hcju/isabelleps huggingface dataset:
valid_selection_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and G :: "nat \<times> ('d \<times> nat \<times> nat) set"   and R :: "('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph"   and f :: "('e \<times> nat) set" shows "valid_selection Rs G R f \<equiv> let wl = worklist G Rs in (nextMax (Range f), R, f) \<in> wl \<and> (\<forall>(N, uu_)\<in>wl. nextMax (Range f) \<le> N) \<and> graph_rule R"
value_ge_add: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and z :: "ant" assumes "valuation K v"   and "\<forall>j\<le>n. f j \<in> carrier K"   and "\<forall>j\<le>n. z \<le> (v \<circ> f) j" shows "z \<le> v (\<Sigma>\<^sub>e K f n)"
gen_inv: fixes x :: "'c"   and G :: "('c, 'd) monoid_scheme"   and gens :: "'c set" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
abs_eq: fixes xa :: "'a \<Rightarrow>\<^sub>0 'b"   and x :: "'a \<Rightarrow>\<^sub>0 'b" shows "PP xa + PP x = PP (xa + x)"
fps_of_poly_quot_normalize_quot': fixes x :: "'b poly \<times> 'b poly" assumes "constant_term (snd x) \<noteq> (0::'b)" shows "fps_of_poly (fst (normalize_quot x)) / fps_of_poly (snd (normalize_quot x)) = fps_of_poly (fst x) / fps_of_poly (snd x)"
max: shows "x * y \<le> max (X1 * Y1) (max (X1 * Y2) (max (X2 * Y1) (X2 * Y2)))"
rel_interior_subset: fixes S :: "'c set" shows "rel_interior S \<subseteq> S"
word_gt_a_gt_0: fixes a :: "'b word"   and n :: "'b word" assumes "a < n" shows "0 < n"
ord_FO_resolution_inferences_between_empty_empty: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset" shows "inference_system.inferences_between (ord_FO_\<Gamma> S) {} {#} = {}"
midpoint_cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B A C" shows "Cong A B B C"
lim: shows "LIM x F. f x / g x :> at_infinity"
that: fixes x :: "'b"   and heap' :: "heap" assumes "execute b heap = Some (x, heap')"   and "P heap'"   and "Q heap'"   and "state_dp_consistency.cmem heap'"   and "R a x" shows "thesis"
nominal_ts_axioms: shows "nominal_ts (\<turnstile>) (\<rightarrow>)"
while_simps_2: fixes s :: "'a" assumes "\<not> guard s" shows "local.while s = return_spmf s"
assms_1: shows "m \<bind> f' \<le> SPEC \<Psi>"
assms_1: shows "\<lbrace>R\<rbrace> f \<lbrace>Q\<rbrace>"
ord_equals: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp"   and "x (Suc n) \<noteq> 0"   and "x n = 0" shows "ord_Zp x = int n"
simps_2: fixes p :: "real"   and M :: "bool stream measure"   and u :: "real"   and d :: "real"   and v :: "real"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "geom_rand_walk u d v (Suc n) = (\<lambda>w. (case w !! n of True \<Rightarrow> u | False \<Rightarrow> d) * geom_rand_walk u d v n w)"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "remove_nth 0 (x # xs) = xs"
setFinal_eq_iff: fixes f :: "face" shows "(setFinal f = f) = final f"
locally_path_connected_space_euclideanreal: shows "locally_path_connected_space euclideanreal"
lookup_eq_zero_in_keys_contradict: fixes f :: "'f \<Rightarrow>\<^sub>0 'e"   and k :: "'f" assumes "lookup f k = (0::'e)" shows "k \<notin> keys f"
image_def: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "\<P> f A = {y. \<exists>x\<in>A. y = f x}"
not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "R j x y" shows "x \<in> carrier"
sel_5: fixes x21 :: "'a"   and x22 :: "('a, 'b) tllist" shows "ttl (TCons x21 x22) = x22"
contour_integral_nearby_loops: fixes S :: "complex set"   and p :: "real \<Rightarrow> complex" assumes "open S"   and "path p"   and "path_image p \<subseteq> S" shows "\<exists>d>0. \<forall>g h. valid_path g \<and> valid_path h \<and> (\<forall>t\<in>{0..1}. cmod (g t - p t) < d \<and> cmod (h t - p t) < d) \<and> pathfinish g = pathstart g \<and> pathfinish h = pathstart h \<longrightarrow> path_image g \<subseteq> S \<and> path_image h \<subseteq> S \<and> (\<forall>f. f holomorphic_on S \<longrightarrow> contour_integral h f = contour_integral g f)"
simps_1: fixes uu :: "'c set" shows "witness_list [] uu = []"
\<phi>_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "'s" assumes "C.ide b"   and "C.ide a"   and "x \<in> Hom.set (b, a)" shows "\<phi> (b, a) (\<psi> (b, a) x) = x"
prems_2: shows "n -a' # as'\<rightarrow>* n''"
abs_summable_on_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(f abs_summable_on A) = integrable (count_space A) f"
\<a>'_def: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" shows "\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] \<equiv> \<alpha>' (\<mu>, \<nu>, \<tau>)"
wf_mult: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "wf (mult r)"
atLeastLessThan_def: fixes l :: "'c"   and u :: "'c" shows "{l..<u} \<equiv> {l..} \<inter> {..<u}"
conversionI': fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>*" shows "(a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*"
wf_mstate_msteps_stateless_UNIV: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and st :: "('a, 'b) mstate"   and \<pi>' :: "(String.literal \<times> event_data list) prefix" assumes "wf_mstate \<phi> \<pi> UNIV st"   and "\<pi> \<le> \<pi>'" shows "flatten_verdicts (msteps_stateless (pdrop (plen \<pi>) \<pi>') st) = M \<pi>' - M \<pi>"
the_paramR: fixes r :: "'a option"   and l :: "'b option"   and R :: "('b \<times> 'a) set" assumes "r \<noteq> None"   and "(l, r) \<in> \<langle>R\<rangle>option_rel" shows "(the l, the r) \<in> R"
CESem_simps_no_tick_4: fixes scrut :: "exp"   and e\<^sub>1 :: "exp"   and e\<^sub>2 :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> C \<rightarrow> CValue"   and r :: "C" shows "(\<N>\<lbrakk> (scrut ? e\<^sub>1 : e\<^sub>2) \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r \<sqsubseteq> CB_project\<cdot>((\<N>\<lbrakk> scrut \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r)\<cdot>((\<N>\<lbrakk> e\<^sub>1 \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r)\<cdot>((\<N>\<lbrakk> e\<^sub>2 \<rbrakk>\<^bsub>\<rho>\<^esub>)\<cdot>r)"
quasi_isometry_onD_2: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "lambda C -quasi_isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "1 / lambda * dist x y - C \<le> dist (f x) (f y)"
inj_on_add': fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and A :: "'b set" assumes "class.cancel_semigroup_add plus" shows "inj_on (\<lambda>b. plus b a) A"
Done_inject: fixes x :: "'d"   and y :: "'d" shows "(Generative_Probabilistic_Value.Done x = Generative_Probabilistic_Value.Done y) = (x = y)"
jnf: shows "\<exists>n_as. jordan_nf A n_as"
retract_of_imp_subset: fixes S :: "'c set"   and T :: "'c set" assumes "S retract_of T" shows "S \<subseteq> T"
xi_1: shows "(\<lambda>n. to_Gromov_completion ((f ^^ r n) basepoint)) \<longlonglongrightarrow> xi"
enumchain_mono: fixes i :: "nat" shows "N \<le> enumchain i \<and> enumchain i < enumchain (Suc i)"
derives_implies_Derivation: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.derives \<NN> \<TT> \<RR> a b" shows "\<exists>D. CFG.Derivation \<NN> \<TT> \<RR> a D b"
skip: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[], t\<rangle> \<Rightarrow> t"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.cod (B.\<alpha>' f) = B.HoHV (B.VVV.cod f)"
rep_eq: shows "coeff 0 = (\<lambda>_. 0::'c)"
cases: fixes a1 :: "'action stream"   and a2 :: "'state" assumes "run a1 a2" obtains a :: "'action" and p :: "'state" and r :: "'action stream" where "a1 = a ## r" and "a2 = p" and "en a p" and "run r (ex a p)"
x: fixes s :: "'a set" assumes "s \<in> S" shows "x \<in> s"
lagrange: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" assumes "Group.group G"   and "subgroup H G" shows "card (rcosets\<^bsub>G\<^esub> H) * card H = order G"
eigen_value_sr_c: shows "eigen_value cA (complex_of_real sr)"
sorted_wrt_iff_nth_less: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" shows "sorted_wrt P xs = (\<forall>i j. i < j \<longrightarrow> j < length xs \<longrightarrow> P (xs ! i) (xs ! j))"
simps_1: fixes ks :: "bool list" shows "isin Lf ks = False"
boundedE: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b + c \<le> a" obtains "b \<le> a" and "c \<le> a"
Lemma_3: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "associative f" shows "map (foldl1 f \<circ> map h) (candidate (@) (map wrap [0..<n + 1])) = candidate f (map h [0..<n + 1])"
assms_3: shows "\<forall>a\<in>A. \<forall>b\<in>A. h a = h b"
H: fixes r :: "'a llist" assumes "r \<in> pfinpref A t"   and "suff A r \<subseteq> u" shows "R"
lift_RealIntVal_def: fixes f :: "int \<Rightarrow> int"   and g :: "real \<Rightarrow> real" shows "lift_RealIntVal f g \<equiv> \<lambda>x. case x of IntVal v \<Rightarrow> IntVal (f v) | RealVal v \<Rightarrow> RealVal (g v)"
list_all2_cons1_def: fixes x :: "'a"   and sg' :: "'s1"   and sh :: "'s2" shows "local.list_all2_cons1 x sg' sh = list_all2 P (x # unstream g sg') (unstream h sh)"
element_ptr_kinds_eq_h: shows "element_ptr_kinds h2 = element_ptr_kinds h |\<union>| {|new_element_ptr|}"
run_bind_writer: fixes m :: "('w, 'a, 'm) writerT"   and f :: "'a \<Rightarrow> ('w, 'a, 'm) writerT" shows "run_writer (local.bind_writer m f) = bind (run_writer m) (\<lambda>(a, ws). bind (run_writer (f a)) (\<lambda>(b, ws'). return (b, ws @ ws')))"
Inf_Sup_aform_err: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "e \<in> UNIV \<rightarrow> {- 1..1}" shows "aform_err e X \<subseteq> {real_of_float (Inf_aform_err p X)..real_of_float (Sup_aform_err p X)}"
add_less_cancel_right: fixes plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and c :: "'g"   and b :: "'g" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
that: fixes b :: "'a set" assumes "b \<in># B"   and "x \<in> b" shows "thesis"
has_contour_integral_negatepath: fixes \<gamma> :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex"   and i :: "complex" assumes "valid_path \<gamma>"   and "((\<lambda>z. f (- z)) has_contour_integral - i) \<gamma>" shows "(f has_contour_integral i) (uminus \<circ> \<gamma>)"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "C.arr f"
Greater_closed: fixes u :: "'a \<Rightarrow> real"   and X :: "'a set"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and k :: "'a \<Rightarrow> nat"   and c :: "'a"   and t :: "real" assumes "u \<in> Regions.region X I r"   and "Regions.valid_region X k I r"   and "c \<in> X"   and "I c = Regions.intv.Greater (k c)"   and "real (k c) < t" shows "u(c := t) \<in> Regions.region X I r"
presheaf_of_rings_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" shows "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> topological_space S is_open \<and> presheaf_of_rings_axioms is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
unit_char': fixes a :: "'b" assumes "B.obj a" shows "GoH.unit a = G (H.unit a) \<cdot>\<^sub>D \<tau>.G.unit (H.map\<^sub>0 a)"
UINF_as_Sup_collect: fixes A :: "'d set"   and f :: "'d \<Rightarrow> ('b, 'c) uexpr" shows "(\<Sqinter> P \<in> A \<bullet> f P) = \<Sqinter>(f ` A)"
is_partial_heap_except_down_def: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and m :: "int"   and r :: "int"   and i :: "int" shows "is_partial_heap_except_down cmp heap l m r i = (\<forall>x. bounded m r x \<longrightarrow> (parent l x \<noteq> i \<longrightarrow> bounded m r (parent l x) \<longrightarrow> cmp (heap (parent l x)) (heap x)) \<and> (parent l x = i \<longrightarrow> bounded m r (parent l (parent l x)) \<longrightarrow> cmp (heap (parent l (parent l x))) (heap x)))"
rbt_inter_with_key_def: fixes f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" shows "rbt_inter_with_key f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k v w)) (rbt_lookup t2 k)) (entries t1)) | compare.GT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k w v)) (rbt_lookup t1 k)) (entries t2)) | compare.EQ \<Rightarrow> rbtreeify (sinter_with f (entries t1) (entries t2)))"
cla_mono: fixes Z :: "('c \<Rightarrow> real) set"   and Z' :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> Z'" shows "Closure\<^sub>\<alpha> Z \<subseteq> Closure\<^sub>\<alpha> Z'"
embranch_exec_0: fixes aa :: "recf"   and xs :: "nat list"   and rgs :: "recf list"   and list :: "recf list"   and a :: "recf" assumes "rec_exec aa xs = 0"   and "zip rgs list \<noteq> []"   and "list_all (\<lambda>rf. primerec rf (length xs)) ([a, aa] @ rgs @ list)" shows "rec_exec (rec_embranch ((a, aa) # zip rgs list)) xs = rec_exec (rec_embranch (zip rgs list)) xs"
kinds_def: fixes xs :: "'edge list" shows "kinds xs \<equiv> map kind xs"
sl_nil: fixes R :: "('a \<times> 'b) set" shows "([], []) \<in> seq_lift R"
ad_not_ae: shows "a - complex_of_real d \<noteq> a + complex_of_real e"
ctr: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" shows "sfilter P s = shd (sdrop_while (Not \<circ> P) s) ## sfilter P (stl (sdrop_while (Not \<circ> P) s))"
wqo_af_conv: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "qo_on P A" shows "wqo_on P A = almost_full_on P A"
reachable_reachable1_trans: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c"   and w :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v"   and "v \<rightarrow>\<^sup>+\<^bsub>G\<^esub> w" shows "u \<rightarrow>\<^sup>+\<^bsub>G\<^esub> w"
qsend_eq_0_iff: fixes cfT :: "(('test, 'atom, 'choice) cmd \<times> 'state) stream" shows "(qsend cfT = 0) = discrCf (shd cfT)"
Minimal_Solutions_length: fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b" shows "length x = length a \<and> length y = length b"
b: shows "Esigma (a * m) = Esigma (a * n)"
liberal_witness_two: fixes x :: "'c"   and y :: "'c"   and A :: "'c set"   and i :: "'d"   and j :: "'d"   and Is :: "'d set" assumes "hasw [x, y] A"   and "hasw [i, j] Is" obtains P :: "'d \<Rightarrow> ('c \<times> 'c) set" where "profile A Is P" and "x \<^bsub>(P i)\<^esub>\<prec> y" and "y \<^bsub>(P j)\<^esub>\<prec> x"
da_weaken: fixes Env :: "env"   and B :: "lname set"   and t :: "Term.term"   and A :: "assigned"   and B' :: "lname set" assumes "Env\<turnstile> B \<guillemotright>t\<guillemotright> A"   and "B \<subseteq> B'" shows "\<exists>A'. Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'"
distinct_1: fixes x1 :: "nonce_t"   and x2 :: "nonce_t" shows "epublK x1 \<noteq> eprivK x2"
measure_space_inter: fixes s :: "'a set"   and t :: "'a set" assumes "s \<in> events"   and "t \<in> events"   and "prob t = prob (space M)" shows "prob (s \<inter> t) = prob s"
Step2: fixes g1 :: "'a"   and g2 :: "'a" assumes "g1 \<in> carrier G"   and "g2 \<in> carrier G" shows "(\<lambda>H\<in>{H. subgroup H G}. g1 \<otimes> g2 <# H #> inv (g1 \<otimes> g2)) = (\<lambda>H\<in>{H. subgroup H G}. g1 <# H #> inv g1) \<otimes>\<^bsub>BijGroup {H. subgroup H G}\<^esub> (\<lambda>H\<in>{H. subgroup H G}. g2 <# H #> inv g2)"
no_outstanding_vol_write_dropWhile_append: fixes sb :: "'b memref list"   and xs :: "'b memref list" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ xs) = dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) xs"
assms: fixes x :: "nat" shows "s (f \<triangleright> Suc x) \<noteq> s (f \<triangleright> x)"
oalist_inv_rawD2: fixes ko :: "'a key_order"   and xs :: "('a \<times> 'b) list" assumes "ko.oalist_inv_raw ko xs" shows "sorted_wrt (ko.lt ko) (map fst xs)"
TokensAt_items_le: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat"   and I :: "item set" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.TokensAt \<TT> Lex Doc k I = LocalLexing.TokensAt \<TT> Lex Doc k (items_le k I)"
b_least2_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat"   and y :: "nat" shows "b_least2 f x y \<equiv> LEAST z. z = y \<or> z < y \<and> f x z \<noteq> 0"
rep_eq: fixes x :: "('a, 'c, 'b) Tree\<^sub>\<alpha>" shows "S_transform_Tree\<^sub>\<alpha> x = S_transform_Tree (rep_Tree\<^sub>\<alpha> x)"
parallel_converter2_comp1_out: fixes conv :: "('y, 'aa, 'ag, 'ah) converter"   and conv' :: "('ag, 'ah, 'ac, 'ae) converter"   and conv'' :: "('z, 'ab, 'ad, 'af) converter" shows "conv \<odot> conv' |\<^sub>= conv'' = (conv |\<^sub>= 1\<^sub>C) \<odot> (conv' |\<^sub>= conv'')"
ide_determines_diagram_2: fixes d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide d" shows "J_C.MkIde (J_C.Map d) = d"
1: fixes i :: "nat" assumes "i < length ws" shows "head (ws ! i) = head w \<and> num_args (ws ! i) = num_args w \<and> (\<forall>k<num_args w. args (subst \<rho> w) ! k \<unrhd>\<^sub>e\<^sub>m\<^sub>b args (ws ! i) ! k) \<and> (Suc i < length ws \<longrightarrow> prio (prio_emb_pos prio (ws ! i) u) = 1)"
measurable_return1: fixes N :: "'c measure"   and x :: "'c"   and L :: "'d measure" shows "return N x \<rightarrow>\<^sub>M L = N \<rightarrow>\<^sub>M L"
seqE: fixes g :: "'a2"   and f :: "'a2" assumes "C2.seq g f" obtains "C2.arr f" and "C2.arr g" and "C2.dom g = C2.cod f"
inputDistinct: fixes \<Psi> :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual" assumes "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<longmapsto> Rs" shows "distinct xvec"
kingman_theorem_AE_nonergodic_ereal: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u" shows "AE x in M. (\<lambda>xa. ereal (u xa x / real xa)) \<longlonglongrightarrow> subcocycle_lim_ereal u x"
ofilter_underS_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wo_rel r" shows "ofilter r A = ((\<exists>a\<in>Field r. A = underS r a) \<or> A = Field r)"
amult_distrib1: fixes z :: "int"   and a :: "ant"   and b :: "ant" assumes "ant z \<noteq> 0" shows "(a + b) * ant z = a * ant z + b * ant z"
raw_match_term_tvs_subset_dom_res: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match_term t u subs = Some subs'" shows "tvs t \<subseteq> dom subs'"
is_rewritableI_is_canon_rewriter: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and b :: "'t \<Rightarrow>\<^sub>0 'b"   and u :: "'t" assumes "rb_aux_inv1 bs"   and "b \<in> set bs"   and "b \<noteq> 0"   and "lt b adds\<^sub>t u"   and "\<not> is_canon_rewriter rword (set bs) u b" shows "is_rewritable bs b u"
case_lstream_conv_case_llist: fixes "done" :: "'c"   and skip :: "'d llist \<Rightarrow> 'c"   and yield :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and xs :: "'d llist" shows "(case lstream xs of Done \<Rightarrow> done | Skip xs' \<Rightarrow> skip xs' | Yield x xs' \<Rightarrow> yield x xs') = (case xs of LNil \<Rightarrow> done | LCons x xs' \<Rightarrow> yield x xs')"
integrable_componentwise_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "(f integrable_on A) = (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
lta_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C D E F" shows "A \<noteq> B \<and> C \<noteq> B \<and> D \<noteq> E \<and> F \<noteq> E \<and> D \<noteq> F"
arr_obj: fixes G :: "prog"   and T :: "ty" shows "G\<turnstile>T.[]\<preceq>Class Object"
ordinal_times_monoL: fixes x :: "ordinal"   and x' :: "ordinal"   and y :: "ordinal" assumes "x \<le> x'" shows "x * y \<le> x' * y"
\<Phi>a_mapsto: fixes f :: "'s" assumes "S.arr f" shows "\<Phi>a f \<in> S'.set (\<Phi>o (S.dom f)) \<rightarrow> S'.set (\<Phi>o (S.cod f))"
i_f_join_iIN: fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and d :: "nat" shows "f \<Join>\<^sub> [n\<dots>,d] = f \<Up> n \<Down> Suc d"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [$Get_global j] : ts _> ts'"
l11_62_existence: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p" shows "\<exists>D. Coplanar A B C D \<and> (\<forall>E. Coplanar A B C E \<longrightarrow> Per E D P)"
T_Tn_T_compose_2: fixes n :: "nat"   and x :: "'a" shows "(T ^^ n) (T x) = (T ^^ Suc n) x"
push_bit_minus_one_eq_not_mask: 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 uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "push_bit n (uminus one) = not (mask n)"
obj_simps_4: fixes a :: "'a" assumes "obj a" shows "local.dom a = a"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat \<Rightarrow>\<^sub>0 'b" shows "lookup (swap0 x xa xb) = (\<lambda>xc. lookup xb (swap x xa xc))"
is_RB_upt_is_sig_GB_upt: fixes d :: "'a \<Rightarrow> nat"   and rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t" assumes "dickson_grading d"   and "is_RB_upt d rword G u" shows "is_sig_GB_upt d G u"