hcju/isabelleps huggingface dataset:
inf_lower: fixes A :: "'a upred set"   and x :: "'a upred" assumes "A \<subseteq> carrier \<P>"   and "x \<in> A" shows "le \<P> (Lattice.inf \<P> A) x"
conf_fs_hext: fixes P :: "jvm_prog"   and h :: "heap"   and sh :: "sheap"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and C :: "char list"   and M :: "char list"   and n :: "nat"   and T\<^sub>r :: "ty"   and frs :: "frame list"   and h' :: "heap" assumes "conf_fs P h sh \<Phi> C M n T\<^sub>r frs"   and "h \<unlhd> h'" shows "conf_fs P h' sh \<Phi> C M n T\<^sub>r frs"
abs_neg_numeral: 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 n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (uminus (numeral.numeral one plus n)) = numeral.numeral one plus n"
divide_octo_def: fixes x :: "octo"   and y :: "octo" shows "x / y = x * inverse y"
dom_cod: fixes f :: "'a" assumes "B.arr f" shows "B.dom (B.cod f) = B.cod f"
simply_connected_retraction_gen: fixes S :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and k :: "'b \<Rightarrow> 'a" assumes "simply_connected S"   and "continuous_on S h"   and "h ` S = T"   and "continuous_on T k"   and "k ` T \<subseteq> S"   and "\<And>y. y \<in> T \<Longrightarrow> h (k y) = y" shows "simply_connected T"
compxEs2_append: fixes es :: "'a expr1 list"   and es' :: "'a expr1 list"   and pc :: "nat"   and d :: "nat" shows "compxEs2 (es @ es') pc d = compxEs2 es pc d @ compxEs2 es' (length (compEs2 es) + pc) (length es + d)"
mono_natI: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n \<le> f (Suc n)" shows "mono f"
e1: shows "edges G - (F - {e}) = insert e (edges G - F)"
sum_three_elements: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" shows "\<exists>x y z. (\<Sum>i\<in>{a, b, c}. f i * i) = x * a + y * b + z * c"
lfinite_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lfinite (xs $ ys) = (lfinite xs \<and> lfinite ys)"
size\<^sub>s\<^sub>t_def: fixes S :: "('a, 'b) strand_step list" shows "size\<^sub>s\<^sub>t S \<equiv> size_list (\<lambda>x. Max (insert 0 (size ` trms\<^sub>s\<^sub>t\<^sub>p x))) S"
resPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and x :: "name" assumes "P \<leadsto><Rel> Q"   and "\<And>P Q x. (P, Q) \<in> Rel \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"   and "Rel \<subseteq> Rel'"   and "eqvt Rel"   and "eqvt Rel'" shows "<\<nu>x>P \<leadsto><Rel'> <\<nu>x>Q"
prems: shows "filter is_Write\<^sub>s\<^sub>b (x # xs) = []"
contour_integral_swap: fixes g :: "real \<Rightarrow> complex"   and h :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex \<Rightarrow> complex" assumes "continuous_on (path_image g \<times> path_image h) (\<lambda>(y1, y2). f y1 y2)"   and "valid_path g"   and "valid_path h"   and "continuous_on {0..1} (\<lambda>t. vector_derivative g (at t))"   and "continuous_on {0..1} (\<lambda>t. vector_derivative h (at t))" shows "contour_integral g (\<lambda>w. contour_integral h (f w)) = contour_integral h (\<lambda>z. contour_integral g (\<lambda>w. f w z))"
map_replicate_const: fixes k :: "'d"   and lst :: "'e list" shows "map (\<lambda>x. k) lst = replicate (length lst) k"
pathstart_rec_join_1: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" shows "pathstart (rec_join ((1, \<gamma>) # l)) = pathstart \<gamma>"
monom_closed: fixes a :: "int"   and n :: "nat" assumes "a \<in> carrier R" shows "monom (UP R) a n \<in> carrier (UP R)"
closed_map_def: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "closed_map X1 X2 f \<equiv> \<forall>U. closedin X1 U \<longrightarrow> closedin X2 (f ` U)"
f14: shows "\<ominus> (s' \<otimes> fst x') \<in> carrier R"
foldD_empty: fixes e :: "'b"   and D :: "'b set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "e \<in> D" shows "foldD D f e {} = e"
A: fixes A :: "'b set" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
ctendsto_componentwise: fixes b :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and F :: "'c filter" assumes "\<And>i j. \<lbrakk>i \<in> CBasis; j \<in> CBasis\<rbrakk> \<Longrightarrow> ((\<lambda>n. cblinfun_apply (b n) j \<bullet>\<^sub>C i) \<longlongrightarrow> cblinfun_apply a j \<bullet>\<^sub>C i) F" shows "(b \<longlongrightarrow> a) F"
G: fixes x :: "'a" shows "\<bar>G x\<bar> \<le> 1"
ln_approx_aux: fixes x :: "real"   and n :: "nat" assumes "0 < x"   and "x \<noteq> 1" shows "inverse (2 * ((x - 1) / (x + 1)) ^ (2 * n + 1)) * (ln x - (\<Sum>k<n. 2 * ((x - 1) / (x + 1)) ^ (2 * k + 1) / real (2 * k + 1))) \<in> {0..1 / (1 - ((x - 1) / (x + 1))\<^sup>2) / real (2 * n + 1)}"
coprime_commute: fixes p :: "'d"   and q :: "'d" shows "comm_monoid_mult_class.coprime p q = comm_monoid_mult_class.coprime q p"
kop_surj: shows "top_pres (\<P> op\<^sub>K)"
S_subset_reduced_imp_S_reduced: fixes S :: "'b set"   and T :: "'b set"   and ts :: "'b list" assumes "PreCoxeterSystemWithDeletion S"   and "T \<in> Pow S"   and "reduced_word T ts" shows "reduced_word S ts"
fmember_is_indexed_fmember: fixes x :: "'b"   and s :: "'b fset" assumes "x |\<in>| s" shows "\<exists>i. x |\<in>|\<^bsub>i\<^esub> s"
three_subset: fixes x :: "'b"   and y :: "'b"   and z :: "'b"   and X :: "'b set" assumes "x \<noteq> y"   and "x \<noteq> z"   and "y \<noteq> z"   and "{x, y, z} \<subseteq> X" shows "3 \<le> card X \<or> infinite X"
closure_iff_nhds_not_empty: fixes x :: "'a"   and X :: "'a set" shows "(x \<in> closure X) = (\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
rep_eq: fixes x :: "'a" shows "count (prime_factorization x) = (\<lambda>p. if prime p then multiplicity p x else 0)"
ta: shows "Partial_Cost_Model.T_on' (rTS h0) ([x, y], h) u = 1"
alb: shows "a \<le> \<lfloor>b\<rfloor>"
fps_conv_radius_inverse: fixes r :: "ereal"   and f :: "complex fps" assumes "\<And>z. ereal (cmod z) < r \<Longrightarrow> eval_fps f z \<noteq> 0" shows "min r (fps_conv_radius f) \<le> fps_conv_radius (inverse f)"
indspaceI: fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set" assumes "f \<in> addfunset FH V"   and "f \<in> aezfun_scalar_mult.smultfunset smult G FH" shows "f \<in> aezfun_scalar_mult.indspace smult G FH V"
closest_valid_1: fixes X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and k :: "'a \<Rightarrow> nat"   and r :: "('a \<times> 'a) set" assumes "{x \<in> X. \<exists>c. I x = Regions.intv.Const c} \<noteq> {}"   and "finite X"   and "Regions.valid_region X k I r" shows "Regions.valid_region X k (\<lambda>x. if x \<notin> {x \<in> X. \<exists>c. I x = Regions.intv.Const c} then I x else if intv_const (I x) = k x then Regions.intv.Greater (k x) else Regions.intv.Intv (intv_const (I x))) (r \<union> {(x, y). x \<in> {x \<in> X. \<exists>c. I x = Regions.intv.Const c} \<and> y \<in> X \<and> intv_const (I x) < k x \<and> Regions.isIntv (if y \<notin> {x \<in> X. \<exists>c. I x = Regions.intv.Const c} then I y else if intv_const (I y) = k y then Regions.intv.Greater (k y) else Regions.intv.Intv (intv_const (I y)))})"
X: shows "next_dim.polyhedral_cone M = next_dim.cone X"
poly_div: fixes q :: "'a poly"   and x :: "'a"   and p :: "'a poly" assumes "poly q x \<noteq> (0::'a)"   and "q dvd p" shows "poly (p div q) x = poly p x / poly q x"
no_fixed_pt: fixes f :: "int \<Rightarrow> int" assumes "f \<in> CofGroups.Ex1"   and "f \<noteq> id" shows "Fix f = {}"
minus_scalar_prod_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>3 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 - v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 - v\<^sub>2 \<bullet> v\<^sub>3"
that: fixes k :: "'b"   and v :: "'c" assumes "m k = Some v"   and "x = f k v" shows "thesis"
Some_those_nth: fixes ys :: "'a option list"   and xs :: "'a list"   and i :: "nat" assumes "those ys = Some xs"   and "i < length xs" shows "Some (xs ! i) = ys ! i"
step: fixes Mkt :: "('b, 'c) discrete_market"   and v :: "real"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and asset :: "'c"   and n :: "nat" shows "remaining_qty Mkt v pf asset (Suc (Suc n)) = (\<lambda>w. remaining_qty Mkt v pf asset (Suc n) w + (cls_val_process Mkt pf (Suc n) w - val_process Mkt pf (Suc n) w) / prices Mkt asset (Suc n) w)"
UP_smult_one: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "\<one> \<odot>\<^bsub>P\<^esub> p = p"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>J) (\<cdot>) \<chi>.A.map (J_C.Map d) (J_C.Map x)"
vector_eq_ldot: fixes y :: "'c"   and z :: "'c" shows "(\<forall>x. x \<bullet> y = x \<bullet> z) = (y = z)"
transpI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
transC_mayDiverge: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>c (c', s')"   and "mayDiverge c' s'" shows "mayDiverge c s"
ran_map_upd: fixes m :: "'b \<Rightarrow> 'a option"   and a :: "'b"   and b :: "'a" assumes "m a = None" shows "ran (m(a \<mapsto> b)) = insert b (ran m)"
select_convs_1: fixes xc :: "int"   and yc :: "int"   and zc :: "int"   and more :: "'a" shows "zc \<lparr>xc = xc, yc = yc, zc = zc, \<dots> = more\<rparr> = zc"
ffUnion_def: shows "ffUnion \<equiv> map_fun ((`) fset \<circ> fset) Abs_fset \<Union>"
obtain_list_of_multiset: fixes m :: "'a multiset" obtains l :: "'a list" where "mset l = m"
simps_17: fixes v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a"   and ve :: "bexp"   and vf :: "'a acom"   and vg :: "'a" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> {vd} WHILE ve DO vf {vg}) = False"
pseudo_divmod_main_hom: fixes lc :: "'a"   and q :: "'a poly"   and r :: "'a poly"   and d :: "'a poly"   and dr :: "nat"   and i :: "nat" shows "pseudo_divmod_main (hom lc) (map_poly hom q) (map_poly hom r) (map_poly hom d) dr i = map_prod (map_poly hom) (map_poly hom) (pseudo_divmod_main lc q r d dr i)"
indRelTEQ_impl_TRel_weakly_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_respects_barbs (indRelTEQ TRel) (STCalWB SWB TWB)" shows "rel_weakly_respects_barbs TRel TWB"
map_def: fixes f :: "'a" shows "hom.map f = (if hom.arr f then f else hom.null)"
complex_sqrt_1: fixes z :: "complex" assumes "z \<noteq> 0" shows "(z = 1 / z) = (z = 1 \<or> z = - 1)"
case_prodE: fixes c :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'a \<times> 'c" assumes "case p of (a, b) \<Rightarrow> c a b" obtains x :: "'a" and y :: "'c" where "p = (x, y)" and "c x y"
assms: shows "exe_osig_conds (exesorts (ExeSignature cto tao sa))"
B_def: shows "B = integral {0..1} (\<lambda>x. x powr (a - 1) * (1 - x) powr (b - 1))"
ex_nat_less: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<exists>m\<le>n. P m) = (\<exists>m\<in>{0..n}. P m)"
abs_conv_abscissa_restrict: fixes P :: "nat \<Rightarrow> bool"   and f :: "'b fds" shows "abs_conv_abscissa (fds_subseries P f) \<le> abs_conv_abscissa f"
simps_6: fixes t :: "'a local.term" shows "Cod \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Trg t \<^bold>\<star> Cod t"
Sum_any_mapping_of: fixes f :: "'r \<Rightarrow> 'q mpoly" shows "(\<Sum>x. mapping_of (f x)) = mapping_of (Sum_any f)"
hinter_iff: fixes u :: "hf"   and x :: "hf"   and y :: "hf" shows "(u \<^bold>\<in> x \<sqinter> y) = (u \<^bold>\<in> x \<and> u \<^bold>\<in> y)"
Cod_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "B.VxV.seq g f" shows "(if B.arr (fst (B.VxV.comp g f)) \<and> B.arr (snd (B.VxV.comp g f)) then (B.cod (fst (B.VxV.comp g f)), B.cod (snd (B.VxV.comp g f))) else (B.null, B.null)) = (if B.arr (fst g) \<and> B.arr (snd g) then (B.cod (fst g), B.cod (snd g)) else (B.null, B.null))"
dghm_HomDom: shows "\<HH>\<lparr>HomDom\<rparr> = \<AA>"
wf_mformula_wf_set: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'" shows "wf_set n (fv \<phi>')"
subst_typ_preserves_Some_typ_of: fixes t :: "Core.term"   and T :: "typ"   and insts :: "((variable \<times> String.literal set) \<times> typ) list" assumes "typ_of t = Some T" shows "typ_of (subst_typ' insts t) = Some (subst_typ insts T)"
assms: fixes p :: "'a poly" assumes "p \<in> set ps" shows "p \<noteq> 0"
token_smallest_accepting_rank: fixes i :: "nat" assumes "\<UU>.smallest_accepting_rank = Some i" shows "\<forall>\<^sub>\<infinity>n. \<forall>x. \<UU>.token_succeeds x = (n < x \<or> (\<exists>j\<ge>i. \<UU>.rank x n = Some j) \<or> \<UU>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
antisympD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "antisymp r"   and "r a b"   and "r b a" shows "a = b"
tensor_mat_add2: fixes m1 :: "'a mat"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "m1 \<in> carrier_mat ps_P.d1 ps_P.d1"   and "m2 \<in> carrier_mat ps_P.d2 ps_P.d2"   and "m3 \<in> carrier_mat ps_P.d2 ps_P.d2" shows "ps_P.tensor_mat m1 (m2 + m3) = ps_P.tensor_mat m1 m2 + ps_P.tensor_mat m1 m3"
\<gamma>_def: shows "\<gamma> \<equiv> (\<beta>, in_doubt_allow)"
equalsffemptyD: fixes A :: "'c fset"   and a :: "'c" assumes "A = {||}" shows "a |\<notin>| A"
diff_add_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus (minus a b) b = a"
empty_def: shows "Height_Balanced_Tree.empty = \<langle>\<rangle>"
relcompE: fixes xz :: "'d \<times> 'e"   and r :: "('d \<times> 'f) set"   and s :: "('f \<times> 'e) set" assumes "xz \<in> r O s" obtains x :: "'d" and y :: "'f" and z :: "'e" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
vsum_def: fixes A :: "V"   and B :: "V" shows "A \<Uplus> B \<equiv> set {0} \<times>\<^sub>\<circ> A \<union>\<^sub>\<circ> set {1} \<times>\<^sub>\<circ> B"
chain_rel_prodD2: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "('b \<times> 'c) set" assumes "Complete_Partial_Order.chain (rel_prod orda ordb) Y" shows "Complete_Partial_Order.chain ordb (snd ` Y)"
aux: fixes a :: "'c"   and c :: "'d"   and R :: "('c \<times> 'd) set"   and b :: "'c" assumes "(a, c) \<in> R"   and "a = b" shows "(b, c) \<in> R"
mkArr_in_hom: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)"   and "B \<subseteq> Univ"   and "|B| <o cardSuc (cmax |UNIV| natLeq)"   and "F \<in> A \<rightarrow> B" shows "\<guillemotleft>mkArr A B F : mkIde A \<rightarrow> mkIde B\<guillemotright>"
assms: shows "(c, d) \<in> set (half_segments_of_aform X)"
atLeastLessThan_empty_iff2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "({} = subset_mset.atLeastLessThan a b) = (\<not> a \<subset># b)"
states_alt_def: fixes r :: "('d \<times> 'c) list"   and p :: "'c" shows "states r p = map snd r"
plus_iarray_def: fixes A :: "'b iarray"   and B :: "'b iarray" shows "A + B = (let length_A = IArray.length A; length_B = IArray.length B; n = max length_A length_B; A' = IArray.of_fun (\<lambda>a. if a < length_A then A !! a else (0::'b)) n; B' = IArray.of_fun (\<lambda>a. if a < length_B then B !! a else (0::'b)) n in IArray.of_fun (\<lambda>a. A' !! a + B' !! a) n)"
IH: fixes v :: "val"   and v' :: "val" assumes "(v, v') \<in> set t" shows "Tf t <:: T v \<rightarrow> T v'"
ide_cod: fixes f :: "'b" assumes "C'.arr f" shows "C'.ide (C'.cod f)"
lcons_def: fixes a :: "'a \<times> 'b"   and t :: "('a, 'b) FingerTree" shows "FingerTree.lcons a t \<equiv> Abs_FingerTree (FingerTreeStruc.lcons a (Rep_FingerTree t))"
strict1: fixes y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "invalid->including\<^sub>S\<^sub>e\<^sub>t(y) = invalid"
simps_3: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "Node x21 x22 x23 x24 x25 \<noteq> Leaf"
zero_on_nonneg: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and x :: "'d" shows "((0::'c) \<le> zero_on f X x) = (x \<notin> X \<longrightarrow> (0::'c) \<le> f x)"
expr_has_density_sound: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and t :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val" assumes "({}, {}, \<Gamma>, \<lambda>_. 1) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "\<Gamma> \<turnstile> e : t"   and "free_vars e = {}" shows "has_subprob_density (expr_sem \<sigma> e) (stock_measure t) (f (\<lambda>_. undefined))"
ZfunD: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and r :: "real" assumes "Zfun f F"   and "0 < r" shows "\<forall>\<^sub>F x in F. norm (f x) < r"
loop_has_cd': fixes \<pi> :: "nat \<Rightarrow> 'n"   and j :: "nat"   and i :: "nat" assumes "is_path \<pi>"   and "j < i"   and "\<pi> i = \<pi> j"   and "\<pi> j \<noteq> local.return" shows "\<exists>k\<in>{j..<i}. i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
abs_power_minus: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "abs (power.power one times (uminus a) n) = abs (power.power one times a n)"
measure_Ici: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf"   and x :: "'b" assumes "rel_pmf R p q"   and "reflp R"   and "transp R" shows "measure_pmf.prob p {y. R x y} \<le> measure_pmf.prob q {y. R x y}"
real_transfer_operator_int_1: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (real_transfer_operator f)"
find_witness_from_frontier: fixes t :: "'t"   and c :: "('loc, 't) configuration"   and loc2 :: "'loc" assumes "t \<in>\<^sub>A frontier (c_imp c loc2)"   and "inv_imps_work_sum c" shows "\<exists>t' loc1 xs. path loc1 loc2 xs \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs)) \<and> (t' \<in>\<^sub>A frontier (c_pts c loc1) \<or> zcount (c_work c loc1) t' < 0)"
elims: fixes x :: "'b stream \<Rightarrow> 'a"   and xa :: "'b stream"   and y :: "'a" assumes "nxt x xa = y" obtains \<phi> :: "'b stream \<Rightarrow> 'a" and xs :: "'b stream" where "x = \<phi>" and "xa = xs" and "y = \<phi> (stl xs)"
LOW_def: fixes X :: "Var set"   and C :: "CtxtProg" shows "LOW X C = (CtxtVars X C \<and> (\<forall>x. x \<in> X \<longrightarrow> CONTEXT x = low))"
Nml_Src: fixes t :: "'a E.term" assumes "E.Nml t" shows "E.Nml (E.Src t)"
fold_map: fixes g :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "fold g (map f xs) = fold (g \<circ> f) xs"
omega_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<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 dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "omega one = bot"
HEndPhase0_41: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s"   and "Inv2c_inner s p" shows "\<exists>D\<in>MajoritySet. \<forall>d\<in>D. mbal (disk s d p) \<le> mbal (dblock s' p) \<and> bal (disk s d p) \<le> bal (dblock s' p)"
CFGExit_axioms: shows "CFGExit sourcenode targetnode kind valid_edge (_Entry_) get_proc get_return_edges procs Main (_Exit_)"
run_state_get: fixes s :: "'a" shows "run_state State_Monad.get s = (s, s)"
add_smult_distrib_left_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and k :: "'a" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "k \<cdot>\<^sub>m (A + B) = k \<cdot>\<^sub>m A + k \<cdot>\<^sub>m B"
repeat_mset_size: fixes n :: "nat"   and A :: "'b multiset" shows "size (repeat_mset n A) = n * size A"
pseudo_proj_True_same_img: fixes n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "pseudo_proj_True n w = pseudo_proj_True n x" shows "stake n w = stake n x"
the_elem'_def: fixes S :: "'b set" shows "the_elem' S = the_elem (set (sorted_list_of_set S))"
\<chi>_def: shows "\<chi> \<equiv> rawpsubstT r (zip (map Var us) (map snd txs))"
Fun_prod: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and x :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "arr g"   and "x \<in> local.set (local.prod (local.dom f) (local.dom g))" shows "Fun (HF'.prod f g) x = UP \<langle>DOWN (Fun f (UP (hfst (DOWN x)))), DOWN (Fun g (UP (hsnd (DOWN x))))\<rangle>"
nfa_accepts_quantify_nfa: fixes A :: "bool list bdd list \<times> bool list"   and n :: "nat"   and i :: "nat"   and bss :: "bool list list" assumes "wf_nfa A (Suc n)"   and "i \<le> n"   and "list_all (is_alph n) bss" shows "nfa_accepts (quantify_nfa i A) bss = (\<exists>bs. nfa_accepts A (insertll i bs bss) \<and> length bs = length bss)"
fun_add_assoc: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a"   and z :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F"   and "z \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> y \<oplus>\<^bsub>F\<^esub> z = x \<oplus>\<^bsub>F\<^esub> (y \<oplus>\<^bsub>F\<^esub> z)"
simps_2: fixes t1 :: "ternaryformula"   and t2 :: "ternaryformula" shows "ternary_ternary_eval (TernaryOr t1 t2) = eval_ternary_Or (ternary_ternary_eval t1) (ternary_ternary_eval t2)"
assms_2: shows "p < \<^bold>|w\<^bold>|"
isMaxim_unique: fixes A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "s.isMaxim A x"   and "s.isMaxim A y" shows "x = y"
no_empty_paths: fixes Q :: "'a set" assumes "Q \<in> \<P>" shows "Q \<noteq> {}"
fix_eq5: fixes f :: "'b \<rightarrow> 'c"   and F :: "('b \<rightarrow> 'c) \<rightarrow> 'b \<rightarrow> 'c"   and x :: "'b" assumes "f = fix\<cdot>F" shows "f\<cdot>x = F\<cdot>f\<cdot>x"
Auth_Fst: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> analz H"   and "Pri_AgrK y \<in> analz H" shows "Pri_AgrK x \<in> analz H"