hcju/isabelleps huggingface dataset:
map_sel: fixes f1 :: "'d \<Rightarrow> 'a"   and f2 :: "'e \<Rightarrow> 'b"   and a :: "('d, 'e, 'c) gpv" shows "the_gpv (map_gpv f1 f2 a) = map_spmf (map_generat f1 f2 ((\<circ>) (map_gpv f1 f2))) (the_gpv a)"
lesseq_a_b: shows "lesseq_b = lesseq_a"
diff_minus_eq_add: fixes a :: "'a"   and b :: "'a" shows "a - - b = a + b"
cindex_poly_smult_1: fixes a :: "real"   and b :: "real"   and c :: "real"   and q :: "real poly"   and p :: "real poly" shows "cindex_poly a b (smult c q) p = sign c * cindex_poly a b q p"
unit_simps_3: shows "cod \<phi> = g \<star> f"
prems_1: shows "q \<in> ctermsl (p1 \<oplus> p2)"
P1: shows "D E TS X Z \<and> D E TS Y Z"
Let_1: shows "atom ` domA \<Delta> \<sharp>* (\<Gamma>, L)"
distinct_11: fixes x1 :: "'a"   and x7 :: "'a" shows "broadcast x1 \<noteq> receive x7"
finpref_LNil: fixes A :: "'b set" shows "finpref A LNil = {LNil}"
in_rtrancl_UnI: fixes x :: "'x \<times> 'x"   and R :: "('x \<times> 'x) set"   and S :: "('x \<times> 'x) set" assumes "x \<in> R\<^sup>* \<or> x \<in> S\<^sup>*" shows "x \<in> (R \<union> S)\<^sup>*"
document_ptr_kinds_eq2_h3: shows "|h3 \<turnstile> document_ptr_kinds_M|\<^sub>r = |h' \<turnstile> document_ptr_kinds_M|\<^sub>r"
simps_8: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and \<phi> :: "'a formula" shows "((w, I) \<Turnstile> FEXISTS \<phi>) = (\<exists>P. finite P \<and> (w, Inr P # I) \<Turnstile> \<phi>)"
rel_cases: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and a :: "'l option"   and b :: "'m option" assumes "rel_option R a b" obtains "a = None" and "b = None"   | x :: "'l" and y :: "'m" where "a = Some x" and "b = Some y" and "R x y"
arctan_double: fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "2 * arctan x = arctan (2 * x / (1 - x\<^sup>2))"
invertible_def: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "Group_Theory.monoid M composition unit"   and "u \<in> M" shows "monoid.invertible M composition unit u = (\<exists>v\<in>M. composition u v = unit \<and> composition v u = unit)"
entries_code: shows "RBT_Impl.entries = RBT_Impl.gen_entries []"
append_single_T_imp_tickFree: fixes t :: "'a event list"   and a :: "'a event"   and P :: "'a process" assumes "t @ [a] \<in> T P" shows "tickFree t"
LowLink_eq_disc_iff_scc_root: fixes v :: "'a" assumes "v \<in> dom (finished s) \<or> stack s \<noteq> [] \<and> v = hd (stack s) \<and> pending s `` {v} = {}" shows "(Min (\<delta> s ` lowlink_set s v) = \<delta> s v) = scc_root s v (scc_of E v)"
not_before0: fixes x :: "'b"   and xs :: "'b list" shows "\<not> x < xs ! 0 in xs"
GB_insert: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_Groebner_basis G"   and "f \<in> pmdl G" shows "is_Groebner_basis (insert f G)"
strict_monoD: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and x :: "'f"   and y :: "'f" assumes "class.order less_eq less"   and "order.strict_mono less f"   and "less x y" shows "f x < f y"
assms_2: fixes n :: "nat" assumes "0 < n" shows "A \<inter> (T ^^ n) -` A \<in> null_sets M"
simps: fixes n :: "nat"   and A :: "nat set"   and t :: "'a option list set"   and pos :: "bool"   and A' :: "nat set"   and t' :: "'a option list set" shows "bin_join n A t pos A' t' = (if t = {} then {} else if t' = {} then if pos then {} else t else if A' = {} then if pos = (replicate n None \<in> t') then t else {} else if A' = A then if pos then t \<inter> t' else t - t' else if A' \<subseteq> A then {as \<in> t. proj_tuple_in_join pos (join_mask n A') as t'} else if A \<subseteq> A' \<and> pos then {as \<in> t'. proj_tuple_in_join pos (join_mask n A) as t} else join t pos t')"
SecurityInvariant_axioms: shows "SecurityInvariant SINVAR_BLPtrusted.sinvar SINVAR_BLPtrusted.default_node_properties True"
compat_asigs_def: fixes a1 :: "'d set \<times> 'd set \<times> 'd set"   and a2 :: "'d set \<times> 'd set \<times> 'd set" shows "compat_asigs a1 a2 \<equiv> outputs a1 \<inter> outputs a2 = {} \<and> internals a1 \<inter> actions a2 = {} \<and> internals a2 \<inter> actions a1 = {}"
Init_subset_Gen: shows "Init \<subseteq> Gen (\<P> 0 0)"
K: shows "\<forall>i\<in>I. K i = inv_into (carrier (Gs i)) (J i)"
normal_iff_subgroup: fixes N :: "'a set" shows "N \<lhd> add_monoid R = subgroup N (add_monoid R)"
I_qe_cooper\<^sub>1: fixes \<phi> :: "atom fm"   and xs :: "int list" assumes "\<forall>a\<in>atoms \<phi>. divisor a \<noteq> 0"   and "\<forall>a\<in>set (atoms\<^sub>0 \<phi>). hd_coeff_is1 a"   and "nqfree \<phi>" shows "Z.I (qe_cooper\<^sub>1 \<phi>) xs = (\<exists>x. Z.I \<phi> (x # xs))"
prob_space_measure_pmf: fixes p :: "'d pmf" shows "prob_space (measure_pmf p)"
BinOpRed1: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e' \<guillemotleft>bop\<guillemotright> e\<^sub>2,s'\<rangle>"
character_double_dual_iso: fixes x :: "('a \<Rightarrow> complex) \<Rightarrow> complex" assumes "x \<in> carrier (Characters (Characters G))" shows "character (Characters (Characters (Characters G))) (G''.double_dual_iso x)"
hitting_paths_def: shows "Q.hitting_paths \<equiv> \<lambda>x. x \<noteq> v0 \<and> ((\<exists>xs\<in>Q. x \<in> set xs) \<or> x = v1)"
map_simp: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and f :: "'a" assumes "constant_transformation A B g"   and "partial_magma.arr A f" shows "constant_transformation.map A B g f = g"
REC_def: shows "REC \<equiv> \<mu> x. mid`?`xa \<rightarrow> right`!`xa \<rightarrow> ack \<rightarrow> x"
are_all_operators_applicable_in_def: fixes s :: "'c \<Rightarrow> 'd option"   and ops :: "('c, 'd) sas_plus_operator list" shows "are_all_operators_applicable_in s ops \<equiv> list_all (SAS_Plus_Representation.is_operator_applicable_in s) ops"
diff_fun_axioms: shows "diff_fun k (charts_submanifold W) f'"
FunctorCompMapsTo: fixes f :: "'a"   and F :: "('b, 'g, 'a, 'h, 'c, 'i) Functor"   and G :: "('g, 'd, 'h, 'e, 'i, 'f) Functor" assumes "f \<in> mor\<^bsub>CatDom (F ;;: G)\<^esub>"   and "F \<approx>>;;; G" shows "G ## F ## f maps\<^bsub>CatCod G\<^esub> G @@ F @@ dom\<^bsub>CatDom F\<^esub> f to G @@ F @@ cod\<^bsub>CatDom F\<^esub> f"
bin_ord_FO_\<Gamma>_def: shows "ord_FO_\<Gamma> S = {uu_. \<exists>CA DA AA A \<sigma> E. uu_ = Infer {#CA#} DA E \<and> ord_resolve_rename S [CA] DA [AA] [A] \<sigma> E}"
app_singleton: fixes x :: "hf"   and y :: "hf" shows "app \<lbrace>\<langle>x, y\<rangle>\<rbrace> x = y"
vars_chop: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "vars (chop t) \<union> vars_hd (head t) = vars t"
simple_function_subalgebra: fixes N :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and M :: "'b measure" assumes "simple_function N f"   and "sets N \<subseteq> sets M"   and "space N = space M" shows "simple_function M f"
comm_groupE_3: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "comm_group G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"
s_r: shows "s r = Some (\<sigma>, \<tau>, \<E> [Ite (VE (CV T)) e1 e2])"
prems_3: fixes w :: "'b" shows "Suspend w \<notin> set (wa # was)"
us_def: shows "us = Abs_upset \<circ> \<up>"
subprob_cexprD: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and \<rho> :: "nat \<Rightarrow> val" assumes "subprob_cexpr V V' \<Gamma> e"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "\<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (merge V V' (x, \<rho>)) e)) \<partial>state_measure V \<Gamma> \<le> 1"
elims_2: fixes x :: "'a list" assumes "sorted x" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys" and "Ball (set ys) ((\<le>) xa) \<and> sorted ys"
set_2: fixes x21 :: "'a"   and x22 :: "'a list" shows "set (x21 \<Colon> x22) = insert x21 (set x22)"
argmin_Max_Args_min_on: fixes i :: "int"   and j :: "int"   and f :: "int \<Rightarrow> 'a" assumes "i \<le> j" shows "argmin f [i..j] = Max (Args_min_on f {i..j})"
h_bigo_self: shows "(\<lambda>n. h (real n)) \<in> O(\<lambda>n. h (real (Suc n)))"
next_ge_fall: fixes S :: "'a fset"   and s :: "'a" assumes "fBall S (\<lambda>y. y \<le> s)" shows "next s |\<notin>| S"
cosh_field_def: fixes z :: "'a" shows "cosh z = (exp z + exp (- z)) / (2::'a)"
uv_cf_edge: shows "(u, v) \<in> cf.E"
ba: fixes y :: "'a set" assumes "y \<in> A" shows "f -` y \<inter> space M \<in> sets M"
seq2: fixes c1 :: "('exp, 'id) MWLfCom"   and m :: "'id \<Rightarrow> 'val"   and c1' :: "('exp, 'id) MWLfCom"   and V :: "('exp, 'id) MWLfCom list"   and m' :: "'id \<Rightarrow> 'val"   and c2 :: "('exp, 'id) MWLfCom" assumes "SRabbr c1 m (c1' # V) m'" shows "SRabbr (c1;c2) m ((c1';c2) # V) m'"
sep: shows "separating (bipartite_web_of \<Gamma>) S"
semilattice_inf_ow_axioms: shows "semilattice_inf_ow U\<^sub>1 inf\<^sub>1 le\<^sub>1 ls\<^sub>1"
from_nat_to_nat: fixes x :: "'b" shows "from_nat (to_nat (x::'b)) = x"
P_2: shows "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'"
r_auxhyp_def: shows "r_auxhyp \<equiv> Cn 1 (r_smn 1 1) [r_const (encode r_prenum), recf.Id 1 0]"
hiding_maintains_is_process: fixes A :: "'b set"   and P :: "'b process" shows "is_process ({(s, X). \<exists>t. s = trace_hide t (ev ` A) \<and> (t, X \<union> ev ` A) \<in> F P} \<union> {(s, X). s \<in> div_hide P A}, div_hide P A)"
ts_distincts: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B TS P Q" shows "A \<noteq> B \<and> A \<noteq> P \<and> A \<noteq> Q \<and> B \<noteq> P \<and> B \<noteq> Q \<and> P \<noteq> Q"
Sup_upper: fixes Inf :: "'k set \<Rightarrow> 'k"   and Sup :: "'k set \<Rightarrow> 'k"   and inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bot :: "'k"   and top :: "'k"   and x :: "'k"   and A :: "'k set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "x \<in> A" shows "less_eq x (Sup A)"
prefix_bottom_iff: fixes xs :: "[:'a:]"   and ys :: "[:'a:]" shows "(prefix\<cdot>xs\<cdot>ys = \<bottom>) = (xs = \<bottom> \<or> ys = \<bottom>)"
assms_4: shows "\<forall>v l. length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) \<le> l \<and> l < length \<pi> + 1 \<longrightarrow> \<A> (State l (index (\<Pi>\<^sub>\<V>) v)) = \<A> (State (length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1) (index (\<Pi>\<^sub>\<V>) v))"
assms_2: shows "F \<in> \<lbrakk>\<H>\<rbrakk>\<^sub>H \<rightarrow> \<lbrakk>\<H>\<rbrakk>\<^sub>H"
st_data0_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and address :: "word32"   and s :: "'b sparc_state" shows "st_data0 instr curr_win rd address s \<equiv> if fst instr \<in> {load_store_type STD, load_store_type STDA} then user_reg_val curr_win (rd AND 30) s else if fst instr \<in> {load_store_type ST, load_store_type STA} then user_reg_val curr_win rd s else if fst instr \<in> {load_store_type STH, load_store_type STHA} then if ucast address = 0 then user_reg_val curr_win rd s << 16 else user_reg_val curr_win rd s else if ucast address = 0 then user_reg_val curr_win rd s << 24 else if ucast address = 1 then user_reg_val curr_win rd s << 16 else if ucast address = 2 then user_reg_val curr_win rd s << 8 else user_reg_val curr_win rd s"
execute_parallel_operator_sas_plus_cons: fixes s :: "'a \<Rightarrow> 'b option"   and op :: "('a, 'b) sas_plus_operator"   and ops :: "('a, 'b) sas_plus_operator list" shows "execute_parallel_operator_sas_plus s (op # ops) = execute_parallel_operator_sas_plus (s ++ map_of (effect_of op)) ops"
mk_poincare_line_cmat_scale: fixes k :: "real"   and A :: "real"   and B :: "complex" shows "cor k *\<^sub>s\<^sub>m mk_poincare_line_cmat A B = mk_poincare_line_cmat (k * A) (cor k * B)"
show9: fixes A :: "real"   and d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and s :: "nat" assumes "1 < A"   and "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "0 < s"   and "(\<lambda>n. real_of_int (a n) powr (1 / real_of_int (2 ^ n))) \<longlonglongrightarrow> A"   and "convergent_prod d"   and "\<forall>n\<ge>s. (\<Prod>j. d (n + j)) < A / (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))" shows "\<forall>m\<ge>s. \<exists>n\<ge>m. d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j))) \<le> real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1)))"
w0_def: shows "w0 \<equiv> lhd (ltl P)"
refl_onD2: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and x :: "'e"   and y :: "'e" assumes "refl_on A r"   and "(x, y) \<in> r" shows "y \<in> A"
quot_dbtm_fresh: fixes s :: "atom"   and t :: "dbtm" shows "s \<sharp> quot_dbtm t"
compute_float_minus: fixes f :: "float"   and g :: "float" shows "f - g = f + - g"
module_vec_simps_3: fixes ty :: "'b itself"   and n :: "nat" shows "carrier (module_vec ty n) = carrier_vec n"
cinner_sum_left: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a" shows "\<langle>sum f A, y\<rangle> = (\<Sum>x\<in>A. \<langle>f x, y\<rangle>)"
ex_terminal_arrow: fixes y :: "'a" assumes "ide y" shows "\<exists>x e. terminal_arrow_from_functor (\<cdot>) (\<cdot>) (\<lambda>x. CCC.prod x b) x y e"
negatively_invariant_restrict_dom: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.negatively_invariant f X M = auto_ll_on_open.negatively_invariant f X (M \<inter> X)"
ALL_MOST: fixes P :: "'b \<Rightarrow> bool" assumes "\<forall>x. P x" shows "\<forall>\<^sub>\<infinity>x. P x"
rep_eq: fixes x :: "nat set"   and xa :: "nat" shows "lookup (powsum_mpoly_aux x xa) = (\<lambda>mon. if infinite x \<or> xa = 0 \<and> mon \<noteq> 0 then 0::'b else if xa = 0 \<and> mon = 0 then of_nat (card x) else if finite x \<and> (\<exists>x\<in>x. mon = Poly_Mapping.single x xa) then 1::'b else (0::'b))"
GreatestI_nat: fixes P :: "nat \<Rightarrow> bool"   and k :: "nat"   and b :: "nat" assumes "P k"   and "\<And>y. P y \<Longrightarrow> y \<le> b" shows "P (Greatest P)"
simps_6: fixes r :: "'a pfm"   and i :: "'a"   and p :: "'a pfm" shows "reduce' r (K\<^sub>! i p) = (r \<^bold>\<longrightarrow>\<^sub>! K\<^sub>! i (reduce' r p))"
right_neutral: fixes a :: "('a \<times> ('b \<times> 'a) list) set" shows "t_prod a t_one = a"
fold_cong: fixes a :: "'c"   and b :: "'c"   and xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "fold f xs a = fold g ys b"
metric_CauchyD: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "uniform_space.Cauchy uniformity X"   and "0 < e" shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
preprocess'_atoms_to_constraints: fixes cs :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat"   and S :: "'a istate"   and ns :: "QDelta ns_constraint"   and q :: "linear_poly"   and a :: "QDelta atom" assumes "preprocess' cs start = S"   and "start = start_fresh_variable cs"   and "ns = (case a of Leq v c \<Rightarrow> LEQ_ns q c | Geq v c \<Rightarrow> GEQ_ns q c)"   and "a \<in> snd ` set (Atoms S)" shows "(atom_var a \<notin> fst ` set (Tableau S) \<longrightarrow> (\<exists>r. r \<noteq> 0 \<and> r *R nsc_of_atom a \<in> snd ` set cs)) \<and> ((atom_var a, q) \<in> set (Tableau S) \<longrightarrow> ns \<in> snd ` set cs)"
close_eq_M_def: fixes M :: "'a atom formula set" shows "close_eq_M M = M \<inter> {Atom (predAtm p xs) |p xs. True} \<union> {Atom (Eq a a) |a. True} \<union> {\<^bold>\<not> (Atom (Eq a b)) |a b. a \<noteq> b}"
Inv_preserves_Can_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Cod C (monoidal_language.Inv C t) = monoidal_language.Dom C t"
punit_min_term: shows "i.punit.min_term = 0"
j: fixes x :: "'M" shows "lift (x, 0) \<bullet> j = 0"
div_less: fixes m :: "nat"   and n :: "nat" assumes "m < n" shows "m div n = 0"
BisT_raw_coind: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" assumes "sym theta"   and "theta \<subseteq> RetrT theta" shows "theta \<subseteq> BisT"
fls_divide_X_intpow': fixes f :: "'a fls"   and i :: "int" assumes "inverse (1::'a) = (1::'a)" shows "f / fls_X_intpow i = fls_shift i f"
segment_2: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
bigo_pos_const: fixes h :: "'b \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a" shows "(\<exists>c. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>) = (\<exists>c>0::'a. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>)"
assms: shows "\<forall>y\<in>set ys. \<forall>z\<in>set zs. y \<le> z"
memo_list_rtrancl_def: fixes r :: "('b \<times> 'b) list" shows "memo_list_rtrancl r = (let tr = rtrancl_list_impl r; rm = map (\<lambda>a. (a, tr [a])) ((remdups \<circ> map fst) r) in (\<lambda>a. case map_of rm a of None \<Rightarrow> [a] | Some as \<Rightarrow> as))"
transdb_inv: fixes e :: "db"   and \<Gamma> :: "nat \<Rightarrow> nat"   and c :: "nat"   and k :: "nat"   and \<Gamma>' :: "nat list" assumes "closed e" shows "transdb (transdb_inv e \<Gamma> c k) \<Gamma>' = e"
stereographic_circline_def: shows "stereographic_circline \<equiv> map_fun rep_plane abs_circline stereographic_circline_pv_clmat"
mbuf2t_take_induct: fixes f :: "event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and buf :: "event_data option list set list \<times> event_data option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "event_data option list set list \<times> event_data option list set list"   and nts' :: "nat list"   and i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and R :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and j :: "nat"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbuf2t_take f z buf nts = (z', buf', nts')"   and "wf_mbuf2 i ja jb P Q buf"   and "list_all2 R [i..<j] nts"   and "ja \<le> j"   and "jb \<le> j"   and "U i z"   and "\<And>k x y t z. \<lbrakk>i \<le> k; Suc k \<le> ja; Suc k \<le> jb; P k x; Q k y; R k t; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f x y t z)" shows "U (min ja jb) z'"
map_sel_1: fixes a :: "('g, 'h, 'i) generat"   and f1 :: "'g \<Rightarrow> 'j"   and f2 :: "'h \<Rightarrow> 'k"   and f3 :: "'i \<Rightarrow> 'l" assumes "is_Pure a" shows "generat.result (map_generat f1 f2 f3 a) = f1 (generat.result a)"
lft_pred_st_def: fixes p :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and u :: "nat \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'c" shows "lft_pred_st p u x = p (u 0) (u 1) (x 0)"
ide_src: fixes \<mu> :: "'a arrow_of_spans_data" assumes "V.arr \<mu>" shows "V.ide (src \<mu>)"
when_cong: fixes P :: "bool"   and Q :: "bool"   and a :: "'a"   and b :: "'a"   and zero :: "'a" assumes "P = Q"   and "Q \<Longrightarrow> a = b" shows "zero.when zero a P = zero.when zero b Q"
assms: shows "xs @ [Broadcast (i, Expunge e mo i)] prefix of j"
distinct_formal_ins: fixes p :: "'f"   and ins :: "'c list"   and outs :: "'c list" assumes "(p, ins, outs) \<in> set procs" shows "distinct ins"
prv_all_neg_imp_neg_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "bprv (imp (all x (neg \<phi>)) (neg (exi x \<phi>)))"
assms_1: shows "x ; 1 \<cdot> y ; 1 \<cdot> y\<^sup>T ; 1 = 0"
satPB_UN: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<Phi> :: "'b \<Rightarrow> ('fsym, 'psym) lit list set"   and I :: "'b set" shows "satPB \<xi> (\<Union> (\<Phi> ` I)) = (\<forall>i\<in>I. satPB \<xi> (\<Phi> i))"
ideal_axioms_def: fixes I :: "'c set"   and R :: "'c set"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "ideal_axioms I R multiplication \<equiv> (\<forall>a b. a \<in> R \<longrightarrow> b \<in> I \<longrightarrow> multiplication a b \<in> I) \<and> (\<forall>a b. a \<in> R \<longrightarrow> b \<in> I \<longrightarrow> multiplication b a \<in> I)"
impl_spec: fixes m :: "('c, 'd) TopoS_packed"   and sinvar_spec :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" assumes "TopoS_modelLibrary m sinvar_spec" shows "TopoS_List_Impl (nm_default m) sinvar_spec (nm_sinvar m) (nm_receiver_violation m) (nm_offending_flows m) (nm_node_props m) (nm_eval m)"
subgroup_indicator_eq_1: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and H :: "'c set" assumes "Group.group G"   and "a \<in> H"   and "a \<in> carrier G" shows "subgroup_indicator G H a = 1"
eqpoll_iff_card_of_ordIso: fixes A :: "'b set"   and B :: "'c set" shows "(A \<approx> B) = (|A| =o |B|)"
injectivity: fixes x :: "'d fps"   and y :: "'d fps" assumes "fps_to_fls x = fps_to_fls y" shows "x = y"
proj_setD_1: fixes y :: "'b"   and x :: "'b"   and A :: "'b set" assumes "y \<in> proj_set x A" shows "y \<in> A"
subset_iT_Div_Int: fixes A :: "nat set"   and B :: "nat set"   and k :: "nat" assumes "A \<subseteq> B" shows "A \<inter> B \<oslash> k = (A \<oslash> k) \<inter> (B \<oslash> k)"
Inf_less_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and S :: "'c set"   and a :: "'c" assumes "class.complete_linorder Inf Sup inf less_eq less sup bot top" shows "less (Inf S) a = (\<exists>x\<in>S. less x a)"
top: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "algebra \<Omega> M" shows "\<Omega> \<in> M"
mkArr_def: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and B :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set"   and F :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr \<Rightarrow> (hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "mkArr A B F = (if A \<subseteq> Univ \<and> |A| <o natLeq \<and> B \<subseteq> Univ \<and> |B| <o natLeq \<and> F \<in> A \<rightarrow> B then THE f. f \<in> hom (mkIde A) (mkIde B) \<and> Fun f = restrict F A else null)"
Gromov_extension_inv_fixed_point: fixes f :: "'b \<Rightarrow> 'b"   and xi :: "'b Gromov_completion" assumes "isometry f"   and "Gromov_extension f xi = xi" shows "Gromov_extension (inv f) xi = xi"
dom_llp2: fixes x :: "'a"   and y :: "'a" assumes "d x \<le> d y" shows "x \<le> d y \<cdot> x"
Union_Un_distrib: fixes A :: "'b set set"   and B :: "'b set set" shows "\<Union> (A \<union> B) = \<Union> A \<union> \<Union> B"
simps_1: fixes x :: "'b" shows "embed_dir (x, sort_dir.Left) = (x, 0)"