hcju/isabelleps huggingface dataset:
e'_def: fixes i :: "nat" shows "e' i = (if i < length xs then e i else 0)"
simps_2: fixes n :: "nat"   and k :: "nat" shows "\<up>\<^sub>\<tau> n k Top = Top"
rel_refl_strong: fixes x :: "'b H"   and Ra :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>z. z \<in> set_H x \<Longrightarrow> Ra z z" shows "rel_H Ra x x"
simps: fixes a :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>" shows "hereditarily_fs a = ((\<exists>tset\<^sub>\<alpha>. a = Conj\<^sub>\<alpha> tset\<^sub>\<alpha> \<and> finite (supp tset\<^sub>\<alpha>) \<and> (\<forall>x. x \<in> set_bset tset\<^sub>\<alpha> \<longrightarrow> hereditarily_fs x)) \<or> (\<exists>t\<^sub>\<alpha>. a = Not\<^sub>\<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>) \<or> (\<exists>f \<phi>. a = Pred\<^sub>\<alpha> f \<phi>) \<or> (\<exists>t\<^sub>\<alpha> f \<alpha>. a = Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> \<and> hereditarily_fs t\<^sub>\<alpha>))"
residual_galois: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f" shows "(f x \<le> y) = (x \<le> residual f y)"
sum_list_mono_gt: fixes xs :: "real list"   and m :: "real" assumes "xs \<noteq> [] \<and> Ball (set xs) ((<) m)" shows "m * real (length xs) < sum_list xs"
IH: shows "(find_indices x Vs = []) = (x \<notin> set Vs)"
primeness_condition_monoid_axioms: shows "primeness_condition_monoid (Ring_Divisibility.mult_of R)"
replicate_MkI_plus_1_2: fixes n :: "int"   and x :: "'b" assumes "0 \<le> n" shows "replicate\<cdot>(MkI\<cdot>(1 + n))\<cdot>x = x : replicate\<cdot>(MkI\<cdot>n)\<cdot>x"
isub_c: shows "induced_subgraph (G \<restriction> {v. u \<rightarrow>\<^sup>* v}) G"
prod_emb_PiE: fixes J :: "'d set"   and I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> E i \<subseteq> space (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) = (\<Pi>\<^sub>E i\<in>I. if i \<in> J then E i else space (M i))"
preserves_arr: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr f" shows "Span.arr (Span.src f)"
Sigma_Union: fixes X :: "'c set set"   and B :: "'c \<Rightarrow> 'd set" shows "Sigma (\<Union> X) B = (\<Union>A\<in>X. Sigma A B)"
step: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow> (c', s'')"
mkPX_cn: fixes x :: "nat"   and P :: "pol"   and Q :: "pol" assumes "x \<noteq> 0"   and "isnorm P"   and "isnorm Q" shows "isnorm (mkPX P x Q)"
marked_deletions_dequeue_Mark: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and r :: "'g"   and fl :: "bool"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h" assumes "sys_mem_store_buffers p s = mw_Mark r fl # ws"   and "mut_m.marked_deletions m s"   and "tso_store_inv s"   and "valid_W_inv s" shows "mut_m.marked_deletions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (obj_mark_update (\<lambda>_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))"
mult_subgroups: fixes H :: "'a set"   and K :: "'a set" assumes "subgroup H G"   and "subgroup K G" shows "subgroup (H <#> K) G"
length_rotate: fixes n :: "nat"   and xs :: "'a list" shows "length (rotate n xs) = length xs"
assms_4: shows "h \<turnstile> get_ancestors_si ancestor \<rightarrow>\<^sub>r ancestor_ancestors"
simps_6: fixes ux :: "'b"   and a :: "('a, 'b) Node"   and m :: "('a, 'b) FingerTreeStruc"   and sf :: "('a, 'b) Digit" shows "FingerTreeStruc.viewLn (Deep ux (One a) m sf) = (case FingerTreeStruc.viewLn m of None \<Rightarrow> Some (a, FingerTreeStruc.digitToTree sf) | Some (b, m2) \<Rightarrow> Some (a, FingerTreeStruc.deep (FingerTreeStruc.nodeToDigit b) m2 sf))"
integral_reflect_and_add: fixes a :: "real"   and f :: "real \<Rightarrow> 'a" assumes "integrable (lebesgue_on {- a..a}) f" shows "integral\<^sup>L (lebesgue_on {- a..a}) f = LINT x|lebesgue_on {0..a}. f x + f (- x)"
prime_def: fixes G :: "('c, 'd) monoid_scheme"   and p :: "'c" shows "prime G p = (p \<notin> Units G \<and> (\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b \<longrightarrow> p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b))"
unitary11_gen_cis_blaschke': fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and a :: "complex"   and b :: "complex" assumes "k \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (- 1, 0, 0, 1) *\<^sub>m\<^sub>m (a, b, cnj b, cnj a)"   and "a \<noteq> 0"   and "mat_det (a, b, cnj b, cnj a) \<noteq> 0" shows "\<exists>k' \<phi> a'. k' \<noteq> 0 \<and> a' * cnj a' \<noteq> 1 \<and> M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (1, - a', - cnj a', 1)"
Fun_ide: fixes a :: "'s" assumes "ide a" shows "Fun a = (\<lambda>x\<in>local.set a. x)"
strict_order_iff: fixes b :: "'a"   and a :: "'a" shows "(b < a) = (a = a \<squnion> b \<and> a \<noteq> b)"
res_units_eq: shows "Units R = {x. 0 < x \<and> x < m \<and> coprime x m}"
disc_1: fixes x1 :: "'b" shows "is_Var (Var x1)"
z_occurs_once_in_cs: shows "count_list (take (Suc i) cs) z = 1"
map_poly_mod: fixes p :: "rat poly"   and q :: "rat poly" shows "map_poly of_rat (p mod q) = map_poly of_rat p mod map_poly of_rat q"
comp: shows "(l\<^sub>1 \<cdot>\<^sub>l \<sigma>)\<^sup>c = l\<^sub>2 \<cdot>\<^sub>l \<sigma>"
\<S>_def: shows "\<S> \<equiv> {(l, R). l \<in> L \<and> R \<in> \<R> \<and> R \<subseteq> {u. u \<turnstile> PTA.inv_of A l}}"
All2_Subset: fixes H :: "fm set"   and l :: "name"   and k :: "tm"   and P :: "fm"   and k' :: "tm"   and P' :: "fm" assumes "H \<turnstile> All2 l k P"   and "H \<turnstile> k' SUBS k"   and "{P} \<turnstile> P'"   and "atom l \<sharp> (k', k)" shows "H \<turnstile> All2 l k' P'"
fold_max_triv: fixes xs :: "'b list"   and k :: "'b" assumes "\<forall>x\<in>set xs. x \<le> k" shows "fold max xs k = k"
errMOD_igWlsAbsIsInBar: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igWlsAbsIsInBar MOD" shows "igWlsAbsIsInBar (errMOD MOD)"
SHBL3: fixes t :: "'trm" assumes "t \<in> trm"   and "FvarsT t = {}" shows "prv (imp (PP t) (PP \<langle>PP t\<rangle>))"
unprot_mcalc: fixes \<sigma> :: "'tp"   and c :: "('fsym, 'psym) lit list" assumes "unprot \<sigma>"   and "c \<in> \<Phi>" shows "\<sigma> \<turnstile> c"
r_update_aux: fixes n :: "nat"   and b :: "nat"   and j :: "nat"   and v :: "nat" assumes "n \<le> e_length b" shows "eval r_update_aux [n, b, j, v] \<down>= Partial_Recursive.list_encode ((take n (Partial_Recursive.list_decode b))[j := v])"
q65_infinite_def: shows "q65 = (\<lambda>k. k\<^sup>2 mod 65) ` {0..}"
not_sdom_not_dom: fixes G :: "nat graph_rec"   and i :: "nat"   and j :: "nat" assumes "cfg_doms G"   and "\<not> cfg_doms.strict_dom G i j"   and "i \<noteq> j" shows "\<not> cfg_doms.dom G i j"
read_only_unowned_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> 'e) list" shows "read_only_unowned \<S> ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> \<O> \<inter> read_only \<S> = {}"
Measure_P_def: fixes vars :: "nat set"   and n :: "nat"   and Ps :: "nat \<Rightarrow> complex mat"   and Cs :: "com list" shows "Measure_P vars n Ps Cs = Measure n (\<lambda>n. mat_extension dims vars (Ps n)) Cs"
multx_def: shows "multx = restrict ((\<otimes>) x) (carrier G)"
rel_mono: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "R \<le> Ra" shows "rel_list_R1 R \<le> rel_list_R1 Ra"
inner_aforms'_inner_lv_rel: fixes a :: "(real \<times> real pdevs) list"   and a' :: "real list set"   and prec :: "nat"   and a'a :: "real list"   and R :: "(real \<times> real pdevs) list"   and x :: "real list" assumes "(a, a') \<in> aforms_rel"   and "inner_aforms' prec a (map pdevs_of_real a'a) = Some R"   and "x \<in> a'" shows "inner_lv_rel x a'a \<in> Affine (hd R)"
ctr_simps_UNIV_I: fixes x :: "'a" shows "x \<in> UNIV"
compare_expansions_MSLCons_eval: fixes C1 :: "'a"   and e1 :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and C2 :: "'a"   and e2 :: "real"   and ys :: "('a \<times> real) msllist"   and g :: "real \<Rightarrow> real" shows "compare_expansions (MS (MSLCons (C1, e1) xs) f) (MS (MSLCons (C2, e2) ys) g) = CMP_BRANCH (COMPARE e1 e2) (LT, 0, 0) (compare_expansions C1 C2) (GT, 0, 0)"
psi_fair_m2: shows "\<turnstile> psi \<longrightarrow> SF(m2)_(x, y)"
initial_starts_with_call: fixes p :: "'b"   and c :: "'a list multiset"   and e :: "('b, 'c) label"   and s' :: "'a list"   and c' :: "'a list multiset" assumes "(([entry fg p], c), e, s', c') \<in> trss fg"   and "initialproc fg p" shows "\<exists>p'. e = LCall p' \<and> isolated_ret fg p'"
ex_commute: fixes x :: "'d \<Longrightarrow> 'e"   and y :: "'f \<Longrightarrow> 'e"   and P :: "'e upred" assumes "x \<bowtie> y" shows "(\<exists> x \<bullet> \<exists> y \<bullet> P) = (\<exists> y \<bullet> \<exists> x \<bullet> P)"
assms_5: shows "x[[top]] \<le> forest_components h * selected_edge h j g * top"
simps_5: fixes v :: "nat" shows "linear (Bound v) = False"
CIComp: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and B :: "ZF"   and d :: "(ZF, ZF) Expression"   and C :: "ZF" assumes "ZFAxioms T"   and "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : A \<turnstile> e : B"   and "Sig aS\<^bsub>T\<^esub> \<triangleright> Vx : B \<turnstile> d : C" shows "[A,e,B]\<^bsub>T\<^esub> ;;\<^bsub>CanonicalCat T\<^esub> [B,d,C]\<^bsub>T\<^esub> = [A,sub e in d,C]\<^bsub>T\<^esub>"
halloc_elim_cases_1: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and oi :: "obj_tag"   and a :: "loc"   and s' :: "state" assumes "G\<turnstile>(Some xc, s) \<midarrow>halloc oi\<succ>a\<rightarrow> s'" obtains "a = undefined" and "s' = (Some xc, s)"
abs_eq: fixes xa :: "'c fset"   and x :: "'d fset" shows "xa |\<times>| x = Abs_fset (fset xa \<times> fset x)"
MapsToI: fixes f :: "'d"   and C :: "('e, 'd, 'f) Category_scheme"   and X :: "'e"   and Y :: "'e" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "dom\<^bsub>C\<^esub> f = X"   and "cod\<^bsub>C\<^esub> f = Y" shows "f maps\<^bsub>C\<^esub> X to Y"
of_nat_mult: fixes m :: "nat"   and n :: "nat" shows "of_nat (m * n) = of_nat m * of_nat n"
upper_bound_left: fixes x :: "'a"   and y :: "'a" shows "- x \<le> - x \<squnion> - y"
game2: shows "\<bar>spmf game2 True - 1 / 2\<bar> \<le> real q1 / 2 ^ len + real q2 / 2 ^ len + \<bar>spmf game3 True - 1 / 2\<bar>"
fm_partial: fixes h :: "'a" shows "((\<lambda>s. \<not> ((\<lambda>s. True) \<longrightarrow>* (\<lambda>s. \<not> \<box> s)) s) \<and>* (\<lambda>s. \<not> ((\<lambda>s. True) \<longrightarrow>* (\<lambda>s. \<not> \<box> s)) s)) h \<longrightarrow> \<not> ((\<lambda>s. True) \<longrightarrow>* (\<lambda>s. \<not> \<box> s)) h"
c_manifold_order_le: fixes l :: "enat" assumes "l \<le> k" shows "c_manifold charts2 l"
inf_closed_cl_op_var: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.clattice_with_clop less_eq less cl_op Inf Sup inf sup bot top"   and "x \<in> range cl_op"   and "y \<in> range cl_op" shows "inf x y \<in> range cl_op"
lebesgue_integrable_real_affine_iff: fixes c :: "real"   and f :: "real \<Rightarrow> 'c"   and t :: "real" assumes "c \<noteq> 0" shows "integrable lebesgue (\<lambda>x. f (t + c * x)) = integrable lebesgue f"
irreducible\<^sub>d_def: fixes p :: "'b poly" shows "irreducible\<^sub>d p = (0 < degree p \<and> (\<forall>q r. degree q < degree p \<longrightarrow> degree r < degree p \<longrightarrow> p \<noteq> q * r))"
induct: fixes P :: "(int \<Rightarrow> (bool \<times> term list) option) \<Rightarrow> int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int \<Rightarrow> (bool \<times> term list) option"   and a1 :: "int"   and a2 :: "int" assumes "\<And>f d i. (\<not> d < i \<Longrightarrow> P f d (i + 1)) \<Longrightarrow> P f d i" shows "P a0 a1 a2"
finite_measure_Union: fixes A :: "real set"   and k :: "nat"   and B :: "real set" assumes "A \<in> \<mu>.events k"   and "B \<in> \<mu>.events k"   and "A \<inter> B = {}" shows "\<mu>.prob k (A \<union> B) = \<mu>.prob k A + \<mu>.prob k B"
word_ao_dist2: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" shows "x && (y || z) = x && y || x && z"
coeffs_pderiv_code: fixes p :: "'b poly" shows "coeffs (pderiv p) = pderiv_coeffs (coeffs p)"
n_de_morgan_var2: fixes x :: "'a"   and y :: "'a" shows "!(!x + !y) = t x \<cdot> t y"
simpleDef_not_phi: fixes n :: "'node"   and g :: "'g"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "v \<in> defs g n" shows "phi g v = None"
call_edges_only: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and a :: "'g"   and Q :: "('i \<Rightarrow> 'j option) \<times> 'k \<Rightarrow> bool"   and r :: "'k"   and p :: "'l"   and fs :: "(('i \<Rightarrow> 'j option) \<Rightarrow> 'j option) list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "\<forall>a'. valid_edge a' \<and> targetnode a' = targetnode a \<longrightarrow> (\<exists>Qx rx fsx. kind a' = Qx:rx\<hookrightarrow>\<^bsub>p\<^esub>fsx)"
Cons_eq_appendI: fixes x :: "'a"   and xs1 :: "'a list"   and ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
refine: fixes R_Iec :: "('c \<times> unit) set" shows "(impl_blue_dfs_ahs, blue_dfs) \<in> bg_impl_rel_ext R_Iec Id \<rightarrow> \<langle>\<langle>\<langle>Id\<rangle>list_rel \<times>\<^sub>r \<langle>Id\<rangle>list_rel\<rangle>option_rel\<rangle>nres_rel"
prime_factorization_prime: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p" shows "factorial_semiring.prime_factorization zero times one normalize p = {#p#}"
nxtActI: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. active c (t i)" shows "n \<le> dynamic_component.nxtAct active c t n \<and> active c (t (dynamic_component.nxtAct active c t n)) \<and> \<not> (\<exists>k\<ge>n. k < dynamic_component.nxtAct active c t n \<and> active c (t k))"
direct_sum_is_module: fixes R :: "('e, 'f) ring_scheme"   and M1 :: "('e, 'g, 'h) module_scheme"   and M2 :: "('e, 'i, 'j) module_scheme" assumes "module R M1"   and "module R M2" shows "module R (direct_sum M1 M2)"
LIMSEQ_D: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c"   and r :: "real" assumes "X \<longlonglongrightarrow> L"   and "0 < r" shows "\<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
trace_def: fixes sig :: "'d signature" shows "trace sig \<equiv> filter (\<lambda>a. a \<in> externals sig) \<circ> schedule"
sublemma: fixes n :: "nat" shows "\<forall>N s. N \<le> n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
assms_1: shows "callee_invariant_on left Il \<I>l"
Skip_tE: fixes s :: "char list \<Rightarrow> int"   and x :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(SKIP, s) \<Rightarrow> x \<Down> t" obtains "x = Suc 0" and "t = s"
t1: shows "{norm (f x) / norm x |x. x \<noteq> (0::'a)} \<union> {norm (f x) / norm x |x. x = (0::'a)} = {norm (f x) / norm x |x. True}"
AYMIN: fixes x :: "('a, 'b) BsSkewElem" assumes "x \<in># image_mset fst (SkewBinomialHeapStruc.queue_to_multiset (map bsmapt qy))" shows "ay \<le> eprio x"
simps_26: fixes f1 :: "'c \<Rightarrow> 'a"   and f2 :: "'d \<Rightarrow> 'b"   and x2 :: "'d" shows "map_tpoly f1 f2 (PNum x2) = PNum (f2 x2)"
split_of_bool: fixes P :: "'a \<Rightarrow> bool"   and p :: "bool" shows "P (of_bool p) = ((p \<longrightarrow> P (1::'a)) \<and> (\<not> p \<longrightarrow> P (0::'a)))"
unit_in_hom_2: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<i>\<^sub>C[a] : a \<star>\<^sub>C a \<Rightarrow>\<^sub>C a\<guillemotright>"
int_nat: fixes z :: "int" assumes "0 < z" shows "int (nat z) = z"
Im_infsetsum: fixes f :: "'c \<Rightarrow> complex"   and A :: "'c set" assumes "f abs_summable_on A" shows "Im (infsetsum f A) = (\<Sum>\<^sub>ax\<in>A. Im (f x))"
r_pdec1_def: shows "r_pdec1 \<equiv> Cn 1 r_sub [recf.Id 1 0, Cn 1 r_triangle [r_maxlt]]"
formulaTrueAssociativity: fixes f1 :: "Literal list list"   and f2 :: "Literal list list"   and f3 :: "Literal list list"   and valuation :: "Literal list" shows "formulaTrue ((f1 @ f2) @ f3) valuation = formulaTrue (f1 @ f2 @ f3) valuation"
exec_stop_vs_control_independence: fixes upd :: "('c \<Rightarrow> 'c) \<Rightarrow> 'd control_state_scheme \<Rightarrow> 'd control_state_scheme"   and f :: "'c \<Rightarrow> 'c"   and \<sigma> :: "'d control_state_scheme" assumes "\<sharp> upd" shows "exec_stop (upd f \<sigma>) = exec_stop \<sigma>"
T_Abs: fixes T\<^sub>1 :: "POPLmark.type"   and \<Gamma> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmark.type" assumes "VarB T\<^sub>1 \<Colon> \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (\<lambda>:T\<^sub>1. t\<^sub>2) : T\<^sub>1 \<rightarrow> \<down>\<^sub>\<tau> 1 0 T\<^sub>2"
loc_imps_fw_conv_path: fixes c :: "('loc, 't) configuration"   and loc1 :: "'loc"   and loc2 :: "'loc"   and M :: "'t zmultiset"   and xs :: "('loc \<times> 'sum \<times> 'loc) list" assumes "loc_imps_fw c loc1 loc2 M xs" shows "path loc1 loc2 xs"
IH: fixes a :: "'b \<Rightarrow> real" assumes "sum a S = 1"   and "\<And>i. i \<in> S \<Longrightarrow> 0 \<le> a i"   and "\<And>i. i \<in> S \<Longrightarrow> y i \<in> C" shows "(\<Sum>j\<in>S. a j *\<^sub>R y j) \<in> C"
well_def_find_zeros_from_lhs_vec: fixes subsets :: "nat list list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list"   and j :: "nat" assumes "length subsets = length signs"   and "invertible_mat (M_mat signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs"   and "j \<in> set (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))" shows "j < length (cols (M_mat signs subsets))"
relcycle_group_restrict: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set" shows "relcycle_group p X (topspace X \<inter> S) = relcycle_group p X S"
assms: fixes y :: "'a" assumes "y \<in> A" shows "y dvd x"
cring_simprules_5: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
max_bot2: fixes x :: "'d upred" shows "ord.max (\<le>) x false = x"
wf_trms_subst_compose: fixes \<theta> :: "'a \<Rightarrow> ('fun, 'a) Term.term"   and \<delta> :: "'a \<Rightarrow> ('fun, 'a) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<theta>)"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range (\<theta> \<circ>\<^sub>s \<delta>))"
insert_mset_to_set: fixes xs' :: "'a list"   and xs :: "'a list"   and x :: "'a" assumes "mset xs' = mset xs + {#x#}" shows "set xs' = set xs \<union> {x}"
assms_3: shows "validate_ground_lit I (subst_lit l \<sigma>)"
plus_adds: fixes s :: "'a"   and t :: "'a"   and u :: "'a" shows "(s + t adds u) = (t adds u \<and> s adds u - t)"
cong_mid2__cong: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a"   and A :: "'a"   and B :: "'a"   and M' :: "'a"   and A' :: "'a"   and B' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M A B"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong M' A' B'"   and "Cong A M A' M'" shows "Cong A B A' B'"
P_new_2: shows "set P_new \<inter> G.second_vertices = {}"
Parallel_empty_lemma: fixes s :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and t :: "'a"   and n :: "nat" assumes "(Parallel [], s) -Pn\<rightarrow> (Parallel Ts, t)" shows "Ts = [] \<and> n = 0 \<and> s = t"
abs_valid_operator_3': fixes name :: "char list"   and pre :: "(nat \<times> nat) list"   and eff :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pre, eff, cost)" shows "list_all (\<lambda>(v, a). ListMem v abs_ast_variable_section) (map (\<lambda>(uu_, v, uu_, a). (v, a)) eff)"
mon_ww_of_foldl: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and ww :: "('c, 'f) label list list" shows "mon_w fg (foldl (@) [] ww) = mon_ww fg ww"
ptensor_mat_add: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and m1 :: "'a mat"   and dims :: "nat list"   and m2 :: "'a mat"   and m3 :: "'a mat" assumes "partial_state2 vars1 vars2"   and "m1 \<in> carrier_mat (partial_state2.d1 dims vars1) (partial_state2.d1 dims vars1)"   and "m2 \<in> carrier_mat (partial_state2.d1 dims vars1) (partial_state2.d1 dims vars1)"   and "m3 \<in> carrier_mat (partial_state2.d2 dims vars2) (partial_state2.d2 dims vars2)" shows "partial_state2.ptensor_mat dims vars1 vars2 (m1 + m2) m3 = partial_state2.ptensor_mat dims vars1 vars2 m1 m3 + partial_state2.ptensor_mat dims vars1 vars2 m2 m3"
one_power2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "power.power one times one 2 = one"
lunit_char_1: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>\<l>[f] : L f \<Rightarrow>\<^sub>S f\<guillemotright>"
are_operators_interfering_def: fixes op\<^sub>1 :: "('a, 'b) strips_operator_scheme"   and op\<^sub>2 :: "('a, 'c) strips_operator_scheme" shows "are_operators_interfering op\<^sub>1 op\<^sub>2 \<equiv> list_ex (\<lambda>v. list_ex ((=) v) (delete_effects_of op\<^sub>1)) (precondition_of op\<^sub>2) \<or> list_ex (\<lambda>v. list_ex ((=) v) (precondition_of op\<^sub>1)) (delete_effects_of op\<^sub>2)"
insert_4: fixes i :: "'i" assumes "i \<in> insert i__ I" shows "integrable (M i) (f i)"
euler_slp_def: shows "euler_slp = slp_of_fas (euler_fas' D)"
SReal_not_Infinitesimal: fixes y :: "real star" assumes "0 < y"   and "y \<in> \<real>" shows "y \<notin> Infinitesimal"
inf_sup_aci_7: fixes 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 x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup" shows "sup x (sup y z) = sup y (sup x z)"
totient_0_iff: fixes n :: "nat" shows "(totient n = 0) = (n = 0)"