hcju/isabelleps huggingface dataset:
vsv_ex1_app1: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" shows "(b = (r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A)"
min_set_obtains_helper: fixes A :: "'b fset"   and B :: "'b fset set" assumes "A \<in> B" shows "\<exists>C. C |\<subseteq>| A \<and> C \<in> min_set B"
not_bot: fixes A :: "'a Predicate.pred" assumes "A \<noteq> \<bottom>" obtains x :: "'a" where "pred.eval A x"
mat_of_vec_mult_vec: fixes a :: "'c vec"   and d :: "nat"   and b :: "'c vec" assumes "a \<in> carrier_vec d"   and "b \<in> carrier_vec d" shows "mat 1 d (\<lambda>(i, j). conjugate a $ j) *\<^sub>v b = vec 1 (\<lambda>k. inner_prod a b)"
less_eq_dual_def_var: fixes x :: "'b dual"   and y :: "'b dual" shows "(x \<le> y) = (\<partial>\<^sup>- y \<le> \<partial>\<^sup>- x)"
adds_canc: fixes s :: "'a"   and u :: "'a"   and t :: "'a" shows "(s + u adds t + u) = (s adds t)"
complex_homo_coords_eq_refl: fixes z :: "complex_homo_coords" shows "z \<approx> z"
comp_arr_dom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f"   and "local.dom f = a" shows "local.comp f a = f"
eval_Or: fixes a :: "atom fm"   and b :: "atom fm"   and L :: "real list" shows "eval (Or a b) L = (eval a L \<or> eval b L)"
norm_conv_dist: fixes x :: "'d" shows "norm x = dist x (0::'d)"
multiplicative_function_divide: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "multiplicative_function f"   and "multiplicative_function g" shows "multiplicative_function (\<lambda>n. f n / g n)"
antisym_conv3: fixes y :: "char"   and x :: "char" assumes "\<not> less_char y x" shows "(\<not> less_char x y) = (x = y)"
lookup_except_min': fixes ko :: "'c nat_term_order"   and xs :: "('c, 'b) oalist_ntm"   and k :: "'c" shows "OAlist_lookup_ntm (OAlist_except_min_ntm ko xs) k = (if k = fst (OAlist_min_key_val_ntm ko xs) then 0::'b else OAlist_lookup_ntm xs k)"
rk2_fas'_def: fixes D :: "nat" shows "rk2_fas' D = map fold_const_fa (rk2_fas (floatarith.Var (2 * D)) (map floatarith.Var [0..<D]) (floatarith.Var (2 * D + 1)) (map floatarith.Var [D..<2 * D]) (floatarith.Var (2 * D + 2)))"
simps_12: fixes x21 :: "'a"   and x22 :: "'a form list" shows "Em \<noteq> Compound x21 x22"
takefill_Suc_cases_2: fixes fill :: "'b"   and n :: "nat"   and x21 :: "'b"   and x22 :: "'b list" shows "takefill fill (Suc n) (x21 # x22) = x21 # takefill fill n x22"
assms: shows "F \<turnstile>\<^sub>F \<Phi>"
not_distinct_map: fixes a :: "'a \<times> 'b \<times> 'a"   and l :: "('a \<times> 'b \<times> 'a) list"   and b :: "'a \<times> 'b \<times> 'a" assumes "a \<in> set l"   and "b \<in> set l"   and "a \<noteq> b"   and "\<alpha> a = \<alpha> b" shows "\<not> distinct (map \<alpha> l)"
cost_simps_1: fixes a :: "'a"   and c :: "nat" shows "cost (a, c) = c"
orthogonal_self: fixes x :: "'b" shows "orthogonal x x = (x = (0::'b))"
ext_per_sum: fixes w :: "'b list"   and p :: "nat"   and q :: "nat" assumes "periodN w p"   and "periodN w q"   and "p \<le> \<^bold>|w\<^bold>|" shows "periodN (take p w \<cdot> w) (p + q)"
approx_scaleR_HFinite: fixes a :: "real star"   and b :: "real"   and c :: "'c star"   and d :: "'c star" assumes "a \<approx> hypreal_of_real b"   and "c \<approx> d"   and "d \<in> HFinite" shows "scaleHR a c \<approx> b *\<^sub>R d"
exec_block: shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> \<Rightarrow> t"
T_on'_embed: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s0 :: "'a list"   and x :: "'b"   and qs :: "'a list" shows "real (Partial_Cost_Model.T_on' A (s0, x) qs) = T\<^sub>p_on_rand' (Partial_Cost_Model.embed A) (return_pmf (s0, x)) qs"
induct: fixes x1 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int) \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {x1} x2 {x3}"   and "\<And>P. P P SKIP P"   and "\<And>P a x. P (\<lambda>s. P (s[a/x])) (x ::= a) P"   and "\<And>P\<^sub>1 c\<^sub>1 P\<^sub>2 c\<^sub>2 P\<^sub>3. \<lbrakk>\<turnstile>\<^sub>t {P\<^sub>1} c\<^sub>1 {P\<^sub>2}; P P\<^sub>1 c\<^sub>1 P\<^sub>2; \<turnstile>\<^sub>t {P\<^sub>2} c\<^sub>2 {P\<^sub>3}; P P\<^sub>2 c\<^sub>2 P\<^sub>3\<rbrakk> \<Longrightarrow> P P\<^sub>1 (c\<^sub>1;; c\<^sub>2) P\<^sub>3"   and "\<And>P b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>t {\<lambda>s. P s \<and> bval b s} c\<^sub>1 {Q}; P (\<lambda>s. P s \<and> bval b s) c\<^sub>1 Q; \<turnstile>\<^sub>t {\<lambda>s. P s \<and> \<not> bval b s} c\<^sub>2 {Q}; P (\<lambda>s. P s \<and> \<not> bval b s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>P c b. \<lbrakk>\<And>n. \<turnstile>\<^sub>t {P (Suc n)} c {P n}; \<And>n. P (P (Suc n)) c (P n); \<forall>n s. P (Suc n) s \<longrightarrow> bval b s; \<forall>s. P 0 s \<longrightarrow> \<not> bval b s\<rbrakk> \<Longrightarrow> P (\<lambda>s. \<exists>n. P n s) (WHILE b DO c) (P 0)"   and "\<And>P' P c Q Q'. \<lbrakk>\<forall>s. P' s \<longrightarrow> P s; \<turnstile>\<^sub>t {P} c {Q}; P P c Q; \<forall>s. Q s \<longrightarrow> Q' s\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
cbox_max_dist: fixes p\<^sub>0 :: "real \<times> real"   and x :: "real"   and y :: "real"   and p\<^sub>1 :: "real \<times> real"   and \<delta> :: "real"   and x\<^sub>0 :: "real"   and y\<^sub>0 :: "real"   and x\<^sub>1 :: "real"   and y\<^sub>1 :: "real" assumes "p\<^sub>0 = (x, y)"   and "p\<^sub>1 = (x + \<delta>, y + \<delta>)"   and "(x\<^sub>0, y\<^sub>0) \<in> cbox p\<^sub>0 p\<^sub>1"   and "(x\<^sub>1, y\<^sub>1) \<in> cbox p\<^sub>0 p\<^sub>1"   and "0 \<le> \<delta>" shows "dist (x\<^sub>0, y\<^sub>0) (x\<^sub>1, y\<^sub>1) \<le> sqrt 2 * \<delta>"
congruent_strQ2: shows "(F2map id (Equiv_Relations.proj (lsbis1 carT1 carT2 strT1 strT2)) (Equiv_Relations.proj (lsbis2 carT1 carT2 strT1 strT2)) \<circ> strT2) respects lsbis2 carT1 carT2 strT1 strT2"
safe_distance_no_collsion_delta_axioms_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" shows "safe_distance_no_collsion_delta_axioms a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta> \<equiv> safe_distance_normal.u a\<^sub>e v\<^sub>e s\<^sub>e \<delta> \<delta> < movement.s a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
hd_dens_ctxt_cong: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<delta>' :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f"   and "\<And>\<sigma>. \<sigma> \<in> space (state_measure (V \<union> V') \<Gamma>) \<Longrightarrow> \<delta> \<sigma> = \<delta>' \<sigma>" shows "(V, V', \<Gamma>, \<delta>') \<turnstile>\<^sub>d e \<Rightarrow> f"
YS2_def: shows "YS2 = {#ys \<in># mset (chop d xs). \<not> med ys < x#}"
no_markers_if_all_snapshotted: fixes t :: "('a, 'b, 'c) event llist"   and i :: "nat"   and j :: "nat"   and c :: "nat" assumes "cotrace init t"   and "i \<le> j"   and "\<forall>p. ps (coS t i) p \<noteq> None"   and "Marker \<notin> set (msgs (coS t i) c)" shows "Marker \<notin> set (msgs (coS t j) c)"
convex_on_ereal_sum: fixes s :: "'b set"   and C :: "'c set"   and f :: "'c \<Rightarrow> ereal"   and a :: "'b \<Rightarrow> real"   and y :: "'b \<Rightarrow> 'c" assumes "finite s"   and "s \<noteq> {}"   and "Lower_Semicontinuous.convex_on C f"   and "convex C"   and "sum a s = 1"   and "\<forall>i. i \<in> s \<longrightarrow> 0 \<le> a i"   and "\<forall>i. i \<in> s \<longrightarrow> y i \<in> C" shows "f (\<Sum>i\<in>s. a i *\<^sub>R y i) \<le> (\<Sum>i\<in>s. ereal (a i) * f (y i))"
hyps_1: shows "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b u"
cases: fixes e :: "'c"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and a1 :: "'a"   and a2 :: "'c" assumes "Rec e r a1 a2" obtains "a1 = zero" and "a2 = e"   | m :: "'a" and n :: "'c" where "a1 = succ m" and "a2 = r m n" and "Rec e r m n"
expandAction: fixes P :: "pi"   and S :: "pi set"   and Q :: "pi"   and Rs :: "residual" assumes "(P, S) \<in> sumComposeSet"   and "Q \<in> S"   and "Q \<longmapsto> Rs" shows "P \<longmapsto> Rs"
red_external_imp_red_external_aggr: fixes P :: "'a prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap" assumes "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>" shows "(ta, va, h') \<in> red_external_aggr P t a M vs h"
invar_finish_imp_correct: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and s :: "'a"   and D :: "'a \<Rightarrow> enat"   and S :: "'a set"   and u :: "'a" assumes "Dijkstra_Invar w s D S"   and "\<forall>u. u \<notin> S \<longrightarrow> D u = \<infinity>" shows "D u = WGraph.\<delta> w s u"
divide_divide_times_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c"   and z :: "'c"   and w :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (divide x y) (divide z w) = divide (times x w) (times y z)"
right_unit: fixes a :: "'a" assumes "a \<in> R" shows "a \<cdot> \<one> = a"
equalityD2: fixes A :: "'b set"   and B :: "'b set" assumes "(A::'b set) = (B::'b set)" shows "(B::'b set) \<subseteq> (A::'b set)"
length_append_singleton: fixes xs :: "'c list"   and x :: "'c" shows "length (xs @ [x]) = Suc (length xs)"
simps_3: fixes a :: "'b"   and b :: "'b"   and p :: "'c" shows "bothNet (AllowPortFromTo a b p) = {a, b}"
univalent_O_converse: fixes R :: "('f \<times> 'e) set" assumes "univalent (R\<inverse>)" shows "R O R\<inverse> = Id_on (Domain R)"
c: shows "c = (m + a) mod q \<and> y1 = (a * x1 + b) mod q"
Vinv3I: fixes x :: "v_state" assumes "ran (decisions x) \<subseteq> locked x" shows "x \<in> Vinv3"
transform_simps_3: fixes a :: "Arity"   and x :: "Vars.var" shows "ccTransform a (Var x) = Var x"
select_convs_5: fixes states :: "'b \<Rightarrow> 'c"   and msgs :: "nat \<Rightarrow> 'd message list"   and process_snapshot :: "'b \<Rightarrow> 'c option"   and channel_snapshot :: "nat \<Rightarrow> 'd list \<times> recording_state"   and more :: "'a" shows "more \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> = more"
Collect_mono: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "p2s P \<subseteq> p2s Q"
lincomb_pad_2: fixes U :: "'a vec set"   and n :: "nat"   and m :: "nat"   and a :: "'a vec \<Rightarrow> 'a" assumes "U \<subseteq> carrier_vec n"   and "finite U" shows "0\<^sub>v m @\<^sub>v lincomb n a U = lincomb (n + m) (a \<circ> unpadl m) ((@\<^sub>v) (0\<^sub>v m) ` U)"
pos_norm_vec_def: fixes D :: "'a vec set"   and x :: "'a vec" shows "pos_norm_vec D x = (let c' = normal_vector D; c = if (0::'a) < c' \<bullet> x then c' else - c' in c)"
lvl_skew: fixes t :: "('b \<times> nat) tree" shows "lvl (skew t) = lvl t"
ncf7: fixes y :: "real" assumes "x \<le> y"   and "y \<le> 0" shows "0 < numer_cf7 y"
flop_gallery: fixes Cs :: "'a set list" assumes "folding.gallery Cs" shows "folding.gallery (map (\<lambda>C. if C \<in> f \<turnstile> folding.\<C> then THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C else f ` C) Cs)"
simps_1: shows "subd 0 = (\<lambda>x. 0)"
finfun_update_twice: fixes f :: "('a, 'b) finfun"   and a :: "'a"   and b :: "'b"   and b' :: "'b" shows "finfun_update (finfun_update f a b) a b' = finfun_update f a b'"
approx_sym: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y" shows "y \<approx> x"
fdj: shows "\<forall>p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
ex_sym: fixes A :: "'p"   and B :: "'p"   and X :: "'p" shows "\<exists>Y. (A B Perp X Y \<or> X = Y) \<and> (\<exists>M. Col A B M \<and> M Midpoint X Y)"
fmap_of_list_SomeD: fixes xs :: "('b \<times> 'a) list"   and k :: "'b"   and v :: "'a" assumes "fmlookup (fmap_of_list xs) k = Some v" shows "(k, v) \<in> set xs"
that: fixes r :: "real"   and b :: "complex" assumes "1 / 12 < r"   and "ball b r \<subseteq> f ` ball a 1" shows "thesis"
simps_3: shows "sch_to_pr Base_fst = c_fst"
rt_fresher_trans_rt_strictly_fresher: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt3 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt2"   and "rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt3" shows "rt1 \<sqsubset>\<^bsub>dip\<^esub> rt3"
init_sim2: fixes b0 :: "('d, 'ls, 'gs) global_config" assumes "b0 \<in> init2" shows "\<exists>a0. a0 \<in> init1 \<and> rel_gc a0 b0"
linorder_linear: fixes x :: "'a"   and y :: "'a" shows "x \<le> y \<or> y \<le> x"
connected_root_iff3: fixes r :: "'a"   and x :: "'a" assumes "point r" shows "connected_root r x = (x\<^sup>T ; 1 = x\<^sup>T\<^sup>+ ; r)"
aform_val_consume_error: fixes err :: "real"   and X :: "real \<times> real pdevs"   and n :: "nat"   and e :: "nat \<Rightarrow> real" assumes "\<bar>err\<bar> \<le> \<bar>pdevs_apply (snd X) n\<bar>" shows "aform_val (e(n := 0)) X + err = aform_val (e(n := err / pdevs_apply (snd X) n)) X"
cod_simps_3: shows "HH\<mu>\<nu>\<pi>.cod.leg1 = HH\<mu>\<nu>\<pi>.leg1"
specify_left: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b nres"   and M :: "'b nres" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> f x \<le> M" shows "m \<bind> f \<le> M"
gen2_unique: fixes i :: "nat"   and j :: "nat"   and A :: "nat"   and B :: "nat"   and a :: "nat list"   and b :: "nat list" assumes "i < j"   and "j < length (Algorithm.gen2 A B a b)" shows "Algorithm.gen2 A B a b ! i \<noteq> Algorithm.gen2 A B a b ! j"
integrable_split_1: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and k :: "'a"   and c :: "real" assumes "f integrable_on cbox a b"   and "k \<in> Basis" shows "f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
Adm_def: fixes \<V> :: "'b V_rec"   and \<rho> :: "'b V_rec \<Rightarrow> 'b set"   and Tr :: "'b list set"   and \<beta> :: "'b list"   and e :: "'b" shows "Adm \<V> \<rho> Tr \<beta> e \<equiv> \<exists>\<gamma>. \<gamma> @ [e] \<in> Tr \<and> \<gamma> \<upharpoonleft> \<rho> \<V> = \<beta> \<upharpoonleft> \<rho> \<V>"
simps_3: fixes x1 :: "real" shows "PInfty \<noteq> ereal x1"
assms_1: shows "gf = [g, f]\<^sub>\<circ>"
finite_Collect_conjI: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
ldropn_0: fixes xs :: "'b llist" shows "ldropn 0 xs = xs"
equals_minim_Under: fixes r :: "('a \<times> 'a) set"   and B :: "'a set"   and a :: "'a" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "a \<in> B"   and "a \<in> Under r B" shows "a = minim r B"
intros_1: fixes G :: "prog"   and membr :: "qtname \<times> memberdecl"   and "class" :: "qtname"   and accclass :: "qtname" assumes "G \<turnstile> membr member_in class"   and "G \<turnstile> membr in class permits_acc_from accclass" shows "G \<turnstile> membr in class dyn_accessible_from accclass"
adds_pp_def: fixes t :: "'a"   and v :: "'a" shows "punit.adds_pp t v = (t adds punit.pp_of_term v)"
nd_fun_ext: fixes f :: "'b nd_fun"   and g :: "'b nd_fun" assumes "\<And>x. (f\<^sub>\<bullet>) x = (g\<^sub>\<bullet>) x" shows "f = g"
borel_measurable_safe_ln: shows "safe_ln \<in> borel_measurable borel"
fmdom_notI: fixes m :: "('d, 'c) fmap"   and x :: "'d" assumes "m $$ x = None" shows "x |\<notin>| fmdom m"
card_set: fixes xs :: "'c list" shows "card (set xs) = length (remdups xs)"
det_row_add: fixes k :: "'d"   and a :: "'d \<Rightarrow> ('c, 'd) vec"   and b :: "'d \<Rightarrow> ('c, 'd) vec"   and c :: "'d \<Rightarrow> ('c, 'd) vec" shows "Determinants.det (\<chi>i. if i = k then a i + b i else c i) = Determinants.det (\<chi>i. if i = k then a i else c i) + Determinants.det (\<chi>i. if i = k then b i else c i)"
inject_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Expression"   and x23 :: "'a"   and y21 :: "'a"   and y22 :: "('a, 'b) Expression"   and y23 :: "'a" shows "(Vx : x21 \<turnstile> x22 : x23 = Vx : y21 \<turnstile> y22 : y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
lfp_Fix: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "lfp f = \<Sqinter> Fix f"
cases: fixes x :: "('b \<times> nat) tree" obtains "x = \<langle>\<rangle>"   | uu_ :: "('b \<times> nat) tree" and uv_ :: "'b" and n :: "nat" and uw_ :: "('b \<times> nat) tree" where "x = \<langle>uu_, (uv_, n), uw_\<rangle>"
T_measure_impl_def: shows "T_measure_impl \<equiv> \<lambda>(Qi, \<pi>i). T_measure2 (Q_\<alpha> Qi) (M_lookup \<pi>i)"
expr_var_inj_term: fixes t :: "expr"   and w :: "Term.var" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
repeat_mset_distrib: fixes m :: "nat"   and n :: "nat"   and A :: "'b multiset" shows "repeat_mset (m + n) A = repeat_mset m A + repeat_mset n A"
Units_minus_one_closed: shows "\<ominus> \<one> \<in> Units R"
tendsto_minus_cancel: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter" assumes "((\<lambda>x. - f x) \<longlongrightarrow> - a) F" shows "(f \<longlongrightarrow> a) F"
less_numeral_extra_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one one"
elims_1: fixes x :: "'e list"   and xa :: "'e list set"   and xb :: "'e list"   and y :: "bool" assumes "mcp x xa xb = y" obtains z :: "'e list" and W :: "'e list set" and p :: "'e list" where "x = z" and "xa = W" and "xb = p" and "y = (prefix p z \<and> p \<in> W \<and> (\<forall>p'. prefix p' z \<and> p' \<in> W \<longrightarrow> length p' \<le> length p))"
gSwapInpPresGWlsInp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSwapInpPresGWlsInp MOD \<equiv> \<forall>zs z1 z2 delta inp' inp. wlsInp delta inp' \<and> gWlsInp MOD delta inp \<longrightarrow> gWlsInp MOD delta (gSwapInp MOD zs z1 z2 inp' inp)"
inst: shows "instance_of\<^sub>l\<^sub>s C\<^sub>1' C\<^sub>1 \<and> instance_of\<^sub>l\<^sub>s C\<^sub>2' C\<^sub>2"
left_mult_zero_mat': fixes A :: "'c mat"   and n :: "nat"   and nr :: "nat" assumes "dim_row A = n" shows "0\<^sub>m nr n * A = 0\<^sub>m nr (dim_col A)"
mono_inv_less': fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'a"   and b2 :: "'a"   and bv :: "bool" assumes "a1 \<le> b1"   and "a2 \<le> b2" shows "inv_less' bv a1 a2 \<le> inv_less' bv b1 b2"
matrix_vector_mul_1: fixes g :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "linear (*s) (*s) g" shows "(*v) (matrix g) = g"
hyps: fixes i :: "nat" assumes "(i::nat) < (j::nat)" shows "(b::'a::refinement_lattice) \<^sup>;^ (j::nat) = (b \<^sup>;^ (i::nat)) ; (b \<^sup>;^ (j - i))"
ordertype_UU_\<omega>2: shows "ordertype UU pair_less = \<omega> \<up> ord_of_nat 2"
case_1: fixes f1 :: "'g"   and f2 :: "nat \<Rightarrow> 'g" shows "(case 0 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f1"
derive_interval_2: fixes x :: "'b sort_key extended \<times> 'b sort_key extended"   and i :: "'b" assumes "snd x \<le> \<lbrakk>Final i\<rbrakk>"   and "is_interval x" shows "is_interval (derive_right x)"
tr_ltr: fixes R :: "('a \<times> 'a) set"   and A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" assumes "trans R" shows "locally_trans R A B C"
elims_1: fixes x :: "('var, 'val, 'ret, 'pname) edge_kind"   and xa :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and y :: "bool" assumes "pred x xa = y" obtains f :: "('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = \<Up>f__" and "xa = cf__ # cfs__" and "y"   | Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = (Q__)\<^sub>\<surd>" and "xa = cfa__ # cfsa__" and "y = Q__ (fst cfa__)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and r :: "'ret" and p :: "'pname" and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Qa__:r\<hookrightarrow>\<^bsub>p__\<^esub>fs" and "xa = cfb__ # cfsb__" and "y = Qa__ (fst cfb__, r)"   | Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool" and p :: "'pname" and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret" and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" where "x = Q\<hookleftarrow>\<^bsub>p\<^esub>f" and "xa = cf # cfs" and "y = (Q cf \<and> cfs \<noteq> [])"   | et :: "('var, 'val, 'ret, 'pname) edge_kind" where "x = et" and "xa = []" and "\<not> y"
measure_spmf_zero_iff: fixes p :: "'h spmf"   and A :: "'h set" shows "(Sigma_Algebra.measure (measure_spmf p) A = 0) = (set_spmf p \<inter> A = {})"
HAStates_def: fixes HA :: "('a, 'b, 'c) hierauto" shows "HAStates HA = \<Union> (States ` SAs HA)"
dickson_grading_adds_imp_le: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and t :: "'a" assumes "dickson_grading d"   and "s adds t" shows "d s \<le> d t"
X1: shows "fst ` X1 \<subseteq> {x \<in> S. s x = 0}"
id_funpow_id: fixes f :: "'e \<Rightarrow> 'e"   and x :: "'e"   and n :: "nat" assumes "f x = x" shows "(f ^^ n) x = x"
computation_swingneg: shows "rat_poly_times B (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) B) = rat_poly_plus (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) (rat_poly_times B (B - rat_poly_times (rat_poly_times A A) A))) (rat_poly_times A (rat_poly_times (B - rat_poly_times (rat_poly_times A A) A) A))"
univ_commute: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (proj r x) = f x"
r_enumdom_nonempty_domain_1: fixes i :: "nat"   and x\<^sub>0 :: "nat"   and y :: "nat" assumes "eval r_phi [i, x\<^sub>0] \<down>" shows "eval r_enumdom [i, y] \<down>"
simple_path_shiftpath: fixes g :: "real \<Rightarrow> 'a"   and a :: "real" assumes "simple_path g"   and "pathfinish g = pathstart g"   and "0 \<le> a"   and "a \<le> 1" shows "simple_path (shiftpath a g)"
prv_neg_suc_zer: fixes t :: "'trm" assumes "t \<in> atrm" shows "prv (neg (eql (suc t) zer))"