hcju/isabelleps huggingface dataset:
minus_eq_plus_uminus: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and is_expansion :: "'a \<Rightarrow> (real \<Rightarrow> real) list \<Rightarrow> bool"   and expansion_level :: "'a itself \<Rightarrow> nat"   and eval :: "'a \<Rightarrow> real \<Rightarrow> real"   and zero_expansion :: "'a"   and const_expansion :: "real \<Rightarrow> 'a"   and powr_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> real \<Rightarrow> 'a"   and power_expansion :: "bool \<Rightarrow> 'a \<Rightarrow> nat \<Rightarrow> 'a"   and trimmed :: "'a \<Rightarrow> bool"   and dominant_term :: "'a \<Rightarrow> real \<times> real list"   and F :: "'a"   and G :: "'a" assumes "class.multiseries divide inverse minus plus times uminus is_expansion expansion_level eval zero_expansion const_expansion powr_expansion power_expansion trimmed dominant_term" shows "minus F G = plus F (uminus G)"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "(x \<le> xa) = (Rep_pos x \<le> Rep_pos xa)"
induct: fixes x1 :: "typ list"   and x2 :: "Core.term"   and x3 :: "typ"   and P :: "typ list \<Rightarrow> Core.term \<Rightarrow> typ \<Rightarrow> bool" assumes "x1 \<turnstile>\<^sub>\<tau> x2 : x3"   and "\<And>uu_ uv_ T. P uu_ (Ct uv_ T) T"   and "\<And>i Ts. i < length Ts \<Longrightarrow> P Ts (Bv i) (Ts ! i)"   and "\<And>uw_ ux_ T. P uw_ (Fv ux_ T) T"   and "\<And>T Ts t T'. \<lbrakk>T # Ts \<turnstile>\<^sub>\<tau> t : T'; P (T # Ts) t T'\<rbrakk> \<Longrightarrow> P Ts (Abs T t) (T \<rightarrow> T')"   and "\<And>Ts u U t T. \<lbrakk>Ts \<turnstile>\<^sub>\<tau> u : U; P Ts u U; Ts \<turnstile>\<^sub>\<tau> t : U \<rightarrow> T; P Ts t (U \<rightarrow> T)\<rbrakk> \<Longrightarrow> P Ts (t $ u) T" shows "P x1 x2 x3"
ennsqrt_ennreal_ennreal_sqrt: fixes t :: "real" assumes "0 \<le> t" shows "ennsqrt (ennreal t) = ennreal (sqrt t)"
fair_chain_def: fixes Rs :: "(('d, 'e) labeled_graph \<times> ('d, 'e) labeled_graph) set"   and S :: "nat \<Rightarrow> ('d, 'f) labeled_graph" shows "fair_chain Rs S \<equiv> chain S \<and> (\<forall>R f i. R \<in> Rs \<and> graph_homomorphism (fst R) (S i) f \<longrightarrow> (\<exists>j. extensible R (S j) f))"
alw_aand: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and \<psi> :: "'a stream \<Rightarrow> bool" shows "alw (\<lambda>xs. \<phi> xs \<and> \<psi> xs) = (\<lambda>xs. alw \<phi> xs \<and> alw \<psi> xs)"
assms_1: shows "\<forall>x\<in>fv t. \<exists>a. \<Gamma> (Var x) = Var a"
config_n_fst_init_length2: fixes s0 :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs). length (fst (snd x)) = length s0"
strict_order_iff: fixes ko :: "'b key_order"   and b :: "'b"   and a :: "'b" shows "ko.lt ko b a = (a = ko.max ko a b \<and> a \<noteq> b)"
assms_7: shows "A \<subseteq> I \<union> J"
cpx_mat_cnj_id: fixes n :: "nat" shows "1\<^sub>m n\<^sup>\<star> = 1\<^sub>m n"
mat_of_rows_row: fixes i :: "nat"   and vs :: "'a vec list"   and n :: "nat" assumes "i < length vs"   and "vs ! i \<in> carrier_vec n" shows "row (mat_of_rows n vs) i = vs ! i"
find_lasso_correct: shows "find_lasso \<le> find_lasso_spec"
polar_abs: fixes v :: "(real, 3) vec" assumes "v \<noteq> 0" shows "polar (proj2_abs v) = proj2_line_abs (M *v v)"
simps_11: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and f3 :: "'i \<Rightarrow> 'g"   and x3 :: "'i" shows "(case sum_bot.Right x3 of Bottom \<Rightarrow> f1 | sum_bot.Left x \<Rightarrow> f2 x | sum_bot.Right x \<Rightarrow> f3 x) = f3 x3"
combining_to_smash: fixes p :: "real poly"   and qs1 :: "real poly list"   and m1 :: "rat mat"   and sub1 :: "nat list list"   and sgn1 :: "rat list list"   and qs2 :: "real poly list"   and m2 :: "rat mat"   and sub2 :: "nat list list"   and sgn2 :: "rat list list" shows "combine_systems p (qs1, m1, sub1, sgn1) (qs2, m2, sub2, sgn2) = smash_systems p qs1 qs2 sub1 sub2 sgn1 sgn2 m1 m2"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>Q Skip Q,A"
simps_19: fixes x21 :: "'a"   and x22 :: "'a llist" shows "lset (LCons x21 x22) = insert x21 (lset x22)"
os_distincts: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A B OS X Y" shows "A \<noteq> B \<and> A \<noteq> X \<and> A \<noteq> Y \<and> B \<noteq> X \<and> B \<noteq> Y"
is_projection_on_def: fixes \<pi> :: "'b \<Rightarrow> 'b"   and M :: "'b set" shows "is_projection_on \<pi> M = (\<forall>h. is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) (\<pi> h))"
deriv_c: shows "\<Gamma>|\<^bsub>M\<^esub>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
card_less: fixes M :: "nat set"   and i :: "nat" assumes "0 \<in> M" shows "card {k \<in> M. k < Suc i} \<noteq> 0"
yins: shows "cs ! i \<in> set (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "is_iso_dghm_axioms \<BB> \<FF>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
LeftDerivationLadder_stepdown: fixes \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder \<alpha> D L \<gamma>"   and "1 < length L" shows "\<exists>L'. LeftDerivationLadder (ladder_stepdown_\<alpha>_0 \<alpha> D L) (drop (ladder_stepdown_diff L) D) L' \<gamma> \<and> length L' = length L - 1 \<and> ladder_i L' 0 = ladder_i L 1 + ladder_ix L 1 \<and> ladder_last_j L' = ladder_last_j L"
rel_funD: fixes A :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and B :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and f :: "'k \<Rightarrow> 'm"   and g :: "'l \<Rightarrow> 'n"   and x :: "'k"   and y :: "'l" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
arg: fixes s :: "('a, 'b) Term.term"   and ss :: "('a, 'b) Term.term list"   and f :: "'a" assumes "s \<in> set ss" shows "Fun f ss \<rhd> s"
diff_union_cancelL: fixes N :: "'k multiset"   and M :: "'k multiset" shows "N + M - N = M"
X_2: fixes e :: "'b" assumes "e \<in> arcs G" shows "head H e \<in> verts G"
coinduct: fixes X :: "'b stream \<Rightarrow> bool"   and x :: "'b stream"   and \<phi> :: "'b stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>xs. x = xs \<and> \<phi> xs \<and> (X (stl xs) \<or> alw \<phi> (stl xs))" shows "alw \<phi> x"
not_outside_1: fixes x :: "'b"   and y :: "'b" assumes "Pareto R x y" shows "x \<in> carrier"
infiter_def: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<infinity> \<equiv> lfp ((;) c)"
ce_rel_lm_25: fixes r :: "(nat \<times> nat) set" assumes "r \<in> ce_rels" shows "r\<inverse> \<in> ce_rels"
simps: fixes a :: "'a"   and P :: "'a set set"   and I :: "'a set" shows "(a \<in> gen P I) = ((\<exists>i. a = i \<and> i \<in> I) \<or> (\<exists>J j0 j. a = j \<and> J \<in> P \<and> j0 \<in> J \<and> j0 \<in> gen P I \<and> j \<in> J))"
greaterThan_non_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.no_top less_eq less" shows "ord.greaterThan less x \<noteq> {}"
ind_in_set_inj': fixes B :: "nat set"   and A :: "nat set" assumes "B \<subseteq> A" shows "inj_on (ind_in_set A) B"
red_preserves_bconf: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "iconf sh e"   and "P,sh \<turnstile>\<^sub>b (e,b) \<surd>" shows "P,sh' \<turnstile>\<^sub>b (e',b') \<surd>"
assms_8: fixes a :: "name"   and x :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<x> \<prec> P'"   and "!P \<longmapsto> a<\<nu>x> \<prec> P''"   and "x \<sharp> P"   and "x \<sharp> C"   and "\<And>C. F C (!P) (a<\<nu>x> \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> <\<nu>x>(P' \<parallel> P''))"
intros: fixes W :: "'a list"   and \<sigma> :: "'a"   and Wtl :: "'a list"   and Wn :: "'a list"   and R :: "('a \<times> 'a) set"   and \<Sigma> :: "'a set"   and W' :: "'a list"   and \<Sigma>' :: "'a set" assumes "W = \<sigma> # Wtl"   and "distinct Wn"   and "set Wn = R `` {\<sigma>} - \<Sigma>"   and "W' = Wn @ Wtl"   and "\<Sigma>' = R `` {\<sigma>} \<union> \<Sigma>" shows "((\<Sigma>, W), \<Sigma>', W') \<in> dfs_step R"
divide_fract_def: fixes q :: "'a fract"   and r :: "'a fract" shows "q div r = q * inverse r"
add: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "additive f" shows "f (plus x y) = plus (f x) (f y)"
max_Var_floatariths_euler_incr_fas: fixes xs :: "floatarith list"   and cxs :: "floatarith list"   and h :: "floatarith"   and z :: "nat" assumes "max_Var_floatariths ode_e \<le> length ode_e"   and "length xs = length ode_e"   and "length cxs = length ode_e"   and "max (max_Var_floatariths xs) (max (max_Var_floatarith h) (max_Var_floatariths cxs)) \<le> z" shows "max_Var_floatariths (euler_incr_fas xs h cxs) \<le> z"
coprime_lead_coeff_factor_1: fixes p :: "int"   and f :: "int poly"   and g :: "int poly" assumes "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff (f * g)) p" shows "comm_monoid_mult_class.coprime (lead_coeff f) p"
Call_\<tau>red1t_obj: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and M :: "String.literal"   and ps :: "'addr expr1 list" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e\<bullet>M(ps), xs) (e'\<bullet>M(ps), xs')"
quot_def: shows "quot = {{x. a \<sim> x} |a. True}"
B: shows "\<integral>\<^sup>+ x. ennreal (- f x * indicat_real {g a..g b} x) \<partial>lborel = \<integral>\<^sup>+ x. ennreal (- f (g x) * g' x * indicat_real {a..b} x) \<partial>lborel"
chaum_ped_asymp_def: fixes \<G> :: "nat \<Rightarrow> 'a cyclic_group" shows "chaum_ped_asymp \<G> \<equiv> \<forall>\<eta>. chaum_ped_\<Sigma> (\<G> \<eta>)"
tauChainRepl: fixes P :: "ccs"   and P' :: "ccs" assumes "P \<parallel> !P \<Longrightarrow>\<^sub>\<tau> P'"   and "P' \<noteq> P \<parallel> !P" shows "!P \<Longrightarrow>\<^sub>\<tau> P'"
simps_3: fixes uu :: "('e set \<times> 'e set) list"   and v :: "('f, 'g, 'e) prot_fun"   and va :: "(('f, 'g, 'e) prot_fun, 'h) Term.term list"   and vb :: "'h" shows "timpls_transformable_to' uu (Fun v va) (Var vb) = False"
snd_grab_subset: fixes N :: "nat set"   and n :: "nat" shows "snd (grab N n) \<subseteq> N"
a_monotone: fixes n :: "nat" shows "a (Suc n) \<le> a n"
env_restr_eq_subset: fixes S :: "'d set"   and S' :: "'d set"   and m1 :: "'d \<Rightarrow> 'e"   and m2 :: "'d \<Rightarrow> 'e" assumes "S \<subseteq> S'"   and "m1 f|` S' = m2 f|` S'" shows "m1 f|` S = m2 f|` S"
intros: fixes x :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "x \<in> S"   and "\<forall>x'\<in>S. f x \<le> f x'" shows "x \<in> least_map f S"
ID_Some: fixes x :: "'a" shows "ID (Some x) = Some x"
dist_les_1: fixes cfun :: "ValD \<rightarrow> ValD" shows "ValF\<cdot>cfun \<notsqsubseteq> ValTT"
normal_changes: fixes f :: "real poly"   and x :: "real" assumes "normal_poly f"   and "0 < x" shows "changes (coeffs (f * [:- x, 1:])) = 1"
pre_poly_lift_hom_is_hom: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e" assumes "cring S"   and "\<phi> \<in> ring_hom R S" shows "ring_hom_ring R (UP S) (to_polynomial S \<circ> \<phi>)"
dense: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "\<exists>z>x. z < y"
freshen_closed: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and f :: "'b"   and a :: "nat" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP"   and "lconsts f = {}" shows "Abstract_Formulas.freshen freshenLC renameLCs a f = f"
in_Res_not_congruent: fixes g :: "'a \<Rightarrow> 'a set"   and x :: "'a"   and b :: "'a"   and y :: "'a" assumes "Complete_set_residues g"   and "x \<in> g b"   and "y \<in> g b"   and "x \<noteq> y" shows "\<not> cong x y b"
simps_2: fixes k :: "nat"   and u :: "nat" shows "\<Z> k (Suc u) = \<Y> (\<Z> k u) (\<P> k u) k"
eqI_snth: fixes u :: "'a stream"   and v :: "'a stream" assumes "\<And>i. u !! i = v !! i" shows "u = v"
disjoint_insert_2: fixes A :: "'b set"   and b :: "'b"   and B :: "'b set" shows "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
Ball_Collect: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
dropWhile_replicate_append: fixes a :: "'b"   and n :: "nat"   and ys :: "'b list" shows "dropWhile ((=) a) (replicate n a @ ys) = dropWhile ((=) a) ys"
bounded_lattice_bot_pair_ow_def: fixes U\<^sub>1 :: "'a set"   and inf\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup\<^sub>1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot\<^sub>1 :: "'a"   and U\<^sub>2 :: "'b set"   and inf\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and le\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ls\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup\<^sub>2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot\<^sub>2 :: "'b" shows "bounded_lattice_bot_pair_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 bot\<^sub>1 U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 bot\<^sub>2 \<equiv> bounded_lattice_bot_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1 sup\<^sub>1 bot\<^sub>1 \<and> bounded_lattice_bot_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2 bot\<^sub>2"
add_diff_cancel_right': fixes a :: "'a"   and b :: "'a" shows "a \<oplus>\<^sub>a b \<ominus>\<^sub>a b = a"
wt: shows "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s"
Lim_emeasure_decseq: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "range A \<subseteq> sets M"   and "decseq A"   and "\<And>i. emeasure M (A i) \<noteq> \<infinity>" shows "(\<lambda>i. emeasure M (A i)) \<longlonglongrightarrow> emeasure M (\<Inter> (range A))"
Sup_inners_def: fixes X :: "real list set"   and y :: "real list" shows "Sup_inners X y = SPEC (\<lambda>r. \<forall>x\<in>X. inner_lv_rel x y \<le> r)"
box_ne_empty_2: fixes a :: "'b"   and b :: "'b" shows "(box a b \<noteq> {}) = (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
H_seq: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "H p x r"   and "H r y q" shows "H p (x \<cdot> y) q"
iso_\<phi>\<^sub>0: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"   and "ide f" shows "local.iso ((d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f])"
add_poly_p'_add_poly_spec: fixes p :: "(char list multiset \<times> int) multiset"   and p' :: "int mpoly"   and q :: "(char list multiset \<times> int) multiset"   and q' :: "int mpoly" assumes "(p, p') \<in> mset_poly_rel"   and "(q, q') \<in> mset_poly_rel" shows "add_poly_p' p q \<le> \<Down> mset_poly_rel (add_poly_spec p' q')"
finite_varsL: fixes l :: "('c, 'd) lit" shows "finite (varsL l)"
assms: fixes i :: "nat" assumes "\<And>j. j \<le> i \<Longrightarrow> \<phi> [sdrop j \<pi>, sdrop j \<pi>']" shows "\<psi> [sdrop i \<pi>, sdrop i \<pi>']"
aux_lemma: fixes k :: "nat" shows "(p ^^ (least_power p a * k)) a = a"
cdens_ctxt_invarD_3: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>" shows "\<Gamma> \<turnstile>\<^sub>c \<delta> : REAL"
MutuallyDistinct_HA: fixes HA :: "('a, 'b, 'c) hierauto" shows "MutuallyDistinct (SAs HA)"
closed_scaling: fixes S :: "'a set"   and c :: "real" assumes "closed S" shows "closed ((*\<^sub>R) c ` S)"
minus_hmultiset_def: fixes A :: "hmultiset"   and B :: "hmultiset" shows "A - B = HMSet (hmsetmset A - hmsetmset B)"
basis_card_eq_dim: fixes B :: "'b set"   and V :: "'b set" assumes "B \<subseteq> V"   and "V \<subseteq> span B"   and "independent B" shows "card B = dim V"
head_\<omega>_numeral: fixes n :: "num" shows "head_\<omega> (numeral n) = 1"
Sup_upper: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> \<Squnion> A"
term_subst_eq_conv: fixes t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and \<tau> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "(t \<cdot> \<sigma> = t \<cdot> \<tau>) = (\<forall>x\<in>fv t. \<sigma> x = \<tau> x)"
better_LAnd1_intro: fixes a :: "coname"   and b1 :: "coname"   and M1 :: "trm"   and b2 :: "coname"   and M2 :: "trm"   and y :: "name"   and x :: "name"   and N :: "trm" assumes "a \<sharp> ([b1].M1, [b2].M2)"   and "y \<sharp> [x].N" shows "Cut <a>.AndR <b1>.M1 <b2>.M2 a y.AndL1 x.N y \<longrightarrow>\<^sub>l Cut <b1>.M1 x.N"
Collect_conv_if: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "_Collect x (x = a \<and> P x) = (if P a then {a} else {})"
statically_complete: fixes B :: "'g"   and N :: "'g set" assumes "B \<in> Bot_G"   and "lifted_q_calc.ground.saturated N"   and "entails_q q N {B}" shows "\<exists>B'\<in>Bot_G. B' \<in> N"
inline1_bind_gpv: fixes gpv :: "('d, 'call, 'ret) gpv"   and f :: "'d \<Rightarrow> ('c, 'call, 'ret) gpv"   and s :: "'s" shows "local.inline1 (gpv \<bind> f) s = local.inline1 gpv s \<bind> (\<lambda>res. case res of Inl (x, s') \<Rightarrow> local.inline1 (f x) s' | Inr (out, rpv', rpv) \<Rightarrow> return_spmf (Inr (out, rpv', rpv \<bind> f)))"
Re_quat_of_real: fixes a :: "real" shows "quat.Re (quat_of_real a) = a"
Res_in_all_clauses: fixes S :: "'b literal set set" shows "res S \<union> S \<subseteq> all_clauses S"
IH_1: shows "\<forall>ys. length ys < length sl \<longrightarrow> wffp ys \<longrightarrow> purgeIdle ys \<noteq> [] \<longrightarrow> wffp (purgeIdle ys)"
assms: shows "(x, y) = split_n_m_x (dim_vec c) (dim_vec b) X"
c_insert_is_pr: shows "c_insert \<in> PrimRec2"
vpair_neq_fst: fixes a :: "V"   and b :: "V" shows "\<langle>a, b\<rangle> \<noteq> a"
hyps: shows "sign_r_pos ([:- a, 1:] ^ n) a"
is_chainD: fixes f :: "nat \<Rightarrow> 'b set"   and x :: "'b"   and m :: "nat"   and n :: "nat" assumes "is_chain f"   and "x \<in> f m" shows "x \<in> f (m + n)"
that: fixes B :: "real" assumes "0 < B"   and "\<And>x. x \<in> {- pi..pi} - {- d<..<d} \<Longrightarrow> \<bar>inverse (sin (x / 2))\<bar> \<le> B" shows "thesis"
distinct_23: fixes x3 :: "'c" shows "NotifyAll x3 \<noteq> WokenUp"
copysign_1_0_2: shows "copysign_double 1 (- 0) = - 1"
vdiffI: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A"   and "x \<notin>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B"
dom_subdist: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set" shows "rdom x \<subseteq> rdom (x \<union> y)"
simps_1: fixes s :: "'s"   and s' :: "'s" shows "local.path s [] s' = (s' = s)"
up_mult_closed: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> up R"   and "q \<in> up R" shows "(\<lambda>n. \<Oplus>i\<in>{..n}. p i \<otimes> q (n - i)) \<in> up R"
less_eq_poly_def: fixes x :: "'a poly"   and y :: "'a poly" shows "(x \<le> y) = (x = y \<or> pos_poly (y - x))"
induct: fixes P :: "int \<times> int \<Rightarrow> poly \<Rightarrow> bool"   and a0 :: "int \<times> int"   and a1 :: "poly" assumes "\<And>y x. P y (C x)"   and "\<And>y c n p. \<lbrakk>P y c; P y p\<rbrakk> \<Longrightarrow> P y (CN c n p)"   and "\<And>y v. P y (Bound v)"   and "\<And>y v va. P y (Add v va)"   and "\<And>y v va. P y (Sub v va)"   and "\<And>y v va. P y (Mul v va)"   and "\<And>y v. P y (Neg v)"   and "\<And>y v va. P y (Pw v va)" shows "P a0 a1"
atLeastLessThan_empty_iff2: fixes a :: "'a"   and b :: "'a" shows "({} = {a..<b}) = (\<not> a < b)"
sort_poly_spec_def: fixes p :: "(char list list \<times> int) list" shows "sort_poly_spec p = SPEC (\<lambda>p'. mset p = mset p' \<and> sorted_wrt (rel2p (Id \<union> term_order_rel)) (map fst p'))"
vcomp_rel_vid_on: fixes r :: "V"   and A :: "V" shows "r \<circ>\<^sub>\<circ> vid_on A = r \<restriction>\<^sup>l\<^sub>\<circ> A"
VA: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and \<omega> :: "(real, 'a) vec \<times> (real, 'a) vec" shows "Vagree \<nu> \<omega> (- UNIV)"
positional_strategy_induction_step: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c" assumes "ParityGame G"   and "v \<in> V\<^bsub>G\<^esub>"   and "\<And>v. v \<in> V\<^bsub>G\<^esub> \<Longrightarrow> \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)"   and "\<And>G v. \<lbrakk>card (\<omega>\<^bsub>G\<^esub> ` V\<^bsub>G\<^esub>) < card (\<omega>\<^bsub>G\<^esub> ` V\<^bsub>G\<^esub>); v \<in> V\<^bsub>G\<^esub>; ParityGame G; \<And>v. v \<in> V\<^bsub>G\<^esub> \<Longrightarrow> \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)\<rbrakk> \<Longrightarrow> \<exists>p. v \<in> ParityGame.winning_region G p" shows "\<exists>p. v \<in> ParityGame.winning_region G p"
fold_A_min_gallery_from1_map: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set"   and Fs :: "'b set list" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "SimplicialComplex.maxsimp X F"   and "F \<in> (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<turnstile> A"   and "SimplicialComplex.min_maxsimpchain X (C # Fs @ [F])" shows "SimplicialComplex.min_maxsimpchain X (C # (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) \<Turnstile> Fs @ [F])"
cases: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" assumes "ord.lexordp_eq less a1 a2" obtains ys :: "'a list" where "a1 = []" and "a2 = ys__"   | x :: "'a" and y :: "'a" and xs :: "'a list" and ys :: "'a list" where "a1 = x__ # xs__" and "a2 = y__ # ysa__" and "less x__ y__"   | x :: "'a" and y :: "'a" and xs :: "'a list" and ys :: "'a list" where "a1 = x # xs" and "a2 = y # ys" and "\<not> less x y" and "\<not> less y x" and "ord.lexordp_eq less xs ys"
filt_def: shows "filt = case_option False (\<lambda>g. dvd_int_poly_non_0 g P \<and> 1 \<le> degree g)"
unit_preserving_hom_axioms: shows "unit_preserving_hom hom"