hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" assumes "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<beta> \<langle>a4, a5\<rangle>" obtains A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)" and l :: "'d" and Z :: "('c \<Rightarrow> real) set" and l' :: "'d" and Z' :: "('c \<Rightarrow> real) set" where "a1 = A" and "a2 = l" and "a3 = Z" and "a4 = l'" and "a5 = Approx\<^sub>\<beta> Z'" and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>"
r_result1_def: shows "r_result1 \<equiv> Cn 3 r_result [recf.Id 3 0, recf.Id 3 1, Cn 3 r_singleton_encode [recf.Id 3 2]]"
Sbis_transC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>s d"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>c (d', t')" and "s' \<approx> t'" and "c' \<approx>s d'"
length_tl_queue_safe_hd: fixes q :: "'a queue"   and a :: "'a"   and q' :: "'a queue" assumes "safe_hd q = (Some a, q')" shows "length (linearize (tl_queue q')) < length (linearize q)"
ordIso_iff_ordLeq: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" shows "(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
iT_Plus_iprev: fixes n :: "nat"   and k :: "nat"   and I :: "nat set" shows "iprev (n + k) (I \<oplus> k) = iprev n I + k"
MtransC_rtranclp_transC: fixes tval :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and aval :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "PL.MtransC tval aval = (PL.transC tval aval)\<^sup>*\<^sup>*"
R0sk1iai_def: shows "R0sk1iai \<equiv> {(s, s'). signals s = signalsInit s'}"
Sigma_mset_cong: fixes A :: "'b multiset"   and B :: "'b multiset"   and C :: "'b \<Rightarrow> 'c multiset"   and D :: "'b \<Rightarrow> 'c multiset" assumes "A = B"   and "\<And>x. x \<in># B \<Longrightarrow> C x = D x" shows "Sigma_mset A C = Sigma_mset B D"
simps_18: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'a"   and V' :: "'e"   and ad :: "'b"   and e :: "('d, 'e, 'b) exp" shows "J1_heap_base.\<tau>move1 typeof_addr P h (insync\<^bsub>V'\<^esub> (ad) e) = J1_heap_base.\<tau>move1 typeof_addr P h e"
landau_pair_def: fixes L :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and l :: "'b filter \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) set"   and L' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and l' :: "'d filter \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> ('d \<Rightarrow> 'c) set"   and Lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and lr :: "'b filter \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> ('b \<Rightarrow> real) set"   and R :: "real \<Rightarrow> real \<Rightarrow> bool" shows "landau_pair L l L' l' Lr lr R \<equiv> ((\<forall>F g. L F g = {f. \<exists>c>0. \<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))}) \<and> (\<forall>F g. l F g = {f. \<forall>c>0. \<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))}) \<and> (\<forall>F' g'. L' F' g' = {f. \<exists>c>0. \<forall>\<^sub>F x in F'. R (norm (f x)) (c * norm (g' x))})) \<and> ((\<forall>F' g'. l' F' g' = {f. \<forall>c>0. \<forall>\<^sub>F x in F'. R (norm (f x)) (c * norm (g' x))}) \<and> (\<forall>F'' g''. Lr F'' g'' = {f. \<exists>c>0. \<forall>\<^sub>F x in F''. R (norm (f x)) (c * norm (g'' x))})) \<and> (\<forall>F'' g''. lr F'' g'' = {f. \<forall>c>0. \<forall>\<^sub>F x in F''. R (norm (f x)) (c * norm (g'' x))}) \<and> (R = (\<le>) \<or> R = (\<lambda>x y. y \<le> x))"
map_sel_1: fixes a :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "a \<noteq> []" shows "lead_coeff (map f a) = f (lead_coeff a)"
simps_2: fixes n :: "nat" shows "num_of_nat (Suc n) = (if 0 < n then Num.inc (num_of_nat n) else num.One)"
ubounds_def: fixes as :: "atom list" shows "ubounds as = concat (map (\<lambda>x. case x of Le i [] \<Rightarrow> [] | Le i (k # ks) \<Rightarrow> if k < 0 then [(i, ks)] else [] | _ \<Rightarrow> []) as)"
FAssRed2: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 v :: "val"   and F :: "char list"   and Cs :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P,E \<turnstile> \<langle>Val v\<bullet>F{Cs} := e,s\<rangle> \<rightarrow> \<langle>Val v\<bullet>F{Cs} := e',s'\<rangle>"
oldBisimBisimEq: shows "\<Union> {Rel. old_bisim Rel} = bisim"
simplePath_empty_conv: fixes s :: "nat"   and t :: "nat" shows "isSimplePath s [] t = (s = t)"
universal_lookup_sem_R: fixes fml :: "nat fml"   and qs :: "real poly list"   and signs :: "rat list list" assumes "\<And>i. i \<in> set_fml fml \<Longrightarrow> i < length qs"   and "set signs = consistent_sign_vectors_R qs UNIV" shows "(\<forall>x. lookup_sem fml (map (\<lambda>p. poly p x) qs)) = list_all (lookup_sem fml) signs"
mult_poly_0_right: fixes p :: "'d poly" shows "p * 0 = 0"
ereal_semiline_unique: fixes a :: "ereal"   and b :: "ereal" shows "({y. a \<le> ereal y} = {y. b \<le> ereal y}) = (a = b)"
2: shows "hbt \<langle>l, (a, b_), r\<rangle>"
prv_eql_reflT: fixes t :: "'trm" assumes "t \<in> trm" shows "prv (eql t t)"
countableE_infinite: fixes S :: "'a set" assumes "countable S"   and "infinite S" obtains e :: "'a \<Rightarrow> nat" where "bij_betw e S UNIV"
The_on_def: fixes U :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "The_on U P = (if \<exists>!x. x \<in> U \<and> P x then Some (THE x. x \<in> U \<and> P x) else None)"
set_plus_elim: fixes x :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "x \<in> A + B" obtains a :: "'d" and b :: "'d" where "x = a + b" and "a \<in> A" and "b \<in> B"
pre_rule': fixes P :: "assn"   and R :: "assn"   and f :: "'c Heap"   and Q :: "'c \<Rightarrow> assn"   and P' :: "assn" assumes "\<not> <P * R> f <Q>"   and "P \<Longrightarrow>\<^sub>A P'" shows "\<not> <P' * R> f <Q>"
is_rbt_rbt_product: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and leq_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and rbt1 :: "('c, 'e) rbt"   and rbt2 :: "('d, 'f) rbt"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> 'g" assumes "class.linorder leq_a less_a"   and "class.linorder leq_b less_b"   and "ord.is_rbt less_a rbt1"   and "ord.is_rbt less_b rbt2" shows "ord.is_rbt (less_prod leq_a less_a less_b) (rbt_product f rbt1 rbt2)"
units_of_units: fixes G :: "('a, 'b) monoid_scheme" shows "Units (units_of G) = Units G"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : u\<^sub>0 \<star> \<chi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
poly_diff_val: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "f \<in> carrier (UP (padic_int p))"   and "a \<in> carrier (padic_int p)"   and "b \<in> carrier (padic_int p)" shows "padic_integers.val_Zp p (a \<ominus>\<^bsub>padic_int p\<^esub> b) \<le> padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) f a \<ominus>\<^bsub>padic_int p\<^esub> UP_cring.to_fun (padic_int p) f b)"
insert: fixes A :: "'a::type set"   and x :: "'a::type" assumes "finite (A::'a::type set)"   and "(A::'a::type set) \<noteq> {}" shows "\<Sqinter>\<^sub>f\<^sub>i\<^sub>n insert (x::'a::type) (A::'a::type set) = x \<sqinter> \<Sqinter>\<^sub>f\<^sub>i\<^sub>n A"
scf_automorphism: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<sigma> permutes alts"   and "is_pref_profile R"   and "image_mset (map ((`) \<sigma>)) (anonymous_profile R) = anonymous_profile R" shows "\<sigma> ` scf R = scf R"
is_extensional: fixes f :: "'c" assumes "\<not> C.arr f" shows "local.map f = D.null"
distinct_3: fixes x1 :: "nat"   and x31 :: "nat"   and x32 :: "exp" shows "EVar x1 \<noteq> ELam x31 x32"
bind_mono_1: fixes M :: "'c nres"   and M' :: "'c nres"   and f :: "'c \<Rightarrow> 'd nres"   and f' :: "'c \<Rightarrow> 'd nres" assumes "M \<le> M'"   and "\<And>x. RETURN x \<le> M \<Longrightarrow> f x \<le> f' x" shows "M \<bind> f \<le> M' \<bind> f'"
res_of_integer_eq: fixes i :: "int" shows "\<guillemotleft>i\<guillemotright> = i mod m"
invar_rderiv_and_add_def: fixes as :: "'a"   and r :: "'b rexp"   and brs :: "bool \<times> 'b rexp list" shows "invar_rderiv_and_add as r brs \<equiv> (if fst brs then True else \<guillemotleft>rderiv as (hd (snd brs))\<guillemotright> \<in> set (snd brs)) \<and> snd brs \<noteq> [] \<and> distinct (snd brs) \<and> (\<forall>i<length (snd brs). snd brs ! i = \<guillemotleft>rderivs (replicate (length (snd brs) - 1 - i) as) r\<guillemotright>)"
ee_fmset: fixes as :: "'a list"   and bs :: "'a list" assumes "essentially_equal G as bs"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "fmset G as = fmset G bs"
step: shows "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (i + 1)"
inf_None_2: fixes x :: "'b option" shows "x \<sqinter> None = None"
pi_half_le_two: shows "pi / 2 \<le> 2"
card_greaterThanAtMost: fixes l :: "nat"   and u :: "nat" shows "card {l<..u} = u - l"
complete_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "complete A r \<equiv> \<forall>x\<in>A. \<forall>y\<in>A - {x}. x \<^bsub>r\<^esub>\<preceq> y \<or> y \<^bsub>r\<^esub>\<preceq> x"
PDis: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<or>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<or>\<^sub>! [r]\<^sub>! q)"
submoduleE_5: fixes H :: "(nat \<Rightarrow> nat \<Rightarrow> int) set"   and aa :: "nat \<Rightarrow> nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "submodule H Zp (UP Zp)"   and "aa \<in> H"   and "b \<in> H" shows "aa \<oplus>\<^bsub>UP Zp\<^esub> b \<in> H"
finite_fresh_set_fresh_all: fixes S :: "'a set"   and a :: "'b" assumes "finite S" shows "atom a \<sharp> S = (\<forall>x\<in>S. atom a \<sharp> x)"
simply_connected_eq_contractible_path: fixes S :: "'a set" shows "simply_connected S = (path_connected S \<and> (\<forall>p. path p \<and> path_image p \<subseteq> S \<and> pathfinish p = pathstart p \<longrightarrow> homotopic_paths S p (linepath (pathstart p) (pathstart p))))"
in_measures_2: fixes x :: "'g"   and y :: "'g"   and f :: "'g \<Rightarrow> nat"   and fs :: "('g \<Rightarrow> nat) list" shows "((x, y) \<in> measures (f # fs)) = (f x < f y \<or> f x = f y \<and> (x, y) \<in> measures fs)"
new_element_types_preserved: fixes h :: "(_) heap"   and h' :: "(_) heap" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
src_simps_3: fixes V\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>B :: "'b \<Rightarrow> 'b"   and src\<^sub>B :: "'b \<Rightarrow> 'b"   and trg\<^sub>B :: "'b \<Rightarrow> 'b"   and F :: "('b bicategorical_language.term, 'b) concrete_category.arr" assumes "strictified_bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"   and "partial_magma.arr (concrete_category.COMP {A. bicategorical_language.Nml V\<^sub>B src\<^sub>B trg\<^sub>B A \<and> bicategorical_language.Ide V\<^sub>B src\<^sub>B trg\<^sub>B A} (\<lambda>A B. {\<mu>. bicategorical_language.Src src\<^sub>B A = bicategorical_language.Src src\<^sub>B B \<and> bicategorical_language.Trg trg\<^sub>B A = bicategorical_language.Trg trg\<^sub>B B \<and> partial_magma.in_hom V\<^sub>B \<mu> (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) A) (Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) B)}) (\<lambda>_ _ _. V\<^sub>B)) F" shows "concrete_category.Map (strictified_bicategory.src V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B F) = Coherence.evaluation_map.eval V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B (\<lambda>\<mu>. if partial_magma.arr V\<^sub>B \<mu> then \<mu> else partial_magma.null V\<^sub>B) (bicategorical_language.Src src\<^sub>B (concrete_category.Dom F))"
ring_hom_in_hom_2: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" assumes "h \<in> ring_hom R S" shows "h \<in> hom (add_monoid R) (add_monoid S)"
elims: fixes x :: "'a list"   and y :: "('a \<times> 'a) list" assumes "vwalk_arcs x = y" obtains "x = []" and "y = []"   | xa :: "'a" where "x = [xa__]" and "y = []"   | xa :: "'a" and ya :: "'a" and xs :: "'a list" where "x = xa # ya # xs" and "y = (xa, ya) # vwalk_arcs (ya # xs)"
ennreal_minus_zero: fixes a :: "ennreal" shows "a - 0 = a"
weak_ranking_aux_wf: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "is_weak_ranking (weak_ranking_aux A)"
T1_def: shows "T1 \<equiv> unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T))"
path_nret_inf_all_cd: fixes \<pi> :: "nat \<Rightarrow> 'n" assumes "is_path \<pi>"   and "\<forall>i. \<pi> i \<noteq> local.return" shows "infinite {j. \<forall>k>j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j}"
res: shows "zassenhaus_reconstruction (hensel_lifting p (find_exponent p (2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound gs))) f gs) p (find_exponent p (2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound gs))) f = fs"
pos_le_divide_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less_eq a (divide b c) = less_eq (times a c) b"
MGF1_def: fixes Z :: "bit list"   and n :: "nat"   and l :: "nat" shows "MGF1 Z n l = take l (MGF2 Z n)"
simps_1: fixes ys :: "udom\<cdot>llist" shows "plusU\<cdot>LNil\<cdot>ys = ys"
simps_2: fixes n :: "nat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "info_2 (Irrational n (p, l, r)) = Inr (p, n)"
cons_chain: fixes x :: "'b"   and S :: "nat \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set" assumes "(x, S 0) \<in> r"   and "\<forall>i. (S i, S (Suc i)) \<in> r" shows "\<forall>i. (case i of 0 \<Rightarrow> x | Suc x \<Rightarrow> S x, case Suc i of 0 \<Rightarrow> x | Suc x \<Rightarrow> S x) \<in> r"
RSpanI_lincomb_arb_len_coeffs: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and rs :: "'a list"   and ms :: "'b list" assumes "RModule R smult M"   and "set rs \<subseteq> R"   and "set ms \<subseteq> M" shows "scalar_mult.lincomb smult rs ms \<in> R_scalar_mult.RSpan R smult ms"
unique_conjugate: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "\<exists>!g. conjugation_pair f g"
multy_def: shows "multy = restrict ((\<otimes>) y) (carrier G)"
lim_stream_eq: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "K.lim_stream x = K x \<bind> (\<lambda>y. K.lim_stream y \<bind> (\<lambda>\<omega>. return (stream_space S) (y ## \<omega>)))"
gt_swap: fixes c :: "'c"   and a :: "'c"   and b :: "'c" assumes "c < a + b" shows "c < b + a"
sublist_iff_sublist_at: fixes xs :: "'a list"   and ys :: "'a list" shows "sublist xs ys = (\<exists>i. sublist_at xs ys i)"
r_triangle_def: shows "r_triangle \<equiv> r_shrink (Pr 1 Z (r_dummy 1 (Cn 2 S [r_add])))"
subgroup_generated_def: fixes S :: "'a set" shows "subgroup_generated S = generate (G \<inter> S)"
inducts: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "flow \<Delta> f" obtains "\<And>e. f e \<le> capacity \<Delta> e" and "\<And>x. \<lbrakk>x \<noteq> source \<Delta>; x \<noteq> sink \<Delta>\<rbrakk> \<Longrightarrow> KIR f x"
HEndPhase2_maxBalInp: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and b :: "nat"   and v :: "InputsOrNi" assumes "HEndPhase2 s s' q"   and "maxBalInp s b v" shows "maxBalInp s' b v"
is_dghm_axioms: shows "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
assms_2: shows "AE x in M. 0 \<le> f x"
cong_imp_coprime: fixes a :: "'a"   and b :: "'a"   and m :: "'a" assumes "[a = b] (mod m)"   and "coprime a m" shows "coprime b m"
static_empty_ord_inter_equiv_static_inter: shows "statically_complete_calculus Bot_FL Inf_FL entails_\<G> Red_I_\<G> Red_F_\<G> = statically_complete_calculus Bot_FL Inf_FL entails_\<G> Red_I_\<G> Red_F_\<G>_empty"
rel_S_def: fixes L1 :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and L2 :: "'d \<Rightarrow> 'k \<Rightarrow> bool"   and Co1 :: "'e \<Rightarrow> 'l \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'm \<Rightarrow> bool"   and Contra1 :: "'g \<Rightarrow> 'n \<Rightarrow> bool"   and Contra2 :: "'h \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_S L1 L2 Co1 Co2 Contra1 Contra2 = vimage2p Rep_S Rep_S (rel_G L1 L2 Co1 Co2 Contra1 Contra2)"
classical_operator_def: fixes \<pi> :: "'b \<Rightarrow> 'c option" shows "classical_operator \<pi> = Let (\<lambda>t. case \<pi> (inv ket t) of None \<Rightarrow> 0 | Some i \<Rightarrow> ket i) (cblinfun_extension (range ket))"
longer: assumes "xs\<^sub>1 = take (length xs\<^sub>1) ys\<^sub>1"   and "xs\<^sub>2 = drop (length xs\<^sub>1) ys\<^sub>1 @ ys\<^sub>2" shows "thesis"
NSDERIV_inverse_fun: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and d :: "'a" assumes "NSDERIV f x :> d"   and "f x \<noteq> (0::'a)" shows "NSDERIV (\<lambda>x. inverse (f x)) x :> - (d * inverse (f x ^ Suc (Suc 0)))"
allDefs_var_disjoint_inv: fixes n :: "'node"   and v :: "'val"   and v' :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> step.allDefs g n"   and "v' \<in> step.allDefs g n"   and "v \<noteq> v'" shows "var g v' \<noteq> var g v"
le_trans: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j"   and "j \<le> k" shows "i \<le> k"
emeasure_point_measure_finite2: fixes X :: "'c set"   and A :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "X \<subseteq> A"   and "finite X" shows "emeasure (point_measure A f) X = sum f X"
intro_nofail_2: fixes S :: "'b nres" shows "(FAIL \<noteq> S) = nofail S"
TraceMsgEq: shows "\<forall>i<length traceMsg - 1. traceMsg ! i = trace' ! i"
assms_3: shows "lose t = [False] \<longrightarrow> ack t = [y] \<and> i1 t = [] \<and> vc t = [] \<and> st_out t = y"
map_sel: fixes f1 :: "'f \<Rightarrow> 'e"   and f2 :: "'g \<Rightarrow> 'd"   and a :: "('f, 'g) nondetT" shows "run_nondet (map_nondetT' f1 f2 a) = f2 (run_nondet a)"
assms_2: shows "Y \<in>\<^sub>\<circ> Vset \<alpha>"
union_single_eq_member: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "add_mset x M = N" shows "x \<in># N"
assms_2: shows "domA \<Delta> \<inter> domA \<Gamma> = {}"
intros_2: 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 u :: "nat" assumes "Network c s t"   and "Height_Bounded_Labeling c s t f l"   and "Network.relabel_precond c t f l u" shows "((f, Network.relabel_effect c f l u), f, l) \<in> Network.pr_algo_rel c s t"
bisimCoinductAux: fixes P :: "pi"   and Q :: "pi"   and X :: "(pi \<times> pi) set" assumes "(P, Q) \<in> X"   and "\<And>P Q. (P, Q) \<in> X \<Longrightarrow> P \<leadsto>[(X \<union> bisim)] Q \<and> (Q, P) \<in> bisim \<union> X" shows "P \<sim> Q"
countableE: fixes S :: "'a set" assumes "countable S" obtains f :: "'a \<Rightarrow> nat" where "inj_on f S"
left_v2: shows "left (len v2 ts c) = right (ext v1)"
partitions_decrease1: fixes p :: "nat \<Rightarrow> nat"   and m :: "nat"   and k :: "nat" assumes "p partitions m"   and "sum p {..m} = k"   and "p 1 = 0" shows "(\<lambda>i. p (i + 1)) partitions m - k"
lborel_distr_uminus: shows "distr lborel borel uminus = lborel"
bisim1BinOpThrow: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and bop :: "bop"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v1 :: "'addr val"   and v2 :: "'addr val" shows "P,e1 \<guillemotleft>bop\<guillemotright> e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v1, v2], xs, length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
bisim1NewThrow: fixes P :: "'a prog"   and h :: "'heap"   and C :: "String.literal"   and xs :: "'addr val list" shows "P,new C,h \<turnstile> (Throw (addr_of_sys_xcpt OutOfMemory), xs) \<leftrightarrow> ([], xs, 0, \<lfloor>addr_of_sys_xcpt OutOfMemory\<rfloor>)"
eq_fold: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and x :: "'b" assumes "semilattice_set f"   and "finite A" shows "semilattice_set.F f (insert x A) = Finite_Set.fold f x A"
chop_empty_left: shows "\<forall>i. N_Chop(i,bot,i)"
cases: fixes a1 :: "'b measure"   and a2 :: "'b measure" assumes "a1 \<le> a2" obtains M :: "'b measure" and N :: "'b measure" where "a1 = Ma__" and "a2 = N__" and "space Ma__ \<subset> space N__"   | M :: "'b measure" and N :: "'b measure" where "a1 = Mb__" and "a2 = Na__" and "space Mb__ = space Na__" and "sets Mb__ \<subset> sets Na__"   | M :: "'b measure" and N :: "'b measure" where "a1 = M" and "a2 = N" and "space M = space N" and "sets M = sets N" and "emeasure M \<le> emeasure N"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>\<^sub>A) (\<cdot>\<^sub>A) A.map A.map A.map"
order_trans: fixes x :: "('d, 'e) Term.term"   and y :: "('d, 'e) Term.term"   and z :: "('d, 'e) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
disc_filtr_prob_space_axioms_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure" shows "disc_filtr_prob_space_axioms M F \<equiv> disc_filtr M F"
cond_spmf_fst_map_inj: fixes f :: "'a \<Rightarrow> 'b"   and p :: "('a \<times> 'c) spmf"   and x :: "'a" assumes "inj f" shows "cond_spmf_fst (map_spmf (apfst f) p) (f x) = cond_spmf_fst p x"
k_def: shows "x (Suc k) \<noteq> 0"
x: shows "range (\<lambda>k. integral s (f k)) = range x"
exhaust: fixes y :: "dir" obtains "y = left"   | "y = right"
openin_inject: fixes x :: "'d topology"   and y :: "'d topology" shows "(openin x = openin y) = (x = y)"
continuous_within_open: fixes a :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "a \<in> A"   and "open A" shows "continuous (at a within A) f = isCont f a"