hcju/isabelleps huggingface dataset:
comm_ring_1_ow_def: fixes U :: "'c set"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" shows "comm_ring_1_ow U times one plus zero minus uminus \<equiv> comm_ring_ow U plus zero minus uminus times \<and> zero_neq_one_ow U one zero \<and> comm_monoid_mult_ow U times one"
S5EnvUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [ENext p]_(e p) \<and> $S5 rmhist p \<longrightarrow> unchanged e p"
elims_1: fixes x :: "'a option"   and y :: "bool" assumes "isSome x = y" obtains uu_ :: "'a" where "x = Some uu_" and "y"   | "x = None" and "\<not> y"
R01iar_def: shows "R01iar \<equiv> {(s, s'). signals s = signalsResp s'}"
single_valued_Mon2Rel: fixes B :: "'a \<Rightarrow> (unit \<times> 'a) option" shows "single_valued (Mon2Rel B)"
simps: fixes e :: "'c"   and a :: "'d"   and r :: "nat"   and ts :: "('c, 'd) SkewBinomialTree list" shows "height_tree (Node e a r ts) = height_queue ts"
elims: fixes x :: "('a \<times> 'b) list"   and y :: "nat" assumes "|x| = y" obtains "x = []" and "y = 0"   | h :: "'a \<times> 'b" and t :: "('a \<times> 'b) list" where "x = h # t" and "y = Suc |(delete t (fst h))|"
trans_Step2: fixes cf :: "('test, 'atom) com \<times> 'state"   and cf' :: "('test, 'atom) com \<times> 'state"   and cf'' :: "('test, 'atom) com \<times> 'state" assumes "cf \<rightarrow>*c cf'"   and "cf' \<rightarrow>c cf''" shows "cf \<rightarrow>*c cf''"
square_part_0: shows "square_part (0::'a) = (0::'a)"
is_const_list: fixes ves :: "e list"   and vs :: "v list" assumes "ves = $$* vs" shows "const_list ves"
\<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G_def: shows "\<sigma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G \<equiv> {([], \<Gamma>\<^sub>Q\<^sub>M\<^sub>S\<^sub>G ())}"
preserves_comp: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VoV.seq g f" shows "VoV.map (VoV.comp g f) = VoV.comp (VoV.map g) (VoV.map f)"
smallomega_1_conv_filterlim: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f \<in> \<omega>[F](\<lambda>_. 1::'c)) = filterlim f at_infinity F"
simps: fixes t1 :: "('b \<times> nat) tree"   and t2 :: "('b \<times> nat) tree" shows "balanced1 t1 t2 = (\<Delta>2 * (size_wbt t2 + 1) \<le> \<Delta>1 * (size_wbt t1 + 1))"
list_emb_append2: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "list_emb P xs ys" shows "list_emb P xs (zs @ ys)"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "f (b1 + b2) = f b1 + f b2"
components_red_rtrancl_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "(red F)\<^sup>*\<^sup>* p q" shows "component_of_term ` keys q \<subseteq> component_of_term ` keys p \<union> component_of_term ` Keys F"
subu_fin: shows "- \<infinity> < subcocycle_avg_ereal u"
\<sigma>'_def: shows "\<sigma>' = \<sigma>_arbitrary(v := w)"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c list"   and b :: "'d list" shows "list_all2 R\<inverse>\<inverse> a b = list_all2 R b a"
in_spanI: fixes v :: "'a vec"   and n :: "nat"   and a :: "'a vec \<Rightarrow> 'a"   and A :: "'a vec set"   and W :: "'a vec set" assumes "v = module.lincomb (module_vec TYPE('a) n) a A"   and "finite A"   and "A \<subseteq> W" shows "v \<in> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) W"
af_G_letter_abs_opt_def: shows "\<up>af\<^sub>G\<^sub>\<UU> \<equiv> af_G_abs_opt.f_abs"
inject_1: fixes x1 :: "'a"   and y1 :: "'a" shows "(Inl x1 = Inl y1) = (x1 = y1)"
has_ell2_norm_def: fixes x :: "'b \<Rightarrow> complex" shows "has_ell2_norm x = bdd_above (sum (\<lambda>i. (cmod (x i))\<^sup>2) ` Collect finite)"
ScopeExt': fixes xvec :: "name list"   and \<Psi>' :: "'b"   and T :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>'"   and "xvec \<sharp>* T" shows "(\<Psi>', \<lparr>\<nu>*xvec\<rparr>S \<parallel> T, (\<lparr>\<nu>*xvec\<rparr>S) \<parallel> T) \<in> Rel'"
union_intersect_unique_imp_sunflower: fixes S :: "'b set set" assumes "\<And>x. x \<in> \<Union> S - \<Inter> S \<Longrightarrow> \<exists>\<^sub>\<le>\<^sub>1 A. A \<in> S \<and> x \<in> A" shows "sunflower S"
fminusE: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A |-| B" obtains "c |\<in>| A" and "c |\<notin>| B"
pre_ranks_Min: fixes x :: "nat" shows "(x < Min (pre_ranks r (w n) q)) = (\<forall>a\<in>pre_ranks r (w n) q. x < a)"
is_map: shows "is_left_adjoint TfHgh_HfHgh.chine"
agree_imp_submap: fixes f1 :: "('d, 'e) fmap"   and f2 :: "('d, 'e) fmap" assumes "f1 \<subseteq>\<^sub>f f2" shows "agree f1 f2"
lt_plus_distinct_eq_max: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "lt p \<noteq> lt q" shows "lt (p + q) = ord_term_lin.max (lt p) (lt q)"
sum_projl_param: fixes s :: "'a + 'b"   and s' :: "'c + 'd"   and Ra :: "('c \<times> 'a) set"   and Rb :: "('d \<times> 'b) set" assumes "is_Inl s"   and "(s', s) \<in> \<langle>Ra, Rb\<rangle>sum_rel" shows "(projl s', projl s) \<in> Ra"
unitary_ex_mat_O: shows "unitary (tensor_P mat_O (1\<^sub>m K))"
remove_child_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> remove_child_locs ptr owner_document"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "type_wf h = type_wf h'"
mat_empty: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v1 :: "v"   and err_v :: "v" shows "evaluate_match ck env s v1 [] err_v (s, Rerr (Rraise err_v))"
llist_inf_simps_1: fixes xs :: "'a llist" shows "LNil \<sqinter> xs = LNil"
compute_set_of_jordan_blocks: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and ev :: "'a" assumes "jordan_nf A n_as" shows "set (local.compute_set_of_jordan_blocks A ev) = set n_as \<inter> UNIV \<times> {ev}"
that: fixes as :: "'edge list" assumes "((ms\<^sub>1', s\<^sub>1'), ms\<^sub>1', transfer (slice_kind S a) s\<^sub>2) \<in> WS S"   and "S,slice_kind S \<turnstile> (ms\<^sub>2,s\<^sub>2) =as @ [a]\<Rightarrow> (ms\<^sub>1',transfer (slice_kind S a) s\<^sub>2)" shows "thesis"
powr_mult: fixes x :: "real"   and y :: "real"   and a :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(x * y) powr a = x powr a * y powr a"
mset_eq_length_filter: fixes xs :: "'k list"   and ys :: "'k list"   and z :: "'k" assumes "mset xs = mset ys" shows "length (filter ((=) z) xs) = length (filter ((=) z) ys)"
sumlist_as_fold: fixes xs :: "'a list" shows "R.sumlist xs = fold (+) xs (0::'a)"
fls_deriv_neg: fixes f :: "'a fls" shows "fls_deriv (- f) = - fls_deriv f"
less_sat_def: fixes x :: "'b sat"   and y :: "'b sat" shows "(x < y) = (nat_of x < nat_of y)"
iptables_goto_bigstep_induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"   and "\<And>t. P [] t t"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Accept\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalAllow)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Drop \<or> a = Reject\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided (Decision FinalDeny)"   and "\<And>m a. \<lbrakk>matches \<gamma> m p; a = Log \<or> a = Empty\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a. \<not> matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs rs\<^sub>1 rs\<^sub>2 t t'. \<lbrakk>rs = rs\<^sub>1 @ rs\<^sub>2; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'; no_matching_Goto \<gamma> p rs\<^sub>1\<rbrakk> \<Longrightarrow> P rs Undecided t'"   and "\<And>m a chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; no_matching_Goto \<gamma> p rs\<^sub>1; P rs\<^sub>1 Undecided Undecided\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>m a chain rs t. \<lbrakk>matches \<gamma> m p; a = Call chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m a] Undecided t"   and "\<And>m a chain rs rest X. \<lbrakk>matches \<gamma> m p; a = Goto chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision X; P rs Undecided (Decision X)\<rbrakk> \<Longrightarrow> P (Rule m a # rest) Undecided (Decision X)"   and "\<And>m a chain rs rest. \<lbrakk>matches \<gamma> m p; a = Goto chain; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided; P rs Undecided Undecided\<rbrakk> \<Longrightarrow> P (Rule m a # rest) Undecided Undecided" shows "P rs s t"
a_2: shows "\<D>\<^sub>\<circ> a = I"
host_root: shows "h \<turnstile> get_root_node (cast\<^sub>e\<^sub>l\<^sub>e\<^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 host) \<rightarrow>\<^sub>r host_root"
Image_empty1: fixes X :: "'b set" shows "{} `` X = {}"
simps_2: fixes v :: "'a"   and w :: "'b"   and v' :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and g :: "('a, 'b) graph" shows "rem_unPath ((v, w, v') # ps) g = rem_unPath ps (del_unEdge v w v' g)"
fun_relD: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'c \<Rightarrow> 'd"   and A :: "('a \<times> 'c) set"   and B :: "('b \<times> 'd) set"   and x :: "'a"   and x' :: "'c" assumes "(f, f') \<in> A \<rightarrow> B"   and "(x, x') \<in> A" shows "(f x, f' x') \<in> B"
seqllsimp: fixes i :: "'a"   and P :: "('b \<times> 'c) \<times> 'd \<times> 'b \<times> 'c \<Rightarrow> bool"   and \<sigma> :: "'a \<Rightarrow> 'b"   and p :: "'c"   and a :: "'d"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and p' :: "'c" shows "seqll i P ((\<sigma>, p), a, \<sigma>', p') = P ((\<sigma> i, p), a, \<sigma>' i, p')"
\<phi>_mapsto: fixes b :: "'d"   and a :: "'d" assumes "D.ide b"   and "D.ide a" shows "\<phi>D (b, a) \<in> D.hom b a \<rightarrow> HomD.set (b, a)"
swap_apply_3: fixes c :: "'c"   and a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "c \<noteq> a"   and "c \<noteq> b" shows "Fun.swap a b f c = f c"
Collect_disj_eq: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "D.dom (D.map f) = D.map (D.dom f)"
left_idem: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<union># (a \<union># b) = a \<union># b"
trans_eq_elapsed: fixes u :: "'c \<Rightarrow> 'b"   and u' :: "'c \<Rightarrow> 'b" assumes "local.trans u u'"   and "u \<in> V" shows "eq_elapsed u u'"
saturated_def: fixes S :: "('a, 'b) fm set" shows "saturated S \<equiv> \<forall>p. (\<^bold>\<not> \<^bold>\<forall> p) \<in> S \<longrightarrow> (\<exists>a. (\<^bold>\<not> p\<langle>\<^bold>\<dagger> a []/0\<rangle>) \<in> S)"
withVar'_def: fixes gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and f :: "integer \<Rightarrow> 'a"   and g :: "'b gState_scheme"   and p :: "pState" shows "withVar' gl v idx f g p = f (the (getVar' gl v idx g p))"
mono_mono_Omega_fun: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "mono f"   and "mono g" shows "mono_mono (Omega_fun f g)"
less_le_not_le: fixes ko :: "'b key_order"   and x :: "'b"   and y :: "'b" shows "ko.lt ko x y = strict (ko.le ko) x y"
funion_commute: fixes A :: "'d fset"   and B :: "'d fset" shows "A |\<union>| B = B |\<union>| A"
support_of_segment_def: shows "support_of_segment = (\<lambda>(a, b). normal_of_segment (a, b) \<bullet> a)"
ipoly_complex_interval_pCons: fixes a :: "int"   and p :: "int poly"   and x :: "complex_interval" shows "ipoly_complex_interval (pCons a p) x = of_int_complex_interval a + x * ipoly_complex_interval p x"
pdouble_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p :: "'b \<times> 'b \<times> 'b" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.pdouble times one plus zero minus a p = (let (x, y, z) = p in if z = zero then p else let l = times (times (numeral.numeral one plus (num.Bit0 num.One)) y) z; m = plus (times (numeral.numeral one plus (num.Bit1 num.One)) (power.power one times x 2)) (times a (power.power one times z 2)) in (times l (minus (power.power one times m 2) (times (times (times (numeral.numeral one plus (num.Bit0 (num.Bit0 num.One))) x) y) l)), minus (times m (minus (times (times (times (numeral.numeral one plus (num.Bit0 (num.Bit1 num.One))) x) y) l) (power.power one times m 2))) (times (times (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times y 2)) (power.power one times l 2)), power.power one times l 3))"
simps_37: fixes x121 :: "expr"   and x122 :: "expr"   and x1 :: "char list" shows "x121;; x122 \<noteq> new x1"
assms_2: fixes m :: "('a, 'b) fmap"   and x :: "'a"   and y :: "'b" assumes "P m"   and "fmlookup m x = None" shows "P (fmupd x y m)"
lesspoll_trans2: fixes X :: "'d set"   and Y :: "'e set"   and Z :: "'f set" assumes "X \<prec> Y"   and "Y \<lesssim> Z" shows "X \<prec> Z"
bsok_def: fixes e :: "'a expr1"   and n :: "nat" shows "bsok e n \<equiv> \<B> e n \<and> expr_locks e = (\<lambda>ad. 0)"
fminus_mono: fixes A :: "'b fset"   and C :: "'b fset"   and D :: "'b fset"   and B :: "'b fset" assumes "A |\<subseteq>| C"   and "D |\<subseteq>| B" shows "A |-| B |\<subseteq>| C |-| D"
red: shows "gram_schmidt_fs.weakly_reduced n (map of_int_hom.vec_hom fs) \<alpha> (Suc i)"
assms_1: shows "\<forall>x\<in>S. is_interval x \<and> x \<subseteq> P"
supp_of_finite_sets: fixes S :: "'b set" assumes "finite S" shows "supp S = \<Union> (supp ` S)"
map_cong0: fixes x :: "('d, 'e, 'f) expr"   and f1 :: "'d \<Rightarrow> 'g"   and g1 :: "'d \<Rightarrow> 'g"   and f2 :: "'e \<Rightarrow> 'h"   and g2 :: "'e \<Rightarrow> 'h"   and f3 :: "'f \<Rightarrow> 'i"   and g3 :: "'f \<Rightarrow> 'i" assumes "\<And>z1. z1 \<in> RID\<^sub>E x \<Longrightarrow> f1 z1 = g1 z1"   and "\<And>z2. z2 \<in> LID\<^sub>E x \<Longrightarrow> f2 z2 = g2 z2"   and "\<And>z3. z3 \<in> set3_expr x \<Longrightarrow> f3 z3 = g3 z3" shows "map_expr f1 f2 f3 x = map_expr g1 g2 g3 x"
trans_gState_progress_rel: fixes prog :: "program" shows "trans (gState_progress_rel prog)"
assms_2: shows "\<rho>' \<equiv> run\<^sub>t (\<delta>\<^sub>m x) q\<^sub>0 w"
mult_imp_le_div_pos: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "(0::'a) < y"   and "z * y \<le> x" shows "z \<le> x / y"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "at_begin_fst_bwtn (as, lm) (s, l, r) ires = (\<exists>lm1 tn rn. lm1 = lm @ 0 \<up> tn \<and> length lm1 = s \<and> (if lm1 = [] then l = Bk # Bk # ires else l = [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> r = Bk \<up> rn)"
simps_1: fixes X :: "'a event set" shows "setinterleaving ([], X, []) = {[]}"
comp_fun_idem_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" shows "comp_fun_idem f \<equiv> comp_fun_commute f \<and> comp_fun_idem_axioms f"
simps_3: fixes d :: "int"   and i :: "int"   and ks :: "int list"   and xs :: "int list" shows "I\<^sub>Z (NDvd d i ks) xs = (\<not> d dvd i + \<langle>ks,xs\<rangle>)"
lift_state_oracle_def: fixes F :: "('j \<Rightarrow> 'h \<Rightarrow> (('i \<times> 'f) \<times> 'j) spmf) \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> (('i \<times> 'f) \<times> 'g) spmf"   and "oracle" :: "'f \<times> 'j \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'f \<times> 'j) spmf" shows "lift_state_oracle F oracle = (\<lambda>(t, s') a. map_spmf rprodl (F ((Pair t ---> id ---> map_spmf lprodr) oracle) s' a))"
lexordp_eq_linear: fixes xs :: "'d word list"   and ys :: "'d word list" shows "signed.lexordp_eq xs ys \<or> signed.lexordp_eq ys xs"
length_negGorbit_list: fixes gs :: "'g list"   and T :: "'a \<Rightarrow> ('f, 'g) aezfun \<Rightarrow> 'v"   and as :: "'a list" shows "length (negHorbit_list gs T as) = length gs"
inject_3: fixes x31 :: "pol"   and x32 :: "nat"   and x33 :: "pol"   and y31 :: "pol"   and y32 :: "nat"   and y33 :: "pol" shows "(PX x31 x32 x33 = PX y31 y32 y33) = (x31 = y31 \<and> x32 = y32 \<and> x33 = y33)"
start_point_at_most_one: fixes x :: "'a" assumes "path x" shows "is_inj (start_points x)"
pair_imageI: fixes a :: "'a"   and b :: "'b"   and A :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
coprime_set_non_empty: fixes N :: "nat" assumes "2 < N" shows "coprime_set N \<noteq> {}"
real_polynomial_function_eq: fixes p :: "'c \<Rightarrow> real" shows "real_polynomial_function p = polynomial_function p"
distinct_count_atmost_1: fixes x :: "'k list" shows "distinct x = (\<forall>a. count (mset x) a = (if a \<in> set x then 1 else 0))"
dynProcModifyReturnSameFaults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s\<in>P. \<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"
d2_def: shows "st2.d2 = prod_list st2.dims2"
leg0_is_map: shows "is_left_adjoint (u\<^sub>0 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0)"
simps_9: fixes op :: "'c" shows "norm_instr (Ubx.instr.IOp op) = Inca.instr.IOp op"
l_adjoint_def: fixes F :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and x :: "'b::refinement_lattice" shows "lower_galois_connections.l_adjoint (F::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (x::'b::refinement_lattice) \<equiv> \<Sqinter> {y::'b::refinement_lattice. x \<sqsubseteq> F y}"
eq_rel_eval_trm: fixes v :: "'a option list"   and n :: "nat"   and t1 :: "'a MFOTL.trm"   and t2 :: "'a MFOTL.trm" assumes "v \<in> eq_rel n t1 t2"   and "trm.is_Const t1 \<or> trm.is_Const t2"   and "\<forall>x\<in>fv_trm t1 \<union> fv_trm t2. x < n" shows "MFOTL.eval_trm (map the v) t1 = MFOTL.eval_trm (map the v) t2"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less (x # ys) = (Ball (set ys) (less x) \<and> linorder.strict_sorted less ys)"
simps_3: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and i :: "'d"   and v :: "'d"   and va :: "'d list"   and x :: "'e"   and z :: "'e" shows "rel_comp_pow r (i # v # va) x z = (\<exists>y. r i x y \<and> rel_comp_pow r (v # va) y z)"
hashed_elgamal_def: fixes \<G> :: "'b cyclic_group" shows "hashed_elgamal \<G> \<equiv> list_cdh \<G>"
subgroupE_3: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" assumes "Group.group G"   and "subgroup H G"   and "a \<in> H" shows "inv\<^bsub>G\<^esub> a \<in> H"
bij_to_nat: shows "bij_betw mod_type_class.to_nat UNIV {0..<CARD('d)}"
append_self_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<cdot> ys = xs) = (ys = \<epsilon>)"
induct: fixes x :: "'b formula"   and P :: "'b formula \<Rightarrow> bool" assumes "x \<in> AX0"   and "\<And>F G. P (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F)"   and "\<And>F G H. P ((F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H)" shows "P x"
simps_5: fixes Fn :: "('a \<rightarrow> 'a) \<rightarrow> 'a"   and Fn_project :: "'a \<rightarrow> 'a \<rightarrow> 'a"   and B :: "bool discr \<rightarrow> 'a"   and B_project :: "'a \<rightarrow> 'a \<rightarrow> 'a \<rightarrow> 'a"   and tick :: "'a \<rightarrow> 'a"   and b :: "bool" shows "semantic_domain.ESem Fn Fn_project B B_project tick (Bool b) = (\<Lambda> \<rho>. tick\<cdot>(B\<cdot>(Discr b)))"
util_def: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "x \<succeq> y = (u y \<le> u x)"
wf_trm_subterm_arity: fixes t :: "('fun, 'b) Term.term"   and f :: "'fun"   and T :: "('fun, 'b) Term.term list" assumes "wf\<^sub>t\<^sub>r\<^sub>m t"   and "Fun f T \<sqsubseteq> t" shows "length T = arity f"
dX_singular: fixes X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "Pdds'.dX X d = {x}"
order_iff_strict: fixes b :: "'c list"   and a :: "'c list" shows "prefix b a = (strict_prefix b a \<or> a = b)"
det_row_sum: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "'e \<Rightarrow> ('c, 'd) vec"   and S :: "'e set" shows "Square_Matrix.det (upd_row M i (sum a S)) = (\<Sum>s\<in>S. Square_Matrix.det (upd_row M i (a s)))"
evA: shows "\<exists>v. dim_vec v = dim_col A \<and> \<not> 0 \<le> inner_prod v (A *\<^sub>v v)"
scene_inter_indep: fixes X :: "'a scene"   and Y :: "'a scene" assumes "idem_scene X"   and "idem_scene Y"   and "X \<bowtie>\<^sub>S Y" shows "X \<sqinter>\<^sub>S Y = \<bottom>\<^sub>S"
B_theorem_2: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "3 \<le> i"   and "i \<le> Suc k" shows "subresultant_prs_locale.\<gamma> n \<delta> f \<beta> i = 1"
zero_less_divide_iff: 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 less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less zero (divide a b) = (less zero a \<and> less zero b \<or> less a zero \<and> less b zero)"