hcju/isabelleps huggingface dataset:
shared_weight_net_witness: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "shared_weight_net (witness Y r rs)"
Image_insert: fixes a :: "'a"   and b :: "'b"   and R :: "('a \<times> 'b) set"   and A :: "'a set" assumes "(a, b) \<in> R"   and "single_valued R" shows "R `` insert a A = insert b (R `` A)"
eval_monom_Nil1: fixes c :: "real"   and basis :: "(real \<Rightarrow> real) list" shows "eval_monom (c, []) basis = (\<lambda>_. c)"
product_code: fixes xs :: "'b list"   and ys :: "'c list" shows "Product_Type.product (set xs) (set ys) = set (concat (map (\<lambda>x. map (Pair x) ys) xs))"
Liminf_llist_lmap_union: fixes xs :: "'b llist"   and g :: "'b \<Rightarrow> 'c set"   and h :: "'b \<Rightarrow> 'c set" assumes "\<forall>x\<in>lset xs. \<forall>Y\<in>lset xs. g x \<inter> h Y = {}" shows "Liminf_llist (lmap (\<lambda>x. g x \<union> h x) xs) = Liminf_llist (lmap g xs) \<union> Liminf_llist (lmap h xs)"
invK_sesK: fixes a :: "rat" shows "invK (sesK a) = sesK a"
Lit: fixes G :: "prog"   and s :: "st"   and v :: "val" shows "G\<turnstile>Norm s \<midarrow>Lit v-\<succ>v\<rightarrow> Norm s"
top_le: fixes a :: "'a" assumes "top \<le> a" shows "a = top"
valid_graph_repr_def: fixes l :: "('a \<times> 'a) list" shows "valid_graph_repr l = (\<forall>(u, v)\<in>set l. u \<noteq> v)"
assms_5: fixes x :: "'a option list" assumes "wf_tuple n C x"   and "P x" shows "P (restrict A x) \<and> P (restrict B x)"
weakenTransitiveWeakCoinduct: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox>\<^sub>w<X> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto>\<^sub>w<{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<sim> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx>\<^sub>w Q"
intro: fixes return :: "int \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "(('c \<Rightarrow> int option) \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "('c \<Rightarrow> int option) \<Rightarrow> 'b \<Rightarrow> 'b"   and altc :: "int cset \<Rightarrow> (int \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b" assumes "monad_state_altc return bind get put altc"   and "memoization return bind get put"   and "monad_fail return bind fail" shows "nondet_fail_exp return bind fail get put altc"
absc_repcs_id: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg" shows "absc (repcs s cfg) = cfg"
I_list_disj: fixes fs :: "atom fm list"   and xs :: "'a list" shows "DLO.I (list_disj fs) xs = (\<exists>\<phi>\<in>set fs. DLO.I \<phi> xs)"
square_free_smult: fixes c :: "'a"   and f :: "'a poly" assumes "c \<noteq> (0::'a)"   and "square_free f" shows "square_free (Polynomial.smult c f)"
suffix_sum: shows "(\<Sum>i = j + 1..<length t. if postrecording_event t i then card {j \<in> {i + 1..<length t}. prerecording_event t j} else 0) = (\<Sum>ia = j + 1..<length t. if postrecording_event (swap_events i j t) ia then card {ja \<in> {ia + 1..<length (swap_events i j t)}. prerecording_event (swap_events i j t) ja} else 0)"
less_le_not_le_classes1: fixes c :: "classes1"   and d :: "classes1" shows "(c < d) = (c \<le> d \<and> \<not> d \<le> c)"
finite'_code_1: fixes xs :: "'a list" shows "Cardinality.finite' (set xs) = True"
Suc: shows "cis (real n * 2 * pi) = 1"
nil_in_permutations: fixes xs :: "'b list" shows "([] \<in> set (permutations xs)) = (xs = [])"
comp_min_basis_nadds: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<in> set (comp_min_basis xs)"   and "q \<in> set (comp_min_basis xs)"   and "p \<noteq> q" shows "\<not> lt q adds\<^sub>t lt p"
inj: fixes a :: "'a"   and b :: "'b"   and a' :: "'a"   and b' :: "'b" assumes "f a b = f a' b'" shows "a = a' \<and> b = b'"
comm_scalar_prod: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> v\<^sub>2 = v\<^sub>2 \<bullet> v\<^sub>1"
prems_2: fixes i :: "nat" assumes "i < length bbs" shows "bbs ! i \<le> (a # as) ! i"
well_analyzed_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" assumes "well_analyzed A"   and "well_analyzed B" shows "well_analyzed (A @ B)"
rtranclp_mult_poly_p_mult_ideal_final: fixes q :: "(char list multiset \<times> int) multiset"   and p :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" assumes "(mult_poly_p q)\<^sup>*\<^sup>* (p, {#}) ({#}, r)" shows "polynomial_of_mset r - polynomial_of_mset p * polynomial_of_mset q \<in> More_Modules.ideal polynomial_bool"
hom_uminus: fixes x :: "'a" shows "hom (- x) = - hom x"
R: shows "yun_gcd.yun_factorization_main gcd (F div gcd F (pderiv F)) (pderiv F div gcd F (pderiv F)) 0 [] = R"
meet_connection2: fixes x :: "int"   and y :: "int" shows "(x \<le> y) = (int.meet y x = x)"
Block: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'d \<Rightarrow> 'd set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and bdy :: "('d, 'e, 'f) com"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A :: "'d set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com"   and Q :: "'d set" assumes "P \<subseteq> {s. init s \<in> P' s}"   and "\<forall>s. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' s) bdy {t. return s t \<in> R s t},{t. return s t \<in> A}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P block init bdy return c Q,A"
distinct_1: fixes x1 :: "'d"   and x2 :: "'d" shows "allow x1 \<noteq> deny x2"
4: shows "contour_integral (linepath a b) f + contour_integral (linepath b a') f + contour_integral (linepath a' b') f + contour_integral (linepath b' a) f = 0"
s\<^sub>2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (the ((\<sigma>;;\<tau>) l))]))"
map_step: fixes g :: "'a \<Rightarrow> 'b"   and vs :: "'a \<Rightarrow> nat"   and vs' :: "'b \<Rightarrow> nat"   and s :: "'c"   and t :: "'c"   and ws :: "'a \<Rightarrow> nat"   and M :: "('c \<times> ('c, 'a) Op) set"   and f :: "'c \<Rightarrow> 'd" assumes "inj g"   and "vs = vs' \<circ> g"   and "((s, vs), t, ws) \<in> step M" shows "((f s, vs'), f t, \<lambda>x. if x \<in> range g then ws (inv g x) else vs' x) \<in> step (map_minsky f g M)"
fold_Cons_rev: fixes xs :: "'a list" shows "fold (#) xs = (@) (rev xs)"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>c. P 0 c"   and "\<And>n c. P n c \<Longrightarrow> P (Suc n) c" shows "P a0 a1"
tendsto_norm_zero_cancel: fixes f :: "'e \<Rightarrow> 'f"   and F :: "'e filter" assumes "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> (0::'f)) F"
ptrm: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and n :: "nat" assumes "Deduct_with_False_Disj_Rename var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "\<sigma> \<in> Deduct_with_False_Disj_Rename.ptrm var fmla Var FvarsT Fvars eql cnj imp all exi prv n" shows "\<sigma> \<in> fmla"
leadsTo_Union: fixes S :: "'b set set"   and F :: "'b program"   and B :: "'b set" assumes "\<And>A. A \<in> S \<Longrightarrow> F \<in> A \<longmapsto> B" shows "F \<in> \<Union> S \<longmapsto> B"
fps_conv_radius_mult: fixes f :: "'a fps"   and g :: "'a fps" shows "min (fps_conv_radius f) (fps_conv_radius g) \<le> fps_conv_radius (f * g)"
in_charts_in_atlas: fixes x :: "('c, 'c) chart"   and k :: "enat" assumes "x \<in> charts_eucl" shows "x \<in> manifold_eucl.atlas k"
imp_refl: fixes P :: "bool" shows "P \<longrightarrow> P"
mult_ge_zero: fixes a :: "'d"   and b :: "'d" assumes "(0::'d) \<le> a"   and "(0::'d) \<le> b" shows "(0::'d) \<le> a * b"
h: shows "h a = \<lfloor>(D', fs)\<rfloor>"
connected_arc_image: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "connected (path_image g)"
lift_Suc_mono_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and n' :: "nat" assumes "class.order less_eq less"   and "\<And>n. less_eq (f n) (f (Suc n))"   and "n \<le> n'" shows "less_eq (f n) (f n')"
cp1: fixes pi1 :: "('b \<times> 'b) list"   and pi2 :: "('c \<times> 'c) list"   and x :: "'a" assumes "cp TYPE('a) TYPE('b) TYPE('c)" shows "pi1 \<bullet> pi2 \<bullet> x = (pi1 \<bullet> pi2) \<bullet> pi1 \<bullet> x"
closest_pair_rec_code_simps: fixes n :: "nat"   and xs :: "(int \<times> int) list" assumes "n = length xs"   and "\<not> n \<le> 3" shows "closest_pair_rec_code xs = (let (xs\<^sub>L, xs\<^sub>R) = split_at (n div 2) xs; l = fst (hd xs\<^sub>R); (ys\<^sub>L, p\<^sub>L) = closest_pair_rec_code xs\<^sub>L; (ys\<^sub>R, p\<^sub>R) = closest_pair_rec_code xs\<^sub>R; ys = Common.merge snd ys\<^sub>L ys\<^sub>R in (ys, combine_code p\<^sub>L p\<^sub>R l ys))"
h3: shows "\<turnstile> \<box>F \<and> G \<longrightarrow> H"
sset_def: fixes xs :: "[:'a:]" shows "sset xs = {x. slistmem x xs}"
ereal_divide_le_pos: fixes x :: "ereal"   and z :: "ereal"   and y :: "ereal" assumes "0 < x"   and "x \<noteq> \<infinity>" shows "(z / x \<le> y) = (z \<le> x * y)"
run_rel_def: fixes R :: "('e \<times> 'f) set" shows "run_rel R \<equiv> {(ra, rb). \<forall>i. (ra i, rb i) \<in> R}"
False: shows "\<not> (p1 = 0 \<or> p2 = 0)"
set_case_def: fixes s :: "'b set"   and c_empty :: "'a"   and c_sing :: "'b \<Rightarrow> 'a"   and c_else :: "'a" shows "set_case s c_empty c_sing c_else = (if s = {} then c_empty else if card s = 1 then c_sing (set_choose s) else c_else)"
mod_diff1_eq: fixes a :: "nat"   and b :: "nat"   and m :: "nat" assumes "a \<le> b" shows "(b - a) mod m = (m + b mod m - a mod m) mod m"
ex_def: fixes P :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "\<^bold>\<exists>x. P x \<equiv> \<lambda>A. \<exists>x. P x A"
insertion_neg: fixes f :: "nat \<Rightarrow> 'a"   and p :: "'a mpoly" shows "insertion f (- p) = - insertion f p"
n_max_eq_sets: fixes A :: "nat set"   and B :: "nat set" assumes "A = B"   and "A \<noteq> {}"   and "\<exists>n. A \<subseteq> {j. j \<le> n}" shows "n_max A = n_max B"
simps_2: fixes S' :: "Runs.var set" shows "in_progressS None S' = False"
bij_betw_exp_times_self_atLeastAtMost': fixes a :: "real"   and b :: "real" assumes "a \<le> b"   and "b \<le> - 1" shows "bij_betw (\<lambda>x. x * exp x) {a..b} {b * exp b..a * exp a}"
sorted1: fixes x :: "'a list" shows "linorder.sorted (\<le>lex) [x] = True"
ide_Ide: fixes a :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "ide (fst a) \<and> ide (snd a)" shows "VxV.ide a"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and point :: "'b point" shows "P (case point of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = ((point = Infinity \<longrightarrow> P f1) \<and> (\<forall>x21 x22. point = Point x21 x22 \<longrightarrow> P (f2 x21 x22)))"
one_index: fixes i :: "'b" shows "1 $h i = (1::'a)"
sup0: fixes a :: "'g" shows "a \<equiv> a \<squnion> \<bottom>"
unat_mask_word32: fixes n :: "nat" shows "unat (mask n) = 2 ^ min n word_bits - 1"
span_union1: fixes U :: "'c set"   and U' :: "'c set"   and W :: "'c set"   and W' :: "'c set" assumes "U \<subseteq> carrier V"   and "U' \<subseteq> carrier V"   and "span U = span U'"   and "W \<subseteq> carrier V"   and "W' \<subseteq> carrier V"   and "span W = span W'" shows "span (U \<union> W) \<subseteq> span (U' \<union> W')"
SN_rel_on_mono: fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and S' :: "('a \<times> 'a) set"   and T :: "'a set" assumes "R \<subseteq> R'"   and "S \<subseteq> S'"   and "SN_rel_on R' S' T" shows "SN_rel_on R S T"
upto_enum_word_nth: fixes i :: "'b word"   and j :: "'b word"   and k :: "nat" assumes "i \<le> j"   and "k \<le> unat (j - i)" shows "[i .e. j] ! k = i + word_of_nat k"
eq_1: shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau> f) (Abs \<tau> g)"
wf_syscls_is_class_Object: fixes P :: "'a prog" assumes "wf_syscls P" shows "is_class P Object"
prems: shows "degree (pCons a2 p2) < degree (pCons a1 p1)"
\<Psi>\<Phi>: fixes f' :: "'t" assumes "S'.arr f'" shows "\<Phi> (\<Psi> f') = f'"
exec_1_eq: fixes P :: "jvm_prog" shows "exec_1 P = {(\<sigma>, \<sigma>'). exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>}"
case_1: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> ('c, 'b) Term.term list \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Var x1 of Var x \<Rightarrow> f1 x | Fun x xa \<Rightarrow> f2 x xa) = f1 x1"
thetaSeqWT_def: shows "thetaSeqWT \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>wT d1 \<and> c2 \<approx>wT d2}"
atLeastLessThan_eq_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a < b"   and "c < d" shows "({a..<b} = {c..<d}) = (a = c \<and> b = d)"
sfilter_not_P: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" assumes "\<not> P (shd s)" shows "sfilter P s = sfilter P (stl s)"
obsf_converter_sel: fixes conv :: "('b, 'a, 'c, 'd) converter"   and x :: "'b" shows "run_converter (obsf_converter conv) x = TRY map_gpv' (\<lambda>y. case y of None \<Rightarrow> (Fault, const_converter Fault) | Some (x, conv') \<Rightarrow> (OK x, obsf_converter conv')) id option_of_exception (gpv_stop (run_converter conv x)) ELSE Done (Fault, const_converter Fault)"
O_mono1: fixes R :: "('a \<times> 'b) set"   and R' :: "('a \<times> 'b) set"   and S :: "('c \<times> 'a) set" assumes "R \<subseteq> R'" shows "S O R \<subseteq> S O R'"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'c list" shows "List.product (x # xs) ys = map (Pair x) ys @ List.product xs ys"
b'_c: shows "Derives1 (b1 @ snd q @ u @ [fst p] @ a2) (j - 1 + length (snd q)) p c"
simps_2: fixes xs :: "'x list"   and n :: "nat" shows "deg_le_sect_pp_aux xs (Suc n) = (let p = deg_le_sect_pp_aux xs n in p + foldr (\<lambda>x. (+) (monom_mult_punit 1 (single_pp x 1) p)) xs 0)"
coerce_u: shows "COERCE('e\<^sub>\<bottom>, 'f\<^sub>\<bottom>) = u_map\<cdot>COERCE('e, 'f)"
dom_char: fixes f :: "'a \<times> 'a" shows "S.VV.dom f = (if S.VV.arr f then S.VxV.dom f else S.VxV.null)"
size_annos: fixes C :: "'b acom" shows "length (annos C) = asize (strip C)"
assms: shows "(a, b) \<in> sset (u ||| v)"
Cod_preserves_Diag: fixes t :: "'a local.term" assumes "Diag t" shows "Diag (Cod t)"
sup_continuous_lfp: fixes F :: "'d \<Rightarrow> 'd" assumes "sup_continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)"
forall: shows "\<forall>y\<in>space M. expl_cond_expect M Y X w * indicat_real (Y -` {Y w} \<inter> space M) y = expl_cond_expect M Y X y * indicat_real (Y -` {Y w} \<inter> space M) y"
wp_under_wlp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp_under_wlp (Embed t)"
exp_float_interval: fixes x :: "float interval"   and p :: "nat" shows "exp ` set_of (real_interval x) \<subseteq> set_of (real_interval (exp_float_interval p x))"
inited_init_class_obj: fixes C :: "qtname"   and G :: "prog"   and s :: "st" shows "inited C (globs (init_class_obj G C s))"
eq_fold: fixes A :: "'k set" shows "sorted_list_of_set A = Finite_Set.fold insort [] A"
FSwImorph_igWlsInp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'i, 'j, 'k) model_scheme"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'i option"   and h :: "'i \<Rightarrow> 'l"   and hA :: "'j \<Rightarrow> 'm"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'h, 'l, 'm, 'n) model_scheme" assumes "igWlsInp MOD delta inp"   and "FSwImorph h hA MOD MOD'" shows "igWlsInp MOD' delta (lift h inp)"
of_bool_eq_2: fixes one :: "'b"   and zero :: "'b" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero True = one"
dim_eq_card: fixes B :: "'d set"   and V :: "'d set" assumes "cspan B = cspan V"   and "cindependent B" shows "cdim V = card B"
compatible_modes_by_case_distinction: fixes mdss :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list"   and X :: "'Var\<^sub>C set" assumes "conc.compatible_modes (conc.restrict_modes mdss X)"   and "conc.compatible_modes (conc.restrict_modes mdss (- X))" shows "conc.compatible_modes mdss"
ltl_const_equiv_no_const_singleton: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "eval \<psi> = Maybe"   and "\<phi> \<sim>\<^sub>C \<psi>" shows "\<phi> = \<psi>"
map_poly_inj_idom_divide_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "map_poly_inj_idom_divide_hom hom \<equiv> inj_idom_divide_hom hom"
S3: shows "c_f_list f \<in> PrimRec2"
interleave_Nil1: fixes xs :: "'a list" shows "[] \<otimes> xs = {xs}"
inject_1: fixes a :: "'e"   and a' :: "'e" shows "(pure a = pure a') = (a = a')"
Abs_inverse: fixes m :: "int" assumes "m \<in> {0..<int CARD('b bit1)}" shows "Rep_bit1 (Abs_bit1 m) = m"
valid_before: shows "valid_mmsaux args cur (ot, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"
hyps_1: shows "\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> e s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. P l' s \<and> (Q l' t \<longrightarrow> Q' l t))"
inf_finite_4_def: fixes x :: "finite_4"   and y :: "finite_4" shows "x \<sqinter> y = (case (x, y) of (a\<^sub>1, xa) \<Rightarrow> a\<^sub>1 | (a\<^sub>2, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>2 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>1 | (a\<^sub>3, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>3 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>4, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>4, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>4, a\<^sub>4) \<Rightarrow> a\<^sub>4)"
is_cons_cluster_def: fixes C :: "'node set" shows "is_cons_cluster C \<equiv> C \<subseteq> W \<and> (\<forall>p\<in>C. \<exists>Q\<subseteq>C. quorum_of p Q) \<and> (\<forall>Q Q'. quorum_of_set C Q \<and> quorum_of_set C Q' \<longrightarrow> W \<inter> Q \<inter> Q' \<noteq> {})"
1: shows "F (\<eta> (G (F b))) = F (G (F (\<eta> b)))"
mult_left_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * (y \<squnion> z) = x * y \<squnion> x * z"
anga_is_ang: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" assumes "QCongAACute a" shows "QCongA a"
one_less_power: fixes a :: "'a"   and n :: "nat" assumes "(1::'a) < a"   and "0 < n" shows "(1::'a) < a ^ n"
bij_betw_points_and_set: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a" shows "bij_betw img (hom unity a) (local.set a)"
canon_ang_eq: fixes \<alpha>\<^sub>1 :: "real"   and \<alpha>\<^sub>2 :: "real" assumes "\<exists>x. \<alpha>\<^sub>1 - \<alpha>\<^sub>2 = 2 * real_of_int x * pi" shows "\<downharpoonright>\<alpha>\<^sub>1\<downharpoonleft> = \<downharpoonright>\<alpha>\<^sub>2\<downharpoonleft>"
le_inf_reflp: fixes w :: "'item stream" shows "w \<preceq>\<^sub>I w"
assms_3: shows "(N + M, M) \<in> mul_eq r"
ltl_equivalence_axioms: shows "ltl_equivalence (\<sim>\<^sub>L)"
rreq_rrep_fresh_def: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and m :: "msg" shows "rreq_rrep_fresh crt m \<equiv> case m of Rreq hopsc x xa xb xc oipc osnc ipcc \<Rightarrow> ipcc \<noteq> oipc \<longrightarrow> oipc \<in> kD crt \<and> (osnc < sqn crt oipc \<or> sqn crt oipc = osnc \<and> the (dhops crt oipc) \<le> hopsc \<and> the (flag crt oipc) = val) | Rrep hopsc dipc dsnc x ipcc \<Rightarrow> ipcc \<noteq> dipc \<longrightarrow> dipc \<in> kD crt \<and> sqn crt dipc = dsnc \<and> the (dhops crt dipc) = hopsc \<and> the (flag crt dipc) = val | _ \<Rightarrow> True"
bv_chop_def: fixes w :: "bit list"   and i :: "nat" shows "bv_chop w i = (let len = length w in (take (len - i) w, drop (len - i) w))"
subgroup_generated_def: fixes G :: "('f, 'g) monoid_scheme"   and S :: "'f set" shows "subgroup_generated G S = G\<lparr>carrier := generate G (carrier G \<inter> S)\<rparr>"
transitive_star: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.transitive x" shows "x\<^sup>\<odot> = mone \<oplus> x"
0: shows "II = {brnL cl (pickFT cl)} \<union> (II - {brnL cl (pickFT cl)})"
get_root_node_parent_same: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr" shows "h \<turnstile> get_root_node (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r root = h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root"
pathfinish_compose: fixes f :: "'b \<Rightarrow> 'a"   and p :: "real \<Rightarrow> 'b" shows "pathfinish (f \<circ> p) = f (pathfinish p)"
cexp_def: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('c \<times> 'c) set" shows "r1 ^c r2 \<equiv> |Func (Field r2) (Field r1)|"
nonpos_linorder_cases: fixes a :: "'a"   and b :: "'a" assumes "a \<le> (0::'a)"   and "b \<le> (0::'a)" obtains "a < b"   | "a = b"   | "b < a"
reduce_specs_def: fixes d :: "nat"   and reach_options_o :: "unit"   and X :: "'b list set" shows "reduce_specs d reach_options_o X = SPEC (\<lambda>R. env_len R d \<and> X \<subseteq> R)"
simps_83: fixes f1 :: "char list \<Rightarrow> 'c trm list \<Rightarrow> 'b"   and f2 :: "'c trm \<Rightarrow> 'c trm \<Rightarrow> 'b"   and f3 :: "'c formula \<Rightarrow> 'b"   and f4 :: "'c formula \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f5 :: "'c formula \<Rightarrow> 'b"   and f6 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f7 :: "\<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f8 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and f9 :: "'c formula \<Rightarrow> \<I> \<Rightarrow> 'c formula \<Rightarrow> 'b"   and x21 :: "'c trm"   and x22 :: "'c trm" shows "(case Eq x21 x22 of Pred x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa | Neg x \<Rightarrow> f3 x | Or x xa \<Rightarrow> f4 x xa | Exists x \<Rightarrow> f5 x | Prev x xa \<Rightarrow> f6 x xa | Next x xa \<Rightarrow> f7 x xa | Since x xa xb \<Rightarrow> f8 x xa xb | Until x xa xb \<Rightarrow> f9 x xa xb) = f2 x21 x22"
triangle_num_le: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "triangle_num m \<le> triangle_num n"
intro: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "wfP R" shows "well_founded R"
tauTransitionChain: fixes P :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<tau> \<prec> P'" shows "P \<Longrightarrow>\<^sub>\<tau> P'"
face_cycle_succ_in_arcsI: fixes a :: "'b" assumes "a \<in> arcs G" shows "face_cycle_succ a \<in> arcs G"