hcju/isabelleps huggingface dataset:
rComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
red_unionI1: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "red F p q" shows "red (F \<union> G) p q"
signed_lconjaction_fst: fixes s :: "'a"   and x :: "'a \<times> bool" shows "fst (signed_lconjaction s x) = lconjby s (fst x)"
set_replicate: fixes n :: "nat"   and x :: "'g" assumes "n \<noteq> 0" shows "set (replicate n x) = {x}"
dghm_proj_snd_is_dghm: fixes i :: "V"   and I :: "V" assumes "i \<in>\<^sub>\<circ> I" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>2 \<AA> \<BB> : \<AA> \<times>\<^sub>D\<^sub>G \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
incomplete_designI: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size"   and "u_block_size < int (card point_set)" shows "incomplete_design point_set block_collection u_block_size"
less_Dom_def: fixes d1 :: "Dom"   and d2 :: "Dom" shows "(d1 < d2) = (if d1 = d2 then False else if d1 = low then True else False)"
psi_l_def: fixes l :: "nat" shows "psi_l l = alpha_l l \<cdot>\<^sub>v \<alpha> + beta_l l \<cdot>\<^sub>v \<beta>"
t_def: shows "t \<equiv> \<lambda>x. if s x \<down> \<and> c (the (s x)) \<down> then Some (the (c (the (s x)))) else None"
simps: fixes \<Sigma> :: "'a set set"   and \<phi> :: "'a ltl" shows "\<A>.F\<^sub>\<A> \<Sigma> \<phi> = {(M_fin \<pi> \<union> \<Union> {\<A>.Acc_fin \<Sigma> \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}, {\<A>.Acc_inf \<pi> \<chi> |\<chi>. \<chi> \<in> dom \<pi>}) |\<pi>. dom \<pi> \<subseteq> \<^bold>G \<phi> \<and> (\<forall>\<chi>\<in>dom \<pi>. the (\<pi> \<chi>) < \<A>.max_rank_of \<Sigma> \<chi>)}"
A: fixes x :: "'a"   and y :: "'b" assumes "(x, y) \<in> set_pmf pq" shows "A x y"
finsert_fminus1: fixes x :: "'c"   and B :: "'c fset"   and A :: "'c fset" assumes "x |\<in>| B" shows "finsert x A |-| B = A |-| B"
rev_SPLIT_reflection: fixes x :: "'a"   and y :: "'a" assumes "x \<equiv> y" shows "Case_Labeling.SPLIT x y"
oalist_inv_rawD2: fixes xs :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs" shows "sorted_wrt tc.lt (map fst xs)"
slp_vp_Append: fixes n :: "'b LDCFG_node"   and as :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n'' :: "'b LDCFG_node"   and as' :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list"   and n' :: "'b LDCFG_node" assumes "CFGExit.same_level_path' n as n''"   and "CFGExit.valid_path' n'' as' n'" shows "CFGExit.valid_path' n (as @ as') n'"
assms_2: fixes J :: "'a set" assumes "J \<in> I" shows "uniform_limit J f g F"
G_is_functor: fixes D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "left_adjoint_functor D C F" shows "functor C D (left_adjoint_functor.G D C F)"
B_def: shows "B = c / real (n - 1) * (1 / (a - l') ^ (n - 1) + 1 / (r' - a) ^ (n - 1))"
simps_2: fixes f :: "'b \<Rightarrow> 'a pred"   and x :: "'b"   and P :: "'b pred" shows "apply f (Insert x P) = Join (f x) (Join (bind P f) Empty)"
EF_lem2b: fixes x :: "'b"   and f :: "'b set" assumes "x \<in> EX' (EF f)" shows "x \<in> EF f"
Fadmit: shows "Fadmit \<sigma> (seq2fml (\<Gamma>, \<Delta>))"
simps_3: fixes n :: "nat"   and uu :: "'b"   and l :: "('a, 'b) Digit"   and t :: "('a, 'b) FingerTreeStruc"   and r :: "('a, 'b) Digit" shows "FingerTreeStruc.is_leveln_ftree n (Deep uu l t r) = (FingerTreeStruc.is_leveln_digit n l \<and> FingerTreeStruc.is_leveln_digit n r \<and> FingerTreeStruc.is_leveln_ftree (Suc n) t)"
integral_spmf_of_set: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" shows "integral\<^sup>L (measure_spmf (spmf_of_set A)) f = sum f A / real (card A)"
length_greaters_le': fixes i :: "nat"   and xs :: "'b list"   and A :: "'b set"   and R :: "('b \<times> 'b) set" assumes "i < length xs"   and "linorder_on A R"   and "set xs \<subseteq> A" shows "length (greaters R (insort_wrt R xs ! i) xs) \<le> length xs - i - 1"
getTvars_def: fixes \<sigma>l :: "'tp list" shows "GE.getTvars \<sigma>l \<equiv> map Var (GE.getVars \<sigma>l)"
lift_Pareto_SD_iff: fixes p :: "'b pmf"   and q :: "'b pmf" assumes "p \<in> lotteries_on alts"   and "q \<in> lotteries_on alts" shows "Pareto (SD \<circ> R') p q = Pareto (SD \<circ> R) p q"
hyps_4: fixes b :: "name"   and bb :: "lam"   and ba :: "name" assumes "b \<sharp> N1"   and "b \<sharp> bb" shows "b \<sharp> N1[ba::=bb]"
size_mset_sum_mset_distrib: fixes A :: "'b multiset multiset" shows "size (\<Sum>\<^sub># A) = \<Sum>\<^sub># (image_mset size A)"
sel_6: fixes x1 :: "real"   and x2 :: "real"   and x3 :: "real"   and x4 :: "real"   and x5 :: "real"   and x6 :: "real"   and x7 :: "real"   and x8 :: "real" shows "Im5 (Octo x1 x2 x3 x4 x5 x6 x7 x8) = x6"
reflE: fixes cl :: "'b potype"   and x :: "'b" assumes "PO cl"   and "x \<in> pset cl" shows "(x, x) \<in> order cl"
map_of_eq_empty_iff: fixes xys :: "('l \<times> 'm) list" shows "(map_of xys = Map.empty) = (xys = [])"
f5: shows "(r' \<otimes> r'', s' \<otimes> s'') \<in> carrier rel"
refl: fixes g :: "'a"   and succs :: "'a \<Rightarrow> 'a list" shows "g [succs]\<rightarrow>* g"
sg20: shows "\<forall>j\<le>Suc d. lose (t + 3 + k + d + j) = [False]"
rep_eq: fixes x :: "'a permutation"   and xa :: "'a permutation" shows "(\<rightarrow>) (x + xa) = (\<rightarrow>) x \<circ> (\<rightarrow>) xa"
split_derivation_leftmost: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat" assumes "derivation_ge D i"   and "\<not> derivation_ge D (Suc i)" shows "\<exists>E F r. D = E @ [(i, r)] @ F \<and> derivation_ge E (Suc i)"
neutral: fixes b :: "real"   and a :: "real" assumes "b \<le> a" shows "g {a..b} = \<^bold>1"
map\<^sub>0_simps_5: fixes a :: "'a" assumes "C.obj a" shows "D.cod (F.map\<^sub>0 a) = F.map\<^sub>0 a"
S_subst: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('f, 'c) Term.term" assumes "S s t" shows "S (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
arr_cod_iff_arr: fixes f :: "'a" shows "Left_a.arr (Left_a.cod f) = Left_a.arr f"
asm: shows "v = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), 0, 0, \<alpha> / complex_of_real (sqrt 2), 0, \<beta> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), 0]]"
conv_dist_sup: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and conv :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv (sup x y) = sup (conv x) (conv y)"
image_inv_f_f: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set" assumes "inj f" shows "inv f ` f ` A = A"
in_lset_lappend_iff: fixes x :: "'c"   and xs :: "'c llist"   and ys :: "'c llist" shows "(x \<in> lset (lappend xs ys)) = (x \<in> lset xs \<or> lfinite xs \<and> x \<in> lset ys)"
remove_nth_sound_r: fixes n :: "nat"   and p :: "nat"   and xs :: "'c list" assumes "n \<le> p"   and "p < length xs" shows "remove_nth n xs ! p = xs ! Suc p"
antec_upRule: fixes Ps :: "'a list"   and \<Phi> :: "'b form multiset"   and \<Psi> :: "'b form multiset" assumes "(Ps,  \<Phi> \<Rightarrow>* \<Psi>) \<in> upRules" shows "\<Phi> = \<Empt> \<or> (\<exists>A. \<Phi> = \<LM> A  \<RM>)"
injective_vimage_restrict: fixes J :: "'d set"   and I :: "'d set"   and A :: "('d \<Rightarrow> 'e) set"   and S :: "'d \<Rightarrow> 'e set"   and B :: "('d \<Rightarrow> 'e) set" assumes "J \<subseteq> I"   and "A \<subseteq> Pi\<^sub>E J S"   and "B \<subseteq> Pi\<^sub>E J S"   and "Pi\<^sub>E I S \<noteq> {}"   and "(\<lambda>x. restrict x J) -` A \<inter> Pi\<^sub>E I S = (\<lambda>x. restrict x J) -` B \<inter> Pi\<^sub>E I S" shows "A = B"
horner_sum_bit_eq_take_bit: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) (map (bit a) [0..<n]) = take_bit n a"
assms_6: shows "h \<turnstile> get_owner_document ptr \<rightarrow>\<^sub>r owner_document"
is_redE: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and a :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F a" obtains b :: "'t \<Rightarrow>\<^sub>0 'c" where "red F a b"
DomainE: fixes a :: "'g"   and r :: "('g \<times> 'h) set" assumes "a \<in> Domain r" obtains b :: "'h" where "(a, b) \<in> r"
Gnat: fixes n :: "nat" shows "good (VNat n) = {BNat n}"
beukers_aux_integral_transform1: fixes z :: "real" assumes "z \<in> {0<..<1}" shows "LBINT (x, y):D. P x * P y / (1 - (1 - x * y) * z) = LBINT (x, y):D. P x * (x * y * z) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * z) ^ (n + 1)"
pv_decomp: shows "awalk_verts u p = xs @ w # ys"
const_polyI: fixes p :: "'b poly" assumes "degree p = 0" shows "[:coeff p 0:] = p"
assms_1: shows "\<Phi>.lang\<^sub>M\<^sub>2\<^sub>L n \<phi> = \<Phi>.lang\<^sub>M\<^sub>2\<^sub>L n \<psi>"
f_def: fixes v :: "'a" shows "f v = (if v \<in> T then f0 v else 0)"
empty_Q0: shows "Q_of_wstate (lhd Sts) = {}"
skip_vals_nonequiv: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and j :: "nat" assumes "vals_nonequiv K (Suc (Suc n)) vv" shows "vals_nonequiv K (Suc n) (compose {l. l \<le> Suc n} vv (skip j))"
finsert_fminus_single: fixes a :: "'a"   and A :: "'a fset" shows "finsert a (A |-| {|a|}) = finsert a A"
less_1_helper: fixes n :: "int"   and m :: "int" assumes "n \<le> m" shows "n - 1 < m"
bind_def: fixes M :: "'b nres"   and f :: "'b \<Rightarrow> 'a nres" shows "M \<bind> f \<equiv> case M of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> Sup (f ` X)"
assms_2: shows "\<^bold>|p\<^bold>| < \<^bold>|u\<^bold>|"
linear_compose_neg: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "linear (\<lambda>x. - f x)"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : \<I> \<otimes> \<I> \<Rightarrow> \<I>\<guillemotright>"
trueReds: shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),(h\<^sub>2, l\<^sub>2, sh\<^sub>2),icheck P D (C\<bullet>\<^sub>sM(map Val vs))\<rangle> \<rightarrow>* \<langle>ef,(h\<^sub>3, override_on (l\<^sub>2 ++ l\<^sub>3) l\<^sub>2 (set pns), sh\<^sub>3),b\<^sub>3\<rangle>"
\<ll>_ide_simp: fixes f :: "'b" assumes "D.ide f" shows "D.\<ll> f = \<l>\<^sub>D[f]"
QE_dnf'_eval: fixes step :: "nat \<Rightarrow> (atom list \<times> atom fm list \<times> nat) list \<Rightarrow> atom fm"   and opt :: "atom fm \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and xs :: "real list" assumes "\<And>amount F \<Gamma>. (\<exists>xs. length xs = amount \<and> eval (list_disj (map (\<lambda>(L, F, n). ExN n (list_conj (map fm.Atom L @ F))) F)) (xs @ \<Gamma>)) = eval (step amount F) \<Gamma>"   and "\<And>xs F. eval (opt F) xs = eval F xs" shows "eval (QE_dnf' opt step \<phi>) xs = eval \<phi> xs"
hyps_3: fixes thesis :: "bool" assumes "\<And>ps T. \<lbrakk>valid_decomp X ps; standard_decomp (poly_deg f0) ps; cone_decomp T ps; (\<And>f. f \<in> F \<Longrightarrow> homogeneous f) \<Longrightarrow> hom_decomp ps; direct_decomp (ideal (insert f0 F) \<inter> P[X]) [ideal {f0} \<inter> P[X], T]\<rbrakk> \<Longrightarrow> thesis"   and "F \<subseteq> P[X]"   and "\<And>f. f \<in> F \<Longrightarrow> poly_deg f \<le> poly_deg f0" shows "thesis"
is_map_from_is_homomorphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (im_sheaf_morphisms U V) (local.im_sheaf U) (local.im_sheaf V)"
map\<^sub>0_def: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>F :: "'d \<times> 'd \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>G :: "'d \<times> 'd \<Rightarrow> 'e"   and H :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>H :: "'c \<times> 'c \<Rightarrow> 'd"   and \<tau>\<^sub>0 :: "'d \<Rightarrow> 'e"   and \<tau>\<^sub>1 :: "'d \<Rightarrow> 'e"   and a :: "'c" assumes "pseudonatural_transformation_whisker_right V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "pseudonatural_transformation_whisker_right.map\<^sub>0 src\<^sub>C H \<tau>\<^sub>0 a = \<tau>\<^sub>0 (weak_arrow_of_homs.map\<^sub>0 src\<^sub>C H a)"
ant_z_in_Ainteg: fixes z :: "int" shows "(z, 0) \<in> Ainteg"
simps_2: fixes d :: "nat"   and ds :: "nat list"   and i :: "nat"   and "is" :: "nat list" shows "digit_decode (d # ds) (i # is) = i + d * digit_decode ds is"
HSem_bot_eq: fixes \<Gamma> :: "(Vars.var \<times> exp) list" shows "\<N>\<lbrace>\<Gamma>\<rbrace> = \<^bold>\<N>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>"
circ_slide: fixes x :: "'a"   and y :: "'a" shows "x * (y * x)\<^sup>\<star> = (x * y)\<^sup>\<star> * x"
allocationUnion: fixes a1 :: "('c \<times> 'd set) set"   and a2 :: "('c \<times> 'd set) set" assumes "a1 \<in> allocationsUniverse"   and "a2 \<in> allocationsUniverse"   and "\<Union> (Range a1) \<inter> \<Union> (Range a2) = {}"   and "Domain a1 \<inter> Domain a2 = {}" shows "a1 \<union> a2 \<in> allocationsUniverse"
\<epsilon>_def: shows "\<epsilon> \<equiv> transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) FG.map \<epsilon>o"
prime_elem_iff_irreducible: fixes x :: "'b" shows "prime_elem x = irreducible x"
Prim_Impl_axioms: shows "Prim_Impl G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin g G_\<alpha>w G_\<alpha>g"
assms_3: fixes x :: "'a" assumes "x \<in> A - B" shows "f x = return_pmf dflt"
npClose: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and n :: "nat" assumes "Ring R"   and "x \<in> carrier R" shows "x^\<^bsup>R n\<^esup> \<in> carrier R"
hom_imp_img_comm_monoid: fixes h :: "'a \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "h \<in> hom G H" shows "Group.comm_monoid (H\<lparr>carrier := h ` carrier G, one := h \<one>\<rparr>)"
d: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j \<le> i" shows "IArray.list_of (IArray.list_of (d\<mu>_impl fs) ! i) ! j = \<mu>' i j"
full_hensels_lemma: fixes p :: "int"   and f :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "hensel p f a" obtains \<alpha> :: "nat \<Rightarrow> int" where "UP_cring.to_fun (padic_int p) f \<alpha> = \<zero>\<^bsub>padic_int p\<^esub>" and "\<alpha> \<in> carrier (padic_int p)" and "padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) a) < padic_integers.val_Zp p (a \<ominus>\<^bsub>padic_int p\<^esub> \<alpha>)" and "padic_integers.val_Zp p (a \<ominus>\<^bsub>padic_int p\<^esub> \<alpha>) = padic_integers.val_Zp p (padic_integers.divide p (UP_cring.to_fun (padic_int p) f a) (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) a))" and "padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) \<alpha>) = padic_integers.val_Zp p (UP_cring.to_fun (padic_int p) (UP_cring.pderiv (padic_int p) f) a)"
\<C>_eq_type_max_eq: fixes t :: "'BExp set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "type_wellformed t"   and "\<forall>x\<in>\<C>. mem\<^sub>1 x = mem\<^sub>2 x" shows "type_max t mem\<^sub>1 = type_max t mem\<^sub>2"
count_image_mset_eq_card_vimage: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and b :: "'d" assumes "finite A" shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow> a'\<guillemotright> \<and> local.iso f)"
\<phi>_simps_2: shows "src r\<^sub>0s\<^sub>1.\<phi> = src p\<^sub>0"
rupture_surj_inj_on: fixes K :: "'a list set"   and p :: "'a list list" assumes "subfield K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "inj_on (UP.rupture_surj K p) ((\<lambda>q. UP.pmod q p) ` carrier (K [X]\<^bsub>K [X]\<^esub>))"
trans: fixes b :: "'e DBMEntry"   and a :: "'e DBMEntry"   and c :: "'e DBMEntry" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
mult2_alt_emptyR: fixes X :: "'a multiset"   and b :: "bool"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "((X, {#}) \<in> mult2_alt b ns s) = (b \<and> X = {#})"
assms_6: fixes t :: "real" assumes "t \<in> {t1<..<t2}" shows "flow0 x t \<notin> {a<--<b}"
psubstEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodEnv rho"   and "goodEnv rho'"   and "freshEnv zs z rho"   and "freshEnv zs z rho'" shows "freshEnv zs z rho &[rho']"
IH: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updTs ts TAS) (upd_invs I P TAS)"
lrev_body2_strict: fixes r :: "'b llist \<rightarrow> 'b llist \<rightarrow> 'b llist" shows "lrev_body2\<cdot>r\<cdot>\<bottom> = \<bottom>"
pair_wf_digraph_subdivide: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "pair_wf_digraph G"   and "e \<in> parcs G"   and "w \<notin> pverts G" shows "pair_wf_digraph (subdivide G e w)"
methd_norec: fixes G :: "prog"   and declC :: "qtname"   and c :: "class"   and sig :: "sig"   and m :: "methd" assumes "class G declC = Some c"   and "ws_prog G"   and "table_of (methods c) sig = Some m" shows "methd G declC sig = Some (declC, m)"
int_encode_inverse: fixes x :: "int" shows "int_decode (int_encode x) = x"
game_multi_def: fixes \<A> :: "'c_a \<Rightarrow> (unit, 'call + 'guess, 'ret + unit) gpv" shows "game_multi \<A> = init \<bind> (\<lambda>(c_o, c_a, s). exec_gpv (\<dagger>(oracle c_o) \<oplus>\<^sub>O eval_oracle c_o c_a) (\<A> c_a) (False, s) \<bind> (\<lambda>(uu_, b, uu_). return_spmf b))"
consistency_def: fixes C :: "('d, 'e) form set set" shows "consistency C = (\<forall>S. S \<in> C \<longrightarrow> (\<forall>p ts. \<not> (Pred p ts \<in> S \<and> Neg (Pred p ts) \<in> S)) \<and> FF \<notin> S \<and> Neg TT \<notin> S \<and> (\<forall>Z. Neg (Neg Z) \<in> S \<longrightarrow> S \<union> {Z} \<in> C) \<and> (\<forall>A B. And A B \<in> S \<longrightarrow> S \<union> {A, B} \<in> C) \<and> (\<forall>A B. Neg (Or A B) \<in> S \<longrightarrow> S \<union> {Neg A, Neg B} \<in> C) \<and> (\<forall>A B. Or A B \<in> S \<longrightarrow> S \<union> {A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (And A B) \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {Neg B} \<in> C) \<and> (\<forall>A B. Impl A B \<in> S \<longrightarrow> S \<union> {Neg A} \<in> C \<or> S \<union> {B} \<in> C) \<and> (\<forall>A B. Neg (Impl A B) \<in> S \<longrightarrow> S \<union> {A, Neg B} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Forall P \<in> S \<longrightarrow> S \<union> {P[t/0]} \<in> C) \<and> (\<forall>P t. closedt 0 t \<longrightarrow> Neg (Exists P) \<in> S \<longrightarrow> S \<union> {Neg (P[t/0])} \<in> C) \<and> (\<forall>P. Exists P \<in> S \<longrightarrow> (\<exists>x. S \<union> {P[App x []/0]} \<in> C)) \<and> (\<forall>P. Neg (Forall P) \<in> S \<longrightarrow> (\<exists>x. S \<union> {Neg (P[App x []/0])} \<in> C)))"
autoref_int_10: shows "(uminus, uminus) \<in> int_rel \<rightarrow> int_rel"
split_list_first_propE: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'g list" and x :: "'g" and zs :: "'g list" where "xs = ys @ x # zs" and "P x" and "\<forall>y\<in>set ys. \<not> P y"
matrix_invertible_gen: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "invertible (matrix f) = (\<exists>g. Vector_Spaces.linear (*s) (*s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
f2r_kcomp_pres: fixes f :: "'a \<Rightarrow> 'c set"   and g :: "'c \<Rightarrow> 'b set" shows "\<R> (f \<circ>\<^sub>K g) = \<R> f ; \<R> g"
Inf_exp_lower: fixes P :: "'a \<Rightarrow> real"   and S :: "('a \<Rightarrow> real) set" assumes "P \<in> S"   and "\<forall>P\<in>S. nneg P" shows "Inf_exp S \<tturnstile> P"