hcju/isabelleps huggingface dataset:
impI_PL: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<union> {p} \<turnstile> q" shows "A \<turnstile> p .\<longrightarrow>. q"
t_def: fixes x :: "nat" shows "t x \<equiv> if s x \<up> then None else if s x \<down>= 0 then Some i\<^sub>0 else Some (the (s x) - 1)"
moebius_pt_cmat_cvec_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex"   and z :: "complex \<times> complex" shows "moebius_pt_cmat_cvec M z = M *\<^sub>m\<^sub>v z"
SatP: fixes b :: "'d"   and ps :: "('c, 'd) fm list"   and a :: "'d"   and branch :: "(('c, 'd) fm list \<times> 'd) list"   and p :: "('c, 'd) fm"   and A :: "'d set"   and n :: "nat" assumes "(\<^bold>@ a p) at b in (ps, a) # branch"   and "new p a ((ps, a) # branch)"   and "A, Suc n \<turnstile> (p # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
a4: fixes t :: "'a" assumes "t \<in> T" shows "\<parallel>t\<parallel> = 1"
s2: shows "(r|labels \<gamma>2|, {#fst \<alpha>_step#}) \<in> mul_eq r"
simplicial_vertex_oriented_simplex: fixes i :: "nat"   and p :: "nat"   and l :: "nat \<Rightarrow> 'c \<Rightarrow> real" shows "simplicial_vertex i (oriented_simplex p l) = (if i \<le> p then l i else undefined)"
typeof_simps_1: fixes b :: "bool" shows "\<tau> boolV b = BoolT"
StackMark_a_def: shows "StackMark_a = (\<lambda>(i, j). case (i, j) of (init, loop) \<Rightarrow> Q1'_a \<sqinter> Q2'_a | (init, _) \<Rightarrow> \<top> | (loop, init) \<Rightarrow> \<top> | (loop, loop) \<Rightarrow> Q3'_a \<sqinter> Q4'_a | (loop, final) \<Rightarrow> Q5'_a | (final, b) \<Rightarrow> \<top>)"
ccSup_least: fixes A :: "'a set"   and z :: "'a" assumes "countable A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "Sup A \<le> z"
Main: fixes n :: "nat" assumes "N \<le> n"   and "L * m n < B (Suc n)" shows "alpha < PS.prob (space M \<inter> (T ^^ n) -` K n)"
OES2: fixes \<Gamma> :: "'b formula list"   and \<BB> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<AA> :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<BB> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<AA> \<^bold>\<or> \<BB> # \<Theta>"
distributed_marginal_eq_joint2: fixes T :: "'b measure"   and S :: "'f measure"   and Y :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'b"   and Py :: "'b \<Rightarrow> ennreal"   and X :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'f"   and Pxy :: "'f \<times> 'b \<Rightarrow> ennreal" assumes "sigma_finite_measure T"   and "sigma_finite_measure S"   and "distributed (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) T Y Py"   and "distributed (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "AE y in T. Py y = \<integral>\<^sup>+ x. Pxy (x, y) \<partial>S"
assms_4: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
canonical_reduced: fixes A :: "('b \<Rightarrow> bool) set"   and \<sigma> :: "nat \<Rightarrow> 'b" assumes "canonical_sequence A \<sigma>" shows "canonical_sequence A (\<natural>\<sigma>)"
rcoset_mult_def: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set"   and A :: "'e set"   and B :: "'e set" shows "[mod I:] A \<Otimes>\<^bsub>R\<^esub> B = (\<Union>a\<in>A. \<Union>b\<in>B. I +>\<^bsub>R\<^esub> a \<otimes>\<^bsub>R\<^esub> b)"
outstanding_refs_dropWhileD: fixes x :: "nat"   and P :: "'b memref \<Rightarrow> bool"   and P' :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" assumes "x \<in> outstanding_refs P (dropWhile P' sb)" shows "x \<in> outstanding_refs P sb"
ide_char: fixes a :: "'c \<times> 'c" shows "C.VV.ide a = (C.VV.arr a \<and> C.VxV.ide a)"
carrier_subgroup_generated_subgroup: shows "carrier (subgroup_generated B K) = K"
Arr_iff_dom_in_domain: fixes f :: "'a" shows "Arr f = (C.dom f \<in> domains f)"
apex_def: shows "SPN_r_SPN_s.apex \<equiv> Maps.dom SPN_r_SPN_s.leg0"
count_of_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "('b \<times> nat) list"   and x :: "'b" shows "count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"
poly_condD_2: fixes p :: "int poly" assumes "poly_cond p" shows "0 < Polynomial.lead_coeff p"
antisym: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le>s a"   and "a \<le>s b" shows "a = b"
simps_1: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and w :: "bool stream" shows "rn_rev_price N der matur 0 w = der w"
same_cs_if_not_recv: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and cid :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "\<not> isRecv ev" shows "fst (cs c cid) = fst (cs c' cid)"
3: assumes "arr \<mu> \<and> B.iso (Map \<mu>)" shows "local.iso \<mu>"
fold_multiset_equiv: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and ys :: "'b list" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs\<rbrakk> \<Longrightarrow> f x \<circ> f y = f y \<circ> f x"   and "mset xs = mset ys" shows "fold f xs = fold f ys"
assms_3: shows "\<forall>v k. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<longrightarrow> \<A> (State k (index (\<Pi>\<^sub>\<V>) v)) = ((trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi> ! k) v = Some True)"
le_supI1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<le> a" shows "x \<le> a + b"
sup_state_opt_Some_Some: fixes P :: "'c prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list" shows "P \<turnstile> \<lfloor>a\<rfloor> \<le>' \<lfloor>b\<rfloor> = P \<turnstile> a \<le>\<^sub>i b"
Rep_uint8_inverse: fixes x :: "uint8" shows "Abs_uint8 (Rep_uint8 x) = x"
map_3: fixes f :: "'c \<Rightarrow> 'b"   and x3 :: "'c" shows "map_ltlc f prop\<^sub>c(x3) = prop\<^sub>c(f x3)"
bl_to_bin_aux_alt: fixes bs :: "bool list"   and w :: "int" shows "bl_to_bin_aux bs w = concat_bit (length bs) (bl_to_bin bs) w"
psym_lens_compl: fixes a :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "psym_lens a" shows "psym_lens (-\<^sub>L a)"
distinct_31: fixes x21 :: "Vars.var"   and x22 :: "exp"   and x23 :: "assn" shows "ANil \<noteq> ACons x21 x22 x23"
are_all_operator_effects_consistent_def: fixes ops :: "('c, 'd) sas_plus_operator list" shows "SAS_Plus_Semantics.are_all_operator_effects_consistent ops \<equiv> list_all (\<lambda>op. list_all (SAS_Plus_Semantics.are_operator_effects_consistent op) ops) ops"
a: shows "a = - (w \<bullet> u / (u \<bullet> u)) \<cdot>\<^sub>v u + b"
functor_axioms: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> G)"
FieldDecls_def: 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 C :: "char list"   and F :: "char list" shows "FieldDecls P C F \<equiv> {(Cs, T). Subobjs P C Cs \<and> (\<exists>Bs fs ms. class P (last Cs) = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> map_of fs F = \<lfloor>T\<rfloor>)}"
inject: fixes x :: "nat"   and ya :: "nat" shows "(atom x = atom ya) = (x = ya)"
upper_bound_left: fixes x :: "'b"   and y :: "'b" shows "- x \<le> - x \<squnion> - y"
cfun_meet_simp: fixes f :: "'c \<rightarrow> 'b"   and g :: "'c \<rightarrow> 'b"   and x :: "'c" shows "(f \<squnion> g)\<cdot>x = f\<cdot>x \<squnion> g\<cdot>x"
sup_idem: fixes x :: "'a" shows "x \<sqinter> x = x"
of_int_div_field: fixes d :: "int"   and n :: "int" assumes "d dvd n" shows "of_int (n div d) = of_int n / of_int d"
smult_assoc: fixes r :: "('f, 'g) aezfun"   and s :: "('f, 'g) aezfun"   and m :: "'v" assumes "r \<in> FG"   and "s \<in> FG"   and "m \<in> V" shows "r \<cdot> s \<cdot> m = (r * s) \<cdot> m"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and generat :: "('c, 'd, 'e) generat" shows "h (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (case generat of Pure x \<Rightarrow> h (f1 x) | IO x1 x2 \<Rightarrow> h (f2 x1 x2))"
islimptI: fixes x :: "'b"   and S :: "'b set" assumes "\<And>T. \<lbrakk>x \<in> T; open T\<rbrakk> \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x" shows "x islimpt S"
lin_dense: fixes p :: "fm"   and l :: "real"   and u :: "real"   and x :: "real"   and bs :: "real list"   and y :: "real" assumes "isrlfm p"   and "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> (\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` set (\<Upsilon> p)"   and "l < x"   and "x < u"   and "Ifm (x # bs) p"   and "l < y"   and "y < u" shows "Ifm (y # bs) p"
hyps_2: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>c\<^sub>2 \<down> Normal s} c\<^sub>2 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
simps: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and x :: "'b"   and t :: "'b btree" assumes "split split" shows "split.delete split k x t = reduce_root (split.del split k x t)"
that: fixes xs' :: "V"   and x :: "V" assumes "xs = xs' #\<^sub>\<circ> x"   and "vfsequence xs'" shows "thesis"
strict_trans2: fixes a :: "complex"   and b :: "complex"   and c :: "complex" assumes "a <\<^sub>\<complex> b"   and "b \<le>\<^sub>\<complex> c" shows "a <\<^sub>\<complex> c"
set_times_mono2: fixes C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and F :: "'b set" assumes "C \<subseteq> D"   and "E \<subseteq> F" shows "C * E \<subseteq> D * F"
"3_2": assumes "x \<noteq> y"   and "rel x y"   and "\<And>x. \<lbrakk>x \<in> set xs; P x\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel xs (y # ys)) = filter P (y # ys)"
sg_subset_elem: fixes H :: "'a set"   and h :: "'a" assumes "G \<guillemotright> H "   and "h \<in> H" shows "h \<in> carrier G"
still_reachable_eq: fixes A :: "'b set"   and Rb :: "('b \<times> 'b) set"   and B :: "'b set"   and Ra :: "('b \<times> 'b) set" assumes "A \<subseteq> Rb\<^sup>* `` B"   and "B \<subseteq> Ra\<^sup>* `` A"   and "\<forall>(x, y)\<in>Ra - Rb. y \<in> Rb\<^sup>* `` B"   and "\<forall>(x, y)\<in>Rb - Ra. y \<in> Ra\<^sup>* `` A" shows "Ra\<^sup>* `` A = Rb\<^sup>* `` B"
connected_conv: fixes G :: "('e, 'f) pre_digraph" shows "Digraph_Component.connected G = (verts G \<noteq> {} \<and> (\<forall>u\<in>verts G. \<forall>v\<in>verts G. (u, v) \<in> rtrancl_on (verts G) ((arcs_ends G)\<^sup>s)))"
is_interpD: fixes I :: "('a, 'b, 'c) interp" assumes "is_interp I" shows "\<forall>x i. (Functions I i has_derivative FunctionFrechet I i x) (at x)"
parts_mult_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and A :: "nat"   and B :: "nat" assumes "s \<turnstile> s'"   and "A \<otimes> B \<in> parts (used s')"   and "A \<otimes> B \<notin> parts (used s)" shows "(\<exists>n SK. A = Auth_PriK n \<and> (Asset n, \<lbrace>Num 2, PubKey B\<rbrace>) \<in> s' \<and> Crypt (SesK SK) (A \<otimes> B) \<in> parts (used s')) \<or> {PriKey A, PriKey B} \<subseteq> spied s'"
finprod_empty: fixes f :: "'b \<Rightarrow> 'a" shows "prod f {} = (1::'a)"
axioms_1: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_linear f" shows "linear f"
crypts_snd: fixes KS :: "key list"   and X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H" shows "foldr Crypt KS Y \<in> crypts H"
m_assoc: fixes x :: "int"   and y :: "int"   and z :: "int" shows "x + y + z = x + (y + z)"
elims_3: fixes x :: "'d mat \<times> 'd mat \<times> 'd mat"   and xa :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec \<times> (('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" assumes "\<not> HMA_M3 x xa" obtains P :: "'d mat" and A :: "'d mat" and Q :: "'d mat" and P' :: "(('d, 'e) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and A' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" and Q' :: "(('d, 'f) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" where "x = (P, A, Q)" and "xa = (P', A', Q')" and "\<not> (Mod_Type_Connect.HMA_M P P' \<and> Mod_Type_Connect.HMA_M A A' \<and> Mod_Type_Connect.HMA_M Q Q')"
subgroup_indicator_pos: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "0 < subgroup_indicator (G\<lparr>carrier := adjoin G H a\<rparr>) H a"
shiftr_eq_drop_bit: fixes a :: "'a"   and n :: "nat" shows "a >> n = drop_bit n a"
curryD: fixes f :: "'c \<times> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "curry f a b" shows "f (a, b)"
assert_bound'_def: fixes dir :: "('i, 'a) Direction"   and i :: "'i"   and x :: "nat"   and c :: "'a"   and s :: "('i, 'a) state" shows "assert_bound' dir i x c s \<equiv> if \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x) then s else let s' = update\<B>\<I> (UBI_upd dir) i x c s in if \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x) then set_unsat [i, LI dir s x] s' else if x \<notin> lvars (\<T> s') \<and> lt dir c (\<langle>\<V> s\<rangle> x) then update x c s' else s'"
star_of_power: fixes x :: "'b"   and n :: "nat" shows "star_of (x ^ n) = star_of x ^ n"
AbstrLevels_S1opt_A11: fixes i :: "AbstrLevelsID" assumes "sS1opt \<in> AbstrLevel i" shows "sA11 \<notin> AbstrLevel i"
ag_inv_inj: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c"   and y :: "'c" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<noteq> y" shows "-\<^sub>a\<^bsub>A\<^esub> x \<noteq> -\<^sub>a\<^bsub>A\<^esub> y"
ord_gt_0_iff: fixes n :: "nat"   and x :: "nat" shows "(0 < ord n x) = coprime n x"
comm_neutral: fixes a :: "'b" shows "a + (0::'b) = a"
unitor_coincidence_2: shows "\<r>\<^sub>C[\<I>\<^sub>C] = \<iota>\<^sub>C"
is2: shows "{xs. Ball X ((!) xs) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m \<and> \<not> xs ! e} = {xs. Ball X ((!) xs) \<and> (\<forall>i\<in>insert e Y. \<not> xs ! i) \<and> length xs = m}"
div_int_poly_step_surjective: fixes a :: "int poly \<times> int poly"   and q :: "int poly"   and b :: "int"   and c :: "(int poly \<times> int poly) option" assumes "Some a = div_int_poly_step q b c" shows "\<exists>c'. c = Some c'"
prod_cases7: fixes y :: "'a \<times> 'b \<times> 'c \<times> 'd \<times> 'e \<times> 'f \<times> 'g" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" and e :: "'e" and f :: "'f" and g :: "'g" where "y = (a, b, c, d, e, f, g)"
approximating_bigstep_fun_seq_semantics: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and s :: "state"   and t :: "state"   and rs\<^sub>2 :: "'c rule list" assumes "\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "approximating_bigstep_fun \<gamma> p (rs\<^sub>1 @ rs\<^sub>2) s = approximating_bigstep_fun \<gamma> p rs\<^sub>2 t"
simps_1: fixes l :: "'b set list" shows "FWNormalisationCore.sorted [] l = True"
last_in_set: fixes as :: "'e list" assumes "as \<noteq> []" shows "last as \<in> set as"
power_Suc2: fixes a :: "'a"   and n :: "nat" shows "power.power 1' (;) a (Suc n) = power.power 1' (;) a n ; a"
aforms_err_to_aforms_def: fixes d :: "nat"   and xs :: "((real \<times> real pdevs) \<times> real) list" shows "aforms_err_to_aforms d xs = map2 (\<lambda>d x. aform_err_to_aform x d) [d..<d + length xs] xs"
trad_foundation: fixes z :: "V" assumes "z \<noteq> 0" shows "\<exists>w. w \<in> elts z \<and> w \<sqinter> z = 0"
intros_1: shows "upOne \<in> CofGroups.Ex1"
inf_absorb1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type"   and y :: "'b::type" assumes "class.semilattice_inf (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)" shows "(inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type) = x"
fbox_whilei: fixes p :: "'b ppath set"   and i :: "'b ppath set"   and t :: "'b ppath set"   and q :: "'b ppath set"   and x :: "'b ppath set" assumes "rdom p \<subseteq> rdom i"   and "rdom i ; pp_a t \<subseteq> rdom q"   and "rdom i ; rdom t \<subseteq> wp x i" shows "rdom p \<subseteq> wp (ppath_aka.whilei t i x) q"
eventually_at_top_linorderI: fixes c :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x. c \<le> x \<Longrightarrow> P x" shows "eventually P at_top"
assms_1: shows "nodes G1 \<inter> nodes G2 = {}"
iso_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.iso C f \<equiv> \<exists>g. category.inverse_arrows C f g"
abs_eq: fixes x :: "'d" shows "- dual x = dual (- x)"
\<psi>_mapsto: fixes b :: "'c"   and a :: "'c" assumes "C.ide b"   and "C.ide a" shows "\<psi>C (b, a) \<in> HomC.set (b, a) \<rightarrow> C.hom b a"
lprefix_llist_ofI: fixes ys :: "'a list"   and xs :: "'a list" assumes "\<exists>zs. ys = xs @ zs" shows "lprefix (llist_of xs) (llist_of ys)"
variant_def: fixes k :: "nat"   and base :: "char list" shows "variant k base = base @ replicate k CHR ''*''"
bind_in_space: fixes A :: "'a \<Rightarrow> 'b measure"   and N :: "'b measure" assumes "random_variable (subprob_algebra N) A" shows "M \<bind> A \<in> space (subprob_algebra N)"
snapshotted_and_not_done_implies_marker_in_channel: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and p :: "'d"   and cid :: "nat"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) \<noteq> Done"   and "i \<le> length t"   and "channel cid = Some (p, q)" shows "Marker \<in> set (msgs (distributed_system.s channel trans send recv init t i) cid)"
fls_const_subdegree: fixes c :: "'a" shows "fls_subdegree (fls_const c) = 0"
isomorphic_implies_ide_2: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "ide g"
sets_into_space: fixes x :: "(nat \<Rightarrow> 'a) set"   and i :: "nat" assumes "x \<in> sets.pullback_algebra M (\<lambda>f. f i) (space P)" shows "x \<subseteq> space P"
connected_component_set_empty: fixes x :: "'c" shows "connected_component_set {} x = {}"
fmfilter_true: fixes m :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x" shows "fmfilter P m = m"
FactRing_simps_1: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
elims_3: fixes x :: "'at set"   and xa :: "'at Literal set" assumes "\<not> x \<Turnstile> xa" obtains I :: "'at set" and C :: "'at Literal set" where "x = I" and "xa = C" and "\<nexists>L. L \<in> C \<and> I \<Turnstile> L"
e: fixes i :: "nat" shows "edges (S i) \<subseteq> edges C"
hol_fn: fixes n :: "nat" shows "f n holomorphic_on S"
is_unit_polyE: fixes p :: "'a poly" assumes "p dvd 1" obtains c :: "'a" where "p = [:c:]" and "c dvd (1::'a)"
Chebyshev_sum_upper_nat: fixes n :: "nat"   and a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" assumes "\<And>i j. \<lbrakk>i \<le> j; j < n\<rbrakk> \<Longrightarrow> a i \<le> a j"   and "\<And>i j. \<lbrakk>i \<le> j; j < n\<rbrakk> \<Longrightarrow> b j \<le> b i" shows "n * (\<Sum>i = 0..<n. a i * b i) \<le> sum a {0..<n} * sum b {0..<n}"
div_le_div_ceiling: fixes x :: "int"   and y :: "int" shows "x div y \<le> div_ceiling x y"
t_def: shows "t = Syntax.Let (Unauth (Auth (Inj1 Unit))) x (Syntax.Let (Unauth (Auth Unit)) y (Var x))"
Inv_preserves_Can_1: fixes t :: "'a local.term" assumes "Can t" shows "Can (Inv t)"
invh_baliR: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invh l"   and "invh r"   and "bheight l = bheight r" shows "invh (baliR l a r)"
msg_ok_other_nos_inc: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg"   and i :: "nat"   and \<sigma>' :: "nat \<Rightarrow> state" assumes "msg_ok \<sigma> m"   and "other nos_inc {i} \<sigma> \<sigma>'" shows "msg_ok \<sigma>' m"
\<D>_3: fixes K :: "'a set" assumes "K \<in> \<D>" shows "K \<noteq> {}"
nonneg_Reals_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "sum f A \<in> \<real>\<^sub>\<ge>\<^sub>0"
wrapS_def: shows "wrapS \<equiv> \<Lambda> s. Abs_cfun ((!!) s)"
order_root: fixes p :: "'c poly"   and a :: "'c" shows "(poly p a = (0::'c)) = (p = 0 \<or> order a p \<noteq> 0)"
characteristic_weak_formula_eqvt: fixes p :: "perm"   and P :: "'state" shows "p \<bullet> characteristic_weak_formula P = characteristic_weak_formula (p \<bullet> P)"
assms_2: shows "SIDE_GEN_ALGO (is_set_to_list A Rs tol)"
max2_equals2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "(r.max2 a b = b) = ((a, b) \<in> r)"
assms_2: shows "w2 \<equiv>\<^sub>U x"
aval_Some: fixes a :: "aexp"   and s :: "char list \<Rightarrow> int option" assumes "vars a \<subseteq> dom s" shows "\<exists>i. Def_Init_Exp.aval a s = Some i"
adjust_zero: fixes us :: "'a vec list"   and w :: "'a vec"   and i :: "nat" assumes "set us \<subseteq> carrier_vec n"   and "corthogonal us"   and "w \<in> carrier_vec n"   and "i < length us" shows "(adjuster n w us + w) \<bullet>c us ! i = (0::'a)"
assms_4: shows "ipresIGSubstAll h' hA' MOD' MOD''"
mod_ring_rel32_def: fixes x :: "uint32"   and y :: "'a mod_ring" shows "mod_ring_rel32 x y = (\<exists>z. urel32 x z \<and> mod_ring_rel z y)"
not_eq_order_implies_strict: fixes a :: "('c, 'd) Term.term"   and b :: "('c, 'd) Term.term" assumes "a \<noteq> b"   and "a \<unrhd> b" shows "a \<rhd> b"
Vr_exp_mem: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier (Vr K v)" shows "x^\<^bsup>K n\<^esup> \<in> carrier (Vr K v)"
nat_mult_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier R" shows "[n] \<cdot> x \<in> carrier R"
moebius_pt_moebius_rotation_inf: fixes v :: "real" shows "moebius_pt (moebius_rotation v) \<infinity>\<^sub>h = \<infinity>\<^sub>h"
dual_star_comp_fix: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (dual_star x) y = sup (times (times x (dual_star x)) y) y"