: "'c set" shows "ring_of_sets sp (Pow sp)"
F1set3_IF1set: fixes a :: "'d IF2"   and x :: "('d, 'd IF1, 'd IF2) F1" assumes "a \<in> F1set3 x" shows "IF2set a \<subseteq> IF1set (ctor1 x)"
one_sheaf_spec_in_sheaf_spec: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and U :: "'c set set" assumes "comm_ring R addition multiplication zero unit"   and "comm_ring.is_zariski_open R addition multiplication zero unit U" shows "comm_ring.one_sheaf_spec R addition multiplication zero unit U \<in> comm_ring.sheaf_spec R addition multiplication zero unit U"
FactGroup_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" shows "G Mod H = \<lparr>carrier = rcosets\<^bsub>G\<^esub> H, monoid.mult = (<#>\<^bsub>G\<^esub>), one = H\<rparr>"
rec_trpl_def: shows "rec_trpl = Cn 3 rec_mult [Cn 3 rec_mult [Cn 3 rec_power [Cn 3 (constn (Pi 0)) [recf.id 3 0], recf.id 3 0], Cn 3 rec_power [Cn 3 (constn (Pi 1)) [recf.id 3 0], recf.id 3 1]], Cn 3 rec_power [Cn 3 (constn (Pi 2)) [recf.id 3 0], recf.id 3 2]]"
hyps: assumes "k \<le> l" shows "concat (take k ([r] \<^sup>@ l)) = r \<^sup>@ k"
quant_eq_iff_mag_eq: fixes x :: "'e['f, 'g]"   and y :: "'e['f, 'g]" shows "(x = y) = (\<lbrakk>x\<rbrakk>\<^sub>Q = \<lbrakk>y\<rbrakk>\<^sub>Q)"
dom_null: shows "Left_a.dom Left_a.null = Left_a.null"
function_uminus_eval: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F" shows "function_uminus S R f a = \<ominus> f a"
right_inverse_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and b :: "'e"   and a :: "'e" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "(divide a b = one) = (a = b)"
maxsimp_singleton: fixes S :: "'b set"   and w :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>" shows "SimplicialComplex.maxsimp (CoxeterComplex.TheComplex S) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w})"
is_interval_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A"   and "bi_total A"   and "(A ===> A ===> (=)) (\<bullet>) (\<bullet>)"   and "rel_set A Basis Basis" shows "(rel_set A ===> (=)) is_interval is_interval"
iso_iff_group_isomorphisms: fixes G :: "('a, 'b) monoid_scheme"   and f :: "'a \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "(f \<in> Group.iso G H) = (\<exists>g. group_isomorphisms G H f g)"
add_add_commute: fixes i1 :: "'d"   and e1 :: "'c"   and i2 :: "'d"   and e2 :: "'c" shows "\<langle>Add i1 e1\<rangle> \<rhd> \<langle>Add i2 e2\<rangle> = \<langle>Add i2 e2\<rangle> \<rhd> \<langle>Add i1 e1\<rangle>"
swapEnv_preserves_freshEnv_distinct: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and ys :: "'varSort"   and x :: "'var"   and y1 :: "'var"   and y2 :: "'var" assumes "goodEnv rho"   and "xs \<noteq> ys \<or> x \<notin> {y1, y2}" shows "freshEnv xs x (rho &[y1 \<and> y2]_ys) = freshEnv xs x rho"
simps_2: fixes x :: "'c"   and xs :: "'c list" shows "rev (x # xs) = rev xs @ [x]"
upper_asymptotic_density_finite: fixes A :: "nat set" assumes "finite A" shows "upper_asymptotic_density A = 0"
simps_9: fixes x1 :: "('a, 'b, 'c, 'd) Tree set['a]"   and x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree" shows "tConj x1 \<noteq> tAct x41 x42 x43"
uconnectedV_Restrcl: fixes F :: "nat uprod set" shows "Restr ((Restr (uconnected F) V)\<^sup>*) V = Restr (uconnected F) V"
ladder_i_of_cut_at_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "L \<noteq> []" shows "ladder_i (ladder_cut L n) 0 = ladder_i L 0"
comp_permute: fixes f :: "'d"   and g :: "'d"   and k :: "'d"   and l :: "'d"   and h :: "'d" assumes "f \<cdot>\<^sub>D g = k \<cdot>\<^sub>D l"   and "D.seq f g"   and "D.seq g h" shows "f \<cdot>\<^sub>D g \<cdot>\<^sub>D h = k \<cdot>\<^sub>D l \<cdot>\<^sub>D h"
idem: fixes a :: "'t" shows "ord_term_lin.min a a = a"
Exist: fixes n :: "nat"   and p :: "pf" shows "dfa_of_pf n (Exist p) = rquot (det_nfa (quantify_nfa 0 (nfa_of_dfa (dfa_of_pf (Suc n) p)))) n"
circ_unfold_sum: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<circ> = x\<^sup>\<circ> \<squnion> x\<^sup>\<circ> * y * (x \<squnion> y)\<^sup>\<circ>"
inject_9: fixes x91 :: "'a MFOTL.formula"   and x92 :: "\<I>"   and x93 :: "'a MFOTL.formula"   and y91 :: "'a MFOTL.formula"   and y92 :: "\<I>"   and y93 :: "'a MFOTL.formula" shows "(formula.Until x91 x92 x93 = formula.Until y91 y92 y93) = (x91 = y91 \<and> x92 = y92 \<and> x93 = y93)"
no_leading_def: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "no_leading P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (hd xs))"
ord_conv_Gcd: fixes x :: "'a" assumes "x \<in> carrier G" shows "ord x = Gcd {n. x [^] n = \<one>}"
Lipschitz_Fourier_convergence_periodic: fixes f :: "real \<Rightarrow> real"   and d :: "real"   and t :: "real"   and M :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "0 < d"   and "\<And>x. \<bar>x - t\<bar> < d \<Longrightarrow> \<bar>f x - f t\<bar> \<le> M * \<bar>x - t\<bar>"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> f t"
idx_sequence_mono: fixes idx :: "nat \<Rightarrow> nat"   and m :: "nat"   and n :: "nat" assumes "idx_sequence idx"   and "m \<le> n" shows "idx m \<le> idx n"
square_integrable_imp_integrable: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S"   and "S \<in> lmeasurable" shows "integrable (lebesgue_on S) f"
pell_valuation_trivial_nat: shows "pell_valuation (case (Suc 0, 0) of (x, y) \<Rightarrow> (int x, y)) = 1"
is_success_def: fixes status :: "status" shows "is_success status = (status = SUCCESS)"
bmg: shows "(\<lambda>x. if x = a then d\<^sup>2 else 0) \<in> borel_measurable (lebesgue_on {a..b})"
simps_2: fixes x2a :: "'b"   and y2a :: "'b" shows "(\<^bold>\<langle>x2a\<^bold>\<rangle> = \<^bold>\<langle>y2a\<^bold>\<rangle>) = (x2a = y2a)"
lenlex_def: fixes r :: "('a \<times> 'a) set" shows "lenlex r = inv_image (less_than <*lex*> lex r) (\<lambda>xs. (length xs, xs))"
simps_2: fixes x :: "real" shows "FVT (Const x) = {}"
assms_1: shows "\<forall>(x, m)\<in>PTA.clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>"
pickFT_pred_def: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and cl :: "('c, 'd, 'e) cmd list"   and n :: "nat" assumes "PL cval" shows "pickFT_pred cl n \<equiv> n < length cl \<and> finished (cl ! n)"
lookup_e_focus: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "g \<in> P[insert x X]" shows "lookup (poly_eval (\<lambda>_. monomial a 0) (focus {x} g)) t = poly_eval (\<lambda>_. a) (lookup (focus X g) t)"
homotopic_with_compose_continuous_right: fixes p :: "('d \<Rightarrow> 'c) \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'b"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and W :: "'d set" assumes "homotopic_with_canon (\<lambda>f. p (f \<circ> h)) X Y f g"   and "continuous_on W h"   and "h ` W \<subseteq> X" shows "homotopic_with_canon p W Y (f \<circ> h) (g \<circ> h)"
class_rec_induct': fixes P :: "prog \<Rightarrow> qtname \<Rightarrow> bool"   and a0 :: "prog"   and a1 :: "qtname" assumes "\<And>G C t f. (\<And>x2. \<lbrakk>class G C = Some x2; ws_prog G; C \<noteq> Object\<rbrakk> \<Longrightarrow> P G (super x2)) \<Longrightarrow> P G C" shows "P a0 a1"
sin_mono_less_eq: fixes x :: "real"   and y :: "real" assumes "- (pi / 2) \<le> x"   and "x \<le> pi / 2"   and "- (pi / 2) \<le> y"   and "y \<le> pi / 2" shows "(sin x < sin y) = (x < y)"
word_sle_def: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le>s b) = (sint a \<le> sint b)"
node_total_order_antisym: fixes e1 :: "('msgid \<times> 'op) event"   and i :: "nat"   and e2 :: "('msgid \<times> 'op) event" assumes "e1 \<sqsubset>\<^sup>i e2"   and "e2 \<sqsubset>\<^sup>i e1" shows "False"
preserves_ide: fixes a :: "'b" assumes "B.ide a" shows "A.ide (local.inv a)"
in_own_span: fixes S :: "rat vec set" assumes "S \<subseteq> Rn" shows "S \<subseteq> gs.span S"
assms_1: shows "set (ms @ n # ns) \<subseteq> M"
continuous_hurwitz_zeta': fixes a :: "real"   and f :: "'a \<Rightarrow> complex"   and s :: "'a"   and A :: "'a set" assumes "0 < a"   and "f s \<noteq> 1"   and "continuous (at s within A) f" shows "continuous (at s within A) (\<lambda>s. hurwitz_zeta a (f s))"
intros_10: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "F, \<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<rightarrow> G, \<Delta>"
ta_seq_consist_convert_RA: fixes vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option"   and ln :: "'l \<Rightarrow>f nat" shows "ta_seq_consist P vs (llist_of (convert_RA ln))"
a: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
ord_isom_inj_on: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "inj_on f (carrier D)"
add_nonpos_eq_0_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less_eq x zero"   and "less_eq y zero" shows "(plus x y = zero) = (x = zero \<and> y = zero)"
set_child_nodes_impl: shows "set_child_nodes = local.a_set_child_nodes"
one_plus_x_nonzero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x < 0" shows "1\<^sub>r \<plusminus> x \<in> carrier K \<and> v (1\<^sub>r \<plusminus> x) < 0"
le: shows "\<forall>x\<in>{i. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i}. \<forall>y\<in>{k}. x < y"
alg_dvd_of_nat: fixes x :: "nat"   and y :: "nat" assumes "x dvd y" shows "of_nat x alg_dvd of_nat y"
assms: shows "interp_ins ops = xs @ [x] @ ys @ [y] @ zs"
getNonWatchedUnfalsifiedLiteralSomeCharacterization_1: fixes clause :: "Literal list"   and w1 :: "Literal"   and w2 :: "Literal"   and M :: "(Literal \<times> bool) list"   and l :: "Literal" assumes "getNonWatchedUnfalsifiedLiteral clause w1 w2 M = Some l" shows "l el clause"
norm_notMT: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "DenyAll \<in> set (policy2list p)" shows "normalizePr p \<noteq> []"
transpose_transpose: fixes xs :: "'a list list" assumes "sorted (rev (map length xs))" shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs"
lsided_times_le_inf: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.ab_quantale times Inf Sup inf less_eq less sup bot top"   and "quantale.lsd times less_eq top x"   and "quantale.lsd times less_eq top y" shows "less_eq (times x y) (inf x y)"
induct: fixes P :: "nat \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "real"   and a2 :: "nat" assumes "\<And>p x. P p x 0"   and "\<And>p x n. \<lbrakk>odd n \<Longrightarrow> P p x (Suc n div 2); \<not> odd n \<Longrightarrow> P p x n\<rbrakk> \<Longrightarrow> P p x (Suc n)" shows "P a0 a1 a2"
dom_lhs: shows "\<D>\<^sub>\<circ> (smc_dg (smc_SemiCAT \<alpha>)) = 4\<^sub>\<nat>"
F_def: shows "F = (\<lambda>x. x powr (1 + s) / (1 + s))"
funionE: fixes c :: "'c"   and A :: "'c fset"   and B :: "'c fset" assumes "c |\<in>| A |\<union>| B" obtains "c |\<in>| A"   | "c |\<in>| B"
sets_pair_eq_sets_fst_snd: fixes A :: "'c measure"   and B :: "'d measure" shows "sets (A \<Otimes>\<^sub>M B) = sets (Sup {vimage_algebra (space A \<times> space B) fst A, vimage_algebra (space A \<times> space B) snd B})"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and R :: "('procS, 'procT) Proc" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L Q"   and "Q \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L R" shows "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L R"
pid_step_def: fixes cstep :: "'b \<Rightarrow> 'f \<Rightarrow> 'b \<Rightarrow> bool"   and en :: "'d \<times> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and ex :: "'d \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'd \<times> 'e" shows "Gen_Scheduler'.pid_step cstep en ex \<equiv> stutter_extend_edges UNIV (rel_of_succ (Gen_Scheduler'.pid_gstep_succ cstep en ex))"
lm153: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "graph X f = Graph f || X"
alt_seq_limit: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_cauchy s" shows "res_lim (alt_seq s) = res_lim s"
coeff_n0_imp_in_Span_others: fixes v :: "'v"   and us :: "'v list"   and vs :: "'v list"   and b :: "'f"   and as :: "'f list"   and w :: "'v"   and bs :: "'f list" assumes "v \<in> V"   and "set us \<subseteq> V"   and "set vs \<subseteq> V"   and "b \<noteq> (0::'f)"   and "length as = length us"   and "w = (as @ b # bs) \<bullet>\<cdot> (us @ v # vs)" shows "v \<in> RSpan (w # us @ vs)"
has_vector_derivative_at_within: fixes f :: "real \<Rightarrow> 'a"   and f' :: "'a"   and x :: "real"   and S :: "real set" assumes "(f has_vector_derivative f') (at x)" shows "(f has_vector_derivative f') (at x within S)"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xa :: "'b set['d]"   and xb :: "'c set['d]" shows "rel_bset x xa xb = rel_set x (set_bset xa) (set_bset xb)"
hyps_8: assumes "cone_decomp T (fst (shift_list (h, U) x qs, B - {(h, U)}))" shows "cone_decomp T (fst (shift1 (shift_list (h, U) x qs, B - {(h, U)})))"
MtransC_rtranclp_transC: shows "(\<rightarrow>*c) = (\<rightarrow>c)\<^sup>*\<^sup>*"