hcju/isabelleps huggingface dataset:
wf_dickson_less_v_aux1: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and seq :: "nat \<Rightarrow> 'c" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "dickson_grading d"   and "\<And>i. gd_term.dickson_less_v pair_of_term ord_term_strict d m (seq (Suc i)) (seq i)" obtains i :: "nat" where "\<And>j. i < j \<Longrightarrow> term_powerprod.component_of_term pair_of_term (seq j) < term_powerprod.component_of_term pair_of_term (seq i)"
scalar_product_def: fixes v :: "'a list"   and w :: "'a list" shows "scalar_product v w = scalar_prodI zer (+) (*) v w"
Ioc_inj: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b = ord.greaterThanAtMost less_eq less c d) = (less_eq b a \<and> less_eq d c \<or> a = c \<and> b = d)"
doubleExist: fixes A :: "atom fm"   and L :: "real list" shows "eval (ExN 2 A) L = eval (ExQ (ExQ A)) L"
reslI: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> z" shows "x \<le> z \<leftarrow> y"
implies_isotone_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> y \<sqinter> z \<le> x \<leadsto> y"
intros_16: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> =n\<Rightarrow> t"
Ln_minus1: shows "Ln (- 1) = \<i> * complex_of_real pi"
setdist_sym: fixes S :: "'a set"   and T :: "'a set" shows "setdist S T = setdist T S"
Lim_transform_within_open: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and T :: "'a set"   and s :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> l) (at a within T)"   and "open s"   and "a \<in> s"   and "\<And>x. \<lbrakk>x \<in> s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x" shows "(g \<longlongrightarrow> l) (at a within T)"
list_of_oalist_of_list: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" shows "list_of_oalist_ntm (OAlist_ntm xs) = sort_oalist_ko_ntm xs"
block_sat_fresh: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and block :: "('c, 'd) fm list \<times> 'd"   and i :: "'d"   and v :: "'b" assumes "M, g \<Turnstile>\<^sub>B block"   and "i \<notin> block_nominals block" shows "M, g(i := v) \<Turnstile>\<^sub>B block"
assms_2: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "key_compare (rep_key_order (snd xs)) (fst (f a)) (fst (f b)) = key_compare (rep_key_order (snd xs)) (fst a) (fst b)"
iPrefix_def: fixes c :: "'d \<Rightarrow> 'c"   and i :: "'d \<times> 'd \<Rightarrow> bool"   and j :: "('c, 'd) action \<Rightarrow> ('c, 'd) action"   and S :: "'d \<Rightarrow> 'c set"   and P :: "('c, 'd) action" shows "iPrefix c i j S P \<equiv> action_of (R (true \<turnstile> \<lambda>(A, A'). do_I c (S (alpha_rp.more A)) (A, A') \<and> alpha_rp.more A' = alpha_rp.more A)) `;` P"
B: fixes z :: "'a" assumes "z \<in> set zs" shows "P z"
pdigraph_base_axioms: shows "pdigraph_base \<alpha> J \<AA>"
countable_SIGMA: fixes I :: "'f set"   and A :: "'f \<Rightarrow> 'g set" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> countable (A i)" shows "countable (Sigma I A)"
new_tv_subst_le: fixes n :: "nat"   and m :: "nat"   and S :: "nat \<Rightarrow> typ" assumes "n \<le> m"   and "new_tv n S" shows "new_tv m S"
nth_weak_ranking_index_1: fixes x :: "'a" assumes "x \<in> carrier" shows "weak_ranking_index x < length (weak_ranking le)"
less_gexp_trans: fixes x :: "'a gexp"   and y :: "'a gexp"   and z :: "'a gexp" assumes "x < y"   and "y < z" shows "x < z"
listToFun_futz: fixes M :: "'a \<Rightarrow> 'b"   and xs :: "('a \<times> 'b list) list"   and x :: "'a" assumes "M \<in> set (listToFun xs)"   and "x \<in> fst ` set xs" shows "M x \<in> {uu_. \<exists>y ys. uu_ = y \<and> (x, ys) \<in> set xs \<and> y \<in> set ys}"
gfp_mono: fixes F :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c" assumes "mono_mono F" shows "mono (gfp F)"
simps_10: fixes x21 :: "'a"   and x22 :: "'a form list" shows "ff \<noteq> Compound x21 x22"
functor_axioms: shows "functor C' (\<cdot>) (\<lambda>i. if C'.arr i then \<phi> i else null)"
distinct_17: fixes x3 :: "'a \<times> nat"   and x5 :: "'a \<times> nat" shows "Formal_out x3 \<noteq> Actual_out x5"
assms_2: shows "h \<in> N \<rightarrow>\<^sub>M M"
block_complement_inv: fixes bl :: "'a set"   and bl2 :: "'a set" assumes "bl \<in># \<B>"   and "bl\<^sup>c = bl2" shows "bl2\<^sup>c = bl"
disc_eq_case_2: fixes option :: "'g option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
inject_3: fixes x3 :: "msg"   and y3 :: "msg" shows "(Hash x3 = Hash y3) = (x3 = y3)"
map_sel_3: fixes a :: "('f, 'g, 'h) Resumption.resumption"   and f1 :: "'f \<Rightarrow> 'i"   and f2 :: "'g \<Rightarrow> 'j" assumes "\<not> Resumption.resumption.is_Done a" shows "Resumption.resumption.resume (Resumption.resumption.map_resumption f1 f2 a) = Resumption.resumption.map_resumption f1 f2 \<circ> Resumption.resumption.resume a"
that: fixes y :: "'a" assumes "\<omega>_limit_set x = {y}"   and "f y = (0::'a)" shows "thesis"
Limsup_le_iff: fixes F :: "'d filter"   and X :: "'d \<Rightarrow> 'b"   and C :: "'b" shows "(Limsup F X \<le> C) = (\<forall>y>C. \<forall>\<^sub>F x in F. X x < y)"
acomplete_optimal: fixes t :: "'b tree"   and t' :: "'c tree" assumes "acomplete t"   and "size t \<le> size t'" shows "height t \<le> height t'"
bdd_below_Ici: fixes a :: "'b multiset" shows "subset_mset.bdd_below (subset_mset.atLeast a)"
formulaContainsItsLiteralsVariable: fixes literal :: "Literal"   and formula :: "Literal list list" assumes "literal el formula" shows "var literal \<in> vars formula"
induct: fixes P :: "('b, 'c) Combinators list \<Rightarrow> 'b set list \<Rightarrow> bool"   and a0 :: "('b, 'c) Combinators list"   and a1 :: "'b set list" assumes "\<And>l. P [] l"   and "\<And>x xs l. \<lbrakk>P (filter (\<lambda>y. \<not> smaller x y l) xs) l; P (filter (\<lambda>y. smaller x y l) xs) l\<rbrakk> \<Longrightarrow> P (x # xs) l" shows "P a0 a1"
sum_squares_bound: fixes x :: "'b"   and y :: "'b" shows "(2::'b) * x * y \<le> x\<^sup>2 + y\<^sup>2"
simps: fixes m :: "('a, 'b) insert_message"   and w :: "('a, 'b option) insert_message list"   and p :: "'a extended"   and s :: "'a extended" shows "integrate_insert m w p s = idx w p \<bind> (\<lambda>l. idx w s \<bind> (\<lambda>u. assert (l < u) \<bind> (\<lambda>_. if Suc l = u then Inr (take l w @ [to_woot_char m] @ drop l w) else mapM (concurrent w l u) (substr w l u) \<bind> (\<lambda>d. assert (concat d \<noteq> []) \<bind> (\<lambda>_. fromSome (find ((\<lambda>x. \<lbrakk>I m\<rbrakk> < x \<or> x = s) \<circ> snd) (zip (p # concat d) (concat d @ [s]))) \<bind> (\<lambda>(x, y). integrate_insert m w x y))))))"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright> \<and> local.iso \<nu>"
sec_prik_rev: shows "Rev_PriK \<notin> range Sec_PriK"
red_in_Hom: fixes a :: "'c local.term" assumes "Ide a" shows "a\<^bold>\<down> \<in> Hom a \<^bold>\<lfloor>a\<^bold>\<rfloor>"
2: shows "\<r>[f \<star> g] \<cdot> \<a>\<^sup>-\<^sup>1[f, g, src g] = f \<star> \<r>[g]"
that: fixes \<pi>' :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>'"   and "\<pi> =\<^bsub>k\<^esub> \<pi>'"   and "\<pi>' n = local.return" shows "thesis"
le_fininfI: fixes v\<^sub>2 :: "'item list"   and w\<^sub>2 :: "'item stream"   and w\<^sub>1 :: "'item list" assumes "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
cases: fixes x :: "'b word \<times> 'b word" obtains ip_start :: "'b word" and ip_end :: "'b word" where "x = (ip_start, ip_end)"
consequence_relation_axioms: shows "consequence_relation {{#}} (\<TTurnstile>e)"
mod_mult_mult2: 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 a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a c) (times b c) = times (modulo a b) c"
ket_k_dim: fixes k :: "nat" shows "ket_k k \<in> carrier_vec K"
proves_eq_abstract_rule_derived_rule: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and x :: "variable"   and \<tau> :: "typ"   and \<Gamma> :: "Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "(x, \<tau>) \<notin> FV \<Gamma>"   and "typ_ok \<Theta> \<tau>"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq s t" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs_fv x \<tau> s) (Abs_fv x \<tau> t)"
steps_imp_seq: fixes a :: "'c"   and b :: "'c"   and ars :: "('c \<times> 'd \<times> 'c) set" assumes "(a, b) \<in> (unlabel ars)\<^sup>*" shows "\<exists>ss\<in>seq ars. fst ss = a \<and> lst ss = b"
inverse_functors_sym: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c" assumes "inverse_functors A B F G" shows "inverse_functors B A G F"
independent_in_carrier: fixes K :: "'a set"   and Us :: "'a list" assumes "R.independent K Us" shows "set Us \<subseteq> carrier R"
K2: shows "norm ((of_real (norm K) + of_real (norm x)) / (2::'a)) < norm K"
ring_simprules_5: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
sym_def: fixes r :: "('c \<times> 'c) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
res1: shows "mat (rat_poly.row_length (blockmat w)) (length (kauff_mat ws)) (kauff_mat (w * ws))"
permutes_inv: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set" assumes "p permutes S" shows "inv p permutes S"
vproduct_def: fixes I :: "V"   and A :: "V \<Rightarrow> V" shows "vproduct I A = set {f. vsv f \<and> \<D>\<^sub>\<circ> f = I \<and> (\<forall>i\<in>elts I. f\<lparr>i\<rparr> \<in>\<^sub>\<circ> A i)}"
rel_with_target_impl_reflC_transC_TRel_is_weak_reduction_correspondence_simulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_correspondence_simulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>*" shows "weak_reduction_correspondence_simulation (TRel\<^sup>*) Target"
assms_2: shows "v \<in> fmdom' (fst a)"
mat_match: shows "M_mat (get_signs (reduce_system p (qs, M_mat signs subsets, subsets, signs))) (get_subsets (reduce_system p (qs, M_mat signs subsets, subsets, signs))) = get_matrix (reduce_system p (qs, M_mat signs subsets, subsets, signs))"
multisetEmptyLeI: fixes a :: "'b multiset"   and r :: "('b \<times> 'b) set" shows "multiset_le {#} a r"
poly_diff_val: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "f \<in> carrier (UP Zp)"   and "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "val_Zp (a \<ominus> b) \<le> val_Zp (f \<bullet> a \<ominus> f \<bullet> b)"
pgalleryD_chamber: fixes X :: "'b set set"   and xs :: "'b set list"   and x :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.pmaxsimpchain X xs"   and "x \<in> set xs" shows "SimplicialComplex.maxsimp X x"
inv_safe: fixes c :: "('loc, 't) configuration" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c" shows "inv_safe c"
submonoid_axioms: shows "submonoid K G (\<cdot>) \<one>"
listrel1_def: fixes r :: "('b \<times> 'b) set" shows "listrel1 r = {(xs, ys). \<exists>us z z' vs. xs = us @ z # vs \<and> (z, z') \<in> r \<and> ys = us @ z' # vs}"
lu_ll: shows "Sup_llist Ns - Rf (Sup_llist Ns) \<subseteq> Liminf_llist Ns"
zero_less_divide_1_iff: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide one a) = less zero a"
incseq_ereal: fixes f :: "nat \<Rightarrow> real" assumes "incseq f" shows "incseq (\<lambda>x. ereal (f x))"
arith: fixes d :: "nat"   and a :: "nat"   and c :: "nat"   and b :: "nat" assumes "d < a"   and "c < b" shows "b * d + c < a * b"
winning_path_drop: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and P :: "'c llist"   and n :: "nat" assumes "ParityGame G"   and "ParityGame.winning_path G p P"   and "enat n < llength P" shows "ParityGame.winning_path G p (ldropn n P)"
mono_Sync_D1: fixes P :: "'c process"   and Q :: "'c process"   and A :: "'c set"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "D (Q \<lbrakk>A\<rbrakk> S) \<subseteq> D (P \<lbrakk>A\<rbrakk> S)"
set_add_def: fixes G :: "('a, 'b) ring_scheme" shows "(<+>\<^bsub>G\<^esub>) = (<#>\<^bsub>add_monoid G\<^esub>)"
contour_integral_join: fixes f :: "complex \<Rightarrow> complex"   and g1 :: "real \<Rightarrow> complex"   and g2 :: "real \<Rightarrow> complex" assumes "f contour_integrable_on g1"   and "f contour_integrable_on g2"   and "valid_path g1"   and "valid_path g2" shows "contour_integral (g1 +++ g2) f = contour_integral g1 f + contour_integral g2 f"
lt_alt: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "lpp p = ordered_powerprod_lin.Max (keys p)"
merge_def: fixes t1 :: "('c, 'd) bintree"   and t2 :: "('c, 'd) bintree" shows "merge t1 t2 = (if priority t1 < priority t2 then Node (priority t1) (val t1) (t2 # children t1) else Node (priority t2) (val t2) (t1 # children t2))"
EVEN_union_ODD: shows "EVEN \<union> ODD = UNIV"
assms_1: fixes n :: "nat" shows "u n \<in> borel_measurable M"
adjacentset_def: fixes x :: "'a set" shows "adjacentset x = {y \<in> X. x \<sim> y}"
is_universal: fixes a' :: "'c"   and \<chi>' :: "'j \<Rightarrow> 'c" assumes "E.cone a' \<chi>'" shows "\<exists>!f. \<guillemotleft>f : a' \<rightarrow>\<^sub>C a\<guillemotright> \<and> E.cones_map f \<chi> = \<chi>'"
LIST_FRAG_DICHOTOMY: fixes l :: "'b list"   and la :: "'b list"   and x :: "'b"   and lb :: "'b list" assumes "sublist l (la @ [x] @ lb)"   and "\<not> ListMem x l" shows "sublist l la \<or> sublist l lb"
listsum_Cons: fixes ds :: "nat list"   and A :: "'b tensor"   and As :: "'b tensor list" shows "listsum ds (A # As) = A + listsum ds As"
hconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "P \<turnstile> h \<surd> \<equiv> (\<forall>a obj. h a = \<lfloor>obj\<rfloor> \<longrightarrow> P,h \<turnstile> obj \<surd>) \<and> preallocated h"
mono_on_greaterD: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'c"   and y :: "'c" assumes "mono_on g A"   and "x \<in> A"   and "y \<in> A"   and "g y < g x" shows "y < x"
simps_2: fixes v :: "ref" shows "isLeaf (Node Tip v Tip) = True"
simps_4: fixes x :: "octo"   and y :: "octo" shows "Im3 (x - y) = Im3 x - Im3 y"
fds_nth_mult_const_left: fixes c :: "'b"   and f :: "'b fds"   and n :: "nat" shows "fds_nth (fds_const c * f) n = c * fds_nth f n"
no_collision_react_initially_strict: assumes "s\<^sub>o \<le> u_max"   and "u_max < other.s_stop" shows "no_collision_react {0<..<\<delta>}"
add_col_sub_row_uppert: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat"   and a :: "'a" assumes "A \<in> carrier_mat n n"   and "i < j"   and "j < n"   and "local.inv_all local.uppert A" shows "local.inv_all local.uppert (add_col_sub_row a i j A)"
measure_pmf_UNIV: fixes p :: "'d pmf" shows "measure_pmf.prob p UNIV = 1"
digraph_op: shows "digraph \<alpha> (op_dg \<CC>)"
rep_list_uminus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (- r) = - rep_list r"
inj_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj f" shows "inj_on f A"
someI_ex: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>x. P x" shows "P (SOME x. P x)"
fresh_string_ordstNS: fixes Y :: "char list"   and Xs :: "char list set" shows "ordstNS Y (fresh_string Xs Y)"
better_sred_Beta: fixes r :: "lam"   and x :: "name"   and s :: "lam"   and ss :: "lam list"   and t :: "lam" assumes "r[x::=s] \<degree>\<degree> ss \<rightarrow>\<^sub>s t" shows "(Lam [x].r) \<degree> s \<degree>\<degree> ss \<rightarrow>\<^sub>s t"
minus_image_eq_vimage: fixes A :: "'a set" shows "uminus ` A = uminus -` A"
prod_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less zero (f a)" shows "less zero (comm_monoid_mult.prod times one f A)"
existence_ivl_zero: fixes x0 :: "'a" assumes "x0 \<in> Y" shows "0 \<in> Y.existence_ivl0 x0"
butlast_def: shows "butlast \<equiv> rec_list [] (\<lambda>x xs xsa. if xs = [] then [] else x # xsa)"
parent_def: fixes l :: "int"   and c :: "int" shows "parent l c = l + (c - l - 1) div 2"
coplanar_4_bis: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and \<beta> :: "Points" assumes "rk {A, B, C} = 3"   and "rk {A, C, \<beta>} = 2" shows "rk {A, B, C, \<beta>} = 3"
g_le_h0: fixes x :: "'v"   and y :: "'v" assumes "edge \<Gamma> x y" shows "g (\<langle>x\<rangle>, \<langle>y\<rangle>) \<le> h 0 (x, y)"
d2_def: shows "stm1.d2 = prod_list stm1.dims2"
counit_is_iso: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and f :: "'c"   and g :: "'c"   and \<eta> :: "'c"   and \<epsilon> :: "'c" assumes "equivalence_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "category.iso V \<epsilon>"
PUpdate_EmptySet: shows "Data2PData \<in> {L |L. PUpdate L}"
check_induct'': fixes s :: "('i, 'a) state"   and P :: "('i, 'a) state \<Rightarrow> bool" assumes "\<nabla> s"   and "\<triangle> (\<T> s)"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<U> s \<Longrightarrow> P s"   and "\<And>s. \<lbrakk>\<nabla> s; \<triangle> (\<T> s); \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<diamond> s; \<not> \<U> s; min_lvar_not_in_bounds s = None\<rbrakk> \<Longrightarrow> P s"   and "\<And>s x\<^sub>i dir I. \<lbrakk>dir = Positive \<or> dir = Negative; \<nabla> s; \<triangle> (\<T> s); \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s; \<diamond> s; \<not> \<U> s; min_lvar_not_in_bounds s = Some x\<^sub>i; \<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i); min_rvar_incdec dir s x\<^sub>i = Inl I\<rbrakk> \<Longrightarrow> P (set_unsat I s)" shows "P (check s)"
simps: fixes N :: "('b literal multiset \<times> nat) multiset"   and P :: "('b literal multiset \<times> nat) multiset"   and Q :: "('b literal multiset \<times> nat) multiset"   and n :: "nat" shows "state_of_wstate (N, P, Q, n) = (set_mset (image_mset fst N), set_mset (image_mset fst P), set_mset (image_mset fst Q))"
inext_singleton_cut_greater_empty_iff: fixes t0 :: "nat"   and I :: "nat set" shows "({inext t0 I} \<down>> t0 = {}) = (I \<down>> t0 = {} \<or> t0 \<notin> I)"
distinct_1: shows "Ubx1 \<noteq> Ubx2"
is_monic_setD: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "punit.is_monic_set B"   and "b \<in> B"   and "b \<noteq> 0" shows "lcf b = (1::'b)"
conjugate_exponent_realI_1: fixes p :: "real"   and q :: "real" assumes "0 < p"   and "0 < q"   and "1 / p + 1 / q = 1" shows "1 < p"
lphil0_transitions_def: fixes s :: "nat" shows "Tr\<^sub>l\<^sub>p s = (if s = 0 then {picks 0 (N - 1)} else if s = 1 then {picks 0 0} else if s = 2 then {putsdown 0 0} else if s = 3 then {putsdown 0 (N - 1)} else {})"
nres_relI: fixes c :: "'e nres"   and R :: "('e \<times> 'f) set"   and a :: "'f nres" assumes "c \<le> \<Down> R a" shows "(c, a) \<in> \<langle>R\<rangle>nres_rel"
GrD2: fixes x :: "'b"   and fx :: "'c"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "(x, fx) \<in> BNF_Def.Gr A f" shows "f x = fx"
1: shows "Suc (card X - Suc j) = card X - j"
assms_2: fixes b :: "'a" assumes "b \<in> A" shows "a dvd b"
Z: shows "(zps, Z) \<in> futures (c_process step out s\<^sub>0) xps"
times_finite_3_def: fixes x :: "finite_3"   and y :: "finite_3" shows "x * y = (case (x, y) of (a\<^sub>1, b) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>2) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>2)"
If_cases: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite A" shows "(\<Prod>x\<in>A. if P x then h x else g x) = prod h (A \<inter> {x. P x}) * prod g (A \<inter> - {x. P x})"
seqsim_trans: fixes s :: "nat \<Rightarrow> 'b"   and t :: "nat \<Rightarrow> 'b"   and z :: "nat \<Rightarrow> 'b" assumes "s \<approx> t"   and "t \<approx> z" shows "s \<approx> z"
strategyproofness: shows "(pmf (sds R1) a < pmf (sds R25) a \<or> pmf (sds R1) a + pmf (sds R1) c < pmf (sds R25) a + pmf (sds R25) c) \<or> pmf (sds R1) a = pmf (sds R25) a \<and> pmf (sds R1) a + pmf (sds R1) c = pmf (sds R25) a + pmf (sds R25) c"
notame7_def: fixes g :: "graph" shows "notame7 g \<equiv> \<not> (tame10ub g \<and> tame11b g \<and> is_tame13a g)"