hcju/isabelleps huggingface dataset:
LT17: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) = ((F \<leadsto> H) \<and> (G \<leadsto> H))"
length_zip: fixes xs :: "'a list"   and ys :: "'b list" shows "length (xs || ys) = min (length xs) (length ys)"
2: shows "minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
State_IdleI: fixes localState :: "'d \<Rightarrow> 'e"   and state :: "'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and output_fun :: "'d \<Rightarrow> 'g message_af" assumes "\<And>c. localState c = state \<Longrightarrow> localState (trans_fun NoMsg c) = state \<and> output_fun (trans_fun NoMsg c) = NoMsg" shows "State_Idle localState output_fun trans_fun state"
induced_vector_indV: fixes v :: "'v" assumes "v \<in> V" shows "induced_vector v \<in> indV"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. \<lbrakk>\<not> y < x \<Longrightarrow> P xs (y # ys); \<not> \<not> y < x \<Longrightarrow> P (x # xs) ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
emsapss_encode_help3_def: fixes H :: "bit list"   and emBits :: "nat" shows "emsapss_encode_help3 H emBits = emsapss_encode_help4 (generate_PS emBits (length H)) H emBits"
equiv_Worder: fixes T :: "('e, 'f) Order_scheme" assumes "Order T"   and "ord_equiv D T" shows "Worder T"
maximalfilters_def: shows "maximalfilters = {F \<in> filters. \<forall>A. A \<in> filters \<and> F \<subseteq> A \<longrightarrow> A = F \<or> A = UNIV}"
polyneg: fixes bs :: "'b list"   and p :: "'b poly" shows "Ipoly bs (~\<^sub>p p) = - Ipoly bs p"
inj_on_mult: 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 a :: "'a"   and A :: "'a set" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "inj_on (times a) A"
diff_add_mset_swap: fixes b :: "'b"   and A :: "'b multiset"   and M :: "'b multiset" assumes "b \<notin># A" shows "add_mset b M - A = add_mset b (M - A)"
key1_issued: fixes s :: "state"   and k :: "key"   and k' :: "key"   and g :: "guest" assumes "s \<in> reach"   and "(k, k') \<in> cards s g" shows "k \<in> issued s"
valid_c_offending_flows: fixes G :: "'v graph" shows "c_offending_flows m G = {F. F \<subseteq> edges G \<and> \<not> c_sinvar m G \<and> c_sinvar m (delete_edges G F) \<and> (\<forall>(e1, e2)\<in>F. \<not> c_sinvar m (add_edge e1 e2 (delete_edges G F)))}"
eqpoll_trans: fixes A :: "'c set"   and B :: "'d set"   and C :: "'e set" assumes "A \<approx> B"   and "B \<approx> C" shows "A \<approx> C"
simps: fixes M :: "('b, 'c) model"   and g :: "'d \<Rightarrow> 'b"   and ps :: "('c, 'd) fm list"   and i :: "'d" shows "(M, g \<Turnstile>\<^sub>B (ps, i)) = (\<forall>p. p on (ps, i) \<longrightarrow> M, g, g i \<Turnstile> p)"
gr0_implies_Suc: fixes n :: "nat" assumes "0 < n" shows "\<exists>m. n = Suc m"
reds_hext_incr: fixes es :: "char list exp list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "char list exp list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "P \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>" shows "h \<unlhd> h'"
nonempty_set_star_has_least: fixes S :: "nat set star" assumes "Iset S \<noteq> {}" shows "\<exists>n\<in>Iset S. \<forall>m\<in>Iset S. n \<le> m"
simps_3: fixes v :: "nat" shows "prev (Suc (Suc v)) 0 = Some (Suc v, Suc v - Suc 0)"
wt_App_plus_\<delta>\<^sub>h_ge: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) + eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef s 0)) * eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t)) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (App s t))) + zhmset_of \<delta>\<^sub>h"
IH: fixes \<nu> :: "variable \<Rightarrow> real" assumes "Uvariation \<nu> (F 0) (U \<union> {RVar x, DVar x})" shows "term_sem I (the (usubstappt \<sigma> (U \<union> {RVar x, DVar x}) \<theta>)) \<nu> = term_sem (USubst.adjoint \<sigma> I (F 0)) \<theta> \<nu>"
monadic_FOREACH_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and c :: "'c \<Rightarrow> bool nres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>0 :: "'c" shows "monadic_FOREACH R \<Phi> S c f \<sigma>0 \<equiv> ASSERT (finite S) \<bind> (\<lambda>_. it_to_sorted_list R S \<bind> (\<lambda>xs0. REC\<^sub>T (\<lambda>W (xs, \<sigma>). ASSERT (FOREACH_inv xs0 \<Phi> (xs, \<sigma>)) \<bind> (\<lambda>_. if xs \<noteq> [] then c \<sigma> \<bind> (\<lambda>b. if b then FOREACH_body f (xs, \<sigma>) \<bind> W else RETURN (xs, \<sigma>)) else RETURN (xs, \<sigma>))) (xs0, \<sigma>0) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>)))"
labels_def: fixes ss :: "'d \<times> ('c \<times> 'd) list" shows "labels ss = map fst (snd ss)"
lfilter_eq_LConsD: fixes P :: "'c \<Rightarrow> bool"   and ys :: "'c llist"   and x :: "'c"   and xs :: "'c llist" assumes "lfilter P ys = LCons x xs" shows "\<exists>us vs. ys = lappend us (LCons x vs) \<and> lfinite us \<and> (\<forall>u\<in>lset us. \<not> P u) \<and> P x \<and> xs = lfilter P vs"
inf_dres_addsimps_1: fixes x :: "'b dres" shows "inf x dSUCCEEDi = dSUCCEEDi"
lookup_mult_vec: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (p ** q) v = lookup (proj_poly (component_of_term v) p * proj_poly (component_of_term v) q) (pp_of_term v)"
simps_6: fixes x61 :: "'c"   and x62 :: "('b, 'c) fm"   and y61 :: "'c"   and y62 :: "('b, 'c) fm" shows "((\<^bold>@ x61 x62) = (\<^bold>@ y61 y62)) = (x61 = y61 \<and> x62 = y62)"
equalOn_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "equalOn A f g = (\<forall>x\<in>A. f x = g x)"
dist: shows "distinct (map fst (step (hd b) (a ! hd b)))"
iIN_pred_insert_conv: fixes n :: "nat"   and d :: "nat" assumes "0 < n" shows "insert (n - Suc 0) [n\<dots>,d] = [n - Suc 0\<dots>,Suc d]"
sd_distr_aux: fixes X :: "('a \<times> 'b) set"   and p :: "'b" shows "\<Squnion>{snd x |x. x \<in> X} \<squnion> \<Squnion>{\<alpha> (fst x) p |x. x \<in> X} = \<Squnion>{snd x \<squnion> \<alpha> (fst x) p |x. x \<in> X}"
sibling_path: fixes G :: "('a, 'b) Graph_scheme"   and root :: "'a"   and v :: "'a"   and w :: "'a" assumes "RootedTree G root"   and "v \<in> V\<^bsub>G\<^esub>"   and "v \<noteq> root"   and "w \<in> V\<^bsub>G\<^esub>"   and "w \<noteq> root"   and "v \<noteq> w"   and "RootedTree.parent G root v = RootedTree.parent G root w" shows "v \<leadsto>\<^bsub>G\<^esub> w = [v, RootedTree.parent G root v, w]"
dres_ne_bot_dWHILEIT: fixes f :: "'b \<Rightarrow> 'b dres"   and I :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "\<And>x. f x \<noteq> dSUCCEED" shows "dWHILEIT I b f s \<noteq> dSUCCEED"
axioms_1: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.comm_monoid G" shows "Group.monoid G"
foldli_transform: fixes iti :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and S0 :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R" shows "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
not_irreducible_one: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero one"
assms_1: shows "A, Suc n \<turnstile> ([], i) # branch"
primitive_iff_separable_lemma: fixes \<chi> :: "nat \<Rightarrow> complex"   and \<Phi> :: "nat \<Rightarrow> complex"   and \<chi>\<^sub>1 :: "nat \<Rightarrow> complex"   and d :: "nat"   and k :: "nat" assumes "(\<forall>n. \<chi> n = \<Phi> n * \<chi>\<^sub>1 n) \<and> primitive_dchar d \<Phi>"   and "1 < d"   and "0 < k"   and "d dvd k"   and "1 < k" shows "(\<Sum>m | m \<in> {1..k} \<and> coprime m k. \<Phi> m * unity_root d (int m)) = of_nat (totient k div totient d) * (\<Sum>m | m \<in> {1..d} \<and> coprime m d. \<Phi> m * unity_root d (int m))"
single_valued_inter1: fixes R :: "('a \<times> 'b) set"   and S :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (R \<inter> S)"
payload_Aenc: fixes X :: "msg"   and Y :: "msg" assumes "Aenc X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
isPath_append: fixes u :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list"   and v :: "nat" shows "isPath u (p1 @ p2) v = (\<exists>w. isPath u p1 w \<and> isPath w p2 v)"
trivial_diagram: fixes \<sigma> :: "'c \<times> ('d \<times> 'c) list"   and ars :: "('c \<times> 'd \<times> 'c) set" assumes "\<sigma> \<in> seq ars" shows "diagram ars (\<sigma>, (fst \<sigma>, []), (lst \<sigma>, []), \<sigma>)"
simps_2: fixes uv :: "'g \<Rightarrow> 'f dres" shows "dSUCCEED \<bind> uv = dSUCCEED"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b list \<Rightarrow> 'a"   and array :: "'b array" shows "P (case array of Array x \<Rightarrow> f x) = (\<forall>x. array = Array x \<longrightarrow> P (f x))"
induct: fixes x1 :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and x2 :: "'a \<Rightarrow> state \<Rightarrow> bool"   and x3 :: "com"   and x4 :: "'a \<Rightarrow> state \<Rightarrow> bool"   and P :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set \<Rightarrow> ('a \<Rightarrow> state \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ('a \<Rightarrow> state \<Rightarrow> bool) \<Rightarrow> bool" assumes "x1 \<turnstile> {x2} x3 {x4}"   and "\<And>C f Pa. P C (\<lambda>z s. \<forall>t\<in>f s. Pa z t) (Do f) Pa"   and "\<And>C Pa c1 Q c2 R. \<lbrakk>C \<turnstile> {Pa} c1 {Q}; P C Pa c1 Q; C \<turnstile> {Q} c2 {R}; P C Q c2 R\<rbrakk> \<Longrightarrow> P C Pa (c1; c2) R"   and "\<And>C Pa b c1 Q c2. \<lbrakk>C \<turnstile> {\<lambda>z s. Pa z s \<and> b s} c1 {Q}; P C (\<lambda>z s. Pa z s \<and> b s) c1 Q; C \<turnstile> {\<lambda>z s. Pa z s \<and> \<not> b s} c2 {Q}; P C (\<lambda>z s. Pa z s \<and> \<not> b s) c2 Q\<rbrakk> \<Longrightarrow> P C Pa (IF b THEN c1 ELSE c2) Q"   and "\<And>C Pa b c. \<lbrakk>C \<turnstile> {\<lambda>z s. Pa z s \<and> b s} c {Pa}; P C (\<lambda>z s. Pa z s \<and> b s) c Pa\<rbrakk> \<Longrightarrow> P C Pa (WHILE b DO c) (\<lambda>z s. Pa z s \<and> \<not> b s)"   and "\<And>C P' c Q' Pa Q. \<lbrakk>C \<turnstile> {P'} c {Q'}; P C P' c Q'; \<forall>s t. (\<forall>z. P' z s \<longrightarrow> Q' z t) \<longrightarrow> (\<forall>z. Pa z s \<longrightarrow> Q z t)\<rbrakk> \<Longrightarrow> P C Pa c Q"   and "\<And>Pa Q. \<lbrakk>{(Pa, CALL, Q)} \<turnstile> {Pa} body {Q}; P {(Pa, CALL, Q)} Pa body Q\<rbrakk> \<Longrightarrow> P {} Pa CALL Q"   and "\<And>Pa Q. P {(Pa, CALL, Q)} Pa CALL Q"   and "\<And>C Pa f c Q g. \<forall>s'. C \<turnstile> {\<lambda>z s. Pa z s' \<and> s = f s'} c {\<lambda>z t. Q z (g s' t)} \<and> P C (\<lambda>z s. Pa z s' \<and> s = f s') c (\<lambda>z t. Q z (g s' t)) \<Longrightarrow> P C Pa (LOCAL f; c; g) Q" shows "P x1 x2 x3 x4"
all_partitions_equivalence': fixes xs :: "'b list" assumes "distinct xs" shows "set (map set (all_partitions_list xs)) = all_partitions (set xs) \<and> (\<forall>ps\<in>set (all_partitions_list xs). distinct ps)"
set_conv_nth: fixes xs :: "'g list" shows "set xs = {xs ! i |i. i < length xs}"
borel_measurable_add: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x + g x) \<in> borel_measurable M"
in_finite_psubset: fixes A :: "'b set"   and B :: "'b set" shows "((A, B) \<in> finite_psubset) = (A \<subset> B \<and> finite B)"
Abs_eq_cong: fixes x :: "'a"   and x' :: "'a"   and y :: "'b"   and y' :: "'b"   and a :: "atom list"   and a' :: "atom list" assumes "\<And>p. (p \<bullet> x = x') = (p \<bullet> y = y')"   and "supp y = supp x"   and "supp y' = supp x'" shows "([a]lst. x = [a']lst. x') = ([a]lst. y = [a']lst. y')"
ker_orthogonal_comp_adjoint: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "f -` {0::'b} = range (adjoint f)\<^sup>\<bottom>"
swap_len: fixes u :: "'b list"   and v :: "'b list" shows "\<^bold>|u \<cdot> v\<^bold>| = \<^bold>|v \<cdot> u\<^bold>|"
thr_conv: fixes ls :: "'b"   and ts :: "'a"   and m :: "'c"   and ws :: "'d \<times> 'e" shows "thr (ls, (ts, m), ws) = ts"
dir_image_alt: fixes r :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'd" shows "dir_image r f = map_prod f f ` r"
finsum_cong': fixes G :: "('h, 'i) ring_scheme"   and A :: "'j set"   and B :: "'j set"   and g :: "'j \<Rightarrow> 'h"   and f :: "'j \<Rightarrow> 'h" assumes "abelian_monoid G"   and "A = B"   and "g \<in> B \<rightarrow> carrier G"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum G f A = finsum G g B"
18: shows "(w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f \<le> top * selected_edge h j g\<^sup>T * ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - ((w \<sqinter> - (top * selected_edge h j g * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * selected_edge h j g * w\<^sup>T\<^sup>\<star>)\<^sup>T) \<sqinter> - forest_components f * selected_edge h j g * top \<sqinter> top * selected_edge h j g\<^sup>T * forest_components f) \<squnion> selected_edge h j g)\<^sup>T\<^sup>\<star>"
P1: shows "Col A B M0 \<and> A B Perp X M0"
w1_def: shows "w1 = get_lower_bound (fst ux) (fst vy)"
eval_Assoc': fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Arr t"   and "Arr u"   and "Arr v"   and "Src t = Trg u"   and "Src u = Trg v" shows "\<lbrace>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<rbrace> = \<a>\<^sup>-\<^sup>1[cod \<lbrace>t\<rbrace>, cod \<lbrace>u\<rbrace>, cod \<lbrace>v\<rbrace>] \<cdot> (\<lbrace>t\<rbrace> \<star> \<lbrace>u\<rbrace> \<star> \<lbrace>v\<rbrace>)"
lower_round_ivl: fixes p :: "nat"   and x :: "float interval" shows "lower (round_interval p x) = float_round_down p (lower x)"
le_t_stop_iff: fixes x :: "real" assumes "a\<^sub>o \<noteq> 0" shows "(x \<le> other.t_stop) = (if a\<^sub>o < 0 then 0 \<le> other.p' x else other.p' x \<le> 0)"
well_foundedI_extremal: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>X. \<forall>y\<in>X. \<not> less y x" shows "well_founded A less"
stieltjes_gamma_aux3: shows "G2 = fps_X * G2' + 1"
constant_square_free_factorization: fixes p :: "'a poly" assumes "degree p = 0" shows "square_free_factorization p (poly.coeff p 0, [])"
i_append_i_take_if: fixes n :: "nat"   and xs :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "(xs \<frown> f) \<Down> n = (if n \<le> length xs then xs \<down> n else xs @ f \<Down> (n - length xs))"
seqll_def: fixes i :: "'e"   and P :: "('f \<times> 'g) \<times> 'h \<times> 'f \<times> 'g \<Rightarrow> bool" shows "seqll i P \<equiv> \<lambda>((\<sigma>, p), a, \<sigma>', p'). P ((\<sigma> i, p), a, \<sigma>' i, p')"
bi_unique_alt_def: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "bi_unique A = (left_unique A \<and> right_unique A)"
basis_expansion_unique: fixes f :: "'d \<Rightarrow> 'c"   and x :: "('c, 'd) vec" shows "((\<Sum>i\<in>UNIV. f i *s axis i (1::'c)) = x) = (\<forall>i. f i = x $ i)"
llist_of_inject: fixes xs :: "'a list"   and ys :: "'a list" shows "(llist_of xs = llist_of ys) = (xs = ys)"
mute_fun_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b" shows "mute_fun f = (\<forall>x y x' y' g. f x y = (x', y', g) \<longrightarrow> y \<noteq> (0::'b) \<longrightarrow> x = x' * g \<and> y * x' = x * y')"
DSTEP_def: fixes G :: "('c \<times> ('c, 'd) RHS) list" shows "DSTEP G = {uu_. \<exists>l N r rhs x. uu_ = (l \<cdot> [Inl N] \<cdot> r, x) \<and> (N, rhs) \<in> set G \<and> (case rhs of Branch A B \<Rightarrow> x = l \<cdot> [Inl A, Inl B] \<cdot> r | Leaf t \<Rightarrow> x = l \<cdot> [Inr t] \<cdot> r)}"
euclid_ext_aux_eq_gcd: fixes r' :: "'a"   and r :: "'a"   and a :: "'a"   and b :: "'a"   and s' :: "'a"   and t' :: "'a"   and s :: "'a"   and t :: "'a"   and x :: "'a"   and y :: "'a"   and c :: "'a" assumes "gcd r' r = gcd a b"   and "s' * a + t' * b = r'"   and "s * a + t * b = r"   and "euclid_ext_aux s' s t' t r' r = ((x, y), c)" shows "c = gcd a b"
induced_automorphism_eq_foldings_on_chambers_fg: fixes C :: "'a set" assumes "C \<in> f \<turnstile> folding_g.\<C>" shows "fun_eq_on \
 g C"
real_of_ereal_le_1: fixes a :: "ereal" assumes "a \<le> 1" shows "real_of_ereal a \<le> 1"
aadd_0_r: fixes x :: "ant" shows "x + 0 = x"
unop_pdevs_raw_def: fixes f :: "'b \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "unop_pdevs_raw f x i = (if x i = (0::'b) then 0::'a else f (x i))"
prems: shows "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (a # A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)"
emeasure_intersection: fixes U :: "nat \<Rightarrow> 'a set"   and e :: "nat \<Rightarrow> real" assumes "\<And>n. U n \<in> events"   and "\<And>n. 0 \<le> e n"   and "summable e"   and "\<And>n. ennreal (1 - e n) \<le> emeasure M (U n)" shows "ennreal (1 - (\<Sum>n. e n)) \<le> emeasure M (\<Inter> (range U))"
cases: fixes x :: "(bool list cstate \<times> (bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)) \<times> bool list" obtains es :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and m :: "bool list" where "x = ((Void, es), m__)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and m :: "bool list" where "x = ((cstate.Fail, va__), ma__)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and m :: "bool list" where "x = ((Store vb__, vaa__), mb__)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and m :: "bool list" where "x = ((cstate.Collect vb, va), m)"
simps: fixes a1 :: "'b"   and a2 :: "'b" shows "((a1, a2) \<in> hull_rel) = (\<exists>p x. a1 = p \<bullet> x \<and> a2 = x)"
exhaust: fixes y :: "('g, 'f) label" obtains x1 :: "'f" where "y = LBase x1"   | x2 :: "'g" where "y = LCall x2"   | "y = LRet"   | x4 :: "'g" where "y = LSpawn x4"
f_eqI: fixes a :: "real"   and b :: "real"   and m :: "real"   and x :: "real"   and y :: "real" assumes "leftcut a b m \<le> x"   and "x \<le> rightcut a b m"   and "leftcut a b m \<le> y"   and "y \<le> rightcut a b m"   and "a \<le> m"   and "m \<le> b"   and "{a..b} \<subseteq> {0..1}" shows "f x = f y"
rwof_reachable: fixes init :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "rwof init cond step s" shows "\<exists>s0. inres init s0 \<and> (s0, s) \<in> (rwof_rel init cond step)\<^sup>*"
strategyproofness: shows "(pmf (sds R9) a < pmf (sds R35) a \<or> pmf (sds R9) a + pmf (sds R9) b < pmf (sds R35) a + pmf (sds R35) b) \<or> pmf (sds R9) a = pmf (sds R35) a \<and> pmf (sds R9) a + pmf (sds R9) b = pmf (sds R35) a + pmf (sds R35) b"
Sum_pmf_def: fixes p :: "real"   and Da :: "'c pmf"   and Db :: "'d pmf" shows "Sum_pmf p Da Db = bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)"
floor_mono: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
normalize_summation: fixes B :: "real"   and f :: "nat \<Rightarrow> real"   and n :: "nat" assumes "B \<noteq> 0" shows "((\<Sum>i = 0..<n + 1. f i * B ^ (n - i)) = 0) = ((\<Sum>i = 0..<n + 1. f i / B ^ i) = 0)"
negated_euclidean_axiom_aux: fixes H :: "circline" assumes "on_circline H (of_complex (1 / 2 + \<i> / 2))"   and "is_poincare_line H"   and "intersects_x_axis_positive H" shows "\<not> intersects_y_axis_positive H"
IH2: shows "(0::'a) \<le> eval_monom_list g m"
same_suffix_nil: fixes ys :: "'c list"   and xs :: "'c list" shows "suffix (ys @ xs) xs = (ys = [])"
mkPoint_img_1: fixes a :: "'s" shows "img \<in> hom unity a \<rightarrow> local.set a"
maximal_independent_subset: fixes V :: "'d set" obtains B :: "'d set" where "B \<subseteq> V" and "cindependent B" and "V \<subseteq> cspan B"
drop_zip: fixes n :: "nat"   and xs :: "'b list"   and ys :: "'c list" shows "drop n (zip xs ys) = zip (drop n xs) (drop n ys)"
pred_mono_strong: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g option"   and Pa :: "'g \<Rightarrow> bool" assumes "pred_option P x"   and "\<And>z. \<lbrakk>z \<in> set_option x; P z\<rbrakk> \<Longrightarrow> Pa z" shows "pred_option Pa x"
boolfun_cases_helper: shows "g = (\<lambda>x. False) \<or> g = (\<lambda>x. x) \<or> g = (\<lambda>x. True) \<or> g = Not"
le_transitivity: 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.Le Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Le Bet Cong C D E F" shows "Tarski_neutral_dimensionless.Le Bet Cong A B E F"
assms_1: shows "\<Gamma>(chain \<mapsto> rs'),\<gamma>,p\<turnstile> \<langle>rs', Undecided\<rangle> \<Rightarrow> Undecided"
vector_affinity_eq: fixes m :: "'a"   and x :: "('a, 'b) vec"   and c :: "('a, 'b) vec"   and y :: "('a, 'b) vec" assumes "m \<noteq> (0::'a)" shows "(m *s x + c = y) = (x = inverse m *s y + - (inverse m *s c))"
G_def: shows "G \<equiv> the_inv_into (ball (x, y) r) H"
S_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and base :: "'c"   and elements :: "'d list"   and n :: "nat" shows "subseqs_impl.S f base elements n = {(foldr f ys base, ys) |ys. subseq_of_length n elements ys}"
step_thread_NoneD: fixes update_state :: "('l, 't, 'x, 'm, 'w, 'o) thread_action \<Rightarrow> 'a"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "step_thread update_state s t = None" shows "t \<notin> active_threads s"
exhaust: fixes y :: "'b discr" obtains x :: "'b" where "y = Discr x"
Affine_def: fixes X :: "'a \<times> 'a pdevs" shows "Affine X = valuate (\<lambda>e. aform_val e X)"
FIXvar: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "PhiWhile b \<Phi> (s, t)" shows "(b, \<Phi>, s, t) \<in> var"
sub: shows "P \<turnstile> D \<preceq>\<^sup>* C"
distinct_3: fixes x1 :: "nat"   and x3 :: "nat" shows "Const x1 \<noteq> Greater x3"