hcju/isabelleps huggingface dataset:
l_one: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> carrier G" shows "\<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"
deriv: fixes x :: "real" assumes "x \<in> {a..b} - (s \<union> {a, b})" shows "((\<lambda>x. integral {c..g x} f) has_vector_derivative g' x *\<^sub>R f (g x)) (at x)"
psimps: fixes A :: "'a mat" assumes "Smith_mxn_dom A" shows "Smith_mxn A = (if dim_row A = 0 \<or> dim_col A = 0 then (1\<^sub>m (dim_row A), A, 1\<^sub>m (dim_col A)) else if dim_row A = 1 then (1\<^sub>m 1, Smith_1xn A) else if dim_row A = 2 then Smith_2xn A else if dim_col A = 1 then let (P, S) = Smith_nx1 A in (P, S, 1\<^sub>m 1) else if dim_col A = 2 then Smith_nx2 A else let A1 = mat_of_row (Matrix.row A 0); A2 = mat_of_rows (dim_col A) (map (Matrix.row A) [1..<dim_row A]); (P1, D1, Q1) = Smith_mxn A2; C = A1 * Q1 @\<^sub>r P1 * A2 * Q1; D = mat_of_rows (dim_col A) [Matrix.row C 0, Matrix.row C 1]; E = mat_of_rows (dim_col A) (map (Matrix.row C) [2..<dim_row A]); (P2, F, Q2) = Smith_2xn D; H = P2 * D * Q2 @\<^sub>r E * Q2; (P_H2, H2) = reduce_column div_op H; (H2_UL, H2_UR, H2_DL, H2_DR) = split_block H2 1 1; (P3, S', Q3) = Smith_mxn H2_DR; S = four_block_mat (Matrix.mat 1 1 (\<lambda>(a, b). H $$ (0, 0))) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_row A - 1) 1) S'; P1' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P1; P2' = four_block_mat P2 (0\<^sub>m 2 (dim_row A - 2)) (0\<^sub>m (dim_row A - 2) 2) (1\<^sub>m (dim_row A - 2)); P3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_row A - 1)) (0\<^sub>m (dim_row A - 1) 1) P3; Q3' = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (dim_col A - 1)) (0\<^sub>m (dim_col A - 1) 1) Q3 in (P3' * P_H2 * P2' * P1', S, Q1 * Q2 * Q3'))"
mult2_altI: fixes X :: "'a multiset"   and X1 :: "'a multiset"   and X2 :: "'a multiset"   and Y :: "'a multiset"   and Y1 :: "'a multiset"   and Y2 :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and b :: "bool"   and s :: "('a \<times> 'a) set" assumes "X = X1 + X2"   and "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "b \<or> Y2 \<noteq> {#}"   and "\<forall>x. x \<in># X2 \<longrightarrow> (\<exists>y. y \<in># Y2 \<and> (x, y) \<in> s)" shows "(X, Y) \<in> mult2_alt b ns s"
del_edge_undirected_degree_minus': fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "finite (edges g)"   and "(v, e, v') \<in> edges g"   and "(v', e, v) \<in> edges g" shows "degree v' (del_unEdge v e v' g) = degree v' g - 1"
space_nonempty: fixes ts :: "traffic"   and v :: "view"   and c :: "cars" shows "left (space ts v c) < right (space ts v c)"
oddpart_leq: fixes n :: "nat" shows "oddpart n \<le> n"
assms_4: shows "fst ` set gs \<subseteq> dgrad_p_set d m"
no_cars_means_free: shows "\<forall>ts v. 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>x. \<nexists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1) \<and> True)) \<longrightarrow> 0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot)"
simps_2: fixes \<alpha> :: "game option" shows "Composeo \<alpha> undefg = undefg"
orderE: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" obtains "a = max a b"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_DF_refine_F: fixes A :: "'a set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F DF A"
ereal_minus_less: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "\<bar>y\<bar> \<noteq> \<infinity>" shows "(x - y < z) = (x < z + y)"
cong: fixes return :: "'d \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c"   and get :: "(('b \<Rightarrow> 'd option) \<Rightarrow> 'c) \<Rightarrow> 'c"   and put :: "('b \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'c" shows "memoization_base.memo return bind get put \<equiv> memoization_base.memo return bind get put"
checkpoint_simplex'_def: shows "checkpoint_simplex' = checkpoint_cs checkpoint_nsc_code"
axioms_3: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "diagram J C D" shows "functor J C D"
ordertype_Field_Restr: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wf r"   and "total_on A r"   and "trans r"   and "small A"   and "Field (Restr r A) = A" shows "ordertype (Field (Restr r A)) (Restr r A) = ordertype A r"
map2_val_compat'D3: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and f :: "('b \<times> 'c) list \<times> 'a \<Rightarrow> ('b \<times> 'd) list \<times> 'a"   and zs :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.map2_val_compat' rep_key_order f"   and "oalist_raw.oalist_inv rep_key_order zs" shows "fst ` set (fst (f zs)) \<subseteq> fst ` set (fst zs)"
AGM_rel_sel_completion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^sub>T\<^sub>R K \<phi> = * \<gamma>\<^sub>A\<^sub>G\<^sub>M K \<phi>"
basis_exists: fixes V :: "('a, 'b) vec set" obtains B :: "('a, 'b) vec set" where "B \<subseteq> V" and "vec.independent B" and "V \<subseteq> vec.span B" and "card B = vec.dim V"
induced_morphism_eq: fixes C :: "('c set \<times> 'd) set" assumes "C \<in> stfx.carrier_stalk" obtains V :: "'c set" and q :: "'d" where "(V, q) \<in> C" and "induced_morphism C = stx.class_of (f \<^sup>\<inverse> X V) (\<phi>\<^sub>f V q)"
INF_union: fixes 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 M :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and B :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf (M ` (A \<union> B)) = inf (Inf (M ` A)) (Inf (M ` B))"
induct: fixes x1 :: "com list"   and x2 :: "state"   and x3 :: "com list"   and x4 :: "state"   and P :: "com list \<Rightarrow> state \<Rightarrow> com list \<Rightarrow> state \<Rightarrow> bool" assumes "(x1, x2) \<rightarrow> (x3, x4)"   and "\<And>t f s cs. t \<in> f s \<Longrightarrow> P (Do f # cs) s cs t"   and "\<And>c1 c2 cs s. P ((c1; c2) # cs) s (c1 # c2 # cs) s"   and "\<And>b s c1 c2 cs. b s \<Longrightarrow> P ((IF b THEN c1 ELSE c2) # cs) s (c1 # cs) s"   and "\<And>b s c1 c2 cs. \<not> b s \<Longrightarrow> P ((IF b THEN c1 ELSE c2) # cs) s (c2 # cs) s"   and "\<And>b s c cs. \<not> b s \<Longrightarrow> P ((WHILE b DO c) # cs) s cs s"   and "\<And>b s c cs. b s \<Longrightarrow> P ((WHILE b DO c) # cs) s (c # (WHILE b DO c) # cs) s"   and "\<And>cs s. P (CALL # cs) s (body # cs) s"   and "\<And>f c g cs s. P ((LOCAL f; c; g) # cs) s (c # Do (\<lambda>t. {g s t}) # cs) (f s)" shows "P x1 x2 x3 x4"
simps_2: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and S :: "'a" shows "sub\<^sub>2 (IF b THEN c1 ELSE c2 {S}) = c2"
mset_factorsD_2: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and F :: "'b multiset"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p" shows "F \<noteq> {#}"
Nil_tl: fixes xs :: "'g list" shows "([] = tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
fps_lr_inverse_unique_ring1_1: fixes f :: "'a fps"   and g :: "'a fps" assumes "f * g = 1"   and "fps_nth g 0 * fps_nth f 0 = (1::'a)" shows "fps_left_inverse g (fps_nth f 0) = f"
dist: fixes p :: "'b list" assumes "awalk u p v"   and "distinct (awalk_verts u p)" shows "P p"
mod_neg_neg_trivial: fixes k :: "int"   and l :: "int" assumes "k \<le> 0"   and "l < k" shows "k mod l = k"
even_of_nat: fixes a :: "nat" shows "even (of_nat a) = even a"
asymp_equiv_refl: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "f \<sim>[F] f"
sub2: shows "(\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE2}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a) \<subseteq> (\<Union>i\<in>{i |i. Inl i \<in> SIGO (OProd ODE1 ODE2)}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
lookupc_fresh: fixes y :: "name"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and a :: "coname" assumes "y \<sharp> \<theta>_n" shows "lookupc y a \<theta>_n = Ax y a"
finite_RF: fixes k :: "nat"   and p :: "nat" shows "finite (RF k p)"
sheaf_spec_is_presheaf: shows "presheaf_of_rings local.local.spectrum local.local.is_zariski_open local.local.sheaf_spec local.local.sheaf_spec_morphisms local.local.\<O>b local.local.add_sheaf_spec local.local.mult_sheaf_spec local.local.zero_sheaf_spec local.local.one_sheaf_spec"
lagrange_finite: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "finite (carrier G)"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = order G"
assms: shows "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')"
cong_exp_iff_simps_7: 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus num.One) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit1 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus n) (numeral.numeral one plus q) = zero)"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and option :: "'c option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
funpow_perm_swap: fixes n :: "nat"   and a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'a" shows "perm_swap a b f ^^ n = perm_swap a b (f ^^ n)"
access_unalive: fixes l :: "Location"   and s :: "Store" assumes "\<not> alive (ref l) s" shows "s@@l = init (ltype l)"
acute_distincts: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Acute Bet Cong A B C" shows "A \<noteq> B \<and> C \<noteq> B"
is_OK_returns_heap_E: fixes h :: "'l"   and f :: "('l, 'm, 'n) prog" assumes "h \<turnstile> ok f" obtains h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>h h'"
tauChainCons: fixes P :: "ccs"   and P' :: "ccs"   and P'' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<longmapsto>\<tau> \<prec> P''" shows "P \<Longrightarrow>\<^sub>\<tau> P''"
accmodi_qmdecl_simp: fixes sig :: "sig"   and m :: "qtname \<times> methd" shows "accmodi (qmdecl sig m) = accmodi m"
m: fixes i :: "nat" assumes "i < dim_row M" shows "map_mat f M $$ (i, i) = f (M $$ (i, i))"
assms_3: fixes n :: "nat" shows "u n \<in> lspace M p"
Eps_\<Phi>_decreasing: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "Eps (\<Phi> (F n)) \<subseteq> Eps (\<Phi> (F m))"
flatten_pi_2_IH_lifting: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" assumes "\<psi> \<in> subfrmlsn \<phi>"   and "suffix i w \<Turnstile>\<^sub>n \<psi>[\<F>\<G> \<psi> (suffix i w)]\<^sub>\<Pi>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>" shows "suffix i w \<Turnstile>\<^sub>n \<psi>[\<F>\<G> \<phi> w]\<^sub>\<Pi>\<^sub>2 = suffix i w \<Turnstile>\<^sub>n \<psi>"
distinct_41: fixes x9 :: "('a, String.literal) phantom" shows "Boolean \<noteq> Enum x9"
finite_iff_ordLess_natLeq: fixes A :: "'a set" shows "finite A = (|A| <o natLeq)"
Hilbert_fun_zero: shows "Hilbert_fun {0} = 0"
p_def: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "p x y z = ((x l\<rightarrow> y) r\<rightarrow> z) \<sqinter> ((z l\<rightarrow> y) r\<rightarrow> x)"
cardinalityOneTheElem: fixes X :: "'c set"   and Y :: "'c set" assumes "card X = 1"   and "X \<subseteq> Y" shows "the_elem X \<in> Y"
open_map_imp_quotient_map: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and T :: "'d set" assumes "continuous_on S f"   and "T \<subseteq> f ` S"   and "\<And>T. openin (top_of_set S) T \<Longrightarrow> openin (top_of_set (f ` S)) (f ` T)" shows "openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T"
equiv_class_eq: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equiv A r"   and "(a, b) \<in> r" shows "r `` {a} = r `` {b}"
sum_character: fixes x :: "i"   and y :: "i" assumes "CEM" shows "\<forall>w. O w (x \<oplus> y) = (O w x \<or> O w y)"
exception_\<I>_def: fixes \<I> :: "('h, 'i) \<I>" shows "exception_\<I> \<I> = map_\<I> id exception_of_option (stop_\<I> \<I>)"
valid_Act: fixes P :: "('state, 'effect) L_state"   and \<alpha> :: "('act, 'effect) L_action"   and x :: "('a, 'pred, ('act, 'effect) L_action) Formula.formula" shows "L_transform.valid P (Formula.Act \<alpha> x) = (\<exists>\<alpha>'' x' P'. Formula.Act \<alpha> x = Formula.Act \<alpha>'' x' \<and> P \<rightarrow>\<^sub>L \<langle>\<alpha>'',P'\<rangle> \<and> L_transform.valid P' x')"
f: shows "lambda C -quasi_isometry_between UNIV UNIV f"
deriv_bound_SN_on: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and n :: "nat" assumes "deriv_bound r a n" shows "SN_on r {a}"
f_shrink_nth: fixes n :: "nat"   and xs :: "'b message_af list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div> k ! n = last_message (xs \<up> (n * k) \<down> k)"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f list" shows "map g (map f v) = map (g \<circ> f) v"
dual_join: fixes x :: "'c"   and y :: "'c" shows "dual (x \<squnion> y) = dual x \<sqinter> dual y"
FDOM_state_succ: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and s :: "('d, 'e) fmap" assumes "fmdom' (snd a) \<subseteq> fmdom' s" shows "fmdom' (state_succ s a) = fmdom' s"
subcomplex_chamber: fixes X :: "'b set set"   and Y :: "'b set set"   and C :: "'b set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y"   and "SimplicialComplex.maxsimp Y C" shows "SimplicialComplex.maxsimp X C"
in_set_subseqs: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<in> set (subseqs ys)) = subseq xs ys"
dist_scaleR: fixes x :: "real"   and a :: "'b"   and y :: "real" shows "dist (x *\<^sub>R a) (y *\<^sub>R a) = \<bar>x - y\<bar> * norm a"
inject_8: fixes x81 :: "msg"   and x82 :: "msg"   and y81 :: "msg"   and y82 :: "msg" shows "(\<lbrace>x81, x82\<rbrace> = \<lbrace>y81, y82\<rbrace>) = (x81 = y81 \<and> x82 = y82)"
monom_mult_uminus_right: fixes c :: "'c"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.monom_mult c t (- p) = - local.punit.monom_mult c t p"
borel_measurable_ennreal_iff: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "((\<lambda>x. ennreal (f x)) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
elims_2: fixes x :: "'b trm"   and xa :: "'b trm" assumes "x \<prec> xa" obtains u :: "'b trm" and M :: "'b trm" and N :: "'b trm" where "x = u" and "xa = M \<cdot> N" and "u = M \<or> u = N \<or> u \<prec> M \<or> u \<prec> N"
level_diff: shows "2 ^ (l' - l) = 2 ^ l' / 2 ^ l"
AE_T_enabled: fixes s :: "'s" shows "almost_everywhere (T s) (enabled s)"
comp_fun_commute_mset_image: fixes f :: "'k \<Rightarrow> 'l" shows "comp_fun_commute (add_mset \<circ> f)"
simps: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'a" shows "r\<inverse>\<inverse> a1 a2 = (\<exists>a b. a1 = b \<and> a2 = a \<and> r a b)"
sup_state_opt_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" shows "sup_state_opt P \<equiv> Opt.le (sup_state P)"
check_ipresIGOp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGOp check check (errMOD MOD) MOD"
ACI_norm_REV: fixes r :: "'c rexp" shows "\<guillemotleft>REV \<guillemotleft>r\<guillemotright>\<guillemotright> = \<guillemotleft>REV r\<guillemotright>"
maddux1f: fixes x :: "'a"   and a :: "'a"   and y :: "'a" shows "x \<sqinter> - (a \<lhd> y) \<rhd> a \<le> x \<rhd> a"
sin_x_ge_neg_x: fixes x :: "real" assumes "0 \<le> x" shows "- x \<le> sin x"
length_prefix_ex': fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "\<exists>ys zs. xs = ys @ xs ! n # zs \<and> length ys = n"
f''_bounded: shows "bounded (f'' ` (T \<times> X))"
inv: fixes i :: "'i"   and n :: "nat" assumes "i \<in> J n" shows "from_nat_into (\<Union> (range J)) (Utn i) = i"
1: shows "\<l>[a] = \<i>[a] \<and> \<r>[a] = \<i>[a]"
assms_2: shows "\<forall>x\<in>A. f x \<le>\<^sub>\<complex> g x"
eq: fixes x :: "'a" shows "p1 x + p2 x = x"
prop_restrict: fixes x :: "'d"   and Z :: "'d set"   and X :: "'d set"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> Z"   and "Z \<subseteq> {x \<in> X. P x}" shows "P x"
le_add_same_cancel1: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># a + b) = (\<Empt> \<subseteq># b)"
resultant_0_2: fixes p :: "'b poly" assumes "0 < degree p" shows "resultant p 0 = (0::'b)"
sdom_singleton: fixes k :: "'c"   and vs :: "'d set" shows "sdom { k := vs}. \<subseteq> {k}"
elims_3: fixes x :: "'a \<Rightarrow> bool"   and xa :: "'a \<Rightarrow> nat"   and xb :: "('a, 'b) Term.term set"   and xc :: "('a, 'b) Term.term" assumes "\<not> intruder_synth' x xa xb xc" obtains pu :: "'a \<Rightarrow> bool" and ar :: "'a \<Rightarrow> nat" and M :: "('a, 'b) Term.term set" and xd :: "'b" where "x = pu__" and "xa = ar__" and "xb = Ma__" and "xc = Var xd" and "Var xd \<notin> Ma__"   | pu :: "'a \<Rightarrow> bool" and ar :: "'a \<Rightarrow> nat" and M :: "('a, 'b) Term.term set" and f :: "'a" and T :: "('a, 'b) Term.term list" where "x = pu" and "xa = ar" and "xb = M" and "xc = Fun f T" and "\<not> (Fun f T \<in> M \<or> pu f \<and> length T = ar f \<and> list_all (intruder_synth' pu ar M) T)"
ia: shows "ia < IArray.length (matrix_to_iarray (divide_by_norm A) !! i)"
on_circline_circline_inversion: fixes H :: "circline"   and z :: "complex_homo" shows "on_circline (circline_inversion H) z = on_circline H (reciprocal (conjugate z))"
process_ret_split_fst_Return: fixes a :: "action"   and m :: "'c match_expr"   and rs :: "'c rule list" assumes "a = Return" shows "process_ret (Rule m a # rs) = add_match (MatchNot m) (process_ret rs)"
relpowp_Suc_I: fixes n :: "nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "(P ^^ n) x y"   and "P y z" shows "(P ^^ Suc n) x z"
lookup_edom: fixes m :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "m x \<noteq> \<bottom>" shows "x \<in> edom m"
freshEnv_swapEnv_exchange1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x2 :: "'var"   and x1 :: "'var" assumes "goodEnv rho" shows "freshEnv xs x2 (rho &[x1 \<and> x2]_xs) = freshEnv xs x1 rho"
elims_2: fixes x :: "primfun"   and xa :: "primfun" assumes "x < xa" obtains x :: "nat" and y :: "nat" where "xa__ = LnChain x" and "xa = LnChain y" and "x < y"
r_match_by_laz: fixes L :: "'a set list"   and l :: "'a list" shows "r_match L l = list_all_zip (\<lambda>Q q. q \<in> Q) L l"
iterates_above_lub: fixes M :: "'a set"   and a :: "'a" assumes "M \<in> Chains leq"   and "M \<noteq> {}"   and "\<And>y. y \<in> M \<Longrightarrow> \<exists>z\<in>M. (y, z) \<in> leq \<and> z \<in> iterates_above a" shows "lub M \<in> iterates_above a"
connectedin_topspace: fixes X :: "'c topology" shows "connectedin X (topspace X) = connected_space X"
subst_comp_type_scheme: fixes R :: "nat \<Rightarrow> typ"   and S :: "nat \<Rightarrow> typ"   and sch :: "type_scheme" shows "$ R ($ S sch) = $ (\<lambda>x. $ R (S x)) sch"
emp_nonsimp: fixes G :: "'b list set" shows "\<not>  \<epsilon> \<in>B G "
transpose_matrix_vector: fixes A :: "(('a, 'b) vec, 'c) vec"   and x :: "('a, 'c) vec" shows "Finite_Cartesian_Product.transpose A *v x = x v* A"
disc_1: fixes x1 :: "'d" shows "isl (Inl x1)"
nextnat_le_unch: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "n < nextnat s" shows "s n = s 0"
eq_ac_1: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (b = a)"
Collect_imp_eq: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "{x. P x \<longrightarrow> Q x} = - {x. P x} \<union> {x. Q x}"
\<Delta>_def: shows "\<Delta> = min (excess f u) (cf (u, v))"
g_def: shows "g = (\<lambda>m. \<Sum>n. f (m, n))"
ran_map_of: fixes xs :: "('b \<times> 'a) list" shows "ran (map_of xs) \<subseteq> snd ` set xs"
Gcd_Lcm: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Gcd A = Lcm {b. \<forall>a\<in>A. dvd.dvd times b a}"
inv_nxt: fixes s :: "'s"   and f :: "'o" assumes "inv s" shows "inv (nxt f s)"
satisfies_properties_sys2: shows "satisfies_properties p qs2 (get_subsets (calculate_data p qs2)) (get_signs (calculate_data p qs2)) (get_matrix (calculate_data p qs2))"
finite_measure_mono_AE: fixes A :: "'s stream set"   and B :: "'s stream set"   and s :: "'s" assumes "AE x\<in>A in T s. x \<in> B"   and "B \<in> T.events s" shows "T.prob s A \<le> T.prob s B"
upd_list_sorted: fixes ps :: "('c \<times> 'd) list"   and a :: "'c"   and b :: "'d"   and x :: "'c"   and y :: "'d"   and qs :: "('c \<times> 'd) list" assumes "sorted1 (ps @ [(a, b)])" shows "upd_list x y (ps @ (a, b) # qs) = (if x < a then upd_list x y ps @ (a, b) # qs else ps @ upd_list x y ((a, b) # qs))"
intro: fixes M :: "'d set"   and composition :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit :: "'d" assumes "\<And>a b. \<lbrakk>a \<in> M; b \<in> M\<rbrakk> \<Longrightarrow> composition a b \<in> M"   and "unit \<in> M"   and "\<And>a b c. \<lbrakk>a \<in> M; b \<in> M; c \<in> M\<rbrakk> \<Longrightarrow> composition (composition a b) c = composition a (composition b c)"   and "\<And>a. a \<in> M \<Longrightarrow> composition unit a = a"   and "\<And>a. a \<in> M \<Longrightarrow> composition a unit = a" shows "Group_Theory.monoid M composition unit"
degree_one_imp_singleton_roots: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "degree p = 1" shows "roots p = {#\<ominus> (inv lead_coeff p \<otimes> const_term p)#}"