hcju/isabelleps huggingface dataset:
H_var2: fixes p :: "'a"   and x :: "'a"   and q :: "'a" shows "H p x q = (t p \<cdot> x = t p \<cdot> x \<cdot> t q)"
init_fin_simulation1: fixes final1 :: "'a \<Rightarrow> bool"   and r1 :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and final2 :: "'g \<Rightarrow> bool"   and r2 :: "('d, 'b, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'b \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and bisim_wait :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'b, 'a, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'b, 'g, 'h, 'e, 'f) \<tau>moves"   and t :: "'b"   and s1 :: "(status \<times> 'a) \<times> 'c"   and s2 :: "(status \<times> 'g) \<times> 'h"   and tl1 :: "('d, 'b, status \<times> 'a, 'c, 'e, 'f action) thread_action"   and s1' :: "(status \<times> 'a) \<times> 'c" assumes "FWdelay_bisimulation_obs final1 r1 final2 r2 bisim bisim_wait \<tau>move1 \<tau>move2"   and "FWbisimulation_base.init_fin_bisim final1 final2 bisim t s1 s2"   and "multithreaded_base.init_fin final1 r1 t s1 tl1 s1'"   and "\<not> \<tau>multithreaded.init_fin_\<tau>move \<tau>move1 s1 tl1 s1'" shows "\<exists>s2' s2'' tl2. (\<tau>trsys.silent_move (multithreaded_base.init_fin final2 r2 t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move2))\<^sup>*\<^sup>* s2 s2' \<and> multithreaded_base.init_fin final2 r2 t s2' tl2 s2'' \<and> \<not> \<tau>multithreaded.init_fin_\<tau>move \<tau>move2 s2' tl2 s2'' \<and> FWbisimulation_base.init_fin_bisim final1 final2 bisim t s1' s2'' \<and> ta_bisim (FWbisimulation_base.init_fin_bisim final1 final2 bisim) tl1 tl2"
simpdvd_def: fixes d :: "int"   and t :: "MIR.num" shows "simpdvd d t \<equiv> let g = numgcd t in if 1 < g then let g' = gcd d g in if g' = 1 then (d, t) else (d div g', reducecoeffh t g') else (d, t)"
vsv_vunion_app_right: fixes r :: "V"   and s :: "V"   and x :: "V" assumes "vsv r"   and "vsv s"   and "\<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> s = []\<^sub>\<circ>"   and "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "(r \<union>\<^sub>\<circ> s)\<lparr>x\<rparr> = s\<lparr>x\<rparr>"
b_App_elim: fixes e\<^sub>1 :: "trm"   and e\<^sub>2 :: "trm"   and e' :: "trm"   and x :: "name" assumes "App e\<^sub>1 e\<^sub>2 \<Down> e'"   and "x \<sharp> (e\<^sub>1, e\<^sub>2, e')" obtains f\<^sub>1 :: "trm" and f\<^sub>2 :: "trm" where "e\<^sub>1 \<Down> Lam [x].f\<^sub>1" and "e\<^sub>2 \<Down> f\<^sub>2" and "f\<^sub>1[x::=f\<^sub>2] \<Down> e'"
is_extensional: fixes f :: "'b" assumes "\<not> D.arr f" shows "src\<^sub>D f = D.null"
nonzeros_unop_pdevs_subset: fixes f :: "'d \<Rightarrow> 'c"   and x :: "nat \<Rightarrow> 'd" shows "{i. unop_pdevs_raw f x i \<noteq> (0::'c)} \<subseteq> {i. x i \<noteq> (0::'d)}"
Skip_reds: fixes s :: "char list \<Rightarrow> val option"   and c' :: "com"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>Skip,s\<rangle> \<rightarrow>* \<langle>c',s'\<rangle>" shows "s = s' \<and> c' = Skip"
c3: fixes a :: "'a"   and as :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "ps = a # as"   and "ls = x # xs"   and "x \<noteq> a" shows "thesis"
g: shows "arr g \<and> local.dom g = c \<and> cod g = d"
inf_absorb2: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "class.semilattice_inf inf less_eq less"   and "less_eq y x" shows "inf x y = y"
five: fixes x :: "'a" shows "5 \<times> x = x \<squnion> x \<squnion> x \<squnion> x \<squnion> x"
le_cases: fixes x :: "'d DBMEntry"   and y :: "'d DBMEntry" obtains "x \<preceq> y"   | "y \<preceq> x"
WT_callee: fixes s :: "'s1" assumes "Il s" shows "\<I>l \<turnstile>c left s \<surd>"
cipher_queriesI: fixes ose :: "'b \<Rightarrow> ('ekey \<times> 'dkey \<times> 'cipher list) option"   and n :: "'b"   and ek :: "'ekey"   and dk :: "'dkey"   and ciphers :: "'cipher list"   and x :: "'cipher" assumes "ose n = Some (ek, dk, ciphers)"   and "x \<in> set ciphers" shows "x \<in> cipher_queries ose"
map_eq_graph: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" shows "(f = g) = (map_graph f = map_graph g)"
natural_transformation_axioms: shows "natural_transformation B D (K \<circ> F) (M \<circ> H) (\<mu>\<sigma>.map \<circ> \<nu>\<tau>.map)"
f_def: shows "f = fds_newman + fds_deriv fds_zeta"
perp_foot_opposite_endpoint_in_S: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "a \<noteq> b" shows "endpoint_in_S (endpoint_in_S a b) (perp_foot c (proj2_line_through a b)) = endpoint_in_S b a"
fls_inverse_of_nat: fixes n :: "nat" shows "inverse (of_nat n) = fls_const (inverse (of_nat n))"
preorder_on_restrict: fixes A :: "'a set" shows "Order_Predicates.preorder_on (carrier \<inter> A \<inter> A) (restrict_relation A (restrict_relation A le))"
ltl_prop_implication_implies_ltl_implication: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and \<psi> :: "'b ltl" assumes "w \<Turnstile> \<phi>"   and "\<phi> \<longrightarrow>\<^sub>P \<psi>" shows "w \<Turnstile> \<psi>"
lc_Abs_iff_body: fixes T :: "Unary_PCF.type"   and t :: "expr" shows "lc (\<Lambda>\<langle>T\<rangle> t) = body T t"
BIT: shows "T\<^sub>p_on_rand' BIT s qs = 75 / 10\<^sup>2 * real (length qs - 1) - 5 / 10"
Sim: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> Rel" shows "R \<leadsto>\<guillemotleft>Rel'\<guillemotright> S"
ext_inject: fixes sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a"   and sp_impl_subj_subj' :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj' :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current' :: "thread_id_t"   and obj' :: "obj_id_t \<Rightarrow> obj_t"   and thread' :: "thread_id_t \<Rightarrow> thread_t"   and more' :: "'a" shows "(\<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj', sp_impl_subj_obj = sp_impl_subj_obj', current = current', obj = obj', thread = thread', \<dots> = more'\<rparr>) = (sp_impl_subj_subj = sp_impl_subj_subj' \<and> sp_impl_subj_obj = sp_impl_subj_obj' \<and> current = current' \<and> obj = obj' \<and> thread = thread' \<and> more = more')"
implies_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<leadsto> x \<le> z \<leadsto> y"
lookup_minus: fixes f :: "'d \<Rightarrow>\<^sub>0 'c"   and g :: "'d \<Rightarrow>\<^sub>0 'c"   and k :: "'d" shows "poly_mapping.lookup (f - g) k = poly_mapping.lookup f k - poly_mapping.lookup g k"
return_refine_prop_return: fixes m :: "'b nres"   and x :: "'c"   and R :: "('c \<times> 'b) set" assumes "nofail m"   and "RETURN x \<le> \<Down> R m" obtains x' :: "'b" where "(x, x') \<in> R" and "RETURN x' \<le> m"
assms_3: shows "f n\<^sub>0 < k \<and> k \<le> n"
radical_sqrt_normal_form_lemma4: fixes z :: "real"   and x :: "real"   and y :: "real" assumes "0 \<le> z"   and "x \<noteq> y * sqrt z" shows "1 / (x + y * sqrt z) = x / (x * x - y * y * z) - y * sqrt z / (x * x - y * y * z)"
assms: shows "valid_unMultigraph.is_trail (rem_unPath p G) v ps v'"
Assert'Precond_3: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<nabla> (assert_bound a s)"
inverse_negative_iff_negative: 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" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse a) zero = less a zero"
power_gt1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a" shows "less one (power.power one times a (Suc n))"
arr_char: fixes f :: "(bool, unit) concrete_category.arr" shows "partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f = (f = concrete_category.MkArr False False () \<or> f = concrete_category.MkArr True True ())"
hyps: fixes Ks :: "'a list" assumes "set Ks \<subseteq> K"   and "combine Ks Us = \<zero>"   and "local.independent K Us" shows "set (take (length Us) Ks) \<subseteq> {\<zero>}"
map_pmf_compose: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "map_pmf (f \<circ> g) = map_pmf f \<circ> map_pmf g"
assms_4: shows "b' \<equiv> snd (normalize_quot (a, b))"
inv_naturality: fixes f :: "'a" assumes "A.arr f" shows "F f \<cdot>\<^sub>B B.inv (\<tau> (A.dom f)) = B.inv (\<tau> (A.cod f)) \<cdot>\<^sub>B G f"
nsub_RI_iconf_aux: fixes e :: "'c exp"   and sh :: "sheap"   and es :: "'c exp list" shows "(\<not> sub_RI e \<longrightarrow> (\<forall>e'. subexp_of e' e \<longrightarrow> \<not> sub_RI e' \<longrightarrow> iconf sh e') \<longrightarrow> iconf sh e) \<and> (\<not> sub_RIs es \<longrightarrow> (\<forall>e'. e' \<in> subexps es \<longrightarrow> \<not> sub_RI e' \<longrightarrow> iconf sh e') \<longrightarrow> iconfs sh es)"
Enc_Exp: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Enc X Y \<noteq> Exp X' Y'"
multi_drop_mem_not_eq: fixes c :: "'d"   and B :: "'d multiset" assumes "c \<in># B" shows "B - {#c#} \<noteq> B"
IH1: assumes "OUadmitFO \<sigma> ODE1 U" shows "osafe (OsubstFO ODE1 \<sigma>)"
ostep_invariantE: fixes A :: "('a \<times> 'b, 'c) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "('a \<times> 'b) \<times> 'c \<times> 'a \<times> 'b \<Rightarrow> bool"   and \<sigma> :: "'a"   and s :: "'b"   and a :: "'c"   and \<sigma>' :: "'a"   and s' :: "'b" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) P"   and "(\<sigma>, s) \<in> oreachable A S U"   and "((\<sigma>, s), a, \<sigma>', s') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" obtains "P ((\<sigma>, s), a, \<sigma>', s')"
inv_into_f_f: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and a :: "'a" assumes "bij_betw f A B"   and "a \<in> A" shows "inv_into A f (f a) = a"
eucl_less_le_not_le: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less" shows "less x y = (less_eq x y \<and> \<not> less_eq y x)"
distr_cong_simp: fixes M :: "'b measure"   and K :: "'b measure"   and N :: "'c measure"   and L :: "'c measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = K"   and "sets N = sets L"   and "\<And>x. x \<in> space M =simp=> f x = g x" shows "distr M N f = distr K L g"
less_linear: fixes x :: "'t"   and y :: "'t" shows "x \<prec>\<^sub>t y \<or> x = y \<or> y \<prec>\<^sub>t x"
main: shows "{x. poly p x = 0} = {- (b / (2 * a)) + e |e. e\<^sup>2 = (b / (2 * a))\<^sup>2 - c / a}"
C_def: shows "C = 2 * (c + ln 2 - 1)"
nonneg_bounded: shows "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
subst_var_inv_def: fixes \<delta> :: "'c \<Rightarrow> ('e, 'c) Term.term"   and X :: "'c set" shows "subst_var_inv \<delta> X \<equiv> \<lambda>x. if Var x \<in> \<delta> ` X then Var (inv_into X \<delta> (Var x)) else Var x"
v'_def: shows "v' = \<lparr>ext = ext v, lan = lan v1 \<squnion> lan v3, own = own v\<rparr>"
SN_on_induct: fixes R :: "('b \<times> 'b) set"   and A :: "'b set"   and s :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "SN_on R A"   and "s \<in> A"   and "\<And>t. (\<And>u. (t, u) \<in> R \<Longrightarrow> P u) \<Longrightarrow> P t" shows "P s"
isolated_ret_def: fixes fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme"   and p :: "'b" shows "isolated_ret fg p \<equiv> (\<forall>u l. (u, l, return fg p) \<notin> edges fg) \<and> entry fg p \<noteq> return fg p"
compl_zero: shows "NOT (0::'b) = - (1::'b)"
first_entrance_set_def: fixes A :: "'a set"   and n :: "nat" shows "first_entrance_set A n = (T ^^ n) --` A - (\<Union>i<n. (T ^^ i) --` A)"
K'_A2': fixes i :: "'a"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! (K\<^sub>! i (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i p \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i q)"
nil_in_A: shows "[] \<in> (rev \<circ> sorted_list_of_set) ` Pow (set nodes)"
abs_triangle_half_l: fixes x :: "'a"   and y :: "'a"   and e :: "'a"   and x' :: "'a" assumes "\<bar>x - y\<bar> < e / (2::'a)"   and "\<bar>x' - y\<bar> < e / (2::'a)" shows "\<bar>x - x'\<bar> < e"
simps_2: fixes n :: "'ubx1" shows "norm_instr (IPushUbx1 n) = Inca.instr.IPush (box_ubx1 n)"
sym_def: fixes r :: "('a \<times> 'a) set" shows "sym r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (y, x) \<in> r)"
sqn_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" assumes "ip \<noteq> dip" shows "sqn (update rt dip (dsn, dsk, flag, hops, nhip, pre)) ip = sqn rt ip"
F_def: shows "F = Abs_fps (\<lambda>n. of_nat (n + 1) ^ k)"
zero_stalk_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c"   and V :: "'c set" assumes "stalk S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I x" shows "stalk.zero_stalk \<FF> \<rho> zero_str I V \<equiv> direct_lim.class_of \<FF> \<rho> I V (zero_str V)"
parts_mpair_key_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and K :: "key" assumes "s \<turnstile> s'"   and "\<lbrace>X, Y\<rbrace> \<in> parts (used s')"   and "\<lbrace>X, Y\<rbrace> \<notin> parts (used s)"   and "X = Key K \<or> Y = Key K \<and> K \<notin> range PubK" shows "{X, Y} \<subseteq> spied s'"
Merge_upper: fixes A :: "'a\<^sub>m set"   and x :: "'a\<^sub>m"   and y :: "'a\<^sub>m" assumes "Merge A = Some x"   and "y \<in> A" shows "bo y x"
R_arg: fixes u :: "'a"   and u' :: "'a"   and t :: "'a" assumes "R u u'" shows "R (app t u) (app t u')"
iso_head_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "iso_head (inv_iso hom) = head G"
rotate_drop_take: fixes n :: "nat"   and xs :: "'a list" shows "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
smallest_accepting_rank_def: shows "smallest_accepting_rank \<equiv> if accept\<^sub>M then Some (LEAST i. finite fail \<and> finite (local.merge i) \<and> infinite (succeed i)) else None"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>src f : src a \<Rightarrow> src b\<guillemotright>"
in_segment_inner_rot: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> {a--b}" shows "(x - a) \<bullet> rot (b - a) = 0"
elims_2: fixes x :: "ct"   and xa :: "ct" assumes "ct_eq x xa" obtains t :: "t" and t' :: "t" where "x = TSome t" and "xa = TSome t'" and "t = t'"   | uu_ :: "ct" where "x = TAny" and "xa = uu_"   | v :: "t" where "x = TSome v" and "xa = TAny"
module_with_def: fixes plus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>M :: "'d"   and minus\<^sub>M :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>M :: "'d \<Rightarrow> 'd"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" shows "module_with plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M scale \<equiv> class.ab_group_add plus\<^sub>M zero\<^sub>M minus\<^sub>M uminus\<^sub>M \<and> module_with_axioms plus\<^sub>M scale"
Esigma_def: fixes n :: "nat" shows "Esigma n \<equiv> \<Sum> (divisor_set n)"
finprod_empty: fixes f :: "'c \<Rightarrow> 'b" shows "prod f {} = (1::'b)"
assms_2: fixes i :: "nat" assumes "i < d" shows "f $ i = (0::'a)"
congruent_rel_def: fixes b :: "'a" shows "congruent_rel b = {(a, c). cong a c b}"
prj0_simps_arr: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and g :: "'a" assumes "elementary_category_with_pullbacks C prj0 prj1" shows "partial_magma.arr C (prj0 f g) = (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.cod C f = partial_magma.cod C g)"
f0: fixes i :: "nat" assumes "i \<noteq> 0" shows "(f i = (0::'a)) = (k < i)"
domains_comp: fixes g :: "('o, 'a) arr"   and f :: "('o, 'a) arr" assumes "seq g f" shows "domains (COMP g f) = domains f"
chart_basis_inv_continuous: shows "continuous_on UNIV local.chart_basis_inv"
set_bd_1: fixes x :: "('o, 'n, 'p) F" shows "|F'set3 x| \<le>o F'bd"
fin: shows "finite ({x. poly (real_of_int_poly p) x = 0} \<times> {x. poly (real_of_int_poly p) x = 0})"
distincts_Cons_remove1: fixes xs :: "'b list"   and xss :: "'b list list" assumes "xs \<in> set xss" shows "distincts (xs # remove1 xs xss) = distincts xss"
shortest_path_pos_cost_axioms_def: fixes G :: "('a, 'b) pre_digraph"   and dist :: "'a \<Rightarrow> ereal"   and c :: "'b \<Rightarrow> real"   and s :: "'a"   and num :: "'a \<Rightarrow> enat" shows "shortest_path_pos_cost_axioms G dist c s num \<equiv> (s \<in> verts G \<and> dist s = 0) \<and> (\<forall>v. v \<in> verts G \<longrightarrow> (dist v = \<infinity>) = (num v = \<infinity>)) \<and> (\<forall>e. e \<in> arcs G \<longrightarrow> 0 \<le> c e)"
polyexp_fps_def: fixes xs :: "('b \<times> nat \<times> 'b) list" shows "polyexp_fps xs = (\<Sum>(a, k, b)\<leftarrow>xs. fps_of_poly (smult a (fps_monom_poly b k)) / (1 - fps_const b * fps_X) ^ (k + 1))"
assms_2: fixes xs :: "'a stream" assumes "\<phi>1 xs" shows "\<phi>2 xs"
dim_eq_card_independent: fixes B :: "'a set" assumes "vector_space_over_itself.independent B" shows "vector_space_over_itself.dim B = card B"
WellDefined: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and \<phi> :: "('a, 'b) Language"   and i :: "('c, 'd) IType" assumes "Interpretation I"   and "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" shows "Sig iS\<^bsub>I\<^esub> \<triangleright> \<phi>"
app_vlrestrictionI: fixes a :: "V"   and A :: "V"   and b :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"
N: fixes i :: "'b" assumes "i \<in> Basis" shows "negligible (N i)"
is_nff_def: fixes dflt :: "'b"   and f :: "nat \<Rightarrow> 'b"   and a :: "'b array" shows "is_nff dflt f a \<equiv> \<exists>\<^sub>Al. a \<mapsto>\<^sub>a l * \<up> (f = nff_\<alpha> dflt l)"
ce_set_lm_5: fixes A :: "nat set"   and p :: "nat \<Rightarrow> nat" assumes "A \<in> ce_sets"   and "p \<in> PrimRec1" shows "{x. p x \<in> A} \<in> ce_sets"
mult_poly_0_left: fixes q :: "'a poly" shows "0 * q = 0"
FGModHom_composite_left: fixes W :: "'w set"   and smult'' :: "('f, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'w \<Rightarrow> 'b" assumes "FGModuleHom G (\<star>) W smult'' S"   and "ImG \<subseteq> W" shows "FGModuleHom G (\<cdot>) V smult'' (S \<circ> T)"
lea_left_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong C B A D E F"
\<psi>\<^sub>1\<^sub>0_is_state: fixes n :: "nat" assumes "1 \<le> n" shows "state n (Matrix.mat (2 ^ n) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ n)))"
fix_imp_eq: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "x \<in> fix f A" shows "f x = x"
pCons_2: shows "map_poly ((\<lambda>a. [:a:]) \<circ> hom) p = map_poly (map_poly hom) (map_poly (\<lambda>a. [:a:]) p)"
vconn_implies_hpar_3: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>B g\<guillemotright>" shows "src\<^sub>B g = src\<^sub>B f"
uminus_one_neq_one_double: shows "- 1 \<noteq> 1"
mat_hom_pow: fixes A :: "'a mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "mat\<^sub>h (A ^\<^sub>m k) = mat\<^sub>h A ^\<^sub>m k"
assms_2: shows "ins = make_insert (fst (interp_ops ops)) val k"
intro: fixes G :: "('c, 'd) monoid_scheme" assumes "comm_monoid_cancel G"   and "primeness_condition_monoid_axioms G" shows "primeness_condition_monoid G"
ipoly_nth_root: fixes n :: "nat"   and p :: "int poly"   and x :: "'a" shows "ipoly (poly_nth_root n p) x = ipoly p (x ^ n)"
frac_less2: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d"   and w :: "'d"   and z :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero x"   and "less_eq x y"   and "less zero w"   and "less w z" shows "less (divide x z) (divide y w)"
parallel_beta_reduces_equivalent: fixes A :: "'a trm"   and B :: "'a trm" shows "A >>\<^sup>* B = A \<rightarrow>\<beta>\<^sup>* B"
collinear_sym2': fixes z1 :: "complex"   and z2 :: "complex"   and z3 :: "complex" assumes "collinear z1 z2 z3" shows "collinear z2 z1 z3"