hcju/isabelleps huggingface dataset:
RedInitProcessing: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Processing)\<rfloor>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle>"
hyps: fixes as :: "nat"   and am :: "nat list" assumes "as < length ap"   and "abc_steps_l (0, lm) ap stp = (as, am)" shows "\<exists>stpaa\<ge>stp. crsp ly (as, am) (steps0 (Suc 0, l, r) tp stpaa) ires"
ext: shows "euclid_ext_poly_dynamic p D H = (S, T)"
positions_in_row: fixes \<Sigma> :: "'b list"   and s :: "('c \<times> bool list) stream"   and i :: "nat" assumes "formula \<Sigma>" shows "positions_in_row s i = {p. snd (s !! p) ! i}"
bernoulli_nat_filtration: fixes N :: "bool stream measure"   and q :: "real" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "0 < p"   and "p < 1" shows "infinite_cts_filtration q N nat_filtration"
subcategory_axioms: shows "subcategory B.VxV.comp (\<lambda>\<mu>\<nu>. B.arr (fst \<mu>\<nu>) \<and> B.arr (snd \<mu>\<nu>) \<and> src\<^sub>B (fst \<mu>\<nu>) = trg\<^sub>B (snd \<mu>\<nu>))"
C_def: shows "C = max 1 ((1 / 2) powr (b - 1))"
entails_frame: fixes P :: "assn"   and Q :: "assn"   and R :: "assn" assumes "P \<Longrightarrow>\<^sub>A Q" shows "P * R \<Longrightarrow>\<^sub>A Q * R"
declared_in_def: fixes G :: "prog"   and m :: "memberdecl"   and C :: "qtname" shows "G\<turnstile> m declared_in C = (case m of fdecl (fn, f) \<Rightarrow> cdeclaredfield G C fn = Some f | mdecl (sig, m) \<Rightarrow> cdeclaredmethd G C sig = Some m)"
cinner_commute: fixes x :: "'a"   and y :: "'a" shows "\<langle>x, y\<rangle> = cnj \<langle>y, x\<rangle>"
append_eq_conv_conj: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
vone_in_omega: shows "1 \<in>\<^sub>\<circ> \<omega>"
dd_lt: shows "\<forall>D\<in>DD. D < C"
game_multi'_def: shows "game_multi' = (\<lambda>c_o c_a s. exec_gpv (\<dagger>(oracle c_o) \<oplus>\<^sub>O eval_oracle' c_o c_a) (\<A> c_a) ((0, None), s) \<bind> (\<lambda>(uu_, (id, j\<^sub>0), s'). return_spmf (j\<^sub>0 \<noteq> None)))"
mod_exE: fixes P :: "'d \<Rightarrow> assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> (\<exists>\<^sub>Ax. P x)" obtains x :: "'d" where "h \<Turnstile> P x"
height_del: fixes x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "del x t = Some t'" shows "height t' \<le> height t"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list \<Rightarrow> ty option"   and a2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a3 :: "expr list"   and a4 :: "ty list" assumes "P,a1,a2 \<turnstile> a3 [:] a4" obtains E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" where "a1 = Ea__" and "a2 = ha__" and "a3 = []" and "a4 = []"   | E :: "char list \<Rightarrow> ty option" and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" and e :: "expr" and T :: "ty" and es :: "expr list" and Ts :: "ty list" where "a1 = E" and "a2 = h" and "a3 = e # es" and "a4 = T # Ts" and "P,E,h \<turnstile> e : T" and "P,E,h \<turnstile> es [:] Ts"
residual_network_sel_2: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v"   and y :: "'v" shows "capacity (res.residual_network f) (x, y) = (if edge (residual_network f) x y then capacity (residual_network f) (x, y) - f (x, y) else if y = source (residual_network f) then 0 else f (y, x))"
polynomial_def: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and p :: "'c list" shows "polynomial\<^bsub>R\<^esub> K p = (p = [] \<or> set p \<subseteq> K \<and> lead_coeff p \<noteq> \<zero>\<^bsub>R\<^esub>)"
create_document_is_weakly_dom_component_safe_step: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr"   and getter :: "(_) RObject_ext \<Rightarrow> 'l" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> create_document \<rightarrow>\<^sub>h h'"   and "ptr \<noteq> cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^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 |h \<turnstile> create_document|\<^sub>r" shows "preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr getter) h h'"
simps_4: fixes uu :: "bool list" shows "internal uu Leaf = False"
obsne_1: shows "obs \<sigma> k \<noteq> obs \<sigma>' k"
finite_extension_finite_dimension_1: fixes K :: "'a list set"   and xs :: "'a list list" assumes "subfield K (A [X])"   and "set xs \<subseteq> carrier (A [X])"   and "\<And>x. x \<in> set xs \<Longrightarrow> (UP_A.algebraic over K) x" shows "UP_A.finite_dimension K (UP_A.finite_extension K xs)"
impl_of_inject: fixes x :: "('a, 'b) RBT.rbt"   and y :: "('a, 'b) RBT.rbt" shows "(impl_of x = impl_of y) = (x = y)"
nonneg: fixes i :: "nat" shows "0 \<le> f i"
slp_of_fa_rev_cnst_def: fixes Const :: "'a"   and Const' :: "floatarith"   and M :: "('a, nat) mapping"   and vs :: "floatarith list"   and vsl :: "nat" shows "slp_of_fa_rev_cnst Const Const' M vs vsl = (Mapping.update Const vsl M, (case Mapping.lookup M Const of None \<Rightarrow> Const' | Some i \<Rightarrow> Var (slp_indexl vsl i)) # vs, Suc vsl)"
assms: shows "set adjs \<subseteq> {(v, d). (u, v) \<in> edges g \<and> w {u, v} = d}"
le_supI: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x"   and "b \<le> x" shows "a + b \<le> x"
isFDERIV_spec_def: fixes N :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and VS :: "real list set" shows "isFDERIV_spec N xs fas VS = SPEC (\<lambda>r. r \<longrightarrow> (\<forall>vs\<in>VS. isFDERIV N xs fas vs))"
scaleR_minus_right: fixes a :: "real"   and x :: "'c" shows "a *\<^sub>R - x = - (a *\<^sub>R x)"
Max_le_imp_subset_atMost: fixes A :: "'a set"   and n :: "'a" assumes "finite A"   and "Max A \<le> n" shows "A \<subseteq> {..n}"
is_unit_hom_if: fixes x :: "'a" assumes "hom x dvd (1::'b)" shows "x dvd (1::'a)"
unitary_sound: fixes P :: "'b \<Rightarrow> real" assumes "unitary P" shows "sound P"
coef_sym\<^sub>h_def: fixes n :: "nat"   and i :: "nat" shows "coef_sym\<^sub>h n i = 1"
pmc_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<not> K \<turnstile> \<phi>" shows "K \<div>\<^bsub>\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi> = K"
fset_eqI: fixes A :: "'a fset"   and B :: "'a fset" assumes "\<And>x. (x |\<in>| A) = (x |\<in>| B)" shows "A = B"
not_distinct_decomp: fixes ws :: "'c list" assumes "\<not> distinct ws" shows "\<exists>xs ys zs y. ws = xs @ [y] @ ys @ [y] @ zs"
ord_spmf_bind_reflI: fixes p :: "'b spmf"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd spmf"   and g :: "'b \<Rightarrow> 'd spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> ord_spmf R (f x) (g x)" shows "ord_spmf R (p \<bind> f) (p \<bind> g)"
closed_segment_PairD: fixes x :: "'a"   and x' :: "'b"   and a :: "'a"   and a' :: "'b"   and b :: "'a"   and b' :: "'b" assumes "(x, x') \<in> closed_segment (a, a') (b, b')" shows "x \<in> closed_segment a b \<and> x' \<in> closed_segment a' b'"
eq_Aeta_expand: fixes a :: "Arity"   and e :: "exp"   and \<rho> :: "Vars.var \<Rightarrow> Value" shows "eq a (\<lbrakk> Aeta_expand a e \<rbrakk>\<^bsub>\<rho>\<^esub>) (\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub>)"
sum_incoming_extend: fixes u :: "nat"   and g :: "nat \<times> nat \<Rightarrow> 'b" assumes "\<And>v. \<lbrakk>v \<in> cf.V; (v, u) \<notin> cf.E\<rbrakk> \<Longrightarrow> g (v, u) = (0::'b)" shows "sum g (cf.incoming u) = (\<Sum>v\<in>cf.V. g (v, u))"
parallel_wiring\<^sub>w_def: shows "parallel_wiring\<^sub>w \<equiv> lassocr\<^sub>w \<circ>\<^sub>w ((id, id) |\<^sub>w swap_lassocr\<^sub>w \<circ>\<^sub>w rassocl\<^sub>w)"
distinct_3: fixes x1 :: "'a"   and x31 :: "'a itrm"   and x32 :: "'a itrm" shows "Opaque x1 \<noteq> x31 \<diamondop> x32"
drlex_pm_lin: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "drlex_pm s t \<or> drlex_pm t s"
even_two_times_div_two: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (numeral.numeral one plus (num.Bit0 num.One))) = a"
\<tau>Exec_movesr_trans: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and es :: "'a expr1 list"   and h :: "'c"   and s :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option"   and s'' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "(JVM_heap_base.\<tau>exec_moves addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t es h)\<^sup>*\<^sup>* s s'"   and "(JVM_heap_base.\<tau>exec_moves addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t es h)\<^sup>*\<^sup>* s' s''" shows "(JVM_heap_base.\<tau>exec_moves addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t es h)\<^sup>*\<^sup>* s s''"
holomorphic_on_ident: fixes s :: "complex set" shows "(\<lambda>x. x) holomorphic_on s"
S_precise: fixes u :: "'v" assumes "u \<in> S" shows "D u = \<delta> s u"
minus_monom: fixes a :: "'a"   and n :: "nat" shows "- Polynomial.monom a n = Polynomial.monom (- a) n"
Inter_iff: fixes A :: "'a"   and C :: "'a set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
assoc_in_hom: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "\<guillemotleft>\<a>\<^sub>D[a, b, c] : (a \<otimes>\<^sub>D b) \<otimes>\<^sub>D c \<rightarrow>\<^sub>D a \<otimes>\<^sub>D b \<otimes>\<^sub>D c\<guillemotright>"
assms_2: assumes "\<not> \<U> s"   and "min_lvar_not_in_bounds s = None" shows "P s"
vmc2_path_axioms: shows "vmc2_path G P v p \<sigma> \<sigma>'"
unat_cong: fixes x :: "'b word"   and y :: "'b word" assumes "x = y" shows "unat x = unat y"
i_last_message_hold_i_drop_mod: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af" assumes "n mod k = 0" shows "f \<Up> n \<longmapsto> k = f \<longmapsto> k \<Up> n"
interaction_bounded_by_Fail: fixes "consider" :: "'d \<Rightarrow> bool"   and n :: "enat" shows "interaction_bounded_by consider Fail n"
cltn2_left_inverse_ex: fixes A :: "cltn2" shows "\<exists>B. cltn2_compose B A = cltn2_id"
is_group_action: shows "group_action (G\<lparr>carrier := P\<rparr>) (conjugation_action (p ^ a)) (subgroups_of_size (p ^ a))"
tminus_left_mono: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a \<le> b" shows "a \<ominus> c \<le> b \<ominus> c"
sorted_quicksort: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (ord.quicksort less xs)"
path_image_circlepath: fixes z :: "complex"   and r :: "real" shows "path_image (circlepath z r) = sphere z \<bar>r\<bar>"
mult_unit_dvd_iff': fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "dvd.dvd times (times a b) c = dvd.dvd times b c"
BodyN: fixes P :: "'a assn"   and pn :: "pname"   and Q :: "'a assn"   and G :: "'a triple set" assumes "insert ({P}. BODY pn .{Q}) G|-{P}. the (body pn) .{Q}" shows "G|-{P}. BODY pn .{Q}"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'mac" shows "cbind (creturn u) f = f u"
hterms_of_fterms_def: fixes ts :: "fterm list" shows "hterms_of_fterms ts \<equiv> map hterm_of_fterm ts"
assms: fixes z :: "complex" assumes "ereal (cmod z) < r" shows "eval_fps f z \<noteq> 0"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "comm_monoid_add_hom hom"   and "inj_monoid_add_hom hom" shows "inj_comm_monoid_add_hom hom"
choice: shows "q = 0 \<or> b \<in> sqrt_irrat"
subcategory_axioms: shows "subcategory (\<cdot>) (left \<omega>)"
norm_f: fixes h :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "h \<in> F"   and "x \<in> cbox a b" shows "norm (h x) \<le> norm (f x)"
b: shows "!z \<cdot> y = y \<cdot> !z"
intros: fixes p :: "'node"   and R :: "'node set" assumes "p \<in> W"   and "\<forall>Sl\<in>slices p. \<exists>q\<in>Sl \<inter> W. q \<in> R \<or> blocking_min R q" shows "blocking_min R p"
Gromov_boundary_extended_product_PInf: fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "(extended_Gromov_product_at e x y = \<infinity>) = (x \<in> Gromov_boundary \<and> y = x)"
ram2_not: shows "ram2 \<noteq> hd (snd (splitAt ram1 (vertices f)) @ fst (splitAt ram1 (vertices f)))"
psubstEnv_idEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" shows "idEnv &[rho] = rho"
zero_in_full_pmdl: fixes K :: "'k set" shows "0 \<in> full_pmdl K"
comp_def: fixes g :: "'a"   and f :: "'a" shows "g \<cdot> f = (if Arr f \<and> Arr g \<and> C.cod f = C.dom g then g \<cdot>\<^sub>C f else C.null)"
has_limits_def: fixes uu :: "'c" shows "A.has_limits uu \<equiv> \<forall>J. category J \<longrightarrow> A.has_limits_of_shape J"
assms_6: fixes p :: "int poly" assumes "p \<in> P" shows "\<beta> p \<noteq> 0"
e1: shows "fst (trunc_bound_eucl p (fst (fst X) + fst (fst Y))) = fst (fst X) + fst (fst Y) + e1"
parSym: fixes P :: "pi"   and Q :: "pi" shows "P \<parallel> Q \<sim>\<^sup>s\<^sub>l Q \<parallel> P"
Sup_lexord: fixes A :: "'b set"   and k :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and c :: "'b set \<Rightarrow> 'b"   and s :: "'b set \<Rightarrow> 'b" assumes "\<And>a S. \<lbrakk>a \<in> A; k a = Sup (k ` A); S = {a' \<in> A. k a' = k a}\<rbrakk> \<Longrightarrow> P (c S)"   and "(\<And>a. a \<in> A \<Longrightarrow> k a \<noteq> Sup (k ` A)) \<Longrightarrow> P (s A)" shows "P (Sup_lexord k c s A)"
4: shows "punit.lt (rep_list a) - punit.lt (rep_list g) + punit.lt (rep_list g) = punit.lt (rep_list a)"
run_of_lasso_def: fixes L :: "'b lasso" shows "run_of_lasso L \<equiv> lasso_reach L \<frown> (lasso_cycle L)\<^sup>\<omega>"
ordering_ord_ijk: fixes f :: "nat \<Rightarrow> 'b"   and ord :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "TernaryOrdering.ordering f ord X"   and "i < j \<and> j < k \<and> (finite X \<longrightarrow> k < card X)" shows "ord (f i) (f j) (f k)"
monom_power: fixes a :: "'b"   and b :: "nat"   and n :: "nat" shows "monom a b ^ n = monom (a ^ n) (b * n)"
InterE: fixes A :: "'k"   and C :: "'k set set"   and X :: "'k set" assumes "A \<in> \<Inter> C" obtains "X \<notin> C"   | "A \<in> X"
generate_pow: fixes a :: "int" assumes "a \<in> carrier (Z n)" shows "generate (Z n) {a} = {a [^]\<^bsub>Z n\<^esub> k |k. k \<in> UNIV}"
USigma_def: fixes \<A> :: "'a set set"   and B :: "'a set \<Rightarrow> 'a set" shows "USigma \<A> B \<equiv> \<Union>X\<in>\<A>. \<Union>y\<in>B X. {insert y X}"
deadends_def: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player" assumes "ParityGame G" shows "ParityGame.deadends G p \<equiv> {v \<in> if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>. \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v \<rightarrow>\<^bsub>G\<^esub> w)}"
LFP_idem: fixes f :: "'a \<Rightarrow> 'a" assumes "f \<in> carrier L \<rightarrow> carrier L"   and "Mono\<^bsub>L\<^esub> f"   and "Idem\<^bsub>L\<^esub> f" shows "LFP\<^bsub>L\<^esub> f .=\<^bsub>L\<^esub> f \<bottom>\<^bsub>L\<^esub>"
dDia_test: fixes p :: "'b upred"   and \<Phi> :: "'b upred" shows "\<^bold><?[p]\<^bold>>\<Phi> = (p \<and> \<Phi>)"
seq_bot_conj: fixes c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" shows "(c::'b::refinement_lattice) ; \<bottom> \<iinter> (d::'b::refinement_lattice) \<sqsubseteq> (c \<iinter> d) ; \<bottom>"
orthonormal_optimal_partial_sum: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and I :: "nat set"   and a :: "nat \<Rightarrow> real" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S"   and "finite I" shows "l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. orthonormal_coeff S w f i * w i x)) \<le> l2norm S (\<lambda>x. f x - (\<Sum>i\<in>I. a i * w i x))"
disjoint_implies_not_part: fixes x :: "i"   and y :: "i" assumes "PM"   and "\<not> O x y" shows "\<not> P x y"
list_take_eq_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs = ys) = (\<forall>n. xs \<down> n = ys \<down> n)"
Nil: fixes A :: "'b set" shows "\<epsilon> \<in> lists A"
LIMSEQ_stirling_integral: fixes s :: "complex"   and n :: "nat" assumes "s \<notin> \<real>\<^sub>\<le>\<^sub>0"   and "0 < n" shows "(\<lambda>x. integral {0..real x} (\<lambda>x. complex_of_real (pbernpoly n x) / (complex_of_real x + s) ^ n)) \<longlonglongrightarrow> stirling_integral n s"
det_interchange_different_rows: fixes i :: "'b"   and j :: "'b"   and A :: "(('c, 'b) vec, 'b) vec" assumes "i \<noteq> j" shows "det (interchange_rows A i j) = - det A"
couSim: shows "weak_barbed_coupled_simulation (indRelTEQ TRel) (STCalWB SWB TWB)"
empty_Collect_eq: fixes P :: "'e \<Rightarrow> bool" shows "({} = Collect P) = (\<forall>x. \<not> P x)"
simps_1: fixes atm :: "'b" shows "noWhile (Atm atm) = True"
PA1: fixes p :: "'b pfm" assumes "ptautology p" shows "\<turnstile>\<^sub>! p"
comm_ring_1_ow_def: fixes U :: "'b set"   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" 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"
preserves_comp: fixes g :: "'a1"   and f :: "'a1" assumes "A1.seq g f" shows "A1.map (g \<cdot>\<^sub>A\<^sub>1 f) = A1.map g \<cdot>\<^sub>A\<^sub>1 A1.map f"
aux0_4: fixes a :: "'b"   and x :: "'b list"   and y :: "'b list" shows "(a \<in> set x \<or> a \<in> set y) = (a \<in> set (x @ y))"
strict_implies_not_eq: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "a \<noteq> b"
sp_append: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es1 :: "'d edge list"   and es2 :: "'d edge list"   and v2 :: "'d" shows "Graph.subpath g v1 (es1 @ es2) v2 = (\<exists>v. Graph.subpath g v1 es1 v \<and> Graph.subpath g v es2 v2)"
fds_truncate_deriv: fixes m :: "nat"   and f :: "'a fds" shows "fds_truncate m (fds_deriv f) = fds_deriv (fds_truncate m f)"
Re_interval_plus: fixes A :: "complex_interval"   and B :: "complex_interval" shows "Re_interval (A + B) = Re_interval A + Re_interval B"
unit_vecs_basis: fixes n :: "nat" shows "vectorspace.basis class_ring (module_vec TYPE('a) n) (set (unit_vecs n))"
basis_in_standard_simplex: fixes i :: "nat"   and p :: "nat" shows "((\<lambda>j. if j = i then 1 else 0) \<in> standard_simplex p) = (i \<le> p)"
intro: fixes b :: "real" assumes "1 < b" shows "information_space_axioms b"
maddux1': fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "conjugate f (x \<sqinter> - f y) \<le> conjugate f x \<sqinter> - y"
enum_eq: fixes j :: "nat" assumes "j < n" shows "s.enum j = t.enum j"
sin_zero_norm_cos_one: fixes x :: "'a" assumes "sin x = (0::'a)" shows "norm (cos x) = 1"
complements_do_not_falsify': fixes l\<^sub>1 :: "fterm literal"   and C\<^sub>1' :: "fterm literal set"   and l\<^sub>2 :: "fterm literal"   and G :: "bool list" assumes "unification"   and "l\<^sub>1 \<in> C\<^sub>1'"   and "l\<^sub>2 \<in> C\<^sub>1'"   and "l\<^sub>1 = l\<^sub>2\<^sup>c"   and "falsifies\<^sub>g G C\<^sub>1'" shows "False"
f_def: fixes x :: "'c" shows "f x = (if Regions.isIntv (I x) then Lt (real (intv_const (I x) + 1)) else if Regions.isConst (I x) then Le (real (intv_const (I x))) else \<infinity>)"
subst_term_preserves_typ_of1: fixes Ts :: "typ list"   and x :: "variable"   and \<tau> :: "typ"   and y :: "variable"   and t :: "Core.term" shows "typ_of1 Ts (subst_term [((x, \<tau>), Fv y \<tau>)] t) = typ_of1 Ts t"
assms_1: fixes x :: "V" assumes "x \<in> X" shows "Ord x"
case_rews_4: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case GT of LT \<Rightarrow> f1 | EQ \<Rightarrow> f2 | GT \<Rightarrow> f3) = f3"
appr: shows "approx_slp' p ord I i ra VS = Some X"
tminus_less: fixes a :: "real"   and b :: "real" assumes "0 \<le> a"   and "0 \<le> b" shows "a \<ominus> b \<le> a"
simps_1: fixes \<alpha> :: "game"   and \<beta> :: "game" shows "Choiceo (Agame \<alpha>) (Agame \<beta>) = Agame (\<alpha> \<union>\<union> \<beta>)"
Suc_1: fixes lholed :: "Lholed"   and LI :: "e list"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "Lfilled i lholed (vs @ [$Return]) LI"   and "return \<C> = Some tcs"   and "length tcs = length vs"   and "\<S>\<bullet>\<C> \<turnstile> LI : ts _> ts'"   and "const_list vs" shows "\<S>\<bullet>\<C>' \<turnstile> vs : [] _> tcs"
psubstAbs_simp: fixes X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodEnv rho"   and "freshEnv xs x rho" shows "Abs xs x X $[rho] = Abs xs x X #[rho]"
Auth_fresh_not_AKcryptSK: fixes authK :: "nat"   and evs :: "event list"   and servK :: "nat" assumes "Key authK \<notin> used evs"   and "evs \<in> kerbIV" shows "\<not> AKcryptSK authK servK evs"
select_convs_4: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cCurrentF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cCurrentF"