hcju/isabelleps huggingface dataset:
C: shows "ns \<noteq> [] \<longrightarrow> Suc 0 < Max (set ns) \<longrightarrow> Max (set ns') < Max (set ns)"
strong_induct: fixes x :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "NF x"   and "\<And>ts x y. listsp (\<lambda>x. NF x \<and> (\<forall>xa. P xa x)) ts \<Longrightarrow> P y (Var x \<degree>\<degree> ts)"   and "\<And>t x y. \<lbrakk>x \<sharp> y; NF t; \<And>z. P z t\<rbrakk> \<Longrightarrow> P y (Lam [x].t)" shows "P z x"
mult_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(a * b = (0::'a)) = (a = (0::'a) \<or> b = (0::'a))"
S_def: shows "S \<equiv> {(l, u). l \<in> L \<and> (\<forall>x\<in>\<X>. 0 \<le> u x) \<and> u \<turnstile> PTA.inv_of A l}"
is_span: shows "C.span \<mu>\<nu>\<pi>.cod.leg0 \<mu>\<nu>\<pi>.cod.leg1"
Nset2_prep1: fixes A :: "'c set"   and n :: "nat" assumes "finite A"   and "card A = Suc (Suc n)" shows "\<exists>x. x \<in> A"
assms_3: shows "\<not> (\<exists>k\<ge>n. enat k < n' \<and> enat k < llength t \<and> \<parallel>c\<parallel>\<^bsub>lnth t k\<^esub>)"
sortedByWeight_Cons_imp_sortedByWeight: fixes t :: "'a tree"   and ts :: "'a tree list" assumes "sortedByWeight (t # ts)" shows "sortedByWeight ts"
C\<^sub>2sat: shows "eval\<^sub>c F G C\<^sub>2"
map_getmin: fixes m :: "'m"   and k :: "'a"   and p :: "'b" assumes "getmin m = (k, p)"   and "invar m"   and "lookup m \<noteq> Map.empty" shows "lookup m k = Some p \<and> (\<forall>p'\<in>ran (lookup m). p \<le> p')"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>v) (blinfun_of_cblinfun x) = (*\<^sub>V) x"
n_init: shows "active_subset (snd (lhd TNs)) = {}"
nz: shows "f (x $ 1) - g (x $ 2) \<noteq> 0"
space_closed: shows "generator \<subseteq> Pow (space (Pi\<^sub>M I M))"
eq_neutr_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<sqinter> b = \<top>) = (a = \<top> \<and> b = \<top>)"
atLeast_def: fixes l :: "'a word" shows "signed.atLeast l \<equiv> {x. l \<le>s x}"
zero_less_binomial_iff: fixes n :: "nat"   and k :: "nat" shows "(0 < n choose k) = (k \<le> n)"
coherence: fixes t :: "'c local.term" assumes "Arr t" shows "coherent t"
exists_in_alphabet: fixes t :: "'a tree" shows "\<exists>a. a \<in> alphabet t"
P2: shows "E \<noteq> D \<and> B \<noteq> A \<and> E \<noteq> F \<and> E \<noteq> P \<and> B \<noteq> C"
finsum_singleton: fixes i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "i \<in> A"   and "finite A"   and "f \<in> A \<rightarrow> carrier R" shows "(\<Oplus>j\<in>A. if i = j then f j else \<zero>) = f i"
indets_monomial_zero: fixes c :: "'c" shows "indets (monomial c 0) = {}"
set_ebounds: fixes as :: "atom list" shows "set (ebounds as) = {k. Eq (Suc k) 0 \<in> set as \<or> Eq 0 (Suc k) \<in> set as}"
splitFace_new_f12: fixes g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVs :: "nat list"   and f12 :: "face"   and f21 :: "face"   and newGraph :: "graph" assumes "pre_splitFace g ram1 ram2 oldF newVs"   and "minGraphProps g"   and "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVs" shows "f12 \<notin> \<F> g"
ring_n_eSum_K_eSumTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "distinct_pds K n P" shows "(\<forall>j\<le>m. f j \<in> carrier (O\<^bsub>K P n\<^esub>)) \<longrightarrow> \<Sigma>\<^sub>e O\<^bsub>K P n\<^esub> f m = \<Sigma>\<^sub>e K f m"
sums_confine_divide: fixes f :: "nat \<Rightarrow> 'b"   and N :: "nat set"   and a :: "'b"   and c :: "'b" assumes "confine f N sums a" shows "confine (\<lambda>n. f n / c) N sums (a / c)"
sample_var_const: fixes x :: "'v"   and m :: "'m" shows "bind (sample_var X x) (\<lambda>_. m) = m"
ran_restrict_def: fixes p :: "'d \<Rightarrow> 'e decision option"   and S :: "'e decision set" shows "p \<triangleright> S \<equiv> \<lambda>x. if p x \<in> Some ` S then p x else \<bottom>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a) (\<Phi>'.Y a) (\<Phi>'.Y a)"
rel_inject_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x2 :: "'c"   and y2 :: "'d" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
connected_in_V_1: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w" shows "v \<in> V"
sep_disj_addI3: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj x (plus y z)"
empty_eq_map_add_iff: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" shows "(Map.empty = f ++ g) = (f = Map.empty \<and> g = Map.empty)"
HSem_Nil: fixes ESem :: "'c \<Rightarrow> ('a \<Rightarrow> 'b) \<rightarrow> 'b"   and \<rho> :: "'a \<Rightarrow> 'b" shows "has_ESem.HSem ESem []\<cdot>\<rho> = \<rho>"
comp_inv_arr: fixes f :: "'b"   and g :: "'b" assumes "D.inverse_arrows f g" shows "g \<cdot>\<^sub>D f = D.dom f"
mirror_D: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "D r \<sigma> \<tau> \<tau>' \<sigma>'"
Max_neq_imp_less: fixes I :: "'a set"   and k :: "'a" assumes "finite I"   and "k \<in> I"   and "k \<noteq> Max I" shows "k < Max I"
ctr_transfer_2: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(R ===> rel_option R) Some Some"
H4: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[_actrans P v \<longrightarrow> P]_v"
small_diff: fixes a :: "V"   and X :: "V set" shows "small (elts a - X)"
inject_1: fixes x11 :: "'b \<times> sort_dir"   and x12 :: "'b sort_key"   and y11 :: "'b \<times> sort_dir"   and y12 :: "'b sort_key" shows "(NonFinal x11 x12 = NonFinal y11 y12) = (x11 = y11 \<and> x12 = y12)"
u_def: fixes t :: "real" shows "u t = (if t \<le> 0 then s\<^sub>e else if t \<le> \<delta> then ego.q t else (ego2.s \<circ> \<tau>) t)"
coeff_mult_prod_sum: fixes f :: "'c \<Rightarrow> 'b poly"   and S :: "'c set" shows "coeff (prod f S) (\<Sum>i\<in>S. degree (f i)) = (\<Prod>i\<in>S. lead_coeff (f i))"
N_def: shows "N = nat \<lceil>exp (exp k)\<rceil>"
toVariable_variable_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and v :: "varDecl" assumes "gState_inv prog g" shows "variable_inv (fst (snd (toVariable g p v)))"
first_def: shows "first \<equiv> \<lfloor>([], (case b of Static \<Rightarrow> [] | NonStatic \<Rightarrow> [OK (Class C)]) @ map OK Ts @ replicate mxl\<^sub>0 Err)\<rfloor>"
subset_finite_imp_finite: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "B \<subseteq> A" shows "finite B"
is_Ghost\<^sub>s\<^sub>b_simps_1: fixes volatile :: "bool"   and a :: "nat"   and t :: "nat"   and v :: "nat" shows "is_Ghost\<^sub>s\<^sub>b (Read\<^sub>s\<^sub>b volatile a t v) = False"
transcendental_complex_logarithm: fixes x :: "complex"   and y :: "complex" assumes "algebraic x"   and "exp y = x"   and "x \<noteq> 1" shows "\<not> algebraic y"
map_undefined: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and T :: "'b set"   and a :: "'a" assumes "map \<alpha> S T"   and "a \<notin> S" shows "\<alpha> a = undefined"
map_gpv_bind_gpv: fixes f :: "'g \<Rightarrow> 'd"   and g :: "'h \<Rightarrow> 'e"   and gpv :: "('i, 'h, 'f) gpv"   and h :: "'i \<Rightarrow> ('g, 'h, 'f) gpv" shows "map_gpv f g (gpv \<bind> h) = map_gpv id g gpv \<bind> (\<lambda>x. map_gpv f g (h x))"
eq2: shows "(bw_diff ^^ j) (\<lambda>z. \<Sum>i = Suc j..card X. z + (int i - int (aa i) - 1) gchoose i + (z + (int i - int (bb i) - 1) gchoose i)) = (\<lambda>z. \<Sum>i = Suc j..card X. z + (int i - int (aa i) - 1) - int j gchoose (i - j) + (z + (int i - int (bb i) - 1) - int j gchoose (i - j)))"
emeasure_eq_measure: fixes A :: "real set" shows "emeasure M A = ennreal (prob A)"
subst_cls_list_def: fixes Cs :: "'a literal multiset list"   and \<sigma> :: "'s" shows "Cs \<cdot>cl \<sigma> = map (\<lambda>A. A \<cdot> \<sigma>) Cs"
ax_ne: shows "cbox a (\<chi>i. if i = m then a $ m + b $ n else b $ i) \<noteq> {}"
tangent_space_restrict: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a" assumes "X \<in> sub.tangent_space p" shows "extensional0 sub.diff_fun_space X"
find_sgas_aux_def: fixes in_list :: "real poly list" shows "find_sgas_aux in_list = concat (map (\<lambda>i. map (insertAt i 0) (find_consistent_signs_at_roots (in_list ! i) (removeAt i in_list))) [0..<length in_list])"
min_int_poly_nonzero: fixes x :: "'a" shows "min_int_poly x \<noteq> 0"
obind_wp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c option" assumes "\<And>r. ovalid (R r) (g r) Q"   and "ovalid P f R" shows "ovalid P (f |>> g) Q"
index_ordered: fixes p :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "infinitely_many p" shows "infinitely_many.index p n < infinitely_many.index p (Suc n)"
fnorm_correct_2: fixes xs :: "'a list"   and e :: "fexpr" assumes "in_carrier xs"   and "nonzero xs (Cond e)" shows "peval xs (Denom e) \<noteq> \<zero>"
assms: fixes a :: "'a" shows "f \<langle>$\<rangle> a = g \<langle>$\<rangle> a"
ty_cases: fixes T :: "ty" shows "(\<exists>T\<^sub>1 T\<^sub>2. T = T\<^sub>1\<rightarrow>T\<^sub>2) \<or> T = TUnit \<or> T = TBase"
f_def: fixes x :: "'a" shows "f x = (if x \<in> A then f' x else undefined)"
sub1_det: fixes A :: "'a mat"   and n :: "nat"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and r'' :: "'a"   and A'' :: "'a mat" assumes "A \<in> carrier_mat n n"   and "local.sub1 q k l (r, A) = (r'', A'')"   and "r \<noteq> (0::'a)"   and "q \<noteq> (0::'a)"   and "l + k < n" shows "r * det A'' = r'' * det A"
while_eval_elim: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and e :: "'d"   and c :: "('a, 'c, 'd) Stmt"   and annos :: "'a ModeUpd list"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_modes ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars"   and "(((sifum_modes.annotate (While e c) annos, mds), mem), (c', mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w ev\<^sub>A ev\<^sub>B" shows "c' = Stmt.If e (c ;; While e c) Stop \<and> mds' = sifum_modes.update_annos mds annos \<and> mem' = mem"
a81: shows "\<Union> (Acc level1 ` DAcc level1 sA81) = Acc level1 sA92 \<union> Acc level1 sA91"
monoid_axioms: shows "Group_Theory.monoid pi.carrier_local_ring_at pi.mult_local_ring_at pi.one_local_ring_at"
D_def: shows "D = denote_while_n_iter M0 M1 DS"
sup_leof_iff: fixes a :: "'b nres"   and b :: "'b nres"   and m :: "'b nres" shows "(sup a b \<le>\<^sub>n m) = (nofail a \<and> nofail b \<longrightarrow> a \<le>\<^sub>n m \<and> b \<le>\<^sub>n m)"
less_eq_vec_def: fixes v :: "'b vec"   and w :: "'b vec" shows "(v \<le> w) = (dim_vec v = dim_vec w \<and> (\<forall>i<dim_vec w. v $ i \<le> w $ i))"
Cesare: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A E B"   and "A Q C" shows "B E C"
row_add_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and b :: "'c"   and q :: "'a" shows "row_add A a b q = (\<chi>i j. if i = a then A $ a $ j + q * A $ b $ j else A $ i $ j)"
2: assumes "P \<turnstile> typeof\<^bsub>h(a \<mapsto> (C, S'))\<^esub> v = \<lfloor>NT\<rfloor>"   and "P \<turnstile> typeof\<^bsub>h\<^esub> v \<noteq> \<lfloor>NT\<rfloor>" shows "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Class C'\<rfloor>"
nb\<^sub>6: shows "set (add_effects_of op') = set (effect_of op)"
Hermite_of_upt_row_preserves_nonzero_rows: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and k :: "nat" assumes "\<not> is_zero_row i A"   and "echelon_form A"   and "ass_function ass"   and "res_function res"   and "k \<le> nrows A" shows "\<not> is_zero_row i (Hermite_of_upt_row_i A k ass res)"
vdomain_vsingleton: fixes a :: "V"   and b :: "V" shows "\<D>\<^sub>\<circ> (ZFC_in_HOL.set {\<langle>a, b\<rangle>}) = ZFC_in_HOL.set {a}"
rbt_lookup_rbt_interwk: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'a" assumes "rbt_sorted t1"   and "rbt_sorted t2" shows "rbt_lookup (rbt_inter_with_key f t1 t2) k = (case rbt_lookup t1 k of None \<Rightarrow> None | Some v \<Rightarrow> case rbt_lookup t2 k of None \<Rightarrow> None | Some w \<Rightarrow> Some (f k v w))"
join_def: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" shows "x \<squnion>\<^bsub>L\<^esub> y = \<Squnion>\<^bsub>L\<^esub>{x, y}"
harry_step_False_True: fixes xs :: "bool list" assumes "True \<in> set xs" shows "harry_step (False # xs @ [True]) = False # harry_step xs @ [True]"
distinct_1: fixes x21 :: "'k"   and x22 :: "'k list" shows "[] \<noteq> x21 # x22"
finite_set_sum: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "finite A"   and "\<forall>i\<in>A. finite (B i)" shows "finite (sum B A)"
reduction_correspondence_simulation_condition_trans: fixes Q :: "'a"   and Cal :: "'a processCalculus"   and P :: "'a"   and Rel :: "('a \<times> 'a) set"   and R :: "'a" assumes "\<forall>Q'. Q \<longmapsto>Cal* Q' \<longrightarrow> (\<exists>P'' Q''. P \<longmapsto>Cal* P'' \<and> Q' \<longmapsto>Cal* Q'' \<and> (P'', Q'') \<in> Rel)"   and "\<forall>R'. R \<longmapsto>Cal* R' \<longrightarrow> (\<exists>Q'' R''. Q \<longmapsto>Cal* Q'' \<and> R' \<longmapsto>Cal* R'' \<and> (Q'', R'') \<in> Rel)"   and "weak_reduction_simulation Rel Cal"   and "trans Rel" shows "\<forall>R'. R \<longmapsto>Cal* R' \<longrightarrow> (\<exists>P'' R''. P \<longmapsto>Cal* P'' \<and> R' \<longmapsto>Cal* R'' \<and> (P'', R'') \<in> Rel)"
combine_rep_number: shows "constant_rep_design \<V>\<^sup>+ \<B>\<^sup>+ (\<r> + \<r>')"
Range_empty_iff: fixes r :: "('d \<times> 'c) set" shows "(Range r = {}) = (r = {})"
colossless_gpv_lossless_spmfD: fixes \<I> :: "('d, 'e) \<I>"   and gpv :: "('f, 'd, 'e) gpv" assumes "colossless_gpv \<I> gpv" shows "lossless_spmf (the_gpv gpv)"
mergeFrameE: fixes F :: "'b frame"   and G :: "'b frame"   and A\<^sub>F\<^sub>G :: "name list"   and \<Psi>\<^sub>F\<^sub>G :: "'b" assumes "F \<otimes>\<^sub>F G = \<langle>A\<^sub>F\<^sub>G, \<Psi>\<^sub>F\<^sub>G\<rangle>"   and "distinct A\<^sub>F\<^sub>G"   and "A\<^sub>F\<^sub>G \<sharp>* F"   and "A\<^sub>F\<^sub>G \<sharp>* G" obtains A\<^sub>F :: "name list" and \<Psi>\<^sub>F :: "'b" and A\<^sub>G :: "name list" and \<Psi>\<^sub>G :: "'b" where "A\<^sub>F\<^sub>G = A\<^sub>F @ A\<^sub>G" and "\<Psi>\<^sub>F\<^sub>G = \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>G" and "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>" and "G = \<langle>A\<^sub>G, \<Psi>\<^sub>G\<rangle>" and "A\<^sub>F \<sharp>* \<Psi>\<^sub>G" and "A\<^sub>G \<sharp>* \<Psi>\<^sub>F"
relation_inter: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "{(x, y). P x y} \<inter> {(x, y). Q x y} = {(x, y). P x y \<and> Q x y}"
IH: assumes "xs \<noteq> []" shows "argmin f xs = xs ! Max {i. i < length xs \<and> (\<forall>x\<in>set xs. f (xs ! i) \<le> f x)}"
fresh_fun_app: fixes h :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp h)"   and "\<exists>a. a \<sharp> (h, h a)"   and "a \<sharp> h" shows "fresh_fun h = h a"
inf_compl_bot_left1: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf (uminus x) (inf x y) = bot"
pseudo_mod_def: fixes f :: "'b poly"   and g :: "'b poly" shows "pseudo_mod f g = snd (pseudo_divmod f g)"
sum_multiset_singleton: fixes A :: "'b set" shows "(\<Sum>n\<in>A. {#n#}) = mset_set A"
False: shows "\<not> (b = a \<or> c = a \<or> c = b)"
divide_nonpos_nonpos: fixes x :: "'a"   and y :: "'a" assumes "x \<le> (0::'a)"   and "y \<le> (0::'a)" shows "(0::'a) \<le> x / y"
rep_eq: fixes x :: "('d, 'e) RBT.rbt" shows "mapping.rep (RBT_Mapping.Mapping x) = RBT.lookup x"
elims: fixes x :: "nat"   and xa :: "('ekey \<times> 'dkey \<times> 'cipher list) option"   and xb :: "'cipher"   and y :: "('plain option \<times> ('ekey \<times> 'dkey \<times> 'cipher list) option) spmf" assumes "decrypt_oracle x xa xb = y" obtains \<eta> :: "nat" and cipher :: "'cipher" where "x = \<eta>'__" and "xa = None" and "xb = ciphera__" and "y = return_spmf (None, None)"   | \<eta> :: "nat" and ekey :: "'ekey" and dkey :: "'dkey" and cstars :: "'cipher list" and cipher :: "'cipher" where "x = \<eta>" and "xa = Some (ekey, dkey, cstars)" and "xb = cipher" and "y = return_spmf (if cipher \<in> set cstars then None else decrypt \<eta> dkey cipher, Some (ekey, dkey, cstars))"
less_le_diff: fixes x :: "nat"   and n :: "nat" assumes "x < n" shows "x \<le> n - Suc 0"
atLeastAtMost_singleton_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "({a..b} = {c}) = (a = b \<and> b = c)"
topological_tendstoI: fixes "open" :: "'b set \<Rightarrow> bool"   and l :: "'b"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c filter" assumes "class.topological_space open"   and "\<And>S. \<lbrakk>open S; l \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> S" shows "filterlim f (topological_space.nhds open l) F"
mask_eq_exp_minus_1: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask n = minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one"
OrR: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta> \<down> Suc n"
reduced_GB_D2: fixes G :: "('a \<Rightarrow>\<^sub>0 'c) set" assumes "punit.is_reduced_GB G" shows "punit.is_auto_reduced G"
runit_simps_1: fixes f :: "'a" assumes "S.ide f" shows "S.arr (S.runit f)"
cart_flip_aux: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and B :: "'c set"   and A :: "'d set" shows "{f (snd p) (fst p) |p. p \<in> B \<times> A} = {f (fst p) (snd p) |p. p \<in> A \<times> B}"
eq_linorder_on_axioms: shows "eq_linorder_on D2 cmp2"
pairwise_orth_fb: shows "pairwise (\<lambda>v j. orthogonal (fb v) (fb j)) B"
snoc: shows "prod_list (nths xs A) = prod_list (nths xs B) * prod_list (nths xs (A - B))"
action_loc_change_prefix: fixes n :: "enat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and P :: "'c prog" assumes "ltake n E [\<approx>] ltake n E'"   and "enat a < n" shows "action_loc P E a = action_loc P E' a"
dfa_accepts_negate: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and bss :: "bool list list" assumes "wf_dfa A n"   and "list_all (is_alph n) bss" shows "dfa_accepts (negate_dfa A) bss = (\<not> dfa_accepts A bss)"
F_minus_D_seq_opt: fixes a :: "'a event list"   and b :: "'a event set"   and P :: "'a process"   and Q :: "'a process" shows "((a, b) \<in> F (P `;` Q)) = (a \<in> D (P `;` Q) \<or> (a, b) \<in> F_minus_D_seq P Q)"
dim_solution_set_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "(vec.dim (solution_set A 0) = 0) = (solution_set A 0 = {0})"
take_append: fixes n :: "nat"   and xs :: "'g list"   and ys :: "'g list" shows "take n (xs @ ys) = take n xs @ take (n - length xs) ys"
old_document: shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node) \<rightarrow>\<^sub>r old_document"
DSourcesA21_L1: shows "DSources level1 sA21 = {sA11}"
inner_prod_with_times_mat: fixes u :: "complex Matrix.vec"   and v :: "complex Matrix.vec" assumes "dim_vec u = dim_vec v" shows "\<langle>u|v\<rangle> = (\<langle>|u\<rangle>| * |v\<rangle>) $$ (0, 0)"
starfunNat_pi_divide_n_Infinitesimal: fixes N :: "nat star" assumes "N \<in> HNatInfinite" shows "(*f* (\<lambda>x. pi / real x)) N \<in> Infinitesimal"
commutative_squareI: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "cospan f g"   and "span h k"   and "local.dom f = cod h"   and "f \<cdot> h = g \<cdot> k" shows "commutative_square f g h k"
B: fixes u :: "nat \<Rightarrow> real" assumes "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> - \<infinity>" shows "Gromov_converging_at_boundary u"
simps_15: fixes v :: "Formula.formula"   and va :: "\<I>"   and vb :: "Formula.formula" shows "remove_neg (formula.Until v va vb) = formula.Until v va vb"
infinite_soundness: fixes t :: "('sequent \<times> 'rule) tree"   and S :: "'structure" assumes "local.wf t"   and "good t"   and "S \<in> structure" shows "sat S (fst (tree.root t))"
le_transitivity: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B Le C D"   and "C D Le E F" shows "A B Le E F"
le: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>real (sum_upto divisor_sum x) - \<zeta> / 2 * x\<^sup>2\<bar> \<le> c2 / 2 + x / 2 + c1 * x * (ln x + 1)"
\<I>_1: shows "\<I> \<Turnstile> \<langle>unlabel \<A>\<rangle>"
zcf_to_poly: fixes a :: "'a" assumes "a \<in> carrier R" shows "zcf (to_polynomial R a) = a"
set_comp_types: fixes U :: "'b set"   and B :: "'b set"   and C :: "'b set"   and A :: "'b set" shows "(\<bullet>\<^bsub>set_cat U\<^esub>) \<in> Hom\<^bsub>set_cat U\<^esub> B C \<rightarrow> Hom\<^bsub>set_cat U\<^esub> A B \<rightarrow> Hom\<^bsub>set_cat U\<^esub> A C"
prod_rel_relcomp_distr: fixes R :: "('k \<times> 'o) set"   and S :: "('l \<times> 'p) set"   and T :: "('o \<times> 'm) set"   and U :: "('p \<times> 'n) set" shows "(R \<times>\<^sub>r S) O (T \<times>\<^sub>r U) = R O T \<times>\<^sub>r S O U"
gauss_jordan_row_echelon: fixes A :: "'c mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'c mat"   and A' :: "'c mat"   and B' :: "'c mat" assumes "A \<in> carrier_mat nr nc"   and "gauss_jordan A B = (A', B')" shows "row_echelon_form A'"
typeof_NT: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>NT\<rfloor>" shows "v = Null"