hcju/isabelleps huggingface dataset:
deterministic: fixes lc :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and lc' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)"   and lc'' :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)" assumes "lc \<leadsto>\<^sub>C lc'"   and "lc \<leadsto>\<^sub>C lc''" shows "lc' = lc''"
distinct_1: fixes x21 :: "tree"   and x22 :: "tree" shows "Leaf \<noteq> Branching x21 x22"
apply_outputs_preserves_length: fixes p :: "'a aexp list"   and s :: "'a \<Rightarrow> value option" shows "length (apply_outputs p s) = length p"
const_in_smallo_const: fixes b :: "'c"   and c :: "'c" shows "((\<lambda>_. b) \<in> o(\<lambda>_. c)) = (b = (0::'c))"
csubspace_space_as_set: fixes S :: "'a ccsubspace" shows "csubspace (space_as_set S)"
get_state_def: fixes f :: "'d \<Rightarrow> ('d, 'e) stateT" shows "get_state f = StateT (\<lambda>s. run_state (f s) s)"
ccBinds_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> cccexp1 e = cccexp2 e"   and "heap1 = heap2" shows "CoCallAnalysis.ccBinds cccexp1 heap1 = CoCallAnalysis.ccBinds cccexp2 heap2"
partition_in_shuffles: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" shows "xs \<in> shuffles (filter P xs) (filter (\<lambda>x. \<not> P x) xs)"
monoF: shows "mono (F::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice)"
add_le_cancel_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus c a) (plus c b) = less_eq a b"
surj_def: fixes f :: "'l \<Rightarrow> 'k" shows "surj f = (\<forall>y. \<exists>x. y = f x)"
andalso_or: fixes t :: "tr"   and s :: "tr" assumes "t \<noteq> \<bottom>" shows "((t andalso s) = FF) = (t = FF \<or> s = FF)"
append_rows_le: fixes A :: "'b mat"   and nr1 :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nr2 :: "nat"   and a :: "'b vec"   and v :: "'b vec"   and b :: "'b vec" assumes "A \<in> carrier_mat nr1 nc"   and "B \<in> carrier_mat nr2 nc"   and "a \<in> carrier_vec nr1"   and "v \<in> carrier_vec nc" shows "((A @\<^sub>r B) *\<^sub>v v \<le> a @\<^sub>v b) = (A *\<^sub>v v \<le> a \<and> B *\<^sub>v v \<le> b)"
basis_expansion_cart_basis'_unique': fixes f :: "'b \<Rightarrow> 'a"   and x :: "('a, 'b) vec" shows "((\<Sum>i\<in>UNIV. f i *s cart_basis' $ i) = x) = (\<forall>i. f i = x $ i)"
type_definition_dnelist: shows "type_definition list_of_dnelist Abs_dnelist {xs. distinct xs \<and> xs \<noteq> []}"
icPar1F: fixes P' :: "pi" assumes "P \<longmapsto> \<alpha> \<prec> P'" shows "F \<alpha> (P' \<parallel> Q)"
feq: fixes i :: "'a" assumes "i \<in> K" shows "f (x i) = y i"
E1: shows "vpeq_obj u (set_object_value x y s1) (set_object_value x y s2)"
assms_1: shows "3 \<le> rk {A, A', C}"
ltake_lappend2: fixes xs :: "'a llist"   and n :: "enat"   and ys :: "'a llist" assumes "llength xs \<le> n" shows "ltake n (lappend xs ys) = lappend xs (ltake (n - llength xs) ys)"
permute_list_def: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'a list" shows "permute_list f xs = map (\<lambda>i. xs ! f i) [0..<length xs]"
a: shows "\<Gamma> \<turnstile> App t1 t2 : T"
type_definition_set_dlist: shows "type_definition list_of_dlist Abs_dlist' {xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None}"
prodinf_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n = g n" shows "prodinf f = prodinf g"
components_are_iso: fixes a :: "'b \<times> 'b" assumes "B.VV.ide a" shows "C.VV.iso (F.FF a)"
TS_index_2: fixes vs :: "'d list"   and xs :: "'d list"   and M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and \<Omega> :: "('d, 'e) ATC set"   and V :: "'d list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs" shows "vs @ xs \<in> C M2 M1 \<Omega> V m (Suc (length xs))"
interaction_bound_Done: fixes "consider" :: "'b \<Rightarrow> bool"   and x :: "'d" shows "interaction_bound consider (Generative_Probabilistic_Value.Done x) = 0"
intro: fixes \<SS> :: "V" assumes "\<And>a b. \<lbrakk>a \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>; b \<in>\<^sub>\<circ> \<SS>\<lparr>\<A>\<rparr>\<rbrakk> \<Longrightarrow> a \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> b = b \<odot>\<^sub>\<circ>\<^bsub>\<SS>\<^esub> a" shows "\<Z>_csgrp_axioms \<SS>"
equalsffemptyD: fixes A :: "'b fset"   and a :: "'b" assumes "A = {||}" shows "a |\<notin>| A"
X_poly_minus_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" shows "X_poly_minus R a = X_poly R \<ominus>\<^bsub>UP R\<^esub> to_polynomial R a"
diagonal_to_Smith_row_i_dvd_jj: fixes a :: "'d"   and i :: "nat"   and A :: "(('a, 'e) vec, 'd) vec"   and b :: "'e" assumes "to_nat a \<in> {i..<min (nrows A) (ncols A)}"   and "to_nat a = to_nat b" shows "Diagonal_to_Smith_row_i A i bezout $ from_nat i $ from_nat i dvd Diagonal_to_Smith_row_i A i bezout $ a $ b"
ih_step: fixes y :: "'b" assumes "y \<in> face_cycle_set x"   and "P y" shows "P (face_cycle_succ y)"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (mul_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (mul_instr_sub1 instr_name result s2))"
MakeCatDom: fixes f :: "'a"   and C :: "('b, 'a, 'c) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> f = dom\<^bsub>MakeCat C\<^esub> f"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "L f = L f'" shows "f = f'"
that: fixes n :: "nat" assumes "m \<le> n"   and "f \<in> dgrad_p_set d n"   and "F \<subseteq> dgrad_p_set d n" shows "thesis"
functor_FF: shows "functor B.VV.comp B.VV.comp I.FF"
build_ip_partition_same_fw_min: fixes A :: "'a wordinterval"   and c :: "parts_connection"   and rs :: "'a simple_rule list"   and B :: "'a wordinterval" assumes "A \<in> set (build_ip_partition c rs)"   and "B \<in> set (build_ip_partition c rs)"   and "A \<noteq> B" shows "\<forall>ip1\<in>wordinterval_to_set A. \<forall>ip2\<in>wordinterval_to_set B. \<not> same_fw_behaviour_one ip1 ip2 c rs"
SemComp: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and c :: "OBJ"   and n :: "nat"   and r :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and d :: "OBJ"   and m :: "nat"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and k :: "nat" assumes " s , c \<rightarrow>\<^sub>n  r "   and " r , d \<rightarrow>\<^sub>m  t "   and "k = max n m + 1" shows " s , Comp c d \<rightarrow>\<^sub>k  t "
P'B'Q': shows "(P', B', Q') = diagonal_to_Smith_i_PQ (j # xs) i bezout (P, B, Q)"
spec_imp_exchange_spec: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "cri.spec (smap exchange_config s)"
comp_char: fixes g :: "'a"   and f :: "'a" shows "V g f = (if f \<in> Collect arr \<and> f = g then f else V.null)"
Step0: fixes g :: "'a" assumes "g \<in> carrier G" shows "(\<lambda>h\<in>carrier G. g \<otimes> h \<otimes> inv g) \<in> Bij (carrier G)"
square_integrable_imp_absolutely_integrable_product: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "(\<lambda>x. f x * g x) absolutely_integrable_on S"
run_detE: fixes c :: "'b Heap"   and \<sigma> :: "heap option"   and \<sigma>' :: "heap option"   and r :: "'b"   and \<tau> :: "heap option"   and s :: "'b" assumes "run c \<sigma> \<sigma>' r"   and "run c \<sigma> \<tau> s"   and "\<not> is_exn \<sigma>" obtains "is_exn \<sigma>'" and "\<sigma>' = \<tau>"   | "\<not> is_exn \<sigma>'" and "\<sigma>' = \<tau>" and "r = s"
rtranclp_sup_rtranclp: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(sup R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*)\<^sup>*\<^sup>* = (sup R S)\<^sup>*\<^sup>*"
hausdorff_distance_def: fixes A :: "'c set"   and B :: "'c set" shows "hausdorff_distance A B = (if A = {} \<or> B = {} \<or> \<not> bounded A \<or> \<not> bounded B then 0 else max (SUP x\<in>A. infdist x B) (SUP x\<in>B. infdist x A))"
uint_shiftl_def: fixes x :: "uint"   and n :: "integer" shows "uint_shiftl x n = (if n < 0 \<or> dflt_size_integer \<le> n then undefined push_bit x n else push_bit (nat_of_integer n) x)"
connected_rel_ind_rule: fixes n :: "nat"   and es :: "(nat \<times> nat) list"   and k :: "nat" assumes "is_valid_graph n (set es)"   and "k \<le> length es" shows "connected_rel_ind n es k = connected_rel n (set (take k es))"
axioms_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "conservative M T" shows "qmpt M T"
ex_extreme_qfp_fp: fixes CC :: "'a set set" assumes "CC-complete A (\<sqsubseteq>)"   and "C \<in> CC"   and "\<forall>q\<in>A. \<forall>x\<in>A. f q \<sim> q \<longrightarrow> x \<sqsubseteq> f q \<longrightarrow> x \<sqsubseteq> q"   and "monotone_on A (\<sqsubseteq>) (\<sqsubseteq>) f" shows "\<exists>c. extreme {q \<in> A. f q \<sim> q \<or> f q = q} (\<lambda>x y. y \<sqsubseteq> x) c"
inv_RBT_Impl_diag_2: fixes t :: "('c, 'd) rbt" assumes "inv1 t"   and "inv2 t" shows "inv2 (RBT_Impl_diag t)"
bh_fact: shows "q.factorization_m f (lead_coeff f, mset gs)"
ics_SDG_path_split: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n'' \<in> set ns" obtains ns' :: "'b SDG_node list" and ns'' :: "'b SDG_node list" where "ns = ns' @ ns''" and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns' n''" and "SDG.intra_call_sum_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns'' n'"
snapshot_pair: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and s :: "('a, 'b) fmap" shows "snapshot PROB s = {(p, e). (p, e) \<in> PROB \<and> agree p s \<and> agree e s}"
nb: shows "\<forall>x\<in>set xs. qfree (f x)"
simps_1: fixes x :: "'b" shows "ins_list x [] = [x]"
snd_fst_flip: fixes xy :: "'b \<times> 'a" shows "snd xy = (fst \<circ>\<circ> case_prod) (\<lambda>x y. (y, x)) xy"
elims: fixes x :: "nat \<Rightarrow> nat"   and xa :: "nat"   and y :: "nat" assumes "f_Max\<^sub>0 x xa = y" obtains f :: "nat \<Rightarrow> nat" where "x = f__" and "xa = 0" and "y = 0"   | f :: "nat \<Rightarrow> nat" and xb :: "nat" where "x = f" and "xa = Suc xb" and "y = max (f (Suc xb)) (f_Max\<^sub>0 f xb)"
int_cases2: fixes z :: "int" obtains n :: "nat" where "z = int na__"   | n :: "nat" where "z = - int n"
record_call: fixes x :: "'a"   and f :: "'a \<Rightarrow> one\<^sub>\<bottom>" shows "(record_call x\<cdot>f) x = two_pred\<cdot>(f x)"
imp_conv_disj: fixes P :: "bool"   and Q :: "bool" shows "(P \<longrightarrow> Q) = (\<not> P \<or> Q)"
not_neutral_obtains_not_neutral: fixes g :: "'d \<Rightarrow> 'a" assumes "G g \<noteq> \<^bold>1" obtains a :: "'d" where "g a \<noteq> \<^bold>1"
stationary_distribution_alt_def: fixes A :: "'a st_mat"   and N :: "'a pmf" shows "MC_syntax.stationary_distribution (transition_of_st_mat A) N = (N = pmf_of_st_vec (A *st st_vec_of_pmf N))"
real_degree_2_factorization_exists_complex: fixes p :: "complex poly" assumes "set (coeffs p) \<subseteq> \<real>" shows "\<exists>qs. p = prod_list qs \<and> (\<forall>q\<in>set qs. set (coeffs q) \<subseteq> \<real> \<and> degree q \<le> 2)"
restrict_map_eq_1: fixes m :: "'b \<Rightarrow> 'a option"   and A :: "'b set"   and k :: "'b" shows "((m |` A) k = None) = (k \<notin> dom m \<inter> A)"
rotate_id: fixes n :: "nat"   and xs :: "'c list" assumes "n mod |xs| = 0" shows "rotate n xs = xs"
itrm_brackets_dist: fixes extra_rule :: "nat itrm \<Rightarrow> nat itrm \<Rightarrow> bool"   and itrm_bracket :: "nat \<Rightarrow> nat itrm \<Rightarrow> nat itrm option"   and vs :: "nat list"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_abstraction extra_rule itrm_bracket"   and "itrm_abstraction.itrm_brackets itrm_bracket vs x = Some x'" shows "itrm_cong (special_idiom.idiom_ext_rule extra_rule) (opaque_dist vs x') x"
cInf_singleton: fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup" shows "Inf {x} = x"
smult_closed: fixes K :: "'a set"   and V :: "'a set"   and R :: "('a, 'b) ring_scheme"   and k :: "'a"   and v :: "'a" assumes "subalgebra K V R"   and "k \<in> K"   and "v \<in> V" shows "k \<otimes>\<^bsub>R\<^esub> v \<in> V"
CombineStrip': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, bool) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "bool set"   and P :: "'e set"   and c' :: "('e, 'd, bool) com"   and Q :: "'e set"   and A :: "'e set"   and c'' :: "('e, 'd, bool) com"   and c :: "('e, 'd, bool) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c' Q,A"   and "\<Gamma>,{}\<turnstile>\<^bsub>/{} \<^esub>P c'' UNIV,UNIV"   and "c'' = mark_guards False (strip_guards (- F) c')"   and "merge_guards c = merge_guards (mark_guards False c')" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c Q,A"
simps: fixes a1 :: "'b list"   and a2 :: "'b list" shows "double a1 a2 = (\<exists>x ys xs. a1 = xs @ ys \<and> a2 = xs @ x # ys \<and> x \<in> set ys)"
winding_number_comp_linear: fixes c :: "complex"   and \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex"   and b :: "complex" assumes "c \<noteq> 0"   and "valid_path \<gamma>"   and "(z - b) / c \<notin> path_image \<gamma>" shows "winding_number ((\<lambda>x. c * x + b) \<circ> \<gamma>) z = winding_number \<gamma> ((z - b) / c)"
disj_lin: fixes p :: "fm"   and q :: "fm" assumes "islin p"   and "islin q" shows "islin (Parametric_Ferrante_Rackoff.disj p q)"
coeff_int_smult: fixes c :: "'b"   and p :: "'b poly"   and i :: "int" shows "coeff_int (smult c p) i = c * coeff_int p i"
fA_rel_inv_def: shows "fA_rel_inv = (\<lambda>s. (\<forall>x. sys_ghost_hs_in_sync x s, sys_ghost_hs_phase s, sys_fA s, gc_fM s, filter is_mw_fA (sys_mem_store_buffers gc s)) \<in> fA_rel)"
wlsFSb_recAll_unique_presCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and us :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set" assumes "wlsFSb MOD"   and "presCons h hA MOD" shows "(wls s X \<longrightarrow> h X = rec MOD X) \<and> (wlsAbs (us, s') A \<longrightarrow> hA A = recAbs MOD A)"
rcos_sum: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "H \<lhd> G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x <#>\<^bsub>G\<^esub> (H #>\<^bsub>G\<^esub> y) = H #>\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> y"
find_non0_1: fixes l :: "nat"   and A :: "'a mat"   and m :: "nat" assumes "det_selection_fun sel_fun"   and "local.find_non0 l A = Some m" shows "A $$ (m, l) \<noteq> (0::'a)"
map_of_leI: fixes l :: "('a \<times> 'b) list"   and m :: "'a \<Rightarrow> 'b option" assumes "distinct (map fst l)"   and "\<And>k v. (k, v) \<in> set l \<Longrightarrow> m k = Some v" shows "map_of l \<subseteq>\<^sub>m m"
negate_negate: fixes P :: "'c \<Rightarrow> bool" shows "\<N> (\<N> P) = P"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, data := SOME x. True, dests := SOME x. True, pre := SOME x. True, dip := SOME x. True, oip := SOME x. True, hops := SOME x. True, dsn := SOME x. True, dsk := SOME x. True, osn := SOME x. True, sip := SOME x. x \<noteq> ip \<xi>\<rparr>"
ltln_Release_alterdef: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = w \<Turnstile>\<^sub>n G\<^sub>n \<psi> or\<^sub>n (\<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>))"
Union_raw: fixes X :: "V set" assumes "X \<in> range elts" shows "\<Union> (elts ` X) \<in> range elts"
mono_dp_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'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 bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and w :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit"   and "relation_algebra.is_direct_product inf top composition converse unit w x"   and "relation_algebra.is_direct_product inf top composition converse unit y z" shows "composition (relation_algebra.Phi inf composition converse w x y z) z = x"
disjoint_sum: fixes B :: "'a set set"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite B"   and "finite A"   and "Congruence.partition A B" shows "sum (sum f) B = sum f A"
wfactorsI: fixes fs :: "'c list"   and G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "\<forall>f\<in>set fs. irreducible G f"   and "foldr (\<otimes>\<^bsub>G\<^esub>) fs \<one>\<^bsub>G\<^esub> \<sim>\<^bsub>G\<^esub> a" shows "wfactors G fs a"
lfilter_cong: fixes xs :: "'a llist"   and ys :: "'a llist"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> lset ys \<Longrightarrow> P x = Q x" shows "lfilter P xs = lfilter Q ys"
ratfps_subdegree_code: fixes x :: "'b ratfps" shows "ratfps_subdegree x = poly_subdegree (fst (quot_of_ratfps x))"
sum_dcharacter_atMost_le: fixes m :: "nat" assumes "\<chi> \<noteq> \<chi>\<^sub>0" shows "cmod (sum \<chi> {..m}) \<le> real (totient n)"
induct: fixes x1 :: "'d option"   and x2 :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option"   and P :: "'d option \<Rightarrow> 'd option \<Rightarrow> bool" assumes "(x1, x2) \<in> option_while' C B"   and "\<And>r. \<not> C r \<Longrightarrow> P (Some r) (Some r)"   and "\<And>r. \<lbrakk>C r; B r = None\<rbrakk> \<Longrightarrow> P (Some r) None"   and "\<And>r r' sr''. \<lbrakk>C r; B r = Some r'; (Some r', sr'') \<in> option_while' C B; P (Some r') sr''\<rbrakk> \<Longrightarrow> P (Some r) sr''" shows "P x1 x2"
simps: fixes x :: "'a"   and n :: "nat" shows "binary_power x n = (if n = 0 then 1::'a else let (d, r) = divmod_nat n 2; rec = binary_power (x * x) d in if r = 0 then rec else rec * x)"
supp_perm: fixes p :: "perm" shows "supp p = {a. p \<bullet> a \<noteq> a}"
listrel_eq_len: fixes xs :: "'d list"   and ys :: "'e list"   and r :: "('d \<times> 'e) set" assumes "(xs, ys) \<in> listrel r" shows "length xs = length ys"
i3: fixes \<eta> :: "nat" shows "\<I>_uniform UNIV (nlists UNIV \<eta>) \<oplus>\<^sub>\<I> \<I>_uniform UNIV (nlists UNIV \<eta>) \<turnstile>res CIPHER.KEY.res \<eta> \<surd>"
pprogress_def: fixes \<phi> :: "'b MFOTL.formula"   and \<pi> :: "(char list \<times> 'b list) prefix" shows "pprogress \<phi> \<pi> = (THE n. \<forall>\<sigma>. prefix_of \<pi> \<sigma> \<longrightarrow> Monitor.progress \<sigma> \<phi> (plen \<pi>) = n)"
conv_abscissa_MInf_iff: fixes f :: "'a fds" shows "(conv_abscissa f = - \<infinity>) = (\<forall>s. fds_converges f s)"
card_bit1: shows "CARD('a bit1) = Suc (2 * CARD('a))"
kuhn_simplex_axioms: shows "kuhn_simplex p n base upd s"
lin_indpt_id: shows "lin_indpt (set (cols (1\<^sub>m n)))"
token_run_enter_final_states: fixes x :: "nat"   and n :: "nat" assumes "token_run x n \<in> F" shows "\<exists>m\<ge>x. token_run x m \<notin> F - {q\<^sub>0} \<and> token_run x (Suc m) \<in> F"
conc_def: fixes A :: "'a list set"   and B :: "'a list set" shows "A @@ B = {xs @ ys |xs ys. xs \<in> A \<and> ys \<in> B}"
S_sub_cc: shows "S \<subseteq> cbox (- c) c"
eval\<^sub>c_def: fixes F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and C :: "fterm literal set" shows "eval\<^sub>c F G C = (\<forall>E. \<exists>l\<in>C. eval\<^sub>l E F G l)"
UN_constant_eq: fixes a :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e set"   and c :: "'e set" assumes "a \<in> A"   and "\<forall>y\<in>A. f y = c" shows "\<Union> (f ` A) = c"
assms_4: fixes n :: "nat" shows "u n \<in> S \<union> T"
insert_def: fixes x :: "'c"   and xs :: "'c list" shows "List.insert x xs = (if x \<in> set xs then xs else x # xs)"
MOST_iff_finiteNeg: fixes P :: "'a \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
expr_has_density_sound_op: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and t :: "pdf_type"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and g :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and t' :: "pdf_type"   and oper :: "pdf_operator" assumes "density_context V V' \<Gamma> \<delta>"   and "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t) f"   and "(\<lambda>(x, y). g x y) \<in> borel_measurable (state_measure V' \<Gamma> \<Otimes>\<^sub>M stock_measure t')"   and "\<And>M \<rho>. has_subprob_density M (stock_measure t) (f \<rho>) \<Longrightarrow> has_density (distr M (stock_measure t') (op_sem oper)) (stock_measure t') (g \<rho>)"   and "\<Gamma> \<turnstile> e : t"   and "op_type oper t = Some t'"   and "free_vars (oper $$ e) \<subseteq> V \<union> V'" shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure (V, V', \<Gamma>, \<delta>) \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> (oper $$ e))) (stock_measure t') g"
open_product_open: fixes A :: "('d \<times> 'e) set" assumes "open A" shows "\<exists>\<U>\<subseteq>{S \<times> T |S T. open S \<and> open T}. \<Union> \<U> = A"
comp_null_2: fixes f :: "'s" shows "f \<cdot>\<^sub>S S.null = S.null"
Compl_anti_mono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "- B \<subseteq> - A"