hcju/isabelleps huggingface dataset:
foldl1_induct: fixes P :: "('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b list \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'b list" assumes "\<And>f x. P f [x]"   and "\<And>f x y. P f [x, y]"   and "\<And>f x y z zs. P f (f x y # z # zs) \<Longrightarrow> P f (x # y # z # zs)"   and "\<And>f. P f []" shows "P f xs"
cong_l_reflexive: fixes F :: "'a set"   and a :: "'a" assumes "F \<in> filters" shows "cong_l F a a"
dgrad_sig_set_closed_mult_scalar: fixes d :: "'a \<Rightarrow> nat"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat" assumes "dickson_grading d"   and "p \<in> punit_dgrad_max_set d"   and "r \<in> dgrad_sig_set' j d" shows "p \<odot> r \<in> dgrad_sig_set' j d"
lex_ord_valI: fixes aux :: "order option"   and k :: "'b"   and xs :: "('b, 'c) oalist_ntm"   and ys :: "('b, 'd) oalist_ntm"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> order option"   and ko :: "'b nat_term_order" assumes "aux \<noteq> Some Eq"   and "k \<in> fst ` set (fst (list_of_oalist_ntm xs)) \<union> fst ` set (fst (list_of_oalist_ntm ys))"   and "aux = f k (OAlist_lookup_ntm xs k) (OAlist_lookup_ntm ys k)"   and "\<And>k'. \<lbrakk>k' \<in> fst ` set (fst (list_of_oalist_ntm xs)) \<union> fst ` set (fst (list_of_oalist_ntm ys)); ko.lt (key_order_of_nat_term_order_inv ko) k' k\<rbrakk> \<Longrightarrow> f k' (OAlist_lookup_ntm xs k') (OAlist_lookup_ntm ys k') = Some Eq" shows "OAlist_lex_ord_ntm ko f xs ys = aux"
is_irred_plus: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<not> is_red F p"   and "\<not> is_red F q" shows "\<not> is_red F (p + q)"
sur_def_var1: fixes x :: "'a" shows "is_sur x = (1 ; x = 1)"
indep_varD: fixes Ma :: "'c measure"   and A :: "'a \<Rightarrow> 'c"   and Mb :: "'c measure"   and B :: "'a \<Rightarrow> 'c"   and Xa :: "'c set"   and Xb :: "'c set" assumes "indep_var Ma A Mb B"   and "Xa \<in> sets Ma"   and "Xb \<in> sets Mb" shows "prob ((\<lambda>x. (A x, B x)) -` (Xa \<times> Xb) \<inter> space M) = prob (A -` Xa \<inter> space M) * prob (B -` Xb \<inter> space M)"
fmeasurable_UN: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c set"   and A :: "'c set"   and M :: "'c measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> A"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> sets M"   and "A \<in> fmeasurable M" shows "\<Union> (F ` I) \<in> fmeasurable M"
residue_holo: fixes s :: "complex set"   and z :: "complex"   and f :: "complex \<Rightarrow> complex" assumes "open s"   and "z \<in> s"   and "f holomorphic_on s" shows "residue f z = 0"
dominates_refl: fixes g :: "'g"   and n :: "'node" assumes "invar g"   and "n \<in> set (\<alpha>n g)" shows "dominates g n n"
elims: fixes x :: "('m, 'n) result"   and xa :: "('m, 'n) result"   and y :: "('m, 'n) result" assumes "x <|> xa = y" obtains x :: "'n" and uu_ :: "('m, 'n) result" where "xa__ = Ok xb__" and "xa = uu_" and "y = Ok xb__"   | v :: "'m" and x :: "'n" where "xa__ = Error v__" and "xa = Ok x" and "y = Ok x"   | e :: "'m" and v :: "'m" where "xa__ = Error e" and "xa = Error v" and "y = Error e"
toEnum_of_nat: fixes n :: "nat" assumes "n < 2 ^ LENGTH('b)" shows "toEnum n = word_of_nat n"
vsvD_2: fixes r :: "V"   and a :: "V"   and b :: "V"   and c :: "V" assumes "vsv r"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> r" shows "b = c"
67762: fixes x :: "'a"   and y :: "'a" shows "- (- - x \<sqinter> (y \<squnion> - y \<sqinter> x)) = - x"
delete_point_strong_block_in: fixes p :: "'a"   and bl :: "'a set" assumes "p \<notin> bl"   and "bl \<in># \<B>" shows "bl \<in># str_del_point_blocks p"
filterlim_iff: fixes f :: "'d \<Rightarrow> 'e"   and F2 :: "'e filter"   and F1 :: "'d filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
cases: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c stream" assumes "Graph_Defs.run E a" obtains x :: "'c" and y :: "'c" and xs :: "'c stream" where "a = x ## y ## xs" and "E x y" and "Graph_Defs.run E (y ## xs)"
integer_of_int_eq_of_int: shows "integer_of_int = of_int"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "x \<in> B"
synth_cut: fixes Y :: "msg"   and X :: "msg"   and H :: "msg set" assumes "Y \<in> synth (insert X H)"   and "X \<in> synth H" shows "Y \<in> synth H"
P_Cons: fixes p :: "pat"   and T :: "POPLmarkRecord.type"   and \<Delta>\<^sub>1 :: "binding list"   and fps :: "(char list \<times> pat) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta>\<^sub>2 :: "binding list"   and l :: "char list" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>\<^sub>1"   and "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>\<^sub>2"   and "fps\<langle>l\<rangle>\<^sub>? = \<bottom>" shows "\<turnstile> (l, p) \<Colon> fps [:] (l, T) \<Colon> fTs \<Rightarrow> \<up>\<^sub>e \<parallel>\<Delta>\<^sub>1\<parallel> 0 \<Delta>\<^sub>2 @ \<Delta>\<^sub>1"
assms: shows "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
inducts: fixes p :: "'c \<times> 'd"   and x :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "sndsp p x"   and "P (snd p)" shows "P x"
ind: fixes tp :: "(action \<times> nat) list" assumes "tp = concat (take as (tms_of ap))"   and "as \<le> length ap" shows "start_of (layout_of ap) as = Suc (length tp div 2)"
le1: shows "adjoint M0 * P * M0 \<le>\<^sub>L adjoint M0 * Q * M0"
individually_rational_on_Cd: fixes ds :: "'b set"   and X :: "'a set"   and d :: "'b" assumes "TruePref_tax.individually_rational_on ds X" shows "TruePref_tax.Cd d X = Singleton_for_d.dX X d"
span_idem: fixes X :: "'c set" assumes "subspace X" shows "span X = X"
append_rows_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A @\<^sub>r B = four_block_mat A (0\<^sub>m (dim_row A) 0) B (0\<^sub>m (dim_row B) 0)"
insert_4: fixes i :: "'b" assumes "i \<in> insert i__ I" shows "(f i \<longlongrightarrow> a i) F"
cong_trivial_identity: fixes A :: "'p"   and B :: "'p" shows "Cong A A B B"
R: fixes U :: "'a set" assumes "openin (top_of_set T) U" shows "openin (top_of_set S) (S \<inter> f -` U)"
telescope_summable': fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "f \<longlonglongrightarrow> c" shows "summable (\<lambda>n. f n - f (Suc n))"
pred_count_space_const1: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and c :: "'e" assumes "f \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "Measurable.pred M (\<lambda>x. f x = c)"
edr_characterization: shows "(\<forall>A. admits_diagonal_reduction A) = ((\<forall>A. admits_triangular_reduction A) \<and> (\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})))"
put_fail: fixes s :: "'v \<Rightarrow> int option" shows "put s fail = fail"
simps_1: fixes m :: "'e" shows "mut_m.mutator_phase_inv_aux m hp_Idle = (\<lambda>s. True)"
rep_eq: shows "Rep_coreflexive 1 = (1::'a)"
simps_12: shows "cospan_shape.comp cospan_shape.BB cospan_shape.AT = cospan_shape.Null"
and_pre_def: fixes P1 :: "'g \<Rightarrow> bool"   and P2 :: "'g \<Rightarrow> bool" shows "and_pre P1 P2 \<equiv> \<lambda>x. P1 x \<and> P2 x"
reflect_tranclp: fixes S :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "\<And>x y. S (f x) (f y) \<Longrightarrow> R x y"   and "bij_on_trancl S f"   and "S\<^sup>+\<^sup>+ (f x) (f y)" shows "R\<^sup>+\<^sup>+ x y"
conv_galois_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(y ; x \<le> z) = (y \<le> - (- z ; x\<^sup>T))"
borel_measurable_simple_function_limit: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<exists>g. (\<forall>n. g n \<in> borel_measurable (lebesgue_on S)) \<and> (\<forall>n. finite (range (g n))) \<and> (\<forall>x. (\<lambda>n. g n x) \<longlonglongrightarrow> f x))"
P10: shows "M Midpoint S R \<and> M Midpoint C C'"
hequiv_def: fixes H :: "(('b, 'c) fm list \<times> 'c) set"   and i :: "'c"   and j :: "'c" shows "hequiv H i j \<equiv> Nom j at i in' H"
right_adjoint_preserved_by_iso: fixes g :: "'a"   and \<phi> :: "'a"   and g' :: "'a" assumes "is_right_adjoint g"   and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>"   and "local.iso \<phi>" shows "is_right_adjoint g'"
handle1: fixes ck :: "bool"   and s1 :: "v sem_env"   and env :: "'b state"   and e :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and pes :: "(pat \<times> exp0) list" assumes "evaluate ck s1 env e (s2, Rval v1)" shows "evaluate ck s1 env (Handle e pes) (s2, Rval v1)"
fmrelD: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'f \<rightharpoonup>\<^sub>f 'd"   and n :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" assumes "fmrel P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
elims: fixes x :: "complex_interval"   and xa :: "complex_interval"   and y :: "complex_interval" assumes "x - xa = y" obtains R :: "real interval" and I :: "real interval" and R' :: "real interval" and I' :: "real interval" where "x = Complex_Interval R I" and "xa = Complex_Interval R' I'" and "y = Complex_Interval (R - R') (I - I')"
left_one_inf: fixes a :: "'a" shows "(1::'a) \<sqinter> a = a"
assms: fixes I :: "'a set" assumes "I \<in> \<I>" shows "I \<lless> J I \<and> finite I"
r\<^sub>0s\<^sub>1_is_cospan: shows "cospan_of_maps_in_bicategory_of_spans (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>1 h) (tab\<^sub>0 g)"
sublist_filter: fixes P :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "subseq (filter P l) l"
distinct_1: fixes x1 :: "'b"   and x2 :: "wake_up_status" shows "InWS x1 \<noteq> PostWS x2"
insert_map_upd: fixes j :: "nat"   and t :: "'c"   and f :: "nat \<Rightarrow> 'c"   and i :: "nat"   and s :: "'c" shows "(insert_map j t f)(i := s) = (if i = j then insert_map i s f else if i < j then insert_map j t (f(i := s)) else insert_map j t (f(i - Suc 0 := s)))"
lookup_operator_find: fixes nme :: "char list" shows "lookup_operator nme = find (\<lambda>op. fst op = nme) ast\<delta>"
simp3: fixes xs :: "wall"   and ys :: "wall"   and x :: "brick list"   and y :: "brick list" assumes "is_tangle_diagram xs \<and> is_tangle_diagram ys \<longrightarrow> is_tangle_diagram (xs \<otimes> ys)" shows "is_tangle_diagram (x * xs) \<and> is_tangle_diagram (y * ys) \<longrightarrow> is_tangle_diagram (x * xs \<otimes> y * ys)"
assms_3: shows "\<beta> \<in>\<^sub>\<circ> \<alpha> + \<omega>"
\<epsilon>o_in_hom: fixes x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide x" shows "\<guillemotleft>Adj.\<epsilon>o x : \<Delta>.map (G x) \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] x\<guillemotright>"
not_subseq_length: fixes ys :: "'c list"   and xs :: "'c list" assumes "length ys < length xs" shows "\<not> subseq xs ys"
lfp_fusion: fixes g :: "'e \<rightarrow> 'd"   and f :: "'e \<rightarrow> 'e"   and h :: "'d \<rightarrow> 'd" assumes "g\<cdot>\<bottom> = \<bottom>"   and "g oo f = h oo g" shows "g\<cdot>(fix\<cdot>f) = fix\<cdot>h"
refs_on_Array_set: fixes p :: "'b array"   and xs :: "'b list"   and h :: "heap"   and t :: "typerep"   and i :: "nat" shows "refs (Array.set p xs h) t i = refs h t i"
FullModel: shows "FFullModel InfModel.intTF local.intFF local.intPF"
T_LUB: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "T (lim_proc (range S)) = \<Inter> (T ` range S)"
basis_in_card: fixes \<E> :: "'a set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.basis_in \<E> B\<^sub>1"   and "\<E>.basis_in \<E> B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
Arr_implies_Ide_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "Ide (Dom t)"
1: shows "fst a \<subseteq>\<^sub>f s2"
poly_at_top_at_top: fixes p :: "real poly" assumes "1 \<le> degree p"   and "0 < lead_coeff p" shows "filterlim (poly p) at_top at_top"
gen_tlength_code_2: fixes n :: "nat"   and x :: "'a"   and xs :: "('a, 'b) tllist" shows "gen_tlength n (TCons x xs) = gen_tlength (n + 1) xs"
right_neutral: fixes a :: "'c" shows "a * (1::'c) = a"
ctr: fixes x :: "complex"   and y :: "complex" shows "x - y = Complex (Re x - Re y) (Im x - Im y)"
bij_betw_inv: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "\<exists>g. bij_betw g B A"
prj'_chine_composite_1: shows "cods.Prj\<^sub>1 \<cdot> \<mu>\<nu>\<pi>.chine = \<mu>.chine \<cdot> doms.Prj\<^sub>1"
S6MemUnch: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> [RNext rmCh mm ires p]_(m p) \<and> $S6 rmhist p \<longrightarrow> unchanged m p"
weight_less_stable: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'c) Term.term" assumes "weight_less s t" shows "weight_less (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
ahm_empty_impl: shows "map_empty ArrayHashMap_Impl.ahm_\<alpha> ArrayHashMap_Impl.ahm_invar ArrayHashMap_Impl.ahm_empty"
simps: fixes prec :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and t :: "taylor_model" shows "tm_round_floats prec I a t = tm_round_floats_of_normed prec I a (tm_norm_poly t)"
intro: fixes \<V> :: "'b set"   and \<B> :: "'b set multiset"   and \<U> :: "'b set"   and \<A> :: "'b set multiset" assumes "incidence_system \<V> \<B>"   and "proper_sub_set_system_axioms \<U> \<A> \<V> \<B>" shows "proper_sub_set_system \<U> \<A> \<V> \<B>"
rep_eq: fixes x :: "integer" shows "int_of_integer (Code_Numeral.dup x) = int_of_integer x + int_of_integer x"
rtranclp_eq_rtrancl_path: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "r\<^sup>*\<^sup>* x y = (\<exists>xs. rtrancl_path r x xs y)"
greaterThanAtMost_eq_atLeastAtMost_diff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less" shows "ord.greaterThanAtMost less_eq less a b = ord.atLeastAtMost less_eq a b - {a}"
select_convs_8: fixes p_iiface :: "char list"   and p_oiface :: "char list"   and p_src :: "'a word"   and p_dst :: "'a word"   and p_proto :: "8 word"   and p_sport :: "16 word"   and p_dport :: "16 word"   and p_tcp_flags :: "tcp_flag set"   and p_payload :: "char list"   and more :: "'b" shows "p_tcp_flags \<lparr>p_iiface = p_iiface::char list, p_oiface = p_oiface::char list, p_src = p_src::'a word, p_dst = p_dst::'a word, p_proto = p_proto::8 word, p_sport = p_sport::16 word, p_dport = p_dport::16 word, p_tcp_flags = p_tcp_flags::tcp_flag set, p_payload = p_payload::char list, \<dots> = more::'b\<rparr> = p_tcp_flags"
left_idem: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<union># (a \<union># b) = a \<union># b"
map_upd_map_add: fixes X :: "'f \<Rightarrow> 'g option"   and V :: "'f"   and v :: "'g" shows "X(V \<mapsto> v) = X ++ [V \<mapsto> v]"
assms_2: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(f has_derivative f' x) (at x within {a..b})"
plus_u_def: fixes x :: "'b\<^sub>\<bottom>"   and y :: "'b\<^sub>\<bottom>" shows "x + y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a + b))\<cdot>y"
distr_id2: fixes M :: "'d measure"   and N :: "'d measure" assumes "sets M = sets N" shows "distr N M (\<lambda>x. x) = N"
atMost_Int_atLeast: fixes n :: "'b" shows "{..n} \<inter> {n..} = {n}"
injectivity: fixes x :: "int poly"   and y :: "int poly" assumes "Ring_Hom_Poly.of_int_poly x = Ring_Hom_Poly.of_int_poly y" shows "x = y"
Cd_greatest: fixes d :: "'b"   and X :: "'a set" shows "Pdds'.Cd d X = greatest (Pd_shuffle_to_top (insert d__ ds') Y d) (X \<inter> Field (Pd_shuffle_to_top (insert d__ ds') Y d))"
runit'_naturality: fixes \<mu> :: "'d" assumes "D.arr \<mu>" shows "(\<mu> \<star>\<^sub>D src\<^sub>D \<mu>) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>] = \<r>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>] \<cdot>\<^sub>D \<mu>"
dims_Aw'_1: shows "dim_row Aw' = prod_list (nths (dims Aw) (Collect even))"
sum_closure: fixes x :: "i"   and y :: "i" assumes "CM" shows "\<exists>z. \<forall>v. O v z = (O v x \<or> O v y)"
simps_1: fixes r :: "bword"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Const r]<>\<nu>) = (Rep_bword r, Rep_bword r)"
s4: fixes x :: "'a"   and y :: "'a" shows "ad x \<cdot> ad (ad x \<cdot> y) = ad x \<cdot> ad y"
IH: assumes "wf\<^sub>c\<^sub>o\<^sub>n\<^sub>s\<^sub>t\<^sub>r S\<^sub>1 \<theta>\<^sub>1"   and "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>i,\<theta>\<^sub>i\<rangle>" shows "\<I> \<Turnstile>\<^sub>c \<langle>S\<^sub>1,\<theta>\<^sub>1\<rangle>"
confT_OK: fixes P :: "'a prog"   and h :: "heap"   and x :: "val"   and T :: "ty" shows "(P,h \<turnstile> x :\<le>\<^sub>\<top> OK T) = (P,h \<turnstile> x :\<le> T)"
first_in_tree_order_def: shows "first_in_tree_order = (\<lambda>ptr f. Heap_Error_Monad.bind (to_tree_order ptr) (\<lambda>tree_order. Heap_Error_Monad.bind (map_filter_M f tree_order) (case_list (return None) (\<lambda>x xa. return (Some x)))))"
assms_2: fixes r :: "'r"   and rs :: "'r list" assumes "set (r # rs) \<subseteq> R"   and "(r # rs) \<bullet>\<cdot> (m # ms) = (0::'m)" shows "r = (0::'r)"
var_eq_reflp: fixes x :: "('e, 'f, 'g) var0" shows "x \<sim> x"
mbisim: fixes t' :: "'t" assumes "t' \<noteq> t" shows "tbisim (ws t' = None) t' (ts1 t') m1 (ts2 t') m2"
wpo_ns_pre_mono: fixes s :: "('f, 'v) Term.term"   and t :: "('f, 'v) Term.term"   and f :: "'f"   and bef :: "('f, 'v) Term.term list"   and aft :: "('f, 'v) Term.term list" assumes "s \<succeq> t" shows "(\<forall>j\<in>set (\<sigma> (f, Suc (length bef + length aft))). Fun f (bef @ s # aft) \<succ> (bef @ t # aft) ! j) \<and> (Fun f (bef @ s # aft), Fun f (bef @ t # aft)) \<in> NS \<and> (\<forall>i<length (\<sigma> (f, Suc (length bef + length aft))). map ((!) (bef @ s # aft)) (\<sigma> (f, Suc (length bef + length aft))) ! i \<succeq> map ((!) (bef @ t # aft)) (\<sigma> (f, Suc (length bef + length aft))) ! i)"
fmrel_on_fsetI: fixes S :: "'c fset"   and P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "('c, 'd) fmap"   and n :: "('c, 'e) fmap" assumes "\<And>x. x |\<in>| S \<Longrightarrow> rel_option P (fmlookup m x) (fmlookup n x)" shows "fmrel_on_fset S P m n"
set_keys_to_list: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "set (punit.keys_to_list p) = keys p"
CatIdDomCod_1: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "dom\<^bsub>C\<^esub> (id\<^bsub>C\<^esub> X) = X"
indRelLST_to_SRel: fixes SP :: "'procS"   and SQ :: "'procS"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "SourceTerm SP \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> SourceTerm SQ" shows "(SP, SQ) \<in> SRel"
InvariantVarsMAfterUnitPropagate: fixes M :: "(Literal \<times> bool) list"   and F0 :: "Literal list list"   and Vbl :: "nat set"   and uLiteral :: "Literal"   and M' :: "(Literal \<times> bool) list" assumes "InvariantVarsM M F0 Vbl"   and "var uLiteral \<in> vars F0 \<union> Vbl"   and "M' = M @ [(uLiteral, False)]" shows "InvariantVarsM M' F0 Vbl"
lr_PE_result_rel: shows "PE_result_rel \<notin> unlr PE.delta"
rank_mono: fixes x :: "hf"   and y :: "hf" assumes "x \<le> y" shows "rank x \<le> rank y"
interval_integral_cong: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b" assumes "\<And>x. x \<in> einterval (min a b) (max a b) \<Longrightarrow> f x = g x" shows "interval_lebesgue_integral lborel a b f = interval_lebesgue_integral lborel a b g"
red_dyn_equiv_red_stat: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F" shows "reducedly_dynamically_complete_calculus Bot Inf entails Red_I Red_F = reducedly_statically_complete_calculus Bot Inf entails Red_I Red_F"
antisym: fixes b :: "'a option \<Rightarrow>\<^sub>0 nat"   and a :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "extended_ord b a"   and "extended_ord a b" shows "a = b"
x: shows "x \<notin> RF (TER (plus_current f g))"
lt_Suc: fixes a :: "nat"   and b :: "nat" shows "(a < b) = (Suc a = b \<or> Suc a < b)"
degen_acc_run_complete_aux2: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat"   and q :: "'Q"   and n :: "nat"   and ofs :: "nat" assumes "num_acc \<noteq> 0"   and "graph_defs.is_run (degeneralize_ext m) r"   and "\<forall>n<num_acc. \<exists>\<^sub>\<infinity>i. n \<in> acc (fst (r i))"   and "r i = (q, n)"   and "ofs < num_acc" shows "\<exists>j\<ge>i. \<exists>q'. r j = (q', (n + ofs) mod num_acc) \<and> (n + ofs) mod num_acc \<in> acc q'"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* C"   and "A\<^sub>Q \<sharp>* xvec"   and "distinct xvec" shows "Prop (P' \<parallel> Q)"
dagger_ext: fixes x :: "'a ppath set" shows "x \<subseteq> pp_star x"
finite_extension_in_carrier: fixes K :: "'a set"   and xs :: "'a list" assumes "K \<subseteq> carrier R"   and "set xs \<subseteq> carrier R" shows "finite_extension K xs \<subseteq> carrier R"
judge_weak_weakening_env: fixes e :: "Syntax.term"   and \<tau> :: "ty"   and \<Gamma> :: "(Syntax.var, ty) fmap" assumes "{$$} \<turnstile>\<^sub>W e : \<tau>" shows "\<Gamma> \<turnstile>\<^sub>W e : \<tau>"
index_mat_addcol_3: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat"   and k :: "nat"   and a :: "'a"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A"   and "k \<noteq> j" shows "addcol a k l A $$ (i, j) = A $$ (i, j)"
t_VAR: fixes \<Gamma> :: "(name \<times> ty) list"   and x :: "name"   and T :: "ty" assumes "valid \<Gamma>"   and "(x, T) \<in> set \<Gamma>" shows "\<Gamma> \<turnstile> VAR x : T"
purgeIdle_Nil_iff: fixes sl :: "('a, 'b, 'c) state list" shows "(GM_sec_model.purgeIdle sl = []) = (\<forall>s\<in>set sl. \<not> isState s)"
is_ub_thelub_ex: fixes S :: "'a set"   and x :: "'a" assumes "\<exists>u. S <<| u"   and "x \<in> S" shows "x \<sqsubseteq> lub S"
simps_2: fixes x :: "'a"   and a :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c" shows "rbt_del_from_right x a y s Empty = Branch R a y s (rbt_del x Empty)"
e_hd_snoc: fixes b :: "nat"   and x :: "nat" assumes "0 < e_length b" shows "e_hd (e_snoc b x) = e_hd b"
simps: fixes a1 :: "'e set"   and a2 :: "'f"   and D :: "'f set"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and e :: "'f" shows "((a1, a2) \<in> foldSetD D f e) = (a1 = {} \<and> a2 = e \<and> e \<in> D \<or> (\<exists>x A y. a1 = insert x A \<and> a2 = f x y \<and> x \<notin> A \<and> f x y \<in> D \<and> (A, y) \<in> foldSetD D f e))"
im_d_gchainsTr1_3: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and k :: "nat" assumes "d_gchain G n f"   and "\<exists>i\<le>n. f i \<in> f ` {i. i \<le> n} - {f 0}"   and "k < (LEAST j. f j \<in> f ` {i. i \<le> n} - {f 0})" shows "f k = f 0"
leD: fixes y :: "'g"   and x :: "'g" assumes "y \<le> x" shows "\<not> x < y"
L3_5ii_lemma1: fixes c1 :: "'a com"   and s1 :: "'a"   and Ts :: "('a ann_com option \<times> 'a set) list"   and s2 :: "'a"   and c2 :: "'a com"   and Ss :: "('a ann_com option \<times> 'a set) list"   and s3 :: "'a" assumes "(c1, s1) -P*\<rightarrow> (Parallel Ts, s2)"   and "All_None Ts"   and "(c2, s2) -P*\<rightarrow> (Parallel Ss, s3)"   and "All_None Ss" shows "(Seq c1 c2, s1) -P*\<rightarrow> (Parallel Ss, s3)"