hcju/isabelleps huggingface dataset:
rtrancl_imp_relpow: fixes p :: "'g \<times> 'g"   and R :: "('g \<times> 'g) set" assumes "p \<in> R\<^sup>*" shows "\<exists>n. p \<in> R ^^ n"
DERIV_cong: fixes f :: "'a \<Rightarrow> 'a"   and X :: "'a"   and F :: "'a filter"   and Y :: "'a" assumes "(f has_field_derivative X) F"   and "X = Y" shows "(f has_field_derivative Y) F"
fract_collapse_1: fixes k :: "'b" shows "Fraction_Field.Fract (0::'b) k = 0"
simp: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "r \<in> complex_rotations"
n_def: shows "n = 1 + nat \<lceil>normf f / e\<rceil>"
diamond_right_2: fixes a :: "'action"   and b :: "'action"   and p :: "'state" assumes "ind a b"   and "en a p"   and "en b (ex a p)" shows "en b p"
distinct_simple_match_to_of_match: fixes ifs :: "char list list"   and m :: "32 simple_match" assumes "distinct ifs" shows "distinct (simple_match_to_of_match m ifs)"
triangle_lemma: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and x :: "'a"   and y :: "'a" assumes "\<And>a b c. \<lbrakk>(a, b) \<in> r; (a, c) \<in> r\<rbrakk> \<Longrightarrow> b = c"   and "(a, x) \<in> r\<^sup>*"   and "(a, y) \<in> r\<^sup>*" shows "(x, y) \<in> r\<^sup>* \<or> (y, x) \<in> r\<^sup>*"
P_LCons': shows "P = LCons v0 (LCons w0 (ltl (ltl P)))"
reject: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
to_stream_nat_case: fixes x :: "'b"   and X :: "nat \<Rightarrow> 'b" shows "to_stream (case_nat x X) = x ## to_stream X"
p_residue_range': fixes m :: "nat"   and n :: "int" shows "p_residue m n \<in> carrier (residue_ring (p ^ m))"
power_inductr: fixes z :: "('a \<times> ('b \<times> 'a) list) set"   and y :: "('a \<times> ('b \<times> 'a) list) set"   and x :: "('a \<times> ('b \<times> 'a) list) set"   and n :: "nat" assumes "z \<union> t_prod y x \<subseteq> y" shows "t_prod z (trace_dioid.power x n) \<subseteq> y"
IH: fixes m :: "nat" shows "lcoeff (gbinomial_series_aux abort x (real m) (x gchoose m)) n = x gchoose (n + m)"
least_n_def: shows "least_n = (LEAST n. A $ i $ n \<noteq> (0::'a))"
a1_not_a2: shows "a1 \<noteq> a2"
m_lcomm: fixes x :: "rat vec"   and y :: "rat vec"   and z :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn"   and "z \<in> Rn" shows "x + (y + z) = y + (x + z)"
B: fixes M :: "ereal" assumes "M < \<infinity>" shows "\<forall>\<^sub>F n in sequentially. M < extended_Gromov_product_at basepoint (to_Gromov_completion (v n)) x"
assms_4: shows "Domainp B = (\<lambda>x. x \<in> \<UU>B)"
simps_1: shows "mht \<langle>\<rangle> = 0"
nat_leq_as_int: shows "(\<le>) = (\<lambda>a b. int a \<le> int b)"
hyps_5: fixes l :: "char list \<Rightarrow> nat" assumes "Q l s\<^sub>3_" shows "\<exists>n. wpw b c n Q l s\<^sub>2_"
liveVal'_suffix: fixes g :: "'g"   and vs :: "'val list"   and vs' :: "'val list" assumes "liveVal' g vs"   and "suffix vs' vs"   and "vs' \<noteq> []" shows "liveVal' g vs'"
pts_pmap_\<Gamma>: fixes f :: "'d set \<Rightarrow> 'c set"   and \<pi> :: "'d prefix" shows "pts (pmap_\<Gamma> f \<pi>) = pts \<pi>"
factors_equal: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a = b"   and "c = d" shows "a * c = b * d"
rb_aux_is_RB_upt: fixes z :: "nat"   and u :: "'t" shows "is_RB_upt dgrad rword (set (fst (fst (rb_aux (([], Koszul_syz_sigs fs, map Inr [0..<length fs]), z))))) u"
inner_ge_zero: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "0 \<le> inner x x"
assms_1: shows "common_boundary_sudivision_exists one_chain1 one_chain2"
simps_2: shows "shows_arctic MinInfty = shows ''-inf''"
distrib_lattice_ow_axioms_def: fixes U :: "'c set"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "distrib_lattice_ow_axioms U inf sup \<equiv> \<forall>x y z. x \<in> U \<longrightarrow> y \<in> U \<longrightarrow> z \<in> U \<longrightarrow> sup x (inf y z) = inf (sup x y) (sup x z)"
Class': fixes A :: "('a, 'b) form"   and G :: "('a, 'b) form list" assumes "Neg A # G \<turnstile> A" shows "G \<turnstile> A"
circ_sub_decompose: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<circ> * y)\<^sup>\<circ> \<le> (x\<^sup>\<star> * y)\<^sup>\<circ> * x\<^sup>\<circ>"
not_alw_iff: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "(\<not> alw P \<omega>) = ev (\<lambda>xs. \<not> P xs) \<omega>"
P2: shows "Bet A' B' A0' \<and> Cong B' A0' A' B'"
wagner_1_var_gen: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> \<cdot> y)\<^sup>\<omega> \<le> (x + y)\<^sup>\<omega>"
no_grey_refs_not_grey_reachableD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and x :: "'h" assumes "no_grey_refs s" shows "\<not> grey_reachable x s"
right_idem: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<inter># b \<inter># b = a \<inter># b"
chain: fixes R :: "'a ring"   and S :: "'a ring" assumes "R \<in> C"   and "S \<in> C" shows "R \<lesssim> S \<or> S \<lesssim> R"
ceiling_le_neg_numeral: fixes x :: "'b"   and v :: "num" shows "(\<lceil>x\<rceil> \<le> - numeral v) = (x \<le> - numeral v)"
hom_u: shows "ring_homomorphism u carrier_direct_lim add_rel mult_rel \<lfloor>V, \<zero>\<^bsub>V\<^esub>\<rfloor> \<lfloor>V, \<one>\<^bsub>V\<^esub>\<rfloor> A add mult zero one"
dagger_ext: fixes x :: "'a" shows "x \<le> x\<^sup>\<star>"
image_poly_index_free: fixes L :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) ring"   and i :: "'c"   and q :: "('c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a) list"   and P :: "'c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a"   and j :: "'c" assumes "field L"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.carrier_coeff P"   and "\<And>P. P \<in> carrier L \<Longrightarrow> M.index_free P i"   and "\<zero>\<^bsub>L\<^esub> = M.indexed_const \<zero>\<^bsub>M\<^esub>"   and "q \<in> carrier (poly_ring L)"   and "P \<in> carrier (image_ring (M.eval_pmod L i q) (poly_ring L))"   and "\<not> M.index_free P j"   and "i \<noteq> j" obtains Q :: "'c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a" where "Q \<in> carrier L" and "\<not> M.index_free Q j"
applicableDecideCharacterization: fixes stateA :: "State"   and decisionVars :: "nat set" shows "applicableDecide stateA decisionVars = (\<exists>l. var l \<in> decisionVars \<and> \<not> l el elements (getM stateA) \<and> \<not> opposite l el elements (getM stateA))"
sympI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
elims: fixes x :: "lit"   and xa :: "lit uprod"   and y :: "lit set \<Rightarrow> lit set" assumes "succs_of_clause x xa = y" obtains l :: "lit" and l1 :: "lit" and l2 :: "lit" where "x = l" and "xa = Upair l1 l2" and "y = (if l = negate l1 then insert l2 else if l = negate l2 then insert l1 else id)"
rel_funE: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and x :: "'a"   and y :: "'b" assumes "rel_fun A B f g"   and "A x y" obtains "B (f x) (g y)"
conf_def: fixes P :: "'b prog" shows "conf P \<equiv> h.conf P undefined"
assms_4: fixes vs :: "nat \<Rightarrow> nat"   and n :: "nat"   and vs' :: "nat \<Rightarrow> nat"   and vs'' :: "nat \<Rightarrow> nat" assumes "vs v = Suc n"   and "P (\<lambda>x. if x = v then n else vs x) vs'"   and "Q vs' vs''" shows "Q vs vs''"
Nat_atleast_pred_NullD: fixes s :: "state"   and a :: "loc"   and n :: "nat" assumes "Null = get_field s a pred"   and "s:Addr a \<ge> n" shows "n = 0"
Cons_eq_appendI: fixes x :: "'c"   and xs1 :: "'c list"   and ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" assumes "x # xs1 = ys"   and "xs = xs1 \<cdot> zs" shows "x # xs = ys \<cdot> zs"
subgroup_generated_by_image: fixes S :: "'a set" assumes "S \<subseteq> carrier B" shows "carrier (subgroup_generated C (g ` S)) = g ` carrier (subgroup_generated B S)"
set_mset_minus_replicate_mset_1: fixes A :: "'b multiset"   and a :: "'b"   and n :: "nat" assumes "count A a \<le> n" shows "set_mset (A - replicate_mset n a) = set_mset A - {a}"
state: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], h) a = ([x, y], rev a @ h)"
intros_2: fixes i :: "nat" shows "Numb i \<in> sexp"
tensor_lookup_eqI: fixes A :: "'a tensor"   and B :: "'a tensor" assumes "dims A = dims B"   and "\<And>is. is \<lhd> dims A \<Longrightarrow> lookup A is = lookup B is" shows "A = B"
one_hmultiset_def: shows "1 = \<omega>^ 0"
\<phi>'_def: shows "\<phi>' = the_inv_into V \<phi>"
gaugeR: shows "gauge (\<lambda>x. ball x (R x))"
vinsert_ident: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<notin>\<^sub>\<circ> A"   and "x \<notin>\<^sub>\<circ> B" shows "(vinsert x A = vinsert x B) = (A = B)"
H: shows "\<exists>\<alpha>. Ord \<alpha> \<and> P \<alpha>"
remove_max_mset: fixes l :: "'a list"   and m :: "'a"   and l' :: "'a list" assumes "l \<noteq> []"   and "(m, l') = remove_max l" shows "add_mset m (mset l') = mset l"
apfst_compose: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'c"   and x :: "'d \<times> 'b" shows "apfst f (apfst g x) = apfst (f \<circ> g) x"
ref_def: fixes v :: "'b" shows "ref v = Heap_Monad.heap (Ref.alloc v)"
fvi_less_nfv: fixes \<phi> :: "formula" shows "\<forall>i\<in>fv \<phi>. i < nfv \<phi>"
left_right_simps_3: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and t :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "pre_term.is_app unapp t" shows "app (pre_term.left unapp t) (pre_term.right unapp t) = t"
axioms_1: fixes V\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>C :: "'c \<Rightarrow> 'c"   and src\<^sub>C :: "'c \<Rightarrow> 'c"   and trg\<^sub>C :: "'c \<Rightarrow> 'c"   and V\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>D :: "'d \<Rightarrow> 'd"   and src\<^sub>D :: "'d \<Rightarrow> 'd"   and trg\<^sub>D :: "'d \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and \<Phi> :: "'c \<times> 'c \<Rightarrow> 'd" assumes "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>" shows "bicategory V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C"
assms_1: shows "class_type_of' T = \<lfloor>C\<rfloor>"
f_i_simps_1: shows "f_i 0 = (\<lambda>_. 0)"
map_disj_None_right': fixes h\<^sub>1 :: "'d \<Rightarrow> 'c option"   and x :: "'d"   and y :: "'c"   and h\<^sub>0 :: "'d \<Rightarrow> 'c option" assumes "h\<^sub>1 x = Some y"   and "h\<^sub>1 \<bottom> h\<^sub>0" shows "h\<^sub>0 x = None"
k_2: fixes i :: "nat"   and G1 :: "'b poly"   and G2 :: "'b poly" assumes "i < ??.Subresultant.k G1 G2" shows "??.Subresultant.F G1 G2 (Suc i) \<noteq> 0"
assms: shows "{x} \<union> set xs = {y} \<union> set ys"
eventually_at_filter: fixes "open" :: "'d set \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and a :: "'d"   and s :: "'d set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<forall>\<^sub>F x in topological_space.nhds open a. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x)"
ordinal_utility_expected_value: shows "ordinal_utility local.\<P> \<R> (\<lambda>x. measure_pmf.expectation x local.ocU)"
Abs_inverse: fixes m :: "int" assumes "m \<in> {0..<int CARD('b bit0)}" shows "Rep_bit0 (Abs_bit0 m) = m"
Cons\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and v :: "val"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and es :: "nat exp list"   and es' :: "nat exp list"   and s\<^sub>2 :: "heap \<times> val list \<times> sheap" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e # es,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>Val v # es',s\<^sub>2\<rangle>"
simps_4: fixes x2 :: "'b equation"   and x1 :: "'b equation" shows "Neg x2 \<noteq> Pos x1"
sel: fixes x :: "'a \<Rightarrow> ('b \<times> ('a, 'b, 'c, 'd) converter, 'c, 'd) gpv" shows "run_converter (Converter x) = x"
tso_store_refs_def: fixes m :: "'e" shows "mut_m.tso_store_refs m = (\<lambda>s. \<Union> (store_refs ` set (sys_mem_store_buffers (mutator m) s)))"
hull_redundant: fixes a :: "'b"   and S :: "'b set \<Rightarrow> bool"   and s :: "'b set" assumes "a \<in> S hull s" shows "S hull insert a s = S hull s"
CFG_SSA_defs_1: fixes g :: "'g"   and n :: "'node" shows "phiDefs g n \<equiv> {v. (n, v) \<in> dom (phis g)}"
vconn_implies_hpar_3: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "S.in_hom \<mu> f g" shows "S.src g = S.src f"
cnf_of_encode_problem_with_operator_interference_exclusion_structure_5: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "cnf (encode_interfering_operator_exclusion \<Pi> t) \<subseteq> cnf (\<Phi>\<^sub>\<forall> \<Pi> t)"
isState_hd_purgeIdle: fixes sl :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "isState (hd sl)" shows "purgeIdle sl \<noteq> [] \<and> hd (purgeIdle sl) = hd sl"
lookup_HSem_other: fixes y :: "'a"   and h :: "('a \<times> 'b) list"   and ESem :: "'b \<Rightarrow> ('a \<Rightarrow> 'c) \<rightarrow> 'c"   and \<rho> :: "'a \<Rightarrow> 'c" assumes "y \<notin> domA h" shows "(has_ESem.HSem ESem h\<cdot>\<rho>) y = \<rho> y"
openin_subtopology_Int_subset: fixes u :: "'a set"   and S :: "'a set"   and v :: "'a set" assumes "openin (top_of_set u) (u \<inter> S)"   and "v \<subseteq> u" shows "openin (top_of_set v) (v \<inter> S)"
foldl1_map_xs: shows "foldl1 f2 (map (h2 i) xs) = One"
bot_fset_def: shows "{||} \<equiv> abs_fset []"
im_sheaf_is_presheaf: shows "presheaf_of_rings S is_open id.im_sheaf id.im_sheaf_morphisms b id.add_im_sheaf id.mult_im_sheaf id.zero_im_sheaf id.one_im_sheaf"
the_inv_G_perm: fixes g :: "'a" assumes "g \<in> G" shows "the_inv (G_perm g) = G_perm (\<ii> g)"
paradox: assumes "\<And>G \<phi>. IsProperInX (\<lambda>x. \<lparr>G,\<^bold>\<iota>y. \<phi> y x\<rparr>)" shows "False"
independent_inj_image: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and S :: "'e set" assumes "module_hom s1 s2 f"   and "\<not> module.dependent s1 S"   and "inj f" shows "\<not> module.dependent s2 (f ` S)"
bottom_left_bottom: fixes a :: "'b" shows "lcm (0::'b) a = (0::'b)"
b_fv: shows "{0..<b} \<subseteq> fv \<phi>"
vimage2p_def: fixes f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> 'f" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
assms: fixes A :: "'a set"   and B :: "'a set" assumes "\<forall>i\<in>S. (i \<in> A) = (i \<in> B)" shows "(A \<in> Rep_formula x) = (B \<in> Rep_formula x)"
unity_def: shows "D.unity \<equiv> D.cod \<iota>\<^sub>D"
var_subterm_trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p_is_vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes x :: "'d"   and a :: "('c, 'd) stateful_strand_step" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p a" shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
validtI: fixes \<Gamma> :: "'g \<Rightarrow> ('e, 'g, 'f) com option"   and c :: "('e, 'g, 'f) com"   and P :: "'e set"   and F :: "'f set"   and Q :: "'e set"   and A :: "'e set" assumes "\<And>s t. \<lbrakk>\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> t; s \<in> P; t \<notin> Fault ` F\<rbrakk> \<Longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A"   and "\<And>s. s \<in> P \<Longrightarrow> \<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
before_dist_fst_fst: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "before vs ram1 ram2"   and "distinct vs" shows "fst (splitAt ram2 (fst (splitAt ram1 vs))) = fst (splitAt ram1 (fst (splitAt ram2 vs)))"
F1set2_IF1set: fixes a :: "'d IF1"   and x :: "('d, 'd IF1, 'd IF2) F1" assumes "a \<in> F1set2 x" shows "IF1set a \<subseteq> IF1set (ctor1 x)"
polya_vinogradov_inequality_primitive: fixes x :: "nat" shows "cmod (sum \<Phi> {1..x}) < sqrt (real c) * ln (real c)"
has_vector_derivative_weaken: fixes f :: "real \<Rightarrow> 'a"   and D :: "'a"   and x :: "real"   and T :: "real set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "(f has_vector_derivative D) (at x within T)"   and "x \<in> S"   and "S \<subseteq> T"   and "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(g has_vector_derivative D) (at x within S)"
isomorphicE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and a' :: "'b" assumes "category C"   and "category.isomorphic C a a'" obtains f :: "'b" where "partial_magma.in_hom C f a a' \<and> category.iso C f"
le_lt_convs_2: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "comparator comp" shows "(comp x y \<noteq> Gt) = le_of_comp comp x y"
IH_2: assumes "opaque y = []" shows "y \<simeq> Pure (unlift_vars 0 y)"
vars_uminus: fixes p :: "'a mpoly" shows "vars (- p) = vars p"
eq_onp_same_args: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a" shows "eq_onp P x x = P x"
mrbst_join_Leaf_left: shows "mrbst_join \<langle>\<rangle> = return_pmf"
length_take: fixes n :: "nat"   and xs :: "'b list" shows "length (take n xs) = min (length xs) n"
isomorphic_implies_hpar_2: fixes f :: "'b"   and f' :: "'b" assumes "f \<cong>\<^sub>D f'" shows "D.ide f'"
trancl_steps_relpow: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and n :: "nat" assumes "a \<subseteq> b\<^sup>+"   and "(x, y) \<in> a ^^ n" shows "\<exists>m\<ge>n. (x, y) \<in> b ^^ m"
cong: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> real" shows "transition_rates.J R \<equiv> transition_rates.J R"
H': shows "mat k n (\<lambda>(i, y). F2 ! i $ y) = P * mat k n (\<lambda>(i, y). F1 ! i $ y)"
simps_1: fixes el :: "'b word"   and s :: "'b word"   and e :: "'b word" shows "wordinterval_element el (WordInterval s e) = (s \<le> el \<and> el \<le> e)"
Setcompr_eq_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" shows "{uu_. \<exists>x. uu_ = f x \<and> x \<in> A} = f ` A"
HF2_def: shows "HF2 \<equiv> Var X0 EQ Eats (Var X1) (Var X2) IFF SyntaxN.All X3 (Var X3 IN Var X0 IFF Var X3 IN Var X1 OR Var X3 EQ Var X2)"
Lu: shows "(\<lambda>n. extended_Gromov_product_at basepoint (to_Gromov_completion (u n)) x) \<longlonglongrightarrow> \<infinity>"
integrable_const: fixes M :: "'b pmf"   and a :: "'c" shows "integrable (measure_pmf M) (\<lambda>x. a)"
subset_insertI: fixes B :: "'e set"   and a :: "'e" shows "B \<subseteq> insert a B"
split_sign_changes_induct: fixes P :: "real poly list \<Rightarrow> real \<Rightarrow> bool"   and x :: "real" assumes "\<And>p x. P [p] x"   and "\<And>p q x. quasi_sturm_seq [p, q] \<Longrightarrow> P [p, q] x"   and "\<And>p q r ps x. \<lbrakk>quasi_sturm_seq (p # q # r # ps); \<lbrakk>poly p x \<noteq> 0; poly q x = 0\<rbrakk> \<Longrightarrow> P (r # ps) x; poly q x \<noteq> 0 \<Longrightarrow> P (q # r # ps) x; poly p x = 0 \<Longrightarrow> P (q # r # ps) x\<rbrakk> \<Longrightarrow> P (p # q # r # ps) x" shows "P ps x"
Cod_in_Obj: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Cod f \<in> Collect B.obj"
edges_def: shows "edges \<equiv> {(u, v). G_\<alpha> g (u, v) \<noteq> \<infinity>}"
divide_strict_left_mono_neg: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < b"   and "c < (0::'a)"   and "(0::'a) < a * b" shows "c / a < c / b"
containsDuplicateEdge_eq: shows "containsDuplicateEdge = containsDuplicateEdge'"
fin: shows "\<not> is_final (steps0 (1, tp) A stp')"
between_translation: fixes a :: "'a"   and y :: "'a"   and z :: "'a"   and x :: "'a" shows "between (a + y, a + z) (a + x) = between (y, z) x"
assms_2: shows "Poly_Mapping.keys a \<subseteq> S"
transpose_mat: fixes n :: "'b" shows "Finite_Cartesian_Product.transpose (mat n) = mat n"
f_def: shows "f \<equiv> \<lambda>x. (1 + 1 / ln (b * x) * ln (1 + hb / b * ln x powr (- 1 - e))) powr (- e / 2)"
siso0_coind: fixes phi :: "('test, 'atom) com \<Rightarrow> bool"   and c :: "('test, 'atom) com" assumes "phi c"   and "\<And>c s c' s'. \<lbrakk>phi c; mustT c s; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> phi c' \<or> siso0 c'"   and "\<And>c s t c' s'. \<lbrakk>phi c; mustT c s; mustT c t; s \<approx> t; (c, s) \<rightarrow>c (c', s')\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>c (c', t')"   and "\<And>c s t s'. \<lbrakk>mustT c s; mustT c t; phi c; s \<approx> t; (c, s) \<rightarrow>t s'\<rbrakk> \<Longrightarrow> \<exists>t'. s' \<approx> t' \<and> (c, t) \<rightarrow>t t'" shows "siso0 c"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "domain_semigroup.ds_ord d (\<cdot>) x y"   and "domain_semigroup.ds_ord d (\<cdot>) y z" shows "domain_semigroup.ds_ord d (\<cdot>) x z"
interleaves_nil_all: fixes P :: "'a \<Rightarrow> 'a list \<Rightarrow> bool"   and xs :: "'a list" shows "xs \<simeq> {[], xs, P}"
seq_char: fixes g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" shows "Maps.seq g f = (Maps.arr f \<and> Maps.arr g \<and> Maps.Dom g = Maps.Cod f)"
fork_transitions_def: fixes N :: "nat"   and i :: "nat"   and s :: "nat" assumes "DiningPhilosophers N" shows "DiningPhilosophers.fork_transitions N i s = (if s = 0 then {picks i i} \<union> {picks ((i + 1) mod N) i} else if s = 1 then {putsdown i i} else if s = 2 then {putsdown ((i + 1) mod N) i} else {})"
between_empty_imp_inext_eq: fixes n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "n \<in> A"   and "n < inext n A"   and "n \<in> B"   and "inext n A \<in> B"   and "B \<down>> n \<down>< inext n A = {}" shows "inext n B = inext n A"
no_change_vs_eff_submap: fixes vs :: "'a set"   and s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" assumes "fmrestrict_set vs s = fmrestrict_set vs (state_succ s a)"   and "fst a \<subseteq>\<^sub>f s" shows "fmrestrict_set vs (snd a) \<subseteq>\<^sub>f fmrestrict_set vs s"
max_step_r: fixes t1 :: "real"   and x :: "real \<Rightarrow> 'b"   and incr :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "nat"   and B :: "real"   and r :: "real"   and L :: "real"   and t :: "nat \<Rightarrow> real"   and s :: "real \<Rightarrow> real \<Rightarrow> 'b \<Rightarrow> 'b"   and s0 :: "'b"   and j :: "nat" assumes "stable_one_step t1 x incr p B r L t s s0"   and "t j \<le> t1" shows "grid.max_stepsize t j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (t1 - t 0) + 1) - 1))"