hcju/isabelleps huggingface dataset:
hextI: fixes typeof_addr :: "'b \<Rightarrow> 'a \<Rightarrow> htype option"   and h :: "'b"   and h' :: "'b" assumes "\<And>a C. typeof_addr h a = \<lfloor>Class_type C\<rfloor> \<Longrightarrow> typeof_addr h' a = \<lfloor>Class_type C\<rfloor>"   and "\<And>a T n. typeof_addr h a = \<lfloor>Array_type T n\<rfloor> \<Longrightarrow> typeof_addr h' a = \<lfloor>Array_type T n\<rfloor>" shows "heap_base.hext typeof_addr h h'"
vertical_composite_axioms: shows "vertical_composite A B G H K \<sigma> \<tau>"
univ_poly_a_inv_def': fixes p :: "'a list" assumes "p \<in> carrier (K [X])" shows "\<ominus>\<^bsub>K [X]\<^esub> p = map (a_inv R) p"
qeq: fixes x :: "'e['f, 'g]"   and y :: "'e['h, 'g]" assumes "x \<cong>\<^sub>Q y" shows "QD('f) = QD('h)"
set_bc: fixes x :: "'Fbc"   and y :: "'Fbc" assumes "Ebc x y" shows "set_Fbc x = set_Fbc y"
assms_2: shows "0 \<le> v\<^sub>e - a\<^sub>e / a\<^sub>o * v\<^sub>o"
simps_1: fixes \<phi> :: "formula" shows "remove_neg (Neg \<phi>) = \<phi>"
order_le_neq_trans: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b"   and "a \<noteq> b" shows "a < b"
flop_chamber: fixes C :: "'a set" assumes "folding.chamber C" shows "folding.chamber (if C \<in> f \<turnstile> folding.\<C> then THE D. D \<in> folding.\<C> - f \<turnstile> folding.\<C> \<and> f ` D = C else f ` C)"
par_strict_not_col_4: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ParStrict Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B D"
function_factors_right: fixes g :: "'d \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'e" shows "(\<forall>x. \<exists>y. g y = f x) = (\<exists>h. f = g \<circ> h)"
Spy_not_see_encrypted_key: fixes B :: "agent"   and NA :: "msg"   and A :: "agent"   and K :: "nat"   and NB :: "msg"   and evs :: "event list" assumes "Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"   and "Notes Spy \<lbrace>NA, NB, Key K\<rbrace> \<notin> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> otway" shows "Key K \<notin> analz (knows Spy evs)"
matrix_sum_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d f n \<in> carrier_mat d d"
assms: shows "[x \<^bold>=\<^sub>\<kappa> y in v]"
range_sum_prod: fixes x :: "int"   and q :: "int"   and y :: "int"   and p :: "int" assumes "x \<in> {0..<q}"   and "y \<in> {0..<p}" shows "x + q * y \<in> {0..<p * q}"
ordinal_exp_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "x ** oSuc y = x ** y * x"
simps: fixes a1 :: "Syntax.term list"   and a2 :: "Syntax.term"   and a3 :: "mode"   and a4 :: "nat"   and a5 :: "Syntax.term list"   and a6 :: "Syntax.term" shows "\<lless>a1, a2\<ggreater> a3\<rightarrow>a4 \<lless>a5, a6\<ggreater> = ((\<exists>\<pi> e uu_. a1 = \<pi> \<and> a2 = e \<and> a3 = uu_ \<and> a4 = 0 \<and> a5 = \<pi> \<and> a6 = e) \<or> (\<exists>\<pi>\<^sub>1 e\<^sub>1 m i \<pi>\<^sub>2 e\<^sub>2 \<pi>\<^sub>3 e\<^sub>3. a1 = \<pi>\<^sub>1 \<and> a2 = e\<^sub>1 \<and> a3 = m \<and> a4 = i + 1 \<and> a5 = \<pi>\<^sub>3 \<and> a6 = e\<^sub>3 \<and> \<lless>\<pi>\<^sub>1, e\<^sub>1\<ggreater> m\<rightarrow>i \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> \<and> \<lless>\<pi>\<^sub>2, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>\<^sub>3, e\<^sub>3\<ggreater>))"
measurable_on_UNIV: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "((\<lambda>x. if x \<in> S then f x else (0::'b)) measurable_on UNIV) = (f measurable_on S)"
ucast_and_neg_mask: fixes x :: "'c word"   and n :: "nat" shows "UCAST('c \<rightarrow> 'b) (x AND NOT (mask n)) = UCAST('c \<rightarrow> 'b) x AND NOT (mask n)"
sp_fundefs_def: fixes F :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option" shows "sp_fundefs F \<equiv> \<forall>f fd. F f = Some fd \<longrightarrow> sp_fundef F fd (body fd) = Ok [None]"
circle_cube_boundary_valid: fixes k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "(k, \<gamma>) \<in> boundary circle_cube" shows "valid_path \<gamma>"
vbrelation_vintersection_vrange: fixes s :: "V" assumes "vdisjnt (\<R>\<^sub>\<circ> r) (\<R>\<^sub>\<circ> s)" shows "vdisjnt r s"
cases_scheme: fixes r :: "'b alpha_d_scheme" obtains ok :: "bool" and more :: "'b" where "r = \<lparr>ok = ok, \<dots> = more\<rparr>"
vprojection_vrange_vsubset: fixes i :: "V"   and I :: "V"   and A :: "V \<Rightarrow> V" assumes "i \<in>\<^sub>\<circ> I" shows "\<R>\<^sub>\<circ> (vprojection I A i) \<subseteq>\<^sub>\<circ> A i"
valid_success': fixes A :: "'c \<Rightarrow> ('b \<times> 'c) option"   and \<sigma> :: "'c"   and b :: "'b"   and \<sigma>' :: "'c"   and M :: "'b \<Rightarrow> 'c \<Rightarrow> (bool \<times> 'c) option" assumes "A \<sigma> = Some (b, \<sigma>')" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M) = (\<sigma>' \<Turnstile> M b)"
map_sorted_list_of_fmap: fixes f :: "'c \<Rightarrow> 'd"   and m :: "('c, 'e) fmap"   and k :: "'c"   and v :: "'e" assumes "inj f"   and "mono f"   and "m $$ k = None" shows "map (\<lambda>(k, v). (f k, v)) (sorted_list_of_fmap (m(k $$:= v))) = insort_key fst (f k, v) (map (\<lambda>(k, v). (f k, v)) (sorted_list_of_fmap m))"
assms: shows "bvars\<^sub>s\<^sub>s\<^sub>t S \<inter> subst_domain \<theta> = {}"
Sup_atMost: fixes y :: "'a" shows "\<Squnion> {..y} = y"
subgroup_of_powers: fixes x :: "'a" assumes "x \<in> carrier G" shows "subgroup (range (([^]) x)) G"
foundation6: fixes \<tau> :: "'a state \<times> 'a state"   and P :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P" shows "\<tau> \<Turnstile> \<delta> P"
Star_unfoldR: fixes r :: "'a language" shows "Star r = Plus One (Times (Star r) r)"
active_thread_ex_red: fixes t :: "'t"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "t \<in> active_threads s" shows "\<exists>taa. Ex (redT_syntax1 s t taa)"
cf_unlift: fixes x :: "'a itrm" assumes "x \<in> CF" shows "CF_pure x \<leftrightarrow> unlift x"
attractor_unfolding: fixes p :: "Player"   and W :: "'a set" shows "attractor p W = W \<union> attractor p W \<union> directly_attracted p (attractor p W)"
RECT_def: fixes B :: "('f \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> 'e"   and x :: "'f" shows "REC\<^sub>T B x \<equiv> if trimono B then flatf_gfp B x else top"
swap_nilpotent: fixes a :: "'d"   and b :: "'d"   and f :: "'d \<Rightarrow> 'e" shows "Fun.swap a b (Fun.swap a b f) = f"
assms_1: fixes n :: "nat" shows "upper_asymptotic_density (A n) \<le> l"
is_icdi_def: fixes n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and n' :: "nat" shows "(n icd\<^bsup>\<pi>\<^esup>\<rightarrow> n') = (is_path \<pi> \<and> n cd\<^bsup>\<pi>\<^esup>\<rightarrow> n' \<and> (\<forall>m\<in>{n'<..<n}. \<not> n cd\<^bsup>\<pi>\<^esup>\<rightarrow> m))"
intersect'_language: fixes A :: "('a, 'b) nba"   and B :: "('a, 'c) nba" shows "NGBA.language (intersect' A B) = NBA.language A \<inter> NBA.language B"
that: fixes x :: "V" assumes "r\<lparr>x\<rparr> = b"   and "x \<in>\<^sub>\<circ> A" shows "thesis"
interchange_rows_iarray_def: fixes A :: "'b iarray iarray"   and a :: "nat"   and b :: "nat" shows "interchange_rows_iarray A a b = IArray.of_fun (\<lambda>n. if n = a then A !! b else if n = b then A !! a else A !! n) (IArray.length A)"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> yahalom"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
hom_grading_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and d :: "'a \<Rightarrow> nat" assumes "class.monoid_add plus zero" shows "monoid_add.hom_grading plus d = (\<exists>f. monoid_add.hom_grading_fun plus d f)"
the_coext_prop_1: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' (C.cod f__) f" shows "\<guillemotleft>x'\<epsilon>.the_coext x' f : x' \<rightarrow>\<^sub>D Go (C.cod f__)\<guillemotright>"
mset_filter: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "mset (filter P xs) = filter_mset P (mset xs)"
inner_sum_Basis: fixes i :: "'b" assumes "i \<in> Basis" shows "One \<bullet> i = 1"
polyfun_finite_roots: fixes c :: "nat \<Rightarrow> 'b"   and n :: "nat" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)} = (\<exists>k\<le>n. c k \<noteq> (0::'b))"
M_def: shows "M \<equiv> distr (Pi\<^sub>M A (\<lambda>_. uniform_measure lborel {a..b})) (count_space (Pow (B \<times> B))) (linorder_from_keys A)"
InvokeElim1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and C :: "Class"   and m :: "Method" assumes "(M, l, (os, S, h), n, ll, t) \<in> Step"   and "get_ins M l = Some (invokeS C m)" shows "\<exists>code l0 v ops hh u k R par. mbody_is C m (par, code, l0) \<and> (os, par, R, ops) \<in> Frame \<and> ((par, code, l0), l0, ([], R, h), k, hh, v) \<in> Exec \<and> n = Suc k \<and> t = (v # ops, S, hh) \<and> ll = l + 1"
field_differentiable_const: fixes c :: "'a"   and F :: "'a filter" shows "(\<lambda>z. c) field_differentiable F"
intro: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "additive_subgroup H G"   and "abelian_group G"   and "abelian_subgroup_axioms H G" shows "abelian_subgroup H G"
similar'_bool: fixes n :: "nat"   and b\<^sub>1 :: "bool discr"   and b\<^sub>2 :: "bool discr" shows "(B\<cdot>b\<^sub>1 \<triangleleft>\<triangleright>\<^bsub>Suc n\<^esub> CB\<cdot>b\<^sub>2) = (b\<^sub>1 = b\<^sub>2)"
semiattractive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "semiattractive B (\<lambda>x y. y \<sqsubseteq> x)"
nn_integral_stretch: fixes f :: "real \<Rightarrow> ennreal"   and c :: "real" assumes "f \<in> borel_measurable borel"   and "c \<noteq> 0" shows "\<integral>\<^sup>+ x. f (c * x) \<partial>lborel = ennreal (1 / \<bar>c\<bar>) * integral\<^sup>N lborel f"
b_e_check_single_weaken_top: fixes \<C> :: "t_context"   and e :: "b_e"   and tn :: "t list"   and tm :: "ct list"   and ts :: "t list" assumes "check_single \<C> e (Type tn) = TopType tm" shows "check_single \<C> e (Type (ts @ tn)) = TopType tm"
chromatic_lb: fixes G :: "nat set \<times> nat set set" assumes "uwellformed G"   and "finite (uverts G)"   and "uverts G \<noteq> {}" shows "ereal (real (card (uverts G))) / ereal_of_enat (\<alpha> G) \<le> ereal_of_enat (chromatic_number G)"
map_of_eq_empty_iff: fixes xys :: "('a \<times> 'b) list" shows "(map_of xys = Map.empty) = (xys = [])"
vs_confD: fixes P :: "'a prog"   and h :: "'heap"   and vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set"   and v :: "'addr val"   and ad :: "'addr"   and al :: "addr_loc" assumes "h.vs_conf P h vs"   and "v \<in> vs (ad, al)" shows "\<exists>T. h.addr_loc_type P h ad al T \<and> h.conf P h v T"
signof_compose: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and q :: "nat \<Rightarrow> nat"   and m :: "nat" assumes "p permutes {0..<n}"   and "q permutes {0..<m}" shows "signof (p \<circ> q) = signof p * signof q"
b: shows "\<forall>x\<in>S. norm x \<le> b"
prems: fixes i :: "nat" shows "coeff (pCons a p) i \<in> R"
func_space_is_module: fixes S :: "'d set" shows "module R (func_space S)"
flow_has_space_derivative: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "((\<lambda>x0. flow0 x0 t) has_derivative blinfun_apply (Dflow x0 t)) (at x0)"
Left_Coset_memI: fixes h :: "'c"   and H :: "'c set"   and composition :: "'e \<Rightarrow> 'c \<Rightarrow> 'd"   and x :: "'e" assumes "h \<in> H" shows "composition x h \<in> coset_notation.Left_Coset composition x H"
start_stepsize_spec_def: shows "start_stepsize_spec = SPEC ((<) 0)"
finprod_multf: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> carrier R"   and "g \<in> A \<rightarrow> carrier R" shows "(\<Oplus>x\<in>A. f x \<oplus> g x) = finsum R f A \<oplus> finsum R g A"
dw_closedD: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "relation.dw_closed r A"   and "a \<in> A"   and "r a b" shows "b \<in> A"
that: fixes ys :: "'a list"   and x :: "'a"   and zs :: "'a list" assumes "xs = ys @ x # zs"   and "P x"   and "\<forall>y\<in>set ys. \<not> P y" shows "thesis"
foldl_Nil: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and a :: "'c" shows "foldl f a [] = a"
of_rat_0: shows "of_rat 0 = (0::'b)"
convert_term_simps_3: fixes t :: "'c"   and u :: "'c" shows "convert_term (app t u) = app (convert_term t) (convert_term u)"
antisym_conv1: fixes x :: "'a"   and y :: "'a" assumes "\<not> x < y" shows "(x \<le> y) = (x = y)"
lemma3: fixes r :: "nat \<Rightarrow> nat \<Rightarrow> b"   and p :: "nat \<times> nat list \<times> nat"   and i :: "nat"   and j :: "nat"   and q :: "nat \<times> nat list \<times> nat"   and k :: "nat" assumes "is_path r p i j i"   and "is_path r q i i k" shows "is_path r (conc p q) (Suc i) j k"
less_num1_def: fixes a :: "1"   and b :: "1" shows "(a < b) = (Rep_num1 a < Rep_num1 b)"
group_isomorphisms_homology_group_prod_retract: fixes S :: "'b set"   and X :: "'b topology"   and p :: "int" assumes "S = {} \<or> S retract_of_space X" obtains \<H> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set set" and \<K> :: "(((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int) set set" where "subgroup \<H> (homology_group p X)" and "subgroup \<K> (homology_group p X)" and "(\<lambda>(x, y). x \<otimes>\<^bsub>homology_group p X\<^esub> y) \<in> Group.iso (subgroup_generated (homology_group p X) \<H> \<times>\<times> subgroup_generated (homology_group p X) \<K>) (homology_group p X)" and "hom_induced p (subtopology X S) {} X {} id \<in> Group.iso (homology_group p (subtopology X S)) (subgroup_generated (homology_group p X) \<H>)" and "hom_induced p X {} X S id \<in> Group.iso (subgroup_generated (homology_group p X) \<K>) (relative_homology_group p X S)"
simps_2: fixes \<mu> :: "'a" shows "E.Nml \<^bold>\<langle>\<mu>\<^bold>\<rangle> = B.arr \<mu>"
vfield_def: fixes D :: "V" shows "vfield D = (\<lambda>r\<in>\<^sub>\<circ>D. \<D>\<^sub>\<circ> r \<union>\<^sub>\<circ> \<R>\<^sub>\<circ> r)"
lm097: fixes X :: "('a \<times> 'b set) set"   and N :: "'a set"   and G :: "'b set"   and bids :: "'a \<times> 'b set \<Rightarrow> nat" assumes "X \<subseteq> N \<times> (Pow G - {{}})" shows "\<forall>pair\<in>X. summedBidVector (real \<circ> bids) N G pair = (real \<circ> summedBidVector bids N G) pair"
assms: shows "\<forall>a\<in>set as. intra_kind (kind a)"
ccw'_sortedP_subsequent: fixes i :: "nat"   and xs :: "((real \<times> real) \<times> real \<times> real) list" assumes "Suc i < length xs"   and "ccw'.sortedP 0 (map dirvec xs)"   and "fst (xs ! Suc i) = snd (xs ! i)" shows "ccw' (fst (xs ! i)) (snd (xs ! i)) (snd (xs ! Suc i))"
funpow_dist_def: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" shows "funpow_dist f x y \<equiv> LEAST n. (f ^^ n) x = y"
less_asym: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and P :: "bool" assumes "class.preorder less_eq less"   and "less x y"   and "\<not> P \<Longrightarrow> less y x" shows "P"
borel_measurable_integrable: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" shows "f \<in> borel_measurable M"
evalS_models: fixes t :: "('a, 'es, 'as) BEState Trace"   and ec :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist \<times> (('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) odlist"   and a :: "'a"   and \<phi> :: "('a, 'p) Kform"   and s :: "('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState" assumes "t \<in> SPR.jkbpC"   and "spr_simAbs ec = SPRdet.sim_equiv_class a t"   and "subjective a \<phi>"   and "s \<in> toSet (snd ec)" shows "evalS (spr_evalProp envVal) (spr_knowledge envObs (fst ec)) (spr_commonKnowledge envObs (fst ec)) (snd ec) \<phi> = spr_repMC (toSet (fst ec)), s \<Turnstile> \<phi>"
simps_5: fixes e1 :: "expr"   and e2 :: "expr" shows "free_vars_expr_code <e1 ,  e2> = free_vars_expr_code e1 \<union> free_vars_expr_code e2"
free_eta: fixes s :: "dB"   and t :: "dB"   and i :: "nat" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "free t i = free s i"
1: shows "s_repr \<in> getOneIp ` set (build_ip_partition c rs)"
allI: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "'a \<Rightarrow> (nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>p. sb.eval cid t t' n (\<gamma> p)" shows "sb.eval cid t t' n (\<forall>\<^sub>bp. \<gamma> p)"
ordered_sel_resolvent_upon_def: fixes Sel :: "'at set"   and P1 :: "'at Literal set"   and P2 :: "'at Literal set"   and C :: "'at Literal set"   and A :: "'at" shows "ordered_sel_resolvent_upon Sel P1 P2 C A \<equiv> C = P1 - {Pos A} \<union> (P2 - {Neg A}) \<and> strictly_maximal_literal P1 (Pos A) \<and> selected_part Sel P1 = {} \<and> (strictly_maximal_literal P2 (Neg A) \<and> selected_part Sel P2 = {} \<or> strictly_maximal_literal (selected_part Sel P2) (Neg A))"
obs: fixes i :: "'a"   and x :: "(real, 'a) vec" shows "(f i)\<^sup>2 * (x $ i)\<^sup>2 \<le> Max {(f i)\<^sup>2 |i. i \<in> UNIV} * (x $ i)\<^sup>2"
multiset_inter_assoc: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset" shows "A \<inter># (B \<inter># C) = A \<inter># B \<inter># C"
inj_on_finite: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
Q''Trans: shows "Q'' \<longmapsto> \<alpha> \<prec> Q'''"
intro: fixes AA :: "('d, 'e, 'f) category_scheme"   and BB :: "('g, 'h, 'i) category_scheme"   and F :: "('d, 'e, 'g, 'h) functor"   and G :: "('d, 'e, 'g, 'h) functor"   and u :: "'d \<Rightarrow> 'h" assumes "Functor F : AA \<longrightarrow> BB"   and "Functor G : AA \<longrightarrow> BB"   and "u \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub>"   and "u \<in> extensional (Ob\<^bsub>AA\<^esub>)"   and "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. u A \<in> Hom\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A G \<^bsub>\<o>\<^esub> A"   and "\<forall>A\<in>Ob\<^bsub>AA\<^esub>. \<forall>B\<in>Ob\<^bsub>AA\<^esub>. \<forall>f\<in>Hom\<^bsub>AA\<^esub> A B. G \<^bsub>\<a>\<^esub> f \<bullet>\<^bsub>BB\<^esub> u A = u B \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f" shows "natural_transformation_axioms AA BB F G u"
obj_def': fixes a :: "'a" shows "C.obj a = (C.arr a \<and> trg\<^sub>C a = a)"
assms_3: assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "P"
nonzero_product: fixes p :: "real poly"   and qs :: "real poly list"   and x :: "real"   and I :: "nat list" assumes "p \<noteq> 0"   and "\<forall>q. List.member qs q \<longrightarrow> coprime p q"   and "x \<in> {x. poly p x = 0}"   and "list_constr I (length qs)" shows "0 < poly (prod_list (retrieve_polys qs I)) x \<or> poly (prod_list (retrieve_polys qs I)) x < 0"
False: shows "(x, y) \<notin> R\<^sup>*"
simps_2: fixes m :: "nat"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" assumes "HBT m" shows "ht \<langle>l, (a, n), r\<rangle> = n"
ts_ord_def: fixes x :: "'a"   and y :: "'a" shows "(x \<sqsubseteq> y) = (n x \<cdot> n y = n y)"
comp_simp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "C1.seq (fst g) (fst f)"   and "C2.seq (snd g) (snd f)" shows "g \<cdot> f = (fst g \<cdot>\<^sub>1 fst f, snd g \<cdot>\<^sub>2 snd f)"
1: fixes \<Gamma>' :: "'a formula list" assumes "set \<Gamma>' \<subseteq> \<Gamma>"   and "\<Turnstile> \<^bold>\<And>\<Gamma>' \<^bold>\<rightarrow> F" shows "thesis"
UINF_all_cong: fixes F :: "'d \<Rightarrow> ('e, 'f) uexpr"   and G :: "'d \<Rightarrow> ('e, 'f) uexpr" assumes "\<And>P. F P = G P" shows "(\<Sqinter> P \<bullet> F P) = (\<Sqinter> P \<bullet> G P)"
poly_deg_monomial: fixes t :: "'c \<Rightarrow>\<^sub>0 'b"   and c :: "'d" shows "poly_deg (monomial c t) = (if c = (0::'d) then 0::'b else deg_pm t)"
MkIde_Dom': fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "MkIde (Dom a) = a"
iAll_iEx_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I. P t) = (\<not> (\<diamond> t I. \<not> P t))"
face_cycle_set_other_da: fixes b :: "'b" assumes "{a, a'} \<inter> G.face_cycle_set b = {}"   and "b \<in> arcs G" shows "H.face_cycle_set b = G.face_cycle_set b"
SN_on_SN_part_conv: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "SN_on r A = (A \<subseteq> SN_part r)"
assms_2: fixes x :: "'a" assumes "x \<notin> orbit f s" shows "f x = x"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Dom \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Dom t \<^bold>\<otimes> Dom u) \<^bold>\<otimes> Dom v"
filterlim_at_bot_nhds: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" assumes "filterlim f at_bot F"   and "(f \<longlongrightarrow> c) F"   and "F \<noteq> bot" shows "False"
geom_rand_walk_pseudo_proj_True: fixes n :: "nat" shows "geom_proc n = geom_proc n \<circ> pseudo_proj_True n"
eq_st_def: fixes S1 :: "(char list \<times> 'a) list"   and S2 :: "(char list \<times> 'a) list" shows "eq_st S1 S2 = (fun_rep S1 = fun_rep S2)"
left_options_plus: fixes u :: "game"   and v :: "game" shows "left_options (plus_game u v) = zunion (zimage (\<lambda>g. plus_game g v) (left_options u)) (zimage (plus_game u) (left_options v))"
enabled_walk: fixes s :: "'s"   and \<omega> :: "('s \<Rightarrow> 's) stream" shows "enabled s (walk s \<omega>)"
compositionality: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and set :: "'d set" shows "f -` g -` set = (g \<circ> f) -` set"
lookupDef_path_conventional: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'var"   and x :: "'node"   and v' :: "'var \<times> 'node \<times> Def" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n = lookupDefNode g m v"   and "n \<notin> set (tl ns)"   and "x \<in> set (tl ns)"   and "v' \<in> braun_ssa.allDefs g x" shows "var g v' \<noteq> v"
polynomial_induct_lemma: fixes f :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier (function_ring (carrier R) R)"   and "a \<in> carrier R" shows "(\<lambda>x\<in>carrier R. a \<oplus> x \<otimes> f x) \<in> carrier (function_ring (carrier R) R)"
sup_plus: fixes R :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "R \<noteq> {}"   and "bdd_above (f ` R)"   and "bdd_above (g ` R)" shows "(SUP x\<in>R. f x + g x) \<le> Sup (f ` R) + Sup (g ` R)"
everything_the_same: fixes x :: "'c"   and y :: "'c" assumes "class.CARD_1 TYPE('c)" shows "x = y"
R_fun: fixes t :: "'a"   and t' :: "'a"   and u :: "'a" assumes "R t t'" shows "R (app t u) (app t' u)"
rel_interior_affine_hull: fixes S :: "'a set" shows "rel_interior (affine hull S) = affine hull S"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "preorder (Br a t1 t2) = [a] @ preorder t1 @ preorder t2"