hcju/isabelleps huggingface dataset:
total_less_nmultiset: fixes X :: "'b nmultiset"   and Y :: "'b nmultiset" assumes "\<not> X < Y"   and "Y \<noteq> X" shows "Y < X"
hnr_If: fixes \<Gamma> :: "assn"   and \<Gamma>1 :: "assn"   and a :: "bool"   and a' :: "bool"   and b' :: "'a Heap"   and \<Gamma>2b :: "assn"   and R :: "'b \<Rightarrow> 'a \<Rightarrow> assn"   and b :: "'b nres"   and c' :: "'a Heap"   and \<Gamma>2c :: "assn"   and c :: "'b nres"   and \<Gamma>' :: "assn" assumes "\<Gamma> \<Longrightarrow>\<^sub>t \<Gamma>1 * hn_val bool_rel a a'"   and "a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') b' \<Gamma>2b R b"   and "\<not> a \<Longrightarrow> hn_refine (\<Gamma>1 * hn_val bool_rel a a') c' \<Gamma>2c R c"   and "\<Gamma>2b \<or>\<^sub>A \<Gamma>2c \<Longrightarrow>\<^sub>t \<Gamma>'" shows "hn_refine \<Gamma> (if a' then b' else c') \<Gamma>' R (if a then b else c)"
distinct_6: fixes x31 :: "(name, (Term_Class.term \<times> sterm) list) fmap"   and x32 :: "name"   and x33 :: "(name, value) fmap"   and x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap" shows "Vrecabs x31 x32 x33 \<noteq> Vabs x21 x22"
not_isTrue_fls: shows "\<not> eval_fm e0 Fls"
ev_mono: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "ev \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "ev \<psi> xs"
order_trans: fixes x :: "'t"   and y :: "'t"   and z :: "'t" assumes "x \<preceq>\<^sub>t y"   and "y \<preceq>\<^sub>t z" shows "x \<preceq>\<^sub>t z"
proj_arcs_G_def: fixes x :: "'b" shows "proj_arcs_G x \<equiv> if x = uv then uw else if x = rev_G uv then vw else x"
ord_of_nat_mult: fixes m :: "nat"   and n :: "nat" shows "(m * n)\<^sub>\<nat> = m\<^sub>\<nat> * n\<^sub>\<nat>"
le_diff_conv2: fixes a :: "'k"   and b :: "'k"   and c :: "'k" assumes "a \<le> b" shows "(c \<le> b - a) = (c + a \<le> b)"
Act_eq_iff_perm_renaming: fixes \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c) formula"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c) formula" shows "(Formula.Act \<alpha>1 x1 = Formula.Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1)"
coll_sum_list: fixes ys :: "(real \<times> real) list"   and x :: "real \<times> real" assumes "\<And>y. y \<in> set ys \<Longrightarrow> coll 0 x y" shows "coll 0 x (sum_list ys)"
weakBisimulationE_2: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<approx> Q" shows "Q \<approx> P"
simps_5: shows "eval_ternary_And TernaryFalse TernaryUnknown = TernaryFalse"
pairwise_rel_prime_1: shows "\<forall>q. List.member qs q \<longrightarrow> coprime p q"
enum: shows "enum = map (\<lambda>ys. the \<circ> map_of (zip enum_a ys)) (List.n_lists (length enum_a) enum_b)"
enabled_stream_trans': fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)"   and xs :: "('s \<times> ('c \<Rightarrow> real)) cfg stream" assumes "cfg \<in> R_G.valid_cfg"   and "abss st = state cfg"   and "MDP.MC.enabled (repcs st cfg) xs" shows "PTA_Reachability.pairwise trans' (smap (snd \<circ> state) xs)"
Iff_I: fixes A :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A H \<turnstile> B"   and "insert B H \<turnstile> A" shows "H \<turnstile> A IFF B"
basis_reduction_add_rows_impl_2: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and state' :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and fs' :: "int vec list" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "LLL_Impl.basis_reduction_add_rows n upw i state = state'"   and "i < m"   and "fs' = fs_state state'" shows "local.basis_reduction_add_rows upw i fs = fs'"
is_monic_set_def: fixes B :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" shows "punit.is_monic_set B \<equiv> \<forall>b\<in>B. b \<noteq> 0 \<longrightarrow> lcf b = (1::'b)"
fls_shift_nth: fixes m :: "int"   and f :: "'a fls"   and n :: "int" shows "fls_nth (fls_shift m f) n = fls_nth f (n + m)"
image_mset_filter_swap2: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a"   and D :: "'b multiset" shows "filter_mset Q (image_mset P D) = image_mset P {#C. C \<in># {#C \<in># D. Q (P C)#}#}"
update_convs_2: fixes edges' :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set"   and nodes :: "'b set"   and edges :: "('b \<times> 'b) set"   and more :: "'c" shows "edges_update edges' \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = \<lparr>nodes = nodes, edges = edges' edges, \<dots> = more\<rparr>"
lprefix_lappend: fixes xs :: "'a llist"   and ys :: "'a llist" shows "lprefix xs (xs $ ys)"
yz: shows "poincare_distance y z = poincare_distance y' z'"
isDERIV_subst_Var_floatarith: fixes fa :: "floatarith"   and vs :: "real list"   and s :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "max_Var_floatarith fa \<le> length vs"   and "\<And>j. j < max_Var_floatarith fa \<Longrightarrow> vs ! s j = vs ! j"   and "\<And>j. (s j = i) = (j = i)" shows "isDERIV i (subst_floatarith (\<lambda>i. Var (s i)) fa) vs = isDERIV i fa vs"
return_time_rec: fixes A :: "'a set"   and n :: "nat" assumes "A \<in> sets M" shows "return_time_function A -` {Suc n} = T --` first_entrance_set A n \<inter> A"
assms_4: fixes i :: "nat" assumes "i < n" shows "v $ i \<le> u $ i"
assms_2: shows "v \<le> 0\<^sub>v n"
is_path_undir_memb: fixes G :: "('c, 'd) graph"   and v :: "'c"   and p :: "('c \<times> 'd \<times> 'c) list"   and v' :: "'c" assumes "valid_graph G"   and "is_path_undir G v p v'" shows "v \<in> nodes G \<and> v' \<in> nodes G"
map_tl: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map f (tl xs) = tl (map f xs)"
eval_wp_Apply: fixes f :: "'c \<Rightarrow> 'c"   and P :: "'c \<Rightarrow> real" shows "wp (Apply f) P = P \<circ> f"
vectorize_atomize_poly: fixes p :: "'k \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "vectorize_poly (atomize_poly p) = p"
defined5: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "\<delta> \<delta> X = true"
unique_servKeys: fixes A :: "agent"   and K :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and K' :: "nat"   and B' :: "agent"   and Ts' :: "msg"   and X' :: "msg" assumes "Says Tgs A (Crypt K \<lbrace>Key servK, Agent B, Ts, X\<rbrace>) \<in> set evs"   and "Says Tgs A' (Crypt K' \<lbrace>Key servK, Agent B', Ts', X'\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "A = A' \<and> B = B' \<and> K = K' \<and> Ts = Ts' \<and> X = X'"
extended_Gromov_product_le_dist': fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "extended_Gromov_product_at e x y \<le> extended_Gromov_distance (to_Gromov_completion e) y"
cSup_mem_eq: fixes c :: "'d"   and X :: "'d set" assumes "c \<in> X"   and "\<forall>x\<in>X. x \<le> c" shows "Sup X = c"
cl_rank_of: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> \<E>.cl X" shows "\<E>.rank_of (insert x X) = \<E>.rank_of X"
assertE_def: fixes P :: "bool" shows "assertE P \<equiv> if P then returnOk () else fail"
stable_pair_on_def: fixes ds :: "D2 set" shows "StableNoDecomp.stable_pair_on ds = (\<lambda>(XD, XH). XD = - Rf StableNoDecomp.CH XH \<and> XH = - Rf (StableNoDecomp.CD_on ds) XD)"
cod_simps_2: shows "dom_\<mu>.cod.leg0 = doms.\<mu>.leg0"
elims_2: fixes ty_ent :: "'b \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and x :: "'b atom formula" assumes "wf_fmla_atom ty_ent x" obtains a :: "predicate" and vs :: "'b list" where "x = Atom (predAtm a vs)" and "wf_pred_atom ty_ent (a, vs)"
Gf: shows "\<guillemotleft>G f : Go (C.dom f) \<rightarrow>\<^sub>D Go (C.cod f)\<guillemotright> \<and> f \<cdot>\<^sub>C \<epsilon>o (C.dom f) = \<epsilon>o (C.cod f) \<cdot>\<^sub>C F (G f)"
path_connected_translation: fixes a :: "'a"   and S :: "'a set" shows "path_connected ((+) a ` S) = path_connected S"
le_mult_floor_Ints: fixes a :: "'b"   and b :: "'b" assumes "(0::'b) \<le> a"   and "a \<in> \<int>" shows "of_int (\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor>) \<le> of_int \<lfloor>a * b\<rfloor>"
assms: shows "set_mset K \<subseteq> r \<down>m J"
tendsto_iff: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and F :: "'c filter" shows "(f \<longlongrightarrow> l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
of_int_uint: fixes w :: "'d word" shows "of_int (uint w) = unsigned w"
length_nth_simps_2: fixes x :: "'a"   and xs :: "'a list" shows "length (x # xs) = Suc (length xs)"
total1E: fixes f :: "nat \<Rightarrow> nat option"   and x :: "nat" assumes "total1 f" shows "f x \<down>"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<rr>"
viewL_correct_1: fixes t :: "('a, 'b) FingerTreeStruc" assumes "FingerTreeStruc.ft_invar t"   and "t = Empty" shows "FingerTreeStruc.viewL t = None"
cr_view\<^sub>h_def: shows "cr_view\<^sub>h \<equiv> \<lambda>x y. x = from_view\<^sub>h y"
filt_equiv_def: fixes F :: "'a \<Rightarrow> 'b measure"   and M :: "'b measure"   and N :: "'b measure" shows "filt_equiv F M N = (sets M = sets N \<and> Filtration.filtration M F \<and> (\<forall>t A. A \<in> sets (F t) \<longrightarrow> (emeasure M A = 0) = (emeasure N A = 0)))"
mult_divide_mult_cancel_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times c b) = divide a b"
simps_8: fixes x3 :: "(Term_Class.term \<times> sterm) list"   and x1 :: "name" shows "Sabs x3 \<noteq> Sconst x1"
relpow_0_E: fixes x :: "'b"   and y :: "'b"   and R :: "('b \<times> 'b) set" assumes "(x, y) \<in> R ^^ 0" obtains "x = y"
mult_le_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less_eq a one"   and "less_eq zero b"   and "less_eq b one" shows "less_eq (times a b) one"
not_empty: shows "\<forall>u. u \<turnstile>\<^bsub>v,n\<^esub> \<lambda>i' j'. if i' = i \<and> j' = j then Le r else if i' = j \<and> j' = i then Le (- r) else M i' j' \<longrightarrow> u \<turnstile>\<^bsub>v,n\<^esub> M"
split_minNoneN1: fixes t :: "'c bro"   and h :: "nat" assumes "t \<in> U h"   and "split_min t = None" shows "t = N1 N0"
AE_sum: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "finite I"   and "\<forall>i\<in>I. AEeq M (f i) (g i)" shows "AEeq M (\<lambda>w. \<Sum>i\<in>I. f i w) (\<lambda>w. \<Sum>i\<in>I. g i w)"
finite_intersection_of_Inter: fixes \<U> :: "'b set set"   and P :: "'b set \<Rightarrow> bool" assumes "finite \<U>"   and "\<And>S. S \<in> \<U> \<Longrightarrow> (finite intersection_of P) S" shows "(finite intersection_of P) (\<Inter> \<U>)"
cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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_inject: fixes x :: "(_) document_ptr"   and y :: "(_) document_ptr" shows "(cast x = cast y) = (x = y)"
mod_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo zero a = zero"
finite_columns: fixes A :: "(('a, 'c) vec, 'b) vec" shows "finite (columns A)"
acquired_False_takeWhile_non_volatile_Write\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set" shows "acquired False (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) A = {}"
minus_div_mult_eq_mod: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (times (divide a b) b) = modulo a b"
simps: fixes K :: "'a set"   and xs :: "'a list" shows "finite_extension K xs = foldr (\<lambda>x K'. simple_extension K' x) xs K"
in_set_spmf_exec_gpv_into_results_gpv: fixes x :: "'a"   and s' :: "('b, 'c) resource"   and gpv :: "('a, 'b, 'c) gpv"   and s :: "('b, 'c) resource"   and \<I> :: "('b, 'c) \<I>" assumes "(x, s') \<in> set_spmf (exec_gpv run_resource gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "\<I> \<turnstile>res s \<surd>" shows "x \<in> results_gpv \<I> gpv"
pdom_member_minus_iff: fixes x :: "'a"   and g :: "('a, 'b) pfun"   and f :: "('a, 'b) pfun" assumes "x \<notin> pdom g" shows "(x \<in> pdom (f - g)) = (x \<in> pdom f)"
diamond_edges_are_valid_2: shows "valid_path (snd diamond_top_right_edge)"
Bonk_Schramm_extension_unfolded_hyperbolic: fixes x :: "'a Bonk_Schramm_extension_unfolded"   and y :: "'a Bonk_Schramm_extension_unfolded"   and z :: "'a Bonk_Schramm_extension_unfolded"   and t :: "'a Bonk_Schramm_extension_unfolded" assumes "x \<in> extended_distance_set"   and "y \<in> extended_distance_set"   and "z \<in> extended_distance_set"   and "t \<in> extended_distance_set" shows "extended_distance x y + extended_distance z t \<le> max (extended_distance x z + extended_distance y t) (extended_distance x t + extended_distance y z) + 2 * deltaG TYPE('a)"
associated_refl: fixes a :: "'a" assumes "a \<in> carrier G" shows "a \<sim> a"
HMA_M_def: shows "Mod_Type_Connect.HMA_M = (\<lambda>a b. a = Mod_Type_Connect.from_hma\<^sub>m b)"
ex_inf: fixes x :: "'b"   and y :: "'b" shows "\<exists>inf. is_inf x y inf"
simps_1: fixes "is" :: "'b set" shows "redT_updIs is [] = is"
append_is_Nil_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs @ ys = []) = (xs = [] \<and> ys = [])"
that_2: fixes x :: "'a"   and y :: "'b" assumes "P x y" shows "Q x y"
Rep_induct: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "type_definition Rep Abs A"   and "y \<in> A"   and "\<And>x. P (Rep x)" shows "P y"
arccos_lt_bounded: fixes y :: "real" assumes "- 1 < y"   and "y < 1" shows "0 < arccos y \<and> arccos y < pi"
infinite_greaterThan_int: fixes n :: "int" shows "infinite {n<..}"
assms: shows "\<forall>l\<in>dom f. \<exists>L. finite L \<and> (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p (f l) l)"
p'_pos_iff: fixes x :: "real" shows "(0 < other.p' x) = (if 0 < a\<^sub>o then - v\<^sub>o / a\<^sub>o < x else if a\<^sub>o < 0 then x < - v\<^sub>o / a\<^sub>o else 0 < v\<^sub>o)"
pos_pf_neg_uvp: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf"   and "d < 1 + r"   and "0 < pf stk (Suc n) (spick w n False)"   and "val_process Mkt pf n (spick w n False) \<le> 0" shows "cls_val_process Mkt pf (Suc n) (spick w n False) < 0"
semiattractive_axioms: shows "semiattractive B (\<sqsubseteq>)"
Step: fixes x :: "name"   and xs :: "name list"   and y :: "name"   and ys :: "name list" shows "composePerm (x # xs) (y # ys) = (x, y) # composePerm xs ys"
cmult_square_le: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "\<kappa> \<le> \<kappa> \<otimes> \<kappa>"
prems_2: shows "\<forall>X\<langle>\<or>\<noteq>: [(t, t')]\<rangle>\<^sub>s\<^sub>t \<in> set (\<forall>Y\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S)"
elims_3: fixes x :: "'c btree" assumes "\<not> nonempty_lasttreebal x" obtains ts :: "('c btree \<times> 'c) list" and t :: "'c btree" where "x = Node ts t" and "\<not> ((\<exists>ls tsub tsep. ts = ls @ [(tsub, tsep)] \<and> height tsub = height t) \<and> nonempty_lasttreebal t)"
pullback_commutes: fixes f :: "'a"   and g :: "'a" assumes "C.cospan f g" shows "C.commutative_square f g \<p>\<^sub>1[f, g] \<p>\<^sub>0[f, g]"
coversallPartitioning: fixes ts :: "'b set list" shows "\<Union> (set ts) = \<Union> (partitioning ts {})"
abs_eq: fixes x :: "circline_mat" shows "pos_oriented (abs_ocircline x) = pos_oriented_clmat x"
assms_1: shows "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common (absorb \<A> cnv) adv"
vec_plus_Cons: fixes a :: "'b"   and as :: "'b list"   and b :: "'b"   and bs :: "'b list" shows "vec_plus (a # as) (b # bs) = (a + b) # vec_plus as bs"
splitAt_distinct_ram_snd: fixes vs :: "'c list"   and ram :: "'c" assumes "distinct vs" shows "ram \<notin> set (snd (splitAt ram vs))"
assms: shows "run_one_step d i (s, vs, ves, $Cvtop t2 x312 t1 sx) = (s', vs', res)"
Alt_eq_map_rexp_iff_2: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e rexp"   and r :: "'d rexp"   and s :: "'d rexp" shows "(map_rexp f x = Alt r s) = (\<exists>r' s'. x = Alt r' s' \<and> map_rexp f r' = r \<and> map_rexp f s' = s)"
aless_zless: fixes m :: "int"   and n :: "int" shows "(ant m < ant n) = (m < n)"
Aexp_edom: fixes e :: "exp"   and a :: "Arity" shows "edom (Aexp_syn' a e) \<subseteq> fv e"
limit_mat_ignore_initial_segment: fixes g :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and d :: "nat"   and k :: "nat" assumes "limit_mat g A d" shows "limit_mat (\<lambda>n. g (n + k)) A d"
inv_unique: fixes y :: "int"   and x :: "int"   and y' :: "int" assumes "y + x = 0"   and "x + y' = 0"   and "x \<in> UNIV"   and "y \<in> UNIV"   and "y' \<in> UNIV" shows "y = y'"
F: shows "\<forall>n\<in>{..length xs}. (xs, \<Union>W\<in>R n. W) \<in> seq_comp_failures P Q"
summable_iff_shift: fixes k :: "nat" shows "summable (\<lambda>n. f (n + k)) = summable f"
le_pre_le: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n - Suc 0" shows "x \<le> n"
cod_char: fixes f :: "(bool, bool list) concrete_category.arr" shows "cod f = (if f = j0 \<or> f = j1 then One else if arr f then f else null)"
typing_appE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and B :: "'a trm"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> App A B : \<sigma>" shows "\<exists>\<tau>. \<Gamma> \<turnstile> A : TArr \<tau> \<sigma> \<and> \<Gamma> \<turnstile> B : \<tau>"
assms_2: shows "to_nat a \<noteq> to_nat b"
suc_max_simp: fixes n :: "nat"   and fft :: "nat" shows "Suc (max (n + 3) fft - Suc (Suc (Suc n))) = max (n + 3) fft - Suc (Suc n)"
antipar_2: shows "src (g \<star> k) = trg (h \<star> f)"
absolutely_integrable_reflect: fixes b :: "'a"   and a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. f (- x)) absolutely_integrable_on cbox (- b) (- a)) = (f absolutely_integrable_on cbox a b)"
prems_3: shows "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)"
sasp_exec_noops: fixes s :: "'a \<Rightarrow> 'b option"   and n :: "nat" shows "execute_serial_plan_sas_plus s (replicate n empty_sasp_action) = s"
of_nat_mod: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "semiring_1.of_nat one plus zero (m mod n) = modulo (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
inter_compl_diff_conv: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> - B = A - B"
range: fixes x :: "nat" shows "range (g x) \<subseteq> (\<lambda>i. f i x) ` Y"
eventually_ball_finite_distrib: fixes A :: "'a set"   and P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and net :: "'b filter" assumes "finite A" shows "(\<forall>\<^sub>F x in net. \<forall>y\<in>A. P x y) = (\<forall>y\<in>A. \<forall>\<^sub>F x in net. P x y)"
T_ex: fixes x :: "'a" assumes "x \<in> {a--b}" shows "{- T..T} \<subseteq> existence_ivl0 x"
nthsSSk2: shows "nths (replicate (Suc (Suc k)) (2::'a)) {0..<Suc k} = replicate (Suc k) (2::'a)"
terminates_def: fixes \<sigma> :: "'var \<Rightarrow> 'val" shows "terminates \<sigma> \<equiv> \<exists>i. path \<sigma> i = local.return"
to_woot_character_def: fixes D :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message" shows "to_woot_character D m = (case m of InsertMessage l i u s \<Rightarrow> InsertMessage l i u (delete_maybe i D s))"
assms_1: shows "\<forall>j. j \<noteq> i \<longrightarrow> quality_increases (\<sigma> j) (\<sigma>' j)"
permutes_inv_inv: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "inv (inv p) = p"
subset_TI: fixes t1 :: "real"   and s :: "real" assumes "t1 \<in> cball t_max et"   and "s \<in> {t_max--t1}" shows "s \<in> cball t_max et"
true_suntil: fixes P :: "'a stream \<Rightarrow> bool" shows "(\<lambda>_. True) suntil P = ev P"
d_idem: fixes x :: "'a" shows "d (d x) = d x"
less_not_sym: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" assumes "x \<prec> y" shows "\<not> y \<prec> x"
Simps_7: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a"   and Y :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>"   and "id\<^bsub>C\<^esub> X = id\<^bsub>C\<^esub> Y" shows "X = Y"
Bossy''_CD_on_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and d1h1 :: "'d"   and d1h2 :: "'d"   and d1h3 :: "'d"   and d2h1 :: "'d"   and d2h2 :: "'d"   and d2h3 :: "'d"   and d3h1 :: "'d"   and d3h2 :: "'d"   and d3h3 :: "'d"   and ds :: "'e set"   and X :: "'d set"   and X' :: "'d set" assumes "ContractsWithBossyConstants Xd Xh Pd Ch d1h1 d1h2 d1h3 d2h1 d2h2 d2h3 d3h1 d3h2 d3h3 ds"   and "X \<subseteq> Contracts.CD_on (ContractsWithBossyConstants.BPd' Xd d1h1 d1h2 d1h3 d2h1 d2h2 d2h3 d3h1 d3h2 d3h3) ds X'" shows "X \<subseteq> set [d1h1, d1h2, d1h3, d3h1, d3h2, d3h3]"
env_delete_fun_upd3: fixes x :: "'a"   and y :: "'a"   and m :: "'a \<Rightarrow> 'b"   and v :: "'b" assumes "x \<noteq> y" shows "env_delete x (m(y := v)) = (env_delete x m)(y := v)"
NEST_def: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set"   and S :: "'b set" assumes "class.arelations meets \<I> e m b ov d s f" shows "arelations.NEST \<I> e m ov d s f S \<equiv> \<exists>i. \<I> i \<and> (S = arelations.BEGIN e m ov d s f i \<or> S = arelations.END e m ov d s f i)"
bij_betw_same_card: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "card A = card B"