hcju/isabelleps huggingface dataset:
IH_2: fixes w :: "nat \<Rightarrow> 'a set" assumes "\<mu>_stable \<phi>2 w"   and "w \<Turnstile>\<^sub>n \<phi>2" shows "\<exists>i. suffix i w \<Turnstile>\<^sub>n af \<phi>2 (prefix i w)[{\<psi>. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)} \<inter> restricted_subformulas \<phi>2]\<^sub>\<nu>"
param_case_list': fixes l :: "'a list"   and l' :: "'b list"   and Rb :: "('a \<times> 'b) set"   and n :: "'c"   and n' :: "'d"   and Ra :: "('c \<times> 'd) set"   and c :: "'a \<Rightarrow> 'a list \<Rightarrow> 'c"   and c' :: "'b \<Rightarrow> 'b list \<Rightarrow> 'd" assumes "(l, l') \<in> \<langle>Rb\<rangle>list_rel"   and "\<lbrakk>l = []; l' = []\<rbrakk> \<Longrightarrow> (n, n') \<in> Ra"   and "\<And>x xs x' xs'. \<lbrakk>l = x # xs; l' = x' # xs'; (x, x') \<in> Rb; (xs, xs') \<in> \<langle>Rb\<rangle>list_rel\<rbrakk> \<Longrightarrow> (c x xs, c' x' xs') \<in> Ra" shows "(case l of [] \<Rightarrow> n | x # xa \<Rightarrow> c x xa, case l' of [] \<Rightarrow> n' | x # xa \<Rightarrow> c' x xa) \<in> Ra"
F3: shows "\<exists>x. (b, x b c R) \<in> R \<or> c = b"
boundedE: fixes b :: "'b"   and c :: "'b"   and a :: "'b" assumes "sup_class.sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
homE_1: fixes A :: "'o"   and B :: "'o"   and f :: "'a" assumes "A \<in> Ob"   and "B \<in> Ob"   and "f \<in> Hom A B" shows "Dom f = A"
fmeasurable_measure_inner_outer: fixes S :: "'a set"   and m :: "real" shows "(S \<in> fmeasurable M \<and> m = Sigma_Algebra.measure M S) = ((\<forall>e>0. \<exists>T\<in>fmeasurable M. T \<subseteq> S \<and> m - e < Sigma_Algebra.measure M T) \<and> (\<forall>e>0. \<exists>U\<in>fmeasurable M. S \<subseteq> U \<and> Sigma_Algebra.measure M U < m + e))"
cSup_singleton: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Sup {x} = x"
pickT: fixes \<sigma> :: "'tp" shows "intT \<sigma> (pickT \<sigma>)"
atLeastLessThan_shift_bounds: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and k :: "nat"   and n :: "nat" shows "sum g {m + k..<n + k} = sum (g \<circ> (+) k) {m..<n}"
fps_mult_equiv_coeff: fixes f :: "'a fps"   and g :: "'a fps"   and p :: "'a"   and h :: "'a fps"   and n :: "nat" assumes "(f, g) \<in> fpsmodrel p" shows "(f * h) $ n mod p = (g * h) $ n mod p"
SUP_least: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and u :: "'c" assumes "\<And>i. i \<in> A \<Longrightarrow> f i \<le> u" shows "\<Squnion>(f ` A) \<le> u"
heun_remainder1_def: fixes x :: "real \<Rightarrow> 'f"   and f :: "real \<times> 'f \<Rightarrow> 'e"   and f' :: "real \<times> 'f \<Rightarrow> (real \<times> 'e) \<Rightarrow>\<^sub>L 'e"   and f'' :: "real \<times> 'f \<Rightarrow> (real \<times> 'e) \<Rightarrow>\<^sub>L (real \<times> 'e) \<Rightarrow>\<^sub>L 'e"   and t :: "real"   and h :: "real"   and s :: "real" shows "heun_remainder1 x f f' f'' t h s = (h ^ 3 / 6) *\<^sub>R (blinfun_apply (blinfun_apply (f'' (h * s + t, x (h * s + t))) (1, f (h * s + t, x (h * s + t)))) (1, f (h * s + t, x (h * s + t))) + blinfun_apply (f' (h * s + t, x (h * s + t))) (0, blinfun_apply (f' (h * s + t, x (h * s + t))) (1, f (h * s + t, x (h * s + t)))))"
set_borel_measurable_subset: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "set_borel_measurable M A f"   and "B \<in> sets M"   and "B \<subseteq> A" shows "set_borel_measurable M B f"
pset_not_is_Zero_is_Plus: fixes r :: "'a rexp" shows "\<forall>x\<in>pset r. \<not> is_Zero x \<and> \<not> is_Plus x"
exhaust: fixes y :: "'b inference" obtains x1 :: "'b list" and x2 :: "'b" where "y = Infer x1 x2"
SORTED_R: shows "sorted_wrt_dist p (nearest_nbors n ps p r)"
cases: fixes n :: "'node"   and S :: "'node \<Rightarrow> bool"   and a :: "'node" assumes "obsp n S a" obtains as :: "'edge list" and n' :: "'node" where "a = n'" and "n -as\<rightarrow>* n'" and "\<forall>nx\<in>set (sourcenodes as). \<not> S nx" and "S n'"
fpairsD_1: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> fpairs r" shows "x \<in>\<^sub>\<circ> r"
simplex_like_below_in_above_pseudominimal_is_top: fixes x :: "'a"   and P :: "'a set"   and t :: "'a" assumes "x \<in> P"   and "simplex_like (P .\<^bold>\<le> x)"   and "t \<in> P .\<^bold>\<le> x"   and "\<And>y. pseudominimal_in (P .\<^bold>\<le> x) y \<Longrightarrow> t \<succeq> y" shows "t = x"
poly_minus_def: fixes p :: "'a list" shows "-- p = - (1::'a) %* p"
get_tag_name_new_character_data: fixes h :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
extended_distance_triang_ineq: fixes x :: "'c Bonk_Schramm_extension_unfolded"   and y :: "'c Bonk_Schramm_extension_unfolded"   and z :: "'c Bonk_Schramm_extension_unfolded" assumes "x \<in> extended_distance_set"   and "y \<in> extended_distance_set"   and "z \<in> extended_distance_set" shows "extended_distance x z \<le> extended_distance x y + extended_distance y z"
extremum: fixes a :: "'b list" shows "\<epsilon> \<le>s a"
Cons_5: assumes "snd b \<noteq> 0"   and "\<not> punit.lt (snd b) adds t" shows "thesis"
expr_sem_rf_eq_on_vars: fixes e :: "expr"   and \<sigma>1 :: "nat \<Rightarrow> val"   and \<sigma>2 :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<sigma>1 x = \<sigma>2 x" shows "expr_sem_rf \<sigma>1 e = expr_sem_rf \<sigma>2 e"
sup_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<squnion> z \<le> y \<squnion> z"
rel_induct: fixes R2 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R3 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('e, 'a, 'c) sp\<^sub>\<mu>"   and y :: "('e, 'b, 'd) sp\<^sub>\<mu>"   and Q :: "('e, 'a, 'c) sp\<^sub>\<mu> \<Rightarrow> ('e, 'b, 'd) sp\<^sub>\<mu> \<Rightarrow> bool" assumes "rel_sp\<^sub>\<mu> R2 R3 x y"   and "\<And>a1 b1. rel_fun (=) Q a1 b1 \<Longrightarrow> Q (Get a1) (Get b1)"   and "\<And>a21 a22 b21 b22. \<lbrakk>R2 a21 b21; R3 a22 b22\<rbrakk> \<Longrightarrow> Q (Put a21 a22) (Put b21 b22)" shows "Q x y"
node_ptr_casts_commute: fixes ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr" shows "(cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = Some node_ptr) = (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_ptr = ptr)"
entropy_distr: fixes MX :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (f x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows "entropy b MX X = - (LINT x|MX. f x * log b (f x))"
b: shows "a * b = pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
uint_of_bl_is_bl_to_bin_drop: fixes l :: "bool list" assumes "length (dropWhile Not l) \<le> LENGTH('a)" shows "uint (of_bl l) = bl_to_bin l"
rv_square: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)" assumes "f \<in> rv M" shows "(\<lambda>w. (f w)\<^sup>2) \<in> rv M"
sup_assn_def: fixes P :: "assn"   and Q :: "assn" shows "sup P Q \<equiv> Abs_assn (\<lambda>h. h \<Turnstile> P \<or> h \<Turnstile> Q)"
runit'_simps_5: fixes f :: "'a" assumes "C.ide f" shows "C.cod \<r>\<^sub>C\<^sup>-\<^sup>1[f] = f \<star>\<^sub>C src\<^sub>C f"
msg_id_unique: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and m1 :: "'c"   and i :: "nat"   and m2 :: "'c"   and j :: "nat" assumes "network history msg_id"   and "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"
assms: shows "fps_to_fls f = fps_to_fls g"
utility_ge: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq> q" shows "measure_pmf.expectation q u \<le> measure_pmf.expectation p u"
map_sum_sel_2: fixes s :: "'h + 'i"   and f :: "'h \<Rightarrow> 'k"   and g :: "'i \<Rightarrow> 'j" assumes "\<not> isl s" shows "projr (map_sum f g s) = g (projr s)"
nat_plus_as_int: shows "(+) = (\<lambda>a b. nat (int a + int b))"
converse_reachable_cases: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v" obtains "u = v" and "u \<in> verts H"   | w :: "'a" where "u \<rightarrow>\<^bsub>H\<^esub> w" and "w \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"
divmod_algorithm_code_2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "divmod m num.One = (numeral.numeral one plus m, zero)"
unat_power_lower: fixes n :: "nat" assumes "n < LENGTH('b)" shows "unat (2 ^ n) = 2 ^ n"
abs_diff_less_iff: 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 x :: "'b"   and a :: "'b"   and r :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (abs (minus x a)) r = (less (minus a r) x \<and> less x (plus a r))"
strict_prefix_setD: fixes xs :: "'a list"   and ys :: "'a list" assumes "strict_prefix xs ys" shows "set xs \<subseteq> set ys"
is_processT6_S1: fixes X :: "'b event set"   and s :: "'b event list"   and P :: "'b process" assumes "tick \<notin> X"   and "(s @ [tick], {}) \<in> F P" shows "(s, X) \<in> F P"
init_fin_descend_state_def: fixes \<sigma> :: "('c, 'd) locks \<times> (('d \<Rightarrow> (('h \<times> 'e) \<times> 'c \<Rightarrow>f nat) option) \<times> 'f) \<times> ('d \<Rightarrow> 'g wait_set_status option) \<times> 'd set" shows "init_fin_descend_state \<sigma> = (locks \<sigma>, (init_fin_descend_thr (thr \<sigma>), shr \<sigma>), wset \<sigma>, interrupts \<sigma>)"
pow_ord_eq_1: fixes x :: "'a" assumes "x \<in> carrier G" shows "x [^] ord x = \<one>"
backward_terminating_iff3: fixes x :: "'a" assumes "path x" shows "backward_terminating x = (x\<^sup>T \<le> x\<^sup>T\<^sup>\<star> ; - (x\<^sup>T ; 1))"
1: shows "(?!) (llist_of_stream (smap int (q ## trace u q))) \<approx> (?!) (llist_of_stream (smap int (q ## trace v q)))"
u_2: shows "measure_pmf.expectation q u \<le> measure_pmf.expectation p u"
abs_eq: shows "Nil_prod = Generator (\<lambda>_. Done)"
isPath_append_edge: fixes v :: "nat"   and p :: "(nat \<times> nat) list"   and v' :: "nat"   and v'' :: "nat" assumes "cf.isPath v p v'"   and "(v', v'') \<in> cf.E" shows "cf.isPath v (p @ [(v', v'')]) v''"
eq_iff: fixes a :: "'g"   and b :: "'g" shows "(a = b) = (b \<le> a \<and> a \<le> b)"
ae: fixes i :: "'c"   and j :: "'c" assumes "i \<in> I"   and "j \<in> I" shows "AE x in M. x \<in> A i \<and> x \<in> A j \<longrightarrow> i = j"
f8: shows "- 1 * 4294967296 = - 4294967296"
fold_insert_idem2: fixes A :: "'a\<^sub>m set"   and z :: "'a\<^sub>m option"   and x :: "'a\<^sub>m" assumes "finite A" shows "Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) z (insert x A) = Finite_Set.fold (\<lambda>x yo. yo \<bind> m x) (z \<bind> m x) A"
Id_on_def: fixes A :: "'c set" shows "Id_on A = \<mu> (\<P> (\<lambda>x. \<eta> (x, x)) A)"
Inter_anti_mono: fixes B :: "'b set set"   and A :: "'b set set" assumes "B \<subseteq> A" shows "\<Inter> A \<subseteq> \<Inter> B"
complete_iff_min_height: fixes t :: "'c tree" shows "complete t = (height t = min_height t)"
wordinterval_union_set_eq: fixes r1 :: "'a wordinterval"   and r2 :: "'a wordinterval" shows "wordinterval_to_set (wordinterval_union r1 r2) = wordinterval_to_set r1 \<union> wordinterval_to_set r2"
smallest_prime_beyond_eq: fixes p :: "nat"   and n :: "nat" assumes "prime p"   and "n \<le> p"   and "\<And>q. \<lbrakk>prime q; n \<le> q\<rbrakk> \<Longrightarrow> p \<le> q" shows "smallest_prime_beyond n = p"
intro: fixes v0 :: "'a"   and G :: "('a, 'b) Graph_scheme"   and v1 :: "'a" assumes "v0 \<in> V\<^bsub>G\<^esub>"   and "v1 \<in> V\<^bsub>G\<^esub>"   and "(v0, v1) \<notin> E\<^bsub>G\<^esub>"   and "v0 \<noteq> v1" shows "v0_v1_Digraph_axioms G v0 v1"
P4: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>F \<longrightarrow> \<box>[F]_v"
assms: shows "\<forall>(uu_, p)\<in>set \<mu>. 0 \<le> p"
Sup_set_fold: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and xs :: "'g list" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (set xs) = fold sup xs bot"
subring_Ring: fixes S :: "('a, 'g) Ring_scheme" assumes "Subring R S" shows "Ring S"
no_grey_refs_no_pending_marks: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state" assumes "no_grey_refs s"   and "valid_W_inv s" shows "\<forall>x. filter is_mw_Mark (sys_mem_store_buffers x s) = []"
relcompp_distrib: fixes R :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and T :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "R OO sup S T = sup (R OO S) (R OO T)"
pmf_expectation_bind_pmf_of_set: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c pmf"   and h :: "'c \<Rightarrow> 'd" assumes "A \<noteq> {}"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> finite (set_pmf (f x))" shows "measure_pmf.expectation (pmf_of_set A \<bind> f) h = (\<Sum>a\<in>A. measure_pmf.expectation (f a) h /\<^sub>R real (card A))"
simps_1: fixes Obj :: "'a set"   and Hom :: "'a \<Rightarrow> 'a \<Rightarrow> 'b set"   and Id :: "'a \<Rightarrow> 'b"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uu :: "'c"   and B :: "'c"   and uv :: "'d" assumes "concrete_category Obj Hom Id Comp" shows "concrete_category.Cod (concrete_category.MkArr uu B uv) = B"
c: shows "\<forall>w. cmod (poly p c) \<le> cmod (poly p w)"
borel_Suc_expectation: fixes f :: "bool stream \<Rightarrow> real"   and n :: "nat"   and g :: "bool stream \<Rightarrow> 'b"   and N :: "'b measure"   and z :: "bool stream" assumes "f \<in> borel_measurable (F (Suc n))"   and "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N"   and "g -` {g z} \<in> sets (F n)"   and "\<forall>y z. g y = g z \<and> y !! n = z !! n \<longrightarrow> f y = f z" shows "expectation (\<lambda>x. f x * indicat_real (g -` {g z}) x) = prob (g -` {g z}) * (p * f (pseudo_proj_True n z) + (1 - p) * f (pseudo_proj_False n z))"
join_restrict: fixes X :: "'b option list set"   and n :: "nat"   and A :: "nat set"   and Y :: "'b option list set"   and B :: "nat set"   and b :: "bool"   and v :: "'b option list" assumes "\<And>v. v \<in> X \<Longrightarrow> wf_tuple n A v"   and "\<And>v. v \<in> Y \<Longrightarrow> wf_tuple n B v"   and "\<not> b \<Longrightarrow> B \<subseteq> A" shows "(v \<in> join X b Y) = (wf_tuple n (A \<union> B) v \<and> restrict A v \<in> X \<and> (if b then restrict B v \<in> Y else restrict B v \<notin> Y))"
restrict_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "restrict f A = (\<lambda>x. if x \<in> A then f x else undefined)"
too_long_lin_dependent: fixes us :: "'b list" assumes "set us \<subseteq> V"   and "dim_R V < length us" shows "\<not> R_lin_independent us"
mem_box_real_2: fixes x :: "real"   and a :: "real"   and b :: "real" shows "(x \<in> cbox a b) = (a \<le> x \<and> x \<le> b)"
inv_right: fixes f :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "bijection f" shows "f (inv f a) = a"
new_element_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter) h h'"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Col of Row \<Rightarrow> f1 | Col \<Rightarrow> f2 | Num \<Rightarrow> f3) = f2"
not_zless: fixes n :: "int"   and m :: "int" shows "(\<not> n < m) = (m \<le> n)"
abs_eq: fixes x :: "int" shows "Word.the_signed_int (Word.Word x) = signed_take_bit (LENGTH('b) - Suc 0) x"
ord_inj_less: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and a :: "'a"   and b :: "'a" assumes "Order E"   and "ord_inj D E f"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "a \<prec> b = f a \<prec>\<^bsub>E\<^esub> f b"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "Right_a.ide a"   and "Right_a.ide a'"   and "Right_a.in_hom g (H\<^sub>R a a') (H\<^sub>R a a)" shows "\<exists>f. Right_a.in_hom f a' a \<and> H\<^sub>R a f = g"
length_MGF2: fixes Z :: "bit list"   and m :: "nat" shows "length (MGF2 Z m) = Suc m * length (sha1 (Z @ nat_to_bv_length m 32))"
substEnv_vsubstEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "goodEnv rho"   and "good Y"   and "y \<noteq> y1" shows "rho &[y1 // y]_ys &[Y / y]_ys = rho &[y1 // y]_ys"
mute_fun_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a" shows "mute_fun f = (\<forall>x y x' y' g. f x y = (x', y', g) \<longrightarrow> y \<noteq> (0::'a) \<longrightarrow> x = x' * g \<and> y * x' = x * y')"
simps_3: fixes F :: "'a formula"   and G :: "'a formula" shows "is_mini_formula (F \<^bold>\<rightarrow> G) = (is_mini_formula F \<and> is_mini_formula G)"
log_le_cancel_iff: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "1 < a"   and "0 < x"   and "0 < y" shows "(log a x \<le> log a y) = (x \<le> y)"
elims: fixes x :: "('e, 'f, 'g) FSM"   and xa :: "('e, 'f, 'h) FSM"   and xb :: "('e, 'f) ATC set"   and xc :: "'e list set"   and xd :: "nat"   and xe :: "nat"   and y :: "'e list set" assumes "TS x xa xb xc xd xe = y" obtains M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" where "x = M2a__" and "xa = M1a__" and "xb = \<Omega>'__" and "xc = Va__" and "xd = ma__" and "xe = 0" and "y = {}"   | M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" where "x = M2b__" and "xa = M1b__" and "xb = \<Omega>''__" and "xc = Vb__" and "xd = mb__" and "xe = Suc 0" and "y = Vb__"   | M2 :: "('e, 'f, 'g) FSM" and M1 :: "('e, 'f, 'h) FSM" and \<Omega> :: "('e, 'f) ATC set" and V :: "'e list set" and m :: "nat" and v :: "nat" where "x = M2" and "xa = M1" and "xb = \<Omega>" and "xc = V" and "xd = m" and "xe = Suc (Suc v)" and "y = TS M2 M1 \<Omega> V m (Suc v) \<union> C M2 M1 \<Omega> V m (Suc (Suc v))"
s_is_partial_execution: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and i :: "nat" shows "s c t i = run_trace (c, take i t)"
sets_range: fixes A :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and M :: "'b measure"   and i :: "'c" assumes "A ` I \<subseteq> sets M"   and "i \<in> I" shows "A i \<in> sets M"
dim_image_eq: fixes f :: "'a \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> real"   and S :: "'a set" assumes "\<forall>x. x \<in> UNIV \<longrightarrow> f x \<in> manifold_eucl.tangent_space \<infinity> a"   and "S \<subseteq> UNIV"   and "linear_on UNIV (manifold_eucl.tangent_space \<infinity> a) (*\<^sub>R) (*\<^sub>R) f"   and "inj_on f (span S)" shows "manifold_eucl.tangent_space.dim \<infinity> a (f ` S) = vs1.dim S"
closed_ereal_atLeast: fixes a :: "ereal" shows "closed {a..}"
dlist_ex_conv_member: fixes P :: "'b \<Rightarrow> bool"   and dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "dlist_ex P dxs = (\<exists>x. DList_Set.member dxs x \<and> P x)"
beukers_integral3_conv_nn_integral: shows "local.beukers_integral3 = enn2real local.beukers_nn_integral3"
trancl_subset_Field2: fixes r :: "('b \<times> 'b) set" shows "r\<^sup>+ \<subseteq> Field r \<times> Field r"
emptyE: fixes a :: "'a"   and P :: "bool" assumes "a \<in> {}" shows "P"
sources_interference_eq: fixes D :: "'d \<Rightarrow> 'c"   and x :: "'d"   and u :: "'c"   and I :: "('c \<times> 'c) set"   and xs :: "'d list" shows "((D x, u) \<in> I \<or> (\<exists>v\<in>sources I D u xs. (D x, v) \<in> I)) = (D x \<in> sources I D u (x # xs))"
cmod_Lam: fixes A :: "real mat"   and n :: "nat"   and m :: "nat"   and lam :: "complex"   and n_as :: "(nat \<times> complex) list"   and i :: "nat" assumes "spectral_radius_1_jnf_max A n m lam n_as"   and "i \<in> spectral_radius_1_jnf_max.I n m n_as" shows "cmod (spectral_radius_1_jnf_max.Lam n_as i) = 1"
axioms_2: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "fully_faithful_functor A B F" shows "full_functor A B F"
kbo_std_axioms: shows "kbo_std ground_heads_var (>\<^sub>s) \<epsilon> \<delta> extf arity_sym arity_var wt_sym"
collapse: fixes option :: "'l option" assumes "option \<noteq> None" shows "Some (the option) = option"
intro_elim_2_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> \<^bold>& \<psi> in v]" shows "[\<psi> in v]"
frequently_at_top_at_topE: fixes P :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>n. frequently (P n) at_top"   and "filterlim g at_top sequentially" obtains s :: "nat \<Rightarrow> 'b" where "\<And>i. P i (s i)" and "filterlim s at_top sequentially"
mod_add_right_eq: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
ZFSucNeq0: fixes x :: "ZF" assumes "Elem x HOLZF.Nat" shows "Nat2nat (SucNat x) \<noteq> 0"
Suc_4: fixes k :: "nat"   and h :: "heap" assumes "I k h"   and "i \<le> k"   and "k < j" shows "success (f k) h"
Shleg_altdef: fixes x :: "real" shows "local.Shleg x = local.Gen_Shleg n x / fact n"