hcju/isabelleps huggingface dataset:
MWLsSteps_det_cases_4: fixes \<iota> :: "nat"   and b :: "'exp"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi) m \<alpha> p m'" obtains "\<alpha> = []" and "p = Some c1" and "m' = m" and "BMap (E b m)"   | "\<alpha> = []" and "p = Some c2" and "m' = m" and "\<not> BMap (E b m)"
stern_brocot_simps_3: shows "right stern_brocot_recurse = pure succ \<diamondop> stern_brocot_recurse"
int_gf_F: fixes y :: "'s"   and A :: "'s set" assumes "y \<in> C"   and "finite A"   and "A \<subseteq> C" shows "((\<lambda>z. measure_pmf.expectation N (\<lambda>x. gf_F x y z * indicat_real A x)) \<longlongrightarrow> measure_pmf.expectation N (\<lambda>x. F x y * indicat_real A x)) (at_left 1)"
T_Var: fixes x :: "vrs"   and T :: "ty"   and \<Gamma> :: "binding list" assumes "VarB x T \<in> set \<Gamma>"   and "\<turnstile> \<Gamma> ok" shows "\<Gamma> \<turnstile> Var x : T"
dres_chain_eq_res: fixes M :: "'b dres set"   and r :: "'b"   and s :: "'b" assumes "is_chain M"   and "dRETURN r \<in> M"   and "dRETURN s \<in> M" shows "r = s"
tendsto_inverse: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter" assumes "(f \<longlongrightarrow> a) F"   and "a \<noteq> (0::'d)" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
snd_conv: fixes x1 :: "'h"   and x2 :: "'g" shows "snd (x1, x2) = x2"
foldseq_distr: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'b"   and v :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "associative f"   and "commutative f" shows "foldseq f (\<lambda>k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)"
assms_6: shows "abs.sound_mode_use (cms\<^sub>A, mem\<^sub>A)"
hom_prod_root_unity: fixes n :: "nat list" shows "map_poly of_real (prod_root_unity n) = prod_root_unity n"
simps_16: fixes x41 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x42 :: "'d"   and x43 :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x31 :: "('a \<Rightarrow> 'b option) \<times> 'c \<Rightarrow> bool"   and x32 :: "'c"   and x33 :: "'d"   and x34 :: "(('a \<Rightarrow> 'b option) \<Rightarrow> 'b option) list" shows "x41\<hookleftarrow>\<^bsub>x42\<^esub>x43 \<noteq> x31:x32\<hookrightarrow>\<^bsub>x33\<^esub>x34"
n: fixes p :: "int"   and f :: "int poly"   and n :: "nat"   and hs :: "int poly list" assumes "prime p"   and "comm_monoid_mult_class.coprime (lead_coeff f) p"   and "poly_mod.square_free_m p f"   and "0 < degree f"   and "berlekamp_hensel p n f = hs"   and "2 * \<bar>lead_coeff f\<bar> * factor_bound f (degree_bound hs) < p ^ n" shows "n \<noteq> 0"
add_diff_cancel_right': fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and a :: "'r"   and b :: "'r" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a b) b = a"
ltrm_deg_0: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "p \<in> carrier (UP Zp)"   and "deg Zp p = 0" shows "monom (UP Zp) (p (deg Zp p)) (deg Zp p) = p"
NonContingent_def: shows "NonContingent \<equiv> \<lambda>F. Necessary F \<^bold>\<or> Impossible F"
Ainker: shows "A \<subseteq> carrier (V.vs kerT)"
card_face_cycle_sets_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "card (pre_digraph_map.face_cycle_sets (app_iso hom G) (map_iso hom)) = card face_cycle_sets"
clkp_set_def: fixes A :: "('g \<times> ('e, 'f) cconstraint \<times> 'h \<times> 'e list \<times> 'g) set \<times> ('g \<Rightarrow> ('e, 'f) cconstraint)" shows "clkp_set A = collect_clki (inv_of A) \<union> collect_clkt (trans_of A)"
assms: shows "SimplicialComplex.min_maxsimpchain (fold_A \<turnstile> A) (C # Cs)"
homotopic_with_imp_continuous: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a set"   and Y :: "'b set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "homotopic_with_canon P X Y f g" shows "continuous_on X f \<and> continuous_on X g"
mod_int_pos_iff: fixes k :: "int"   and l :: "int" shows "(0 \<le> k mod l) = (l dvd k \<or> l = 0 \<and> 0 \<le> k \<or> 0 < l)"
integral_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>L (restrict_space M \<Omega>) f = LINT x|M. indicat_real \<Omega> x *\<^sub>R f x"
add_def: fixes d :: "nat"   and dip :: "nat"   and store :: "nat \<Rightarrow> (p \<times> nat list) option" shows "add d dip store \<equiv> case store dip of None \<Rightarrow> store(dip \<mapsto> (req, [d])) | Some (p, q) \<Rightarrow> store(dip \<mapsto> (p, q @ [d]))"
add_strict_right_mono: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subset># b" shows "a + c \<subset># b + c"
tpOf_rOfWax: fixes \<sigma> :: "'tp" shows "TE.tpOf (rOfWax \<sigma>) = \<sigma>"
fds_nth_def: shows "fds_nth \<equiv> map_fun Rep_fds id (case_nat (0::'b))"
exact_def: fixes k :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "exact k qs = exact_aux k (card X) qs"
n_3: shows "a <\<^bsub>F (\<lambda>i. if h i < n + 1 then Lab else Lba)\<^esub> b"
carrier_possible_subset: fixes t :: "'a ttree"   and A :: "'a set"   and x :: "'a" assumes "carrier t \<subseteq> A"   and "possible t x" shows "x \<in> A"
trm_eqI: fixes t :: "'a"   and f :: "'a" assumes "\<guillemotleft>t : local.dom f \<rightarrow> \<one>\<guillemotright>" shows "t = \<t>[f]"
assms: shows "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet> `\<^sub>\<bullet> A"
cinner_same_CBasis: fixes u :: "'a" assumes "u \<in> CBasis" shows "\<langle>u, u\<rangle> = 1"
vblcase_zeroX: fixes a :: "'a"   and b :: "vbl \<Rightarrow> 'a" shows "(case zeroX of zeroX \<Rightarrow> a | nextX x \<Rightarrow> b x) = a"
Radius: fixes Y :: "nat set" assumes "Y \<in> Roots'" shows "cmod (root' Y) \<le> Radius"
assms_2: shows "\<phi>\<^sub>a \<in> a"
head_app_iso: fixes hom :: "('e, 'f, 'd, 'c) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" shows "head (app_iso hom G) = iso_head hom"
rsp: shows "realrel (\<lambda>n. 0) (\<lambda>n. 0)"
Rats_inverse: fixes a :: "'a" assumes "a \<in> \<rat>" shows "inverse a \<in> \<rat>"
l8: shows "fact (p - 1) = \<Prod>local.S1"
noreps_cons: fixes x :: "'b"   and xs :: "'b list" shows "noreps (x # xs) = (noreps xs \<and> (xs = [] \<or> x \<noteq> hd xs))"
list_of_vec_map: fixes xs :: "'a Matrix.vec" shows "list_of_vec xs = map (($) xs) [0..<dim_vec xs]"
same_messages_imply_same_resulting_messages: fixes c :: "('p, 's, 'm) configuration"   and i :: "nat"   and d :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration" assumes "msgs c i = msgs d i"   and "c \<turnstile> ev \<mapsto> c'"   and "d \<turnstile> ev \<mapsto> d'"   and "regular_event ev" shows "msgs c' i = msgs d' i"
lhfr: shows "(\<exists>x. Ifm (x # bs) p) = (Ifm (x # bs) (minusinf (rlfm (simpfm p))) \<or> Ifm (x # bs) (plusinf (rlfm (simpfm p))) \<or> (\<exists>a\<in>set (uset (rlfm (simpfm p))). \<exists>b\<in>set (uset (rlfm (simpfm p))). Ifm (x # bs) (usubst (rlfm (simpfm p)) (case (a, b) of (x, xa) \<Rightarrow> (case x of (t, n) \<Rightarrow> \<lambda>(s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) xa))))"
clearjunk_restrict: fixes A :: "'c set"   and al :: "('c \<times> 'd) list" shows "AList.clearjunk (AList.restrict A al) = AList.restrict A (AList.clearjunk al)"
test_num': fixes i :: "int"   and a :: "'a" shows "test_num' i a = (i \<in> \<gamma> a)"
adjust_halt_eq: fixes l :: "cell list"   and r :: "cell list"   and ap :: "(action \<times> nat) list"   and stp :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "steps0 (1, l, r) ap stp = (0, l', r')"   and "tm_wf0 ap" shows "\<exists>stp. steps0 (Suc 0, l, r) (adjust0 ap) stp = (Suc (length ap div 2), l', r')"
assms: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
rel_fun_def: fixes A :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'j \<Rightarrow> bool" shows "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
PerpAt_def: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "X PerpAt A B C D  \<equiv> A \<noteq> B \<and> C \<noteq> D \<and> Col X A B \<and> Col X C D \<and> (\<forall>U V. Col U A B \<and> Col V C D \<longrightarrow> Per U X V)"
Re_det_sgn_congruence: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and H :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det M \<noteq> 0" shows "sgn (Re (mat_det (congruence M H))) = sgn (Re (mat_det H))"
a_owner_document_valid_def: fixes h :: "(_) heap" shows "local.a_owner_document_valid h = (\<forall>node_ptr\<in>fset (node_ptr_kinds h). (\<exists>document_ptr. document_ptr |\<in>| document_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_disconnected_nodes document_ptr|\<^sub>r) \<or> (\<exists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r))"
1: shows "path A (stake (length r) (sdrop (Suc k) w) || r) q = gupath A w ([Suc (Suc k)..<Suc k + length (q # r)] || r) (Suc k, q)"
antisym_conv: fixes y :: "'c list"   and x :: "'c list" assumes "suffix y x" shows "suffix x y = (x = y)"
assms: shows "\<forall>\<^sub>F x in F. f x = g x"
irc_on_discard: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set"   and C :: "'d set"   and B :: "'d set" assumes "irc_on A f"   and "finite C"   and "B \<union> C \<subseteq> A"   and "f (B \<union> C) \<inter> C = {}" shows "f (B \<union> C) = f B"
flows_state_IFS: shows "flows_state (generate_valid_stateful_policy_IFSACS_2 G M edgesList) = set (filter_compliant_stateful_ACS G M edgesList)"
bl2nat_cons_oc: fixes ks :: "cell list" shows "bl2nat (ks @ [Oc]) 0 = bl2nat ks 0 + 2 ^ length ks"
left_translations_of_group_def: fixes G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "left_translations_of_group G composition unit \<equiv> group G composition unit"
trans: fixes b :: "'a list"   and a :: "'a list"   and c :: "'a list" assumes "sublist b a"   and "sublist c b" shows "sublist c a"
nth_ucast: fixes w :: "'c word"   and n :: "nat" shows "ucast w !! n = (w !! n \<and> n < LENGTH('b))"
geom_rand_walk_strictly_positive: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real" assumes "prob_grw p M geom_proc u d init"   and "0 < init"   and "0 < d"   and "d < u" shows "\<forall>n w. 0 < geom_proc n w"
path_split_Cons: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" obtains a' :: "'a" and as' :: "'a list" where "as = a' # as'" and "n = sourcenode a'" and "valid_edge a'" and "CFG.path sourcenode targetnode valid_edge (targetnode a') as' n'"
cases: fixes a1 :: "('a, 'b, 'c) com"   and a2 :: "('a, 'b, 'c) com" assumes "com_relp a1 a2" obtains c1 :: "('a, 'b, 'c) com" and c2 :: "('a, 'b, 'c) com" where "a1 = c1__" and "a2 = Seq c1__ c2__"   | c2 :: "('a, 'b, 'c) com" and c1 :: "('a, 'b, 'c) com" where "a1 = c2a__" and "a2 = Seq c1a__ c2a__"   | c1 :: "('a, 'b, 'c) com" and b :: "'a set" and c2 :: "('a, 'b, 'c) com" where "a1 = c1b__" and "a2 = Cond b__ c1b__ c2b__"   | c2 :: "('a, 'b, 'c) com" and b :: "'a set" and c1 :: "('a, 'b, 'c) com" where "a1 = c2c__" and "a2 = Cond ba__ c1c__ c2c__"   | c :: "('a, 'b, 'c) com" and b :: "'a set" where "a1 = c__" and "a2 = While b c__"   | c :: "'a \<Rightarrow> ('a, 'b, 'c) com" and x :: "'a" where "a1 = ca__ x" and "a2 = DynCom ca__"   | c :: "('a, 'b, 'c) com" and f :: "'c" and g :: "'a set" where "a1 = c" and "a2 = Guard f g c"   | c1 :: "('a, 'b, 'c) com" and c2 :: "('a, 'b, 'c) com" where "a1 = c1d__" and "a2 = Catch c1d__ c2d__"   | c2 :: "('a, 'b, 'c) com" and c1 :: "('a, 'b, 'c) com" where "a1 = c2" and "a2 = Catch c1 c2"
semi_complete_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "semi_complete_on r A = (WN_on r A \<and> CR_on r A)"
root_shorter_eq: fixes u :: "'b list"   and r :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> r*" shows "\<^bold>|r\<^bold>| \<le> \<^bold>|u\<^bold>|"
MStep_Compose_Aux: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and l1 :: "int"   and r :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, s, n, l1, r) \<in> MStep" shows "\<forall>k t l2. (M, l1, r, k, l2, t) \<in> MStep \<longrightarrow> (M, l, s, n + k, l2, t) \<in> MStep"
eqk: fixes k :: "nat" shows "matrix_sum d (\<lambda>k. D0 k \<rho>) (Suc k) = M0 * \<rho> * adjoint M0 + matrix_sum d (\<lambda>k. D0 k (DS (M1 * \<rho> * adjoint M1))) k"
image_mset_id: fixes x :: "'a multiset" shows "image_mset id x = x"
locally_empty: fixes P :: "'a set \<Rightarrow> bool" shows "locally P {}"
csl': shows "cs\<^bsup>\<pi> @\<^bsup>l\<^esup> \<pi>l\<^esup> l = cs\<^bsup>\<pi>\<^esup> l"
null_eqI: fixes n :: "'a \<times> 'a" assumes "\<And>f. VV.comp n f = n \<and> VV.comp f n = n" shows "n = VV.null"
spr_sim_tFirst_tLast: fixes t :: "('a, 'es, 'as) BEState Trace"   and s :: "('a, 'es, 'as) spr_simWorld" assumes "spr_sim t = s"   and "t \<in> SPR.jkbpC" shows "(sprFst s, sprLst s) \<in> sprCRel s"
bex_triv_one_point2: fixes A :: "'b set"   and a :: "'b" shows "(\<exists>x\<in>A. a = x) = (a \<in> A)"
sign_permut: fixes p :: "'c \<Rightarrow> 'c"   and q :: "'b poly" shows "degree (of_int (sign p) * q) = degree q"
axioms: fixes k :: "enat"   and charts :: "('c, 'd) chart set"   and f :: "'c \<Rightarrow> 'e" assumes "diff_fun k charts f" shows "diff k charts charts_eucl f"
cblinfun_extension_existsI: fixes S :: "'c set"   and B :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and \<phi> :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> S \<Longrightarrow> B *\<^sub>V x = \<phi> x" shows "cblinfun_extension_exists S \<phi>"
powr_ln_powr_upper_bound: fixes a :: "real"   and x :: "real"   and b :: "real"   and p' :: "real" assumes "1 < a"   and "a \<le> x"   and "x \<le> b" shows "x powr p * ln x powr p' \<le> max (a powr p) (b powr p) * max (ln a powr p') (ln b powr p')"
exec2: fixes q :: "nat" assumes "q \<notin> set (map fst (step (hd w) (ss ! hd w)))"   and "q < length ss" shows "ssa ! q = ss ! q"
c: shows "Ide c \<and> Arr c \<and> Dom c = c \<and> Cod c = c \<and> ide \<lbrace>c\<rbrace> \<and> ide \<lbrace>\<^bold>\<lfloor>c\<^bold>\<rfloor>\<rbrace> \<and> \<lbrace>c\<^bold>\<down>\<rbrace> \<in> hom \<lbrace>c\<rbrace> \<lbrace>\<^bold>\<lfloor>c\<^bold>\<rfloor>\<rbrace>"
iT_Plus_neg_cut_eq: fixes t :: "nat"   and k :: "nat"   and I :: "nat set" assumes "t \<le> k" shows "I \<down>\<ge> t \<oplus>- k = I \<oplus>- k"
inner_verts_conv: fixes p :: "'b list"   and u :: "'a" shows "inner_verts p = butlast (tl (awalk_verts u p))"
degree_m_mult_eq: fixes p :: "int poly"   and q :: "int poly" assumes "monic p"   and "lead_coeff q mod p ^ l \<noteq> 0"   and "1 < p ^ l" shows "pl.degree_m (p * q) = degree p + degree q"
trms_list\<^sub>s\<^sub>t_def: fixes S :: "('c, 'd) strand_step list" shows "trms_list\<^sub>s\<^sub>t S \<equiv> remdups (concat (map trms_list\<^sub>s\<^sub>t\<^sub>p S))"
cont: fixes f :: "real \<Rightarrow> real" assumes "antimono_fun_sum_integral_diff f" shows "continuous_on {0..} f"
pullback_sums_map_double_middle2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and F :: "'b \<Rightarrow> 'c"   and xs :: "'b list"   and ds :: "'c list"   and d :: "'c"   and e :: "'c"   and es :: "'c list"   and f :: "'c"   and g :: "'c"   and gs :: "'c list" assumes "class.monoid_add plus zero"   and "map F (monoid_add.sums plus zero xs) = ds @ [d, e] @ es @ [f, g] @ gs" shows "\<exists>as a bs b cs. xs = as @ [a] @ bs @ [b] @ cs \<and> d = F (monoid_add.sum_list plus zero as) \<and> e = F (monoid_add.sum_list plus zero (as @ [a])) \<and> f = F (monoid_add.sum_list plus zero (as @ [a] @ bs)) \<and> g = F (monoid_add.sum_list plus zero (as @ [a] @ bs @ [b]))"
26279: fixes x :: "'a"   and y :: "'a" shows "- - (x \<sqinter> y) \<le> - - x"
Abs_List_inverse: fixes y :: "('b, unit) Old_Datatype.node set" assumes "y \<in> List" shows "Rep_List (Abs_List y) = y"
prems_1: shows "iconf (shp s) (RI (C,e) ; [] \<leftarrow> e')"
before_or: fixes ram1 :: "'a"   and vs :: "'a list"   and ram2 :: "'a" assumes "ram1 \<in> set vs"   and "ram2 \<in> set vs"   and "ram1 \<noteq> ram2" shows "before vs ram1 ram2 \<or> before vs ram2 ram1"
assms_8: shows "pathfinish g $ 2 = a $ 2"
flip_def: fixes tup :: "'d \<times> 'c" shows "flip tup = (snd tup, fst tup)"
map_prod_def: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'f \<Rightarrow> 'h" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
f: shows "f \<in> N \<rightarrow>\<^sub>M subprob_algebra B"
fmlookup_ran'_iff: fixes y :: "'d"   and m :: "'e \<rightharpoonup>\<^sub>f 'd" shows "(y \<in> fmran' m) = (\<exists>x. fmlookup m x = Some y)"
simps: fixes AB :: "('b \<times> bal) tree"   and c :: "'b"   and bc :: "bal"   and C :: "('b \<times> bal) tree" shows "balL AB c bc C = (case bc of Lh \<Rightarrow> case AB of \<langle>A, (a, Lh), B\<rangle> \<Rightarrow> \<langle>A, (a, Bal), \<langle>B, (c, Bal), C\<rangle>\<rangle> | \<langle>A, (a, Bal), B\<rangle> \<Rightarrow> \<langle>A, (a, Rh), \<langle>B, (c, Lh), C\<rangle>\<rangle> | \<langle>A, (a, Rh), B\<rangle> \<Rightarrow> rot2 A a B c C | Bal \<Rightarrow> \<langle>AB, (c, Lh), C\<rangle> | Rh \<Rightarrow> \<langle>AB, (c, Bal), C\<rangle>)"
le_add_same_cancel1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus a b) = less_eq zero b"
inf_def: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linorder_lattice inf sup less_eq less" shows "inf x y = (if less_eq x y then x else y)"
f3: assumes "triangle_circumscribes_triangle A' B' C' A B C" shows "col M N P"
gs1: shows "gs1 = filter (\<lambda>gi. p.dvdm gi h) gs"
nn_integral_T: fixes f :: "'s stream \<Rightarrow> ennreal"   and s :: "'s" assumes "f \<in> borel_measurable (stream_space (count_space UNIV))" shows "integral\<^sup>N (T s) f = \<integral>\<^sup>+ t. \<integral>\<^sup>+ \<omega>. f (t ## \<omega>) \<partial>T t \<partial>measure_pmf (K s)"
induct: fixes x1 :: "conc_interface"   and x2 :: "conc_interface"   and P :: "conc_interface \<Rightarrow> conc_interface \<Rightarrow> bool" assumes "x1 \<simeq> x2"   and "\<And>Pa. P Pa Pa"   and "\<And>Pa Q. \<lbrakk>Pa \<simeq> Q; P Pa Q\<rbrakk> \<Longrightarrow> P Q Pa"   and "\<And>Pa Q R. \<lbrakk>Pa \<simeq> Q; P Pa Q; Q \<simeq> R; P Q R\<rbrakk> \<Longrightarrow> P Pa R"   and "\<And>Pa Q P'. \<lbrakk>Pa \<simeq> Q; P Pa Q\<rbrakk> \<Longrightarrow> P (P' \<otimes>\<^sub>c Pa) (P' \<otimes>\<^sub>c Q)"   and "\<And>Pa Q P'. \<lbrakk>Pa \<simeq> Q; P Pa Q\<rbrakk> \<Longrightarrow> P (Pa \<otimes>\<^sub>c P') (Q \<otimes>\<^sub>c P')"   and "\<And>Pa Q x. \<lbrakk>Pa \<simeq> Q; P Pa Q\<rbrakk> \<Longrightarrow> P (Exists_int_conc x Pa) (Exists_int_conc x Q)"   and "\<And>Pa Q R. P (Pa \<otimes>\<^sub>c (Q \<otimes>\<^sub>c R)) ((Pa \<otimes>\<^sub>c Q) \<otimes>\<^sub>c R)"   and "\<And>Pa Q. P (Pa \<otimes>\<^sub>c Q) (Q \<otimes>\<^sub>c Pa)"   and "\<And>Pa. P (\<epsilon>\<^sub>c \<otimes>\<^sub>c Pa) Pa"   and "\<And>Pa. P (Pa \<otimes>\<^sub>c \<epsilon>\<^sub>c) Pa" shows "P x1 x2"
finfun_rec_unique: fixes f :: "('c, 'a) finfun \<Rightarrow> 'b"   and cnst :: "'a \<Rightarrow> 'b"   and upd :: "('c, 'a) finfun \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "('c, 'a) finfun \<Rightarrow> 'b" assumes "\<And>c. f (finfun_const c) = cnst c"   and "\<And>g a b. f (finfun_update g a b) = upd g a b (f g)"   and "\<And>c. f' (finfun_const c) = cnst c"   and "\<And>g a b. f' (finfun_update g a b) = upd g a b (f' g)" shows "f = f'"
preserves_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (\<ll> f)"
alphaBoundOutput: fixes y :: "name"   and B :: "('d, 'e, 'f) boundOutput"   and x :: "name" assumes "y \<sharp> B" shows "\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>y\<rparr>([(x, y)] \<bullet> B)"
lg_sym: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p" assumes "QCong l"   and "l A B" shows "l B A"
unforwarded_non_volatile_reads_antimono: fixes W :: "nat set"   and W' :: "nat set"   and xs :: "'b memref list" assumes "W \<subseteq> W'" shows "unforwarded_non_volatile_reads xs W' \<subseteq> unforwarded_non_volatile_reads xs W"
minus_add_uminus_vec: fixes v :: "'a vec"   and n :: "nat"   and w :: "'a vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "v - w = v + - w"
subst_of_dom_subset: fixes L :: "('a \<times> ('b, 'a) Term.term) list" shows "subst_domain (subst_of L) \<subseteq> set (map fst L)"
env_axioms: shows "env substTerm substAssert substCond (\<otimes>) SImp' (\<one>) SChanEq'"
ListSum_V_F_eq_ListSum_F: fixes g :: "graph"   and V :: "nat list"   and P :: "face \<Rightarrow> bool"   and w :: "face \<Rightarrow> nat" assumes "Invariants.inv g"   and "separated g (set V)"   and "distinct V"   and "set V \<subseteq> \<V> g"   and "\<forall>f\<in>Collect P. |vertices f| \<le> 4" shows "(\<Sum>\<^bsub>v\<in>V\<^esub> ListSum (filter P (facesAt g v)) w) = ListSum (filter (\<lambda>f. \<exists>v\<in>set V. f \<in> set (facesAt g v) \<inter> Collect P) (faces g)) w"
set_mkIde: fixes A :: "'s set" assumes "A \<subseteq> Univ"   and "|A| <o cardSuc (cmax |UNIV| natLeq)" shows "local.set (mkIde A) = A"
select_convs_3: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and more :: "'b" shows "g_V0 \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V0"
arg_apps_Nil: fixes s :: "('a, 'b) tm" shows "arg (apps s []) = arg s"
subst_bv2_subst_bv2: fixes i :: "nat"   and j :: "nat"   and t :: "Core.term"   and v :: "Core.term"   and u :: "Core.term" assumes "i < j + 1" shows "subst_bv2 (subst_bv2 t (Suc j) (lift v i)) i (subst_bv2 u j v) = subst_bv2 (subst_bv2 t i u) j v"
adjoint_transpose_left_3: fixes u :: "'a"   and v :: "'a"   and \<mu> :: "'a" assumes "ide u"   and "ide v"   and "src k = trg v"   and "src k\<^sup>* = trg u"   and "\<guillemotleft>\<mu> : k \<star> v \<Rightarrow> u\<guillemotright>" shows "k.trnl\<^sub>\<epsilon> u (k.trnl\<^sub>\<eta> v \<mu>) = \<mu>"