hcju/isabelleps huggingface dataset:
c_manifolds_def: fixes k :: "enat"   and charts1 :: "('c, 'd) chart set"   and charts2 :: "('f, 'g) chart set" shows "c_manifolds k charts1 charts2 \<equiv> c_manifold charts1 k \<and> c_manifold charts2 k"
cring_lcf_pow: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P"   and "p (deg R p) [^] n \<noteq> \<zero>" shows "(p [^]\<^bsub>P\<^esub> n) (deg R (p [^]\<^bsub>P\<^esub> n)) = p (deg R p) [^] n"
hyps: assumes "\<forall>a\<in>set pi. case a of (a, b) \<Rightarrow> a \<sharp> x \<and> b \<sharp> x" shows "pi \<bullet> x = x"
le_1: shows "A *\<^sub>v x \<le> b"
axis_def: fixes k :: "'b"   and x :: "'a" shows "axis k x = (\<chi>i. if i = k then x else (0::'a))"
that: fixes \<xi> :: "real" assumes "a < \<xi>"   and "\<xi> < b"   and "f b - f a = f' \<xi> (b - a)" shows "thesis"
\<rho>'_ide_simp: fixes a :: "'c local.term set" assumes "ide a" shows "\<rho>' a = runit' a"
newton_seq_fact5: fixes l :: "nat" shows "local.divide (f \<bullet> ns l) (f' \<bullet> ns l) \<in> carrier Zp"
uniform_discrete_def: fixes S :: "'b set" shows "uniform_discrete S = (\<exists>e>0. \<forall>x\<in>S. \<forall>y\<in>S. dist x y < e \<longrightarrow> x = y)"
uniform_limit_iff: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and S :: "'b set"   and l :: "'b \<Rightarrow> 'c"   and F :: "'a filter" shows "uniform_limit S f l F = (\<forall>e>0. \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e)"
assms_3: shows "lesubs \<subseteq>\<^sub>m subs"
d: shows "M.sumlist (map (\<lambda>j. - \<mu> i j \<cdot>\<^sub>v gso j) [0..<i]) \<in> carrier_vec n"
label_all: fixes t :: "'L tree" assumes "t \<in> ranked_trees A" shows "\<exists>q\<in>Q. accs \<delta> t q"
adjoint_pair_antipar_3: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g" shows "Span.src f = Span.trg g"
deterministic: fixes lc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and lc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and lc'' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "lc \<leadsto>\<^sub>A lc'"   and "lc \<leadsto>\<^sub>A lc''" shows "lc' = lc''"
has_field_derivative_stirling_sum_complex: fixes x :: "complex"   and j :: "nat"   and m :: "nat" assumes "0 < Re x" shows "(stirling_sum j m has_field_derivative stirling_sum (Suc j) m x) (at x)"
sparsegrid_subset: fixes p :: "(nat \<times> int) list"   and dm :: "nat"   and lm :: "nat" assumes "p \<in> sparsegrid dm lm" shows "p \<in> sparsegrid' dm"
jkbpCn_jkbpC_inc: fixes t :: "('a, 'es, 'as) BEState Trace"   and n :: "nat" assumes "t \<in> SPR.jkbpCn n" shows "t \<in> SPR.jkbpC"
abs_eq: fixes x :: "circline_mat" shows "opposite_ocircline (abs_ocircline x) = abs_ocircline (opposite_ocircline_clmat x)"
i2: shows "PROP Q \<Longrightarrow> PROP R"
card_of_least: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "well_order_on A r" shows "ordLeq3 (card_of A) r"
mod_ring_plus32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (plus_p32 pp) (+)"
replicate_mset_eq_iff: fixes m :: "nat"   and x :: "'a"   and n :: "nat"   and y :: "'a" shows "(replicate_mset m x = replicate_mset n y) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
weak_inf_of_singleton: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<Sqinter>{x} .= x"
mod_mult_div_eq: 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 a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times b (divide a b)) = a"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> Q"   and "Q \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> R" shows "P \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> R"
eqvt_def: fixes f :: "'a" shows "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
aezfun_setspan_projD2: fixes a :: "'a"   and A :: "'a set"   and x :: "('b, 'a) aezfun" assumes "a \<notin> A" shows "aezfun (aezfun_setspan_proj A x) a = (0::'b)"
c_sum_is_sum: fixes u :: "nat" shows "c_sum u = c_fst u + c_snd u"
mult_left_le_one_le: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) \<le> y"   and "y \<le> (1::'a)" shows "y * x \<le> x"
r_null: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp" shows "x \<otimes> \<zero> = \<zero>"
is_wrong_def: fixes behaviour :: "'a behaviour" shows "is_wrong behaviour = (\<exists>x3. behaviour = Goes_wrong x3)"
least_Upper_above: fixes L :: "('c, 'd) gorder_scheme"   and s :: "'c"   and A :: "'c set"   and x :: "'c" assumes "is_lub L s A"   and "x \<in> A"   and "A \<subseteq> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> s"
dmdInitD: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>Init F \<equiv> \<diamond>F"
ch2ch_lambda: fixes S :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>x. chain (\<lambda>i. S i x)" shows "chain S"
final_threadE: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c" assumes "final_thread.final_thread final s t" obtains x :: "'a" where "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "final x" and "wset s t = None"
less_Suc_unat_less_bound: fixes n :: "nat"   and x :: "'a word" assumes "n < Suc (unat x)" shows "n < 2 ^ LENGTH('a)"
A: shows "padic_val p a \<le> padic_val p (\<ominus>\<^bsub>padic_int p\<^esub> a)"
the_only_singleton: fixes default :: "unit \<Rightarrow> 'a"   and xq :: "'a seq" shows "the_only default xq = singleton default (pred_of_seq xq)"
sum_cong_Suc: fixes A :: "nat set"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "0 \<notin> A"   and "\<And>x. Suc x \<in> A \<Longrightarrow> f (Suc x) = g (Suc x)" shows "sum f A = sum g A"
diagonal_imp_submatrix_element_not0: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and I :: "nat set"   and k :: "nat"   and J :: "nat set"   and b :: "nat" assumes "diagonal_mat A"   and "A \<in> carrier_mat n m"   and "card I = k"   and "card J = k"   and "I \<subseteq> {0..<n}"   and "J \<subseteq> {0..<m}"   and "b < k"   and "\<exists>i<k. submatrix A I J $$ (i, b) \<noteq> (0::'b)" shows "\<exists>!i. i < k \<and> submatrix A I J $$ (i, b) \<noteq> (0::'b)"
complement_antitone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "- y \<le> - x"
that: fixes x :: "'a"   and T :: "'a set"   and U :: "'b set"   and q :: "'b \<Rightarrow> 'a" assumes "p x = y"   and "x \<in> T"   and "openin (top_of_set c) T"   and "y \<in> U"   and "openin (top_of_set S) U"   and "homeomorphism T U p q" shows "thesis"
elims_3: fixes x :: "'a st option"   and xa :: "char list set" assumes "\<not> top_on\<^sub>o x xa" obtains S :: "'a st" and X :: "char list set" where "x = Some S" and "xa = X" and "\<not> top_on\<^sub>s S X"
arr_Rel_length: shows "vcard S = 3\<^sub>\<nat>"
push_pres_Labeling: fixes e :: "nat \<times> nat" assumes "push_precond (push_effect f e) l e" shows "Labeling c s t (push_effect (push_effect f e) e) l"
find_ce_correct: shows "find_ce \<le> find_ce_spec"
causal_program_history_Write: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and is\<^sub>s\<^sub>b :: "instr list"   and sb :: "'b memref list"   and v :: "nat" assumes "causal_program_history (Write volatile a sop A L R W # is\<^sub>s\<^sub>b) sb" shows "causal_program_history is\<^sub>s\<^sub>b (sb @ [Write\<^sub>s\<^sub>b volatile a sop v A L R W])"
xs: shows "xs \<equiv> lconjseq (alternating_list (2 * k) s t)"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "poly.coeff (q.Mp f) i = q.M (poly.coeff f i)"
IR2_1: shows "callee_invariant_on (callee_of_rest rest2) IR2 (\<I>RA \<oplus>\<^sub>\<I> \<I>RU)"
IH'': fixes \<nu>' :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "dterm_sem I (Tsubst \<theta> \<sigma>) \<nu>' = dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> \<nu>'"
map_index_no_index: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map_index (\<lambda>n. f) xs = map f xs"
set_rt_F\<^sub>m_eq: fixes A :: "'c set"   and B :: "'d set" shows "{x. set1_blindable\<^sub>m x \<bind> fsts \<subseteq> A \<and> set1_blindable\<^sub>m x \<bind> snds \<bind> set \<subseteq> B} = {x. set1_blindable\<^sub>m x \<subseteq> {x. fsts x \<subseteq> A \<and> snds x \<subseteq> {x. set x \<subseteq> B}}}"
path_connected_Ioo: fixes a :: "real"   and b :: "real" shows "path_connected {a<..<b}"
list_all_fusion_def: fixes g :: "('c, 'd) generator"   and P :: "'c \<Rightarrow> bool"   and s :: "'d" shows "list_all_fusion g P s = list_all P (list.unfoldr g s)"
process_ret_splitcases: fixes rs :: "'c rule list" obtains "process_ret rs = rs"   | rs\<^sub>1 :: "'c rule list" and rs\<^sub>2 :: "'c rule list" and m :: "'c match_expr" where "rs = rs\<^sub>1 @ [Rule m Return] @ rs\<^sub>2" and "process_ret rs = rs\<^sub>1 @ process_ret ([Rule m Return] @ rs\<^sub>2)"
assms_1: shows "NT \<eta> : F \<Longrightarrow> G"
mult_onel: fixes x :: "'a" shows "(1::'a) \<cdot> x = x"
pmc_tautology: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A"   and "\<phi> \<in> Cn {}" shows "Tarskian_logic.meet_contraction K selection \<phi> = K"
conj_commute: fixes x :: "'b"   and y :: "'b" shows "x AND y = y AND x"
sasp_op_to_strips_set_delete_effects_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "set (delete_effects_of (\<phi>\<^sub>O \<Psi> op)) = (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
normal_form_normal_form: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "F \<subseteq> P[X]" shows "normal_form F (normal_form F p) = normal_form F p"
xz1: shows "- det2 a11 a12 a21 a22 * x1 + det2 a12 a13 a22 a23 * z1 = (0::'a)"
assms: shows "fmdom' (fmrestrict_set vs (snd a)) \<noteq> {}"
Paa_def: shows "Paa = Pos ''a'' [Fun ''a'' []]"
exec_gpv_def: fixes callee :: "'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'e) spmf" shows "exec_gpv callee \<equiv> curry (fixp_fun (\<lambda>exec_gpv (c, s). the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y))))"
One_Var: fixes x :: "name"   and M :: "lam" assumes " Var x \<longrightarrow>\<^sub>1 M" shows "M = Var x"
f'a_not_infinite: shows "val_Zp (f' \<bullet> a) \<noteq> \<infinity>"
f: shows "f \<in> [U]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}"
var_closed_1: fixes K :: "('c multiset \<Rightarrow> 'a) set" assumes "subring K L" shows "X\<^bsub>L\<^esub> \<in> carrier (K [X]\<^bsub>L\<^esub>)"
distinct_5: fixes x2 :: "int"   and x31 :: "'b expr'"   and x32 :: "'b expr'" shows "Lit' x2 \<noteq> Add' x31 x32"
bernoulli_pmf_Not: fixes p :: "real" assumes "p \<in> {0..1}" shows "bernoulli_pmf p = map_pmf Not (bernoulli_pmf (1 - p))"
ord_equiv_sym: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" assumes "Order D"   and "Order E"   and "ord_equiv D E" shows "ord_equiv E D"
imp_def: fixes \<gamma> :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "\<gamma> \<longrightarrow>\<^sup>c \<gamma>' \<equiv> \<lambda>t n. \<gamma> t n \<longrightarrow> \<gamma>' t n"
fp_1: shows "yfp \<in> \<omega>_limit_set x"
dense: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.dense_order less_eq less"   and "less x y" shows "\<exists>z. less x z \<and> less z y"
Restr_subset: fixes A :: "'b set"   and B :: "'b set"   and r :: "('b \<times> 'b) set" assumes "A \<subseteq> B" shows "Restr (Restr r B) A = Restr r A"
gso_dim: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs" shows "dim_vec (gram_schmidt_fs.gso n fs i) = n"
l: shows "(f \<circ> (r1 \<circ> r2)) \<longlonglongrightarrow> (l1, l2)"
norm_triangle_le: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (plus x y) \<le> e"
dom_cod: fixes f :: "'b" assumes "B.arr f" shows "B.dom (B.cod f) = B.cod f"
sig_red_zero_regularI_adds: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "p \<noteq> 0"   and "sig_red_zero (\<prec>\<^sub>t) G p"   and "lt p adds\<^sub>t lt q" shows "sig_red_zero (\<prec>\<^sub>t) G q"
arrow_determines_cone_1: fixes d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "'c"   and x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide d"   and "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a d x" shows "Limit.cone (\<cdot>\<^sub>J) (\<cdot>) (\<chi>.\<Phi>.Ya.Cop_S.Map d) a (\<chi>.\<Phi>.Ya.Cop_S.Map x)"
bf_cong: fixes u :: "nat \<times> nat"   and q' :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and M :: "nat set" assumes "snd u \<le> fst u"   and "\<And>y x. \<lbrakk>x \<le> y; y \<le> fst u\<rbrakk> \<Longrightarrow> q' y x = q y x" shows "bf M q u = bf M q' u"
nearly_healthy_wlp_Skip: shows "nearly_healthy (wlp Skip)"
l_get_disconnected_document_axioms: shows "l_get_disconnected_document get_disconnected_document get_disconnected_document_locs"
True: shows "bounded A \<and> bounded B \<and> bounded C \<and> bounded D"
in_acquired_no_pending_write_outstanding_write: fixes a :: "nat"   and sb :: "'b memref list"   and A :: "nat set" assumes "a \<in> acquired False sb A" shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}"
inverse_arrows_char: fixes S :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and \<AA> :: "('ah \<times> 'ah) set"   and f :: "'ag"   and g :: "'ag" assumes "set_category S \<AA>" shows "category.inverse_arrows S f g = ((partial_magma.arr S (S g f) \<and> partial_magma.arr S (S f g)) \<and> compose (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f)) (set_category_given_img.Fun S (set_category.some_img S \<AA>) g) (set_category_given_img.Fun S (set_category.some_img S \<AA>) f) = (\<lambda>x\<in>set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S f). x) \<and> compose (set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S g)) (set_category_given_img.Fun S (set_category.some_img S \<AA>) f) (set_category_given_img.Fun S (set_category.some_img S \<AA>) g) = (\<lambda>y\<in>set_category_data.set S (set_category.some_img S \<AA>) (partial_magma.dom S g). y))"
image_mset_union: fixes f :: "'d \<Rightarrow> 'c"   and M :: "'d multiset"   and N :: "'d multiset" shows "image_mset f (M + N) = image_mset f M + image_mset f N"
poly_mult_is_polynomial: fixes K :: "'a set"   and p1 :: "'a list"   and p2 :: "'a list" assumes "subring K R"   and "set p1 \<subseteq> K"   and "set p2 \<subseteq> K" shows "polynomial K (poly_mult p1 p2)"
ENC_def: fixes \<Sigma> :: "'b list"   and n :: "nat"   and V :: "nat set" assumes "formula \<Sigma>" shows "formula.ENC n V = INTERSECT (sorted_list_of_set (formula.valid_ENC n ` V))"
semiring_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "semiring_isom hom \<equiv> inj_semiring_hom hom \<and> bijective hom"
lower_split_interval2: fixes X :: "'b interval"   and m :: "'b" shows "lower (snd (split_interval X m)) = min (upper X) m"
simps_1: fixes x :: "'a"   and t :: "'a tree \<times> nat \<times> nat" shows "nxt (Insert x) t = insert3_d x t"
assms: assumes "\<Phi>" shows "S \<le> ASSERT \<Phi> \<bind> (\<lambda>_. S')"
nlex: shows "list_all (\<lambda>x. Counterclockwise_2D_Arbitrary.lex x 0) xs"
elims_2: fixes x :: "'edge list"   and xa :: "'edge list" assumes "same_level_path_aux x xa" obtains cs :: "'edge list" where "x = cs__" and "xa = []"   | cs :: "'edge list" and a :: "'edge" and as :: "'edge list" where "x = cs" and "xa = a # as" and "case kind a of Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs \<Rightarrow> same_level_path_aux (a # cs) as | Q\<hookleftarrow>\<^bsub>p\<^esub>f \<Rightarrow> case cs of [] \<Rightarrow> False | c' # cs' \<Rightarrow> a \<in> get_return_edges c' \<and> same_level_path_aux cs' as | _ \<Rightarrow> same_level_path_aux cs as"
Dirichlet_kernel_continuous_strong: fixes n :: "nat" shows "continuous_on {- (2 * pi)<..<2 * pi} (Dirichlet_kernel n)"
of_nat_eq_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "(semiring_1.of_nat one plus zero m = semiring_1.of_nat one plus zero n) = (m = n)"
sel_4: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree" shows "r (Node x21 x22 x23 x24 x25) = x22"
associated_eqI: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "dvd.dvd times b a"   and "normalize a = a"   and "normalize b = b" shows "a = b"
rq: fixes i :: "nat" assumes "i < max_Var_floatarith f" shows "rs ! i = qs ! i"
Nml_Nmlize_5: fixes t :: "'a local.term" assumes "Arr t" shows "Cod \<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
hom_0_iff: fixes x :: "'a poly" shows "(pCons (0::'a) x = 0) = (x = 0)"
normalize_right_idem: fixes a :: "'b"   and b :: "'b" shows "gcd a (normalize b) = gcd a b"
assms_2: shows "\<forall>i. B \<le> X i"
exhaust_disc: fixes unboxed :: "('a, 'b, 'c) unboxed" obtains "is_dyn_operand unboxed"   | "is_OpUbx1 unboxed"   | "is_OpUbx2 unboxed"
get_root_node_root_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root" shows "root |\<in>| object_ptr_kinds h"
elims_3: fixes x :: "('state, 'effect) L_state"   and xa :: "'pred" assumes "\<not> x \<turnstile>\<^sub>L xa" obtains uu_ :: "'effect" and uv_ :: "'effect fs_set" and P :: "'state" and \<phi> :: "'pred" where "x = AC (uu_, uv_, P)" and "xa = \<phi>'__" and "\<not> P \<turnstile> \<phi>'__"   | uw_ :: "'effect fs_set \<times> 'state" and \<phi> :: "'pred" where "x = EF uw_" and "xa = \<phi>"
mem\<^sub>2'_def: shows "mem\<^sub>2' = subst (g2 (to_partial mem\<^sub>2 |` differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k)) mem\<^sub>h"
r_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<oplus> \<zero> = x"
schnorr_sigma_protocol_axioms: shows "schnorr_sigma_protocol \<G>"
Arity_transformation_correct: fixes a :: "Arity"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "eq a (\<lbrakk> \<T>\<^bsub>a\<^esub> e \<rbrakk>\<^bsub>\<rho>\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)"
weight_finite: fixes x :: "'v" shows "weight \<Gamma> x \<noteq> \<top>"
comm_mult_strict_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "(0::'a) < c" shows "c * a < c * b"
ptr_kinds_M_ok: fixes h :: "(_) heap" shows "h \<turnstile> ok object_ptr_kinds_M"
rtrancl_converseD: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set" assumes "(x, y) \<in> (r\<inverse>)\<^sup>*" shows "(y, x) \<in> r\<^sup>*"
implies_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y \<le> z) = (x \<le> y \<leadsto> z)"
fi_is_sum_of_mu_gso: fixes i :: "nat" assumes "i < length fs" shows "fs ! i = M.sumlist (map (\<lambda>j. \<mu> i j \<cdot>\<^sub>v gso j) [0..<Suc i])"
d'_red: shows "reduce [C] D D' \<noteq> D'"
freshness_lemma: fixes h :: "'b \<Rightarrow> 'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp h)"   and "\<exists>a. a \<sharp> (h, h a)" shows "\<exists>fr. \<forall>a. a \<sharp> h \<longrightarrow> h a = fr"
redundant_infer_def: fixes N :: "'a literal multiset set"   and \<gamma> :: "'a Inference_System.inference" shows "src.redundant_infer N \<gamma> = (\<exists>DD. set_mset DD \<subseteq> N \<and> (\<forall>I. I \<TTurnstile>m DD + old_side_prems_of \<gamma> \<longrightarrow> I \<TTurnstile> old_concl_of \<gamma>) \<and> (\<forall>D. D \<in># DD \<longrightarrow> D < old_main_prem_of \<gamma>))"
lfinite_finite_index: fixes xs :: "'c llist" assumes "lfinite xs" shows "finite {n. enat n < llength xs}"
conseqPrePost: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and P' :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q' :: "'f set"   and A' :: "'f set"   and P :: "'f set"   and Q :: "'f set"   and A :: "'f set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q',A'"   and "P \<subseteq> P'"   and "Q' \<subseteq> Q"   and "A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
index_less_size_conv: fixes xs :: "'c list"   and x :: "'c" shows "(index xs x < length xs) = (x \<in> set xs)"
to_tree_order_ptr_in_result: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and nodes :: "(_) object_ptr list" assumes "l_to_tree_order_wf heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_child_nodes"   and "h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r nodes" shows "ptr \<in> set nodes"