hcju/isabelleps huggingface dataset:
cond_mono: fixes P\<^sub>2 :: "('a, 'b) uexpr"   and P\<^sub>1 :: "('a, 'b) uexpr"   and Q\<^sub>2 :: "('a, 'b) uexpr"   and Q\<^sub>1 :: "('a, 'b) uexpr"   and b :: "'b upred" assumes "P\<^sub>1 \<sqsubseteq> P\<^sub>2"   and "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2" shows "P\<^sub>1 \<triangleleft> b \<triangleright> Q\<^sub>1 \<sqsubseteq> P\<^sub>2 \<triangleleft> b \<triangleright> Q\<^sub>2"
zip_append1: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'d list" shows "zip (xs @ ys) zs = zip xs (take (length xs) zs) @ zip ys (drop (length xs) zs)"
Zfun_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "Zfun f F = (\<forall>r>0. \<forall>\<^sub>F x in F. norm (f x) < r)"
fds_eq_iff: fixes f :: "'a fds"   and g :: "'a fds" shows "(f = g) = (\<forall>n>0. fds_nth f n = fds_nth g n)"
numeral_power_le_of_int_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "num"   and n :: "nat"   and a :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (power.power one times (numeral.numeral one plus x) n) (ring_1.of_int one plus zero minus a) = (numeral x ^ n \<le> a)"
a_diff_kde': shows "a - complex_of_real kde \<in> closed_segment (a - complex_of_real d) (a + complex_of_real kde)"
uncountable_closed_interval: fixes a :: "real"   and b :: "real" shows "uncountable {a..b} = (a < b)"
ustutter: fixes \<xi> :: "'a" shows "U \<xi> \<xi>"
constrains_weaken_L: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set" assumes "F \<in> A co A'"   and "B \<subseteq> A" shows "F \<in> B co A'"
hyps_1: shows "\<Gamma>\<turnstile> (c\<^sub>1, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
foldSetD_closed: fixes B :: "'b set"   and D :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'b set"   and z :: "'c"   and e :: "'c" assumes "LCD B D f"   and "(A, z) \<in> foldSetD D f e" shows "z \<in> D"
sets_Sup_eq: fixes M :: "'d measure set"   and X :: "'d set" assumes "\<And>m. m \<in> M \<Longrightarrow> space m = X"   and "M \<noteq> {}" shows "sets (Sup M) = sigma_sets X (\<Union> (sets ` M))"
map_graph_edge_preserving: fixes f :: "('e \<times> 'f) set"   and G :: "('g, 'e) labeled_graph" shows "edge_preserving f (edges G) (edges (map_graph f G))"
rep_eq: shows "generator local.upt_generator = (\<lambda>n. n < bound, \<lambda>n. (n, Suc n))"
closedin_Un_complement_component: fixes u :: "'b set"   and S :: "'b set"   and c :: "'b set" assumes "locally connected u"   and "closedin (top_of_set u) S"   and "c \<in> components (u - S)" shows "closedin (top_of_set u) (S \<union> c)"
subst2_1: fixes x :: "name"   and Tvec :: "'a list"   and xvec :: "name list"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> Tvec"   and "x \<sharp> xvec"   and "x \<sharp> P" shows "x \<sharp> P[xvec::=Tvec]"
proj_Cons_6: fixes b :: "'a strand_label \<times> 'b"   and n :: "'a"   and A :: "('a strand_label \<times> 'b) list" assumes "fst b \<noteq> \<star>"   and "fst b \<noteq> ln n" shows "proj n (b # A) = proj n A"
remove_max_is_heap: fixes l :: "'b"   and m :: "'a"   and l' :: "'b" assumes "\<not> is_empty l"   and "is_heap (as_tree l)"   and "(m, l') = remove_max l" shows "is_heap (as_tree l')"
res_closure: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "closure (residual f \<circ> f)"
IH: assumes "\<forall>as' a as''. asx = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n'"   and "asx \<noteq> [] \<longrightarrow> n' = last (targetnodes asx)" shows "(V, asx, asx) \<in> dependent_live_vars n'"
trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "prefix a b"   and "prefix b c" shows "prefix a c"
sorted_list_of_fset_simps_2: fixes S :: "'c fset" shows "fset_of_list (sorted_list_of_fset S) = S"
step_redexes_Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and r :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and r' :: "('b, 'a, 'c) com"   and s' :: "('b, 'c) xstate"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and c :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile> (r, s) \<rightarrow> (r', s')"   and "Seq r c\<^sub>2 \<in> redexes c" shows "\<exists>c'. \<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s') \<and> Seq r' c\<^sub>2 \<in> redexes c'"
Pi_anti_mono: fixes A' :: "'a set"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "A' \<subseteq> A" shows "Pi A B \<subseteq> Pi A' B"
L_distinct: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right" shows "distinct (L_left @ L_right)"
record_simps_29: fixes f :: "tid_or_exn set \<Rightarrow> tid_or_exn set"   and x :: "'b state" shows "defined_mods (update_defined_types f x) = defined_mods x"
less_eq_fract_respect: fixes b :: "'a"   and b' :: "'a"   and d :: "'a"   and d' :: "'a"   and a :: "'a"   and a' :: "'a"   and c :: "'a"   and c' :: "'a" assumes "b \<noteq> (0::'a)"   and "b' \<noteq> (0::'a)"   and "d \<noteq> (0::'a)"   and "d' \<noteq> (0::'a)"   and "a * b' = a' * b"   and "c * d' = c' * d" shows "(a * d * (b * d) \<le> c * b * (b * d)) = (a' * d' * (b' * d') \<le> c' * b' * (b' * d'))"
join_ge2: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and join :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Top :: "'b"   and y :: "'b"   and x :: "'b" assumes "class.SL_top le join Top" shows "le y (join x y)"
eqvt: fixes x1 :: "(name \<times> ty) list"   and x2 :: "trm"   and x3 :: "trm"   and x4 :: "ty"   and pi :: "(name \<times> name) list" assumes "x1 \<turnstile> x2 \<Leftrightarrow> x3 : x4" shows "pi \<bullet> x1 \<turnstile> pi \<bullet> x2 \<Leftrightarrow> pi \<bullet> x3 : pi \<bullet> x4"
abs_minus: fixes m :: "nat"   and n :: "nat" shows "abs_nat m - abs_nat n = abs_nat (m - n)"
N1_def: shows "N1 = H1 <#> H \<inter> K1"
basis_reduction_add_rows_def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "basis_reduction_add_rows upw i fs = (if upw then basis_reduction_add_rows_loop i fs i else fs)"
psi_l_le_id: fixes l :: "nat" shows "proj_psi_l l \<le>\<^sub>L 1\<^sub>m N"
inv: shows "\<forall>p. outpt s p \<in> {chosen s, NotAnInput}"
pointbased_implies_continuous: fixes f :: "'b set \<Rightarrow> 'c set" assumes "pointbased f" shows "continuous f"
distinct_drop: fixes xs :: "'a list"   and i :: "nat" assumes "distinct xs" shows "distinct (drop i xs)"
simps_1: fixes \<theta>1 :: "('a, 'c) trm"   and \<theta>2 :: "('a, 'c) trm"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Fsubst (Geq \<theta>1 \<theta>2) \<sigma> = Geq (Tsubst \<theta>1 \<sigma>) (Tsubst \<theta>2 \<sigma>)"
Right_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> G" shows "K |\<cdot> g = Class g"
oghoare_intros_26: fixes P :: "'a set"   and P' :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set"   and a :: "('a, 'b, 'c) ann"   and c :: "('a, 'b, 'c) com"   and Q' :: "'a set"   and A' :: "'a set"   and Q :: "'a set"   and A :: "'a set" assumes "P \<subseteq> P'"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P' a c Q', A'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P a c Q, A"
homogeneous_idealE: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F" obtains F' :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" and q :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" where "finite F'" and "F' \<subseteq> F" and "p = (\<Sum>f\<in>F'. q f * f)" and "\<And>f. poly_deg (q f * f) \<le> poly_deg p" and "\<And>f. f \<notin> F' \<Longrightarrow> q f = 0"
right_neutral: fixes a :: "'a" shows "a + (0::'a) = a"
mod_size_less: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and a :: "'b" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "euclidean_size (modulo a b) < euclidean_size b"
oid_greatest: fixes i :: "'a" assumes "i \<in> set (interp_rga xs)" shows "i < oid"
step_wf: shows "wf {(s', s). s \<in> wa_invar WAA \<and> s \<in> wa_cond WAA \<and> (s, s') \<in> wa_step WAA}"
Group_VectorSpaceHomSet: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and W :: "'c set" assumes "VectorSpace smult V"   and "VectorSpace smult' W" shows "Group (VectorSpaceHomSet smult V smult' W)"
surj_to_test: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B"   and "\<forall>b\<in>B. \<exists>a\<in>A. f a = b" shows "surj_to f A B"
fmul_def: fixes m :: "roundmode"   and a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fmul m a b = (if is_nan a \<or> is_nan b \<or> is_zero a \<and> is_infinity b \<or> is_infinity a \<and> is_zero b then some_nan else if is_infinity a \<or> is_infinity b then if sign a = sign b then \<infinity> else - \<infinity> else zerosign (if sign a = sign b then 0 else 1) (IEEE.round m (valof a * valof b)))"
simps: fixes ts :: "traffic"   and a :: "traffic" shows "ts \<^bold>\<Rightarrow> a = (a = ts \<or> (\<exists>ts' ts''. a = ts'' \<and> ts \<^bold>\<Rightarrow> ts' \<and> ts' \<^bold>\<leadsto> ts'') \<or> (\<exists>ts' ts''. a = ts'' \<and> ts \<^bold>\<Rightarrow> ts' \<and> (\<exists>c n. ts' \<^bold>\<midarrow>c( c, n ) \<^bold>\<rightarrow> ts'')) \<or> (\<exists>ts' ts''. a = ts'' \<and> ts \<^bold>\<Rightarrow> ts' \<and> (\<exists>c. ts' \<^bold>\<midarrow>wdc( c ) \<^bold>\<rightarrow> ts'')) \<or> (\<exists>ts' ts''. a = ts'' \<and> ts \<^bold>\<Rightarrow> ts' \<and> (\<exists>c. ts' \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts'')) \<or> (\<exists>ts' ts''. a = ts'' \<and> ts \<^bold>\<Rightarrow> ts' \<and> (\<exists>c n. ts' \<^bold>\<midarrow>wdr( c, n ) \<^bold>\<rightarrow> ts'')))"
distinct_7: fixes x1 :: "'b"   and x51 :: "'b formula"   and x52 :: "'b formula" shows "Atom x1 \<noteq> x51 \<^bold>\<or> x52"
path_weight_def: fixes p :: "('d \<times> 'c \<times> 'd) list" shows "path_weight p \<equiv> sum_list (map (fst \<circ> snd) p)"
iLast_False: fixes t0 :: "nat"   and I :: "nat set" shows "\<not> (\<ominus> t t0 I. False)"
euler_genus_def: shows "euler_genus \<equiv> (int (2 * card sccs) - int (card isolated_verts) - euler_char) div 2"
path_in_edges: fixes G :: "('a, 'b) graph"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail G v ps v'" shows "set ps \<subseteq> edges G"
Max_torder_in_Domain: fixes B :: "'a set" assumes "finite B"   and "B \<noteq> {}"   and "B \<subseteq> A" shows "Max_torder r B \<in> A"
perp_col0: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and X :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"   and "X \<noteq> Y"   and "Tarski_neutral_dimensionless.Col Bet A B X"   and "Tarski_neutral_dimensionless.Col Bet A B Y" shows "Tarski_neutral_dimensionless.Perp Bet Cong C D X Y"
alwaysSFI: fixes A :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "\<turnstile> SF(A)_v \<longrightarrow> \<box>SF(A)_v"
well_order_on_domain: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "well_order_on A r"   and "(a, b) \<in> r" shows "a \<in> A \<and> b \<in> A"
arrI: fixes f :: "'a" assumes "left g f" shows "L.arr f"
preserves_hom: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>" shows "\<guillemotleft>FG.\<eta> f : B.map a \<rightarrow>\<^sub>B FG.GF.map b\<guillemotright>"
lossless_xor_funct: fixes S1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> unit \<Rightarrow> 'a spmf"   and R1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'a spmf"   and S2_14_OT :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'b spmf"   and R2_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'b spmf"   and protocol_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and adv_14_OT :: "real"   and A :: "bool \<times> bool"   and B :: "bool \<times> bool" assumes "gmw_base S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT" shows "lossless_spmf (xor_funct A B)"
blinding_of_blindable_simps_1: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "('c, 'd) blindable\<^sub>m" shows "blinding_of_blindable h bo (Unblinded x) y = (\<exists>y. y = Unblinded y \<and> bo x y)"
p_int_pow_nonzero: fixes n :: "int" assumes "0 \<le> n" shows "[p] \<cdot> \<one> [^] n \<in> nonzero Zp"
tNode_2: fixes p :: "ref" assumes "x \<in> set_of rt"   and "x \<noteq> Null"   and "high x \<noteq> Null"   and "Dag p low high rt" shows "high x \<in> set_of rt"
eq: fixes x :: "nat" shows "(if x < i then if q x < j then q x else Suc (q x) else if x = i then j else if q (x - 1) < j then q (x - 1) else Suc (q (x - 1))) = (if x < i then if q' x < j then q' x else Suc (q' x) else if x = i then j else if q' (x - 1) < j then q' (x - 1) else Suc (q' (x - 1)))"
size_mset_set_eq_card: fixes A :: "'b set" assumes "finite A" shows "size (mset_set A) = card A"
max_step: shows "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (T - t 0) + 1) - 1))"
randomBidsAlg_def: fixes N :: "integer set"   and \<Omega> :: "'c list"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and random :: "integer" shows "randomBidsAlg N \<Omega> b random \<equiv> resolvingBidAlg (N \<union> {seller}) \<Omega> b random"
subgroup_inverse_iff: fixes x :: "'a" assumes "additive.invertible x"   and "x \<in> R" shows "(- x \<in> I) = (x \<in> I)"
msg_id_unique: fixes m1 :: "'msg"   and i :: "nat"   and m2 :: "'msg"   and j :: "nat" assumes "Broadcast m1 \<in> set (history i)"   and "Broadcast m2 \<in> set (history j)"   and "msg_id m1 = msg_id m2" shows "i = j \<and> m1 = m2"
simps_3: fixes l1 :: "value"   and v :: "'a aexp"   and va :: "'a aexp" shows "local.less_aexp_aux (L l1) (Plus v va) = True"
float_defs_2: fixes a :: "('a, 'b) IEEE.float" shows "is_infinity a = (IEEE.exponent a = emax TYPE(('a, 'b) IEEE.float) \<and> fraction a = 0)"
simps_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "[] \<noteq> x21 # x22"
qmodule_def: fixes M :: "('h, 'i, 'j) Module_scheme"   and H :: "'h set" shows "M /\<^sub>m H = \<lparr>carrier = set_mr_cos M H, pop = mr_cospOp M H, mop = mr_cosmOp M H, zero = H, sprod = mr_cos_sprod M H\<rparr>"
fw_mono: 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 i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "i \<le> n"   and "j \<le> n" shows "less_eq (linordered_ab_monoid_add.fw plus less_eq m n k i j) (m i j)"
simps_62: fixes x6 :: "'a OCL_Types.type"   and x5 :: "'a OCL_Types.type" shows "OrderedSet x6 \<noteq> Set x5"
globEANow: fixes n :: "nat"   and i :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> i"   and "active c (t i)"   and "dynamic_component.eval tCMP active c t t' n (dynamic_component.glob \<gamma>)" shows "dynamic_component.eval tCMP active c t t' i \<gamma>"
gbinomial_reduce_nat: fixes k :: "nat"   and a :: "'a" assumes "0 < k" shows "a gchoose k = a - (1::'a) gchoose (k - 1) + (a - (1::'a) gchoose k)"
fst_rec1_pair: fixes ga :: "('d, 'd IF1 \<times> 'e, 'd IF2 \<times> 'f) F1 \<Rightarrow> 'e"   and g :: "('d, 'd IF1 \<times> 'e, 'd IF2 \<times> 'f) F2 \<Rightarrow> 'f" shows "fst \<circ> fold1 <ctor1 \<circ> F1map id fst fst , ga> <ctor2 \<circ> F2map id fst fst , g> = id"
in_homE: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and f :: "'ag"   and a :: "'ag"   and b :: "'ag" assumes "category C"   and "partial_magma.in_hom C f a b" obtains "partial_magma.arr C f" and "partial_magma.dom C f = a" and "partial_magma.cod C f = b"
eval_monom_mono_1: fixes g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and m :: "'c monom" assumes "\<And>x. g x \<le> f x"   and "\<And>x. (0::'d) \<le> g x" shows "eval_monom g m \<le> eval_monom f m"
less: shows "root p (real_of_int x) \<le> real_of_int (start_value x p)"
Fun_mapsto: fixes f :: "'s" assumes "S.arr f" shows "S.Fun f \<in> extensional (S.Dom f) \<inter> (S.Dom f \<rightarrow> S.Cod f)"
c1_implies_local_lipschitz: fixes T :: "real set"   and X :: "'a set"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and f' :: "real \<times> 'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a" assumes "\<And>t x. \<lbrakk>t \<in> T; x \<in> X\<rbrakk> \<Longrightarrow> (f t has_derivative blinfun_apply (f' (t, x))) (at x)"   and "continuous_on (T \<times> X) f'"   and "open T"   and "open X" shows "local_lipschitz T X f"
lemma_4_5_ii: fixes a :: "'a"   and b :: "'a" shows "(a r\<rightarrow> b) \<squnion>2 (b r\<rightarrow> a) = (1::'a)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a"   and a :: "'a"   and \<chi> :: "'b \<Rightarrow> 'a" assumes "category C"   and "category J"   and "diagram J C D"   and "constant_functor J C a"   and "natural_transformation J C (constant_functor.map J C a) D \<chi>" shows "Limit.cone J C D a \<chi>"
TC_def: shows "TC \<equiv> transrec (\<lambda>f x. x \<squnion> \<Squnion> (f ` elts x))"
fv_subset_if_in_strand_ik: fixes t :: "('a, 'b) Term.term"   and S :: "('a, 'b) strand_step list" assumes "t \<in> ik\<^sub>s\<^sub>t S" shows "fv t \<subseteq> \<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S))"
bdd_above_infdist_aux: fixes A :: "'b set"   and B :: "'b set" assumes "bounded A"   and "bounded B" shows "bdd_above ((\<lambda>x. infdist x B) ` A)"
c_dom_sinks: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and xps :: "('f \<times> 'g) list" shows "sinks I (c_dom D) u xps = sinks I D u (map fst xps)"
asymp_equiv_imp_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<sim>[F] g" shows "f \<in> \<Theta>[F](g)"
type_definition_lexlist: shows "type_definition unlex Lex {xs. True}"
intro: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"   and "tiebreaker_lifting_axioms Prec_F_g" shows "tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F_g"
snd_initmissing: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and x :: "(nat \<Rightarrow> 'e option) \<times> 'g" assumes "openproc np onp sr" shows "snd (openproc.initmissing np sr x) = snd x"
mset_subset_eq_add_left: fixes A :: "'k multiset"   and B :: "'k multiset" shows "A \<subseteq># A + B"
split_sel: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and prod :: "'d \<times> 'e" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (prod = (fst prod, snd prod) \<longrightarrow> P (f (fst prod) (snd prod)))"
upd_locks_empty_conv: fixes t :: "'d"   and ls :: "('c, 'd) locks" shows "(\<lambda>(l, las). upd_locks l t las) \<circ>$ ($ls, K$ []$) = ls"
closed_open: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a * e + (b * e + c) = (a + b) * e + c"
set_sel_1: fixes a :: "'g list" assumes "a \<noteq> []" shows "hd a \<in> set a"
abs_pat_psubst: fixes p :: "pat"   and \<theta> :: "(name \<times> trm) list"   and t :: "btrm" assumes "supp p \<sharp>* \<theta>" shows "\<theta>\<lparr>\<lambda>[p]. t\<rparr>\<^sub>b = (\<lambda>[p]. \<theta>\<lparr>t\<rparr>\<^sub>b)"
simps_25: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (Times e1 e2) (Minus v va) = False"
eq: fixes x :: "'b" shows "indicator (\<Union> (X ` I)) x = \<integral>\<^sup>+ i. indicator (X i) x \<partial>count_space I"
conjunctSomeProtoAnyD: fixes a :: "protocol"   and b :: "8 word" assumes "Some ProtoAny = simple_proto_conjunct a (Proto b)" shows "False"
induct: fixes x1 :: "com"   and x2 :: "state"   and P :: "com \<Rightarrow> state \<Rightarrow> bool" assumes "x1 \<down> x2"   and "\<And>f s. f s \<noteq> {} \<Longrightarrow> P (Do f) s"   and "\<And>c\<^sub>1 s\<^sub>0 c\<^sub>2. \<lbrakk>c\<^sub>1 \<down> s\<^sub>0; P c\<^sub>1 s\<^sub>0; \<forall>s\<^sub>1. s\<^sub>0 -c\<^sub>1\<rightarrow> s\<^sub>1 \<longrightarrow> c\<^sub>2 \<down> s\<^sub>1 \<and> P c\<^sub>2 s\<^sub>1\<rbrakk> \<Longrightarrow> P (c\<^sub>1; c\<^sub>2) s\<^sub>0"   and "\<And>b s c\<^sub>1 c\<^sub>2. \<lbrakk>b s; c\<^sub>1 \<down> s; P c\<^sub>1 s\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s"   and "\<And>b s c\<^sub>2 c\<^sub>1. \<lbrakk>\<not> b s; c\<^sub>2 \<down> s; P c\<^sub>2 s\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s"   and "\<And>b s c. \<not> b s \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>b s c. \<lbrakk>b s; c \<down> s; P c s; \<forall>t. s -c\<rightarrow> t \<longrightarrow> WHILE b DO c \<down> t \<and> P (WHILE b DO c) t\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s"   and "\<And>c f s g. \<lbrakk>c \<down> f s; P c (f s)\<rbrakk> \<Longrightarrow> P (LOCAL f; c; g) s" shows "P x1 x2"
r_neg: fixes x :: "'c" assumes "x \<in> carrier H" shows "x \<oplus>\<^bsub>H\<^esub> \<ominus>\<^bsub>H\<^esub> x = \<zero>\<^bsub>H\<^esub>"