hcju/isabelleps huggingface dataset:
connected_iff_seq: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "arr \<mu>"   and "arr \<nu>" shows "(sources \<nu> \<inter> targets \<mu> \<noteq> {}) = (src \<nu> = trg \<mu>)"
intros_23: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool" assumes "P,E,h,sh \<turnstile> e : T"   and "\<forall>C'\<in>set (C # Cs). is_class P C'"   and "\<not> sub_RI e"   and "\<forall>C'\<in>set (tl Cs). \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>"   and "b \<longrightarrow> (\<forall>C'\<in>set Cs. \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>)"   and "distinct Cs"   and "supercls_lst P Cs" shows "P,E,h,sh \<turnstile> INIT C (Cs,b) \<leftarrow> e : T"
sub_from_counter_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3 :: "'b state \<times> (v, v) result"   and extra3 :: "nat" assumes "evaluate ck env s e r3" shows "\<forall>count count' s' r'. clock s = count + extra3 \<and> r3 = (s', r') \<and> clock s' = count' + extra3 \<and> ck = True \<longrightarrow> evaluate True env (update_clock (\<lambda>_. count) s) e (update_clock (\<lambda>_. count') s', r')"
lookup_dflt_eq_lookup_pair: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs" shows "lookup_dflt xs = lookup_pair xs"
char_to_diget_inv: fixes c :: "char" assumes "is_digit_char c" shows "digit_to_char (char_to_digit c) = c"
seqE: fixes g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
fresh_swap_id: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "good X"   and "fresh xs x1 X"   and "fresh xs x2 X" shows "X #[x1 \<and> x2]_xs = X"
get_scdom_component_ptr_in_heap2: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_scdom_component ptr" shows "ptr |\<in>| object_ptr_kinds h"
proj_eq_sym: fixes p :: "'a \<times> 'a \<times> 'a"   and p' :: "'a \<times> 'a \<times> 'a" assumes "proj_eq p p'" shows "proj_eq p' p"
simps: fixes st :: "nat"   and ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_fourtimes_case_inv st ires rs (l, r) = (if st = Suc 0 then wcode_on_left_moving_2 ires rs (l, r) else if st = Suc (Suc 0) then wcode_on_checking_2 ires rs (l, r) else if st = 7 then wcode_goon_checking ires rs (l, r) else if st = 8 then wcode_right_move ires rs (l, r) else if st = 9 then wcode_erase2 ires rs (l, r) else if st = 10 then wcode_on_right_moving_2 ires rs (l, r) else if st = 11 then wcode_goon_right_moving_2 ires rs (l, r) else if st = 12 then wcode_backto_standard_pos_2 ires rs (l, r) else if st = t_twice_len + 14 then wcode_before_fourtimes ires rs (l, r) else False)"
tiebids'_def: shows "tiebids' = (\<lambda>allocation N G. summedBidVector (pseudoAllocation allocation <| (N \<times> finestpart G)) N G)"
Unifier_ground_rm_vars: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and X :: "'a set"   and t' :: "('b, 'a) Term.term" assumes "ground (subst_range s)"   and "Unifier (rm_vars X s) t t'" shows "Unifier s t t'"
disj: shows "ODE_dom ODE1 \<inter> ODE_dom ODE2 = {}"
eventually_at_top_not_equal: fixes c :: "'b" shows "\<forall>\<^sub>F x in at_top. x \<noteq> c"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and catch :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_catch return bind fail catch" shows "monad_fail return bind fail"
in_S: fixes x :: "'c stream" shows "x \<in> space (stream_space (count_space UNIV))"
accessible_bounds: fixes s :: "('i, 'a) state" shows "\<B>\<^sub>i ` {s'. (s, s') \<in> {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}\<^sup>*} = {\<B>\<^sub>i s}"
compat: fixes m :: "mini_alg" shows "ma_compatible (ma_of_rat (rat_of_int (sgn a) * rat_of_nat sq / rat_of_int b)) m = True"
H3: fixes x12 :: "nat"   and q :: "int mpoly"   and r :: "int mpoly" assumes "x12 \<in># dom_m A"   and "the (fmlookup A x12) * q - r \<in> More_Modules.ideal polynomial_bool"   and "r - spec \<in> More_Modules.ideal polynomial_bool" shows "spec \<in> pac_ideal (set_mset B)"
update_convs_2: fixes refs' :: "(typerep \<Rightarrow> nat \<Rightarrow> nat) \<Rightarrow> typerep \<Rightarrow> nat \<Rightarrow> nat"   and arrays :: "typerep \<Rightarrow> nat \<Rightarrow> nat list"   and refs :: "typerep \<Rightarrow> nat \<Rightarrow> nat"   and lim :: "nat"   and more :: "'b" shows "refs_update refs' \<lparr>arrays = arrays, refs = refs, lim = lim, \<dots> = more\<rparr> = \<lparr>arrays = arrays, refs = refs' refs, lim = lim, \<dots> = more\<rparr>"
times_divide_eq_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide b c) a = divide (times b a) c"
cases: fixes a :: "atom"   and H :: "chmsg set" assumes "a \<in> atoms H" obtains c :: "secprop set" and A :: "agent" and B :: "agent" and M :: "atom list" and At :: "atom" where "a = Ata__" and "StatCh c__ A B (Msg M__) \<in> H" and "Ata__ \<in> set M__"   | c :: "secprop set" and K :: "key" and M :: "atom list" and At :: "atom" where "a = At" and "DynCh c K (Msg M) \<in> H" and "At \<in> set M"
f_ac: fixes x :: "'a"   and y :: "'a"   and z :: "'b" shows "f x (f y z) = f y (f x z)"
stabilizer_is_subgroup: fixes m :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)" shows "subgroup (conjG.stabilizer m) G"
P: fixes J :: "'a set"   and H :: "'a set" assumes "J \<subseteq> H"   and "finite H"   and "H \<subseteq> A" shows "Pi\<^sub>M J (\<lambda>_. M) = distr (Pi\<^sub>M H (\<lambda>_. M)) (Pi\<^sub>M J (\<lambda>_. M)) (\<lambda>f. restrict f J)"
right_total_rel_set: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A" shows "right_total (rel_set A)"
pred_inject: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and aa :: "'d stream" shows "pred_stream P (a ## aa) = (P a \<and> pred_stream P aa)"
bit_seq_of_nat_set_inj: shows "inj bit_seq_of_nat_set"
adjacent_card: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y" shows "card x = card y"
rbt_sorted_fold_rbt_insertwk: fixes t :: "('a, 'c) rbt"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "('a \<times> 'c) list" assumes "rbt_sorted t" shows "rbt_sorted (List.fold (\<lambda>(k, v). rbt_insert_with_key f k v) xs t)"
bij_h: shows "bij_betw (\<lambda>j. j + i) {0..<Suc (degree f)} {i..<Suc (degree f + i)}"
inext_eq_fin: fixes a :: "nat"   and I :: "nat set"   and b :: "nat" assumes "inext a I = inext b I"   and "finite I"   and "a \<noteq> Max I"   and "b \<noteq> Max I" shows "a = b"
lfilledk_def_1: shows "Lfilled j lholed' (vs' @ [$Return]) lfilledk"
that: fixes s' :: "'state" assumes "(c, s) \<rightarrow>*t s'"   and "s' \<approx> t'" shows "thesis"
multiset_prod_fact_insert: fixes A :: "'b multiset"   and x :: "'b" shows "(\<Prod>y\<in>set_mset (A + {#x#}). fact (count (A + {#x#}) y)) = (count A x + 1) * (\<Prod>y\<in>set_mset A. fact (count A y))"
A72_DAcc_level1: shows "DAcc level1 sA72 = {sA82}"
add_right_strict_mono: fixes n :: "'a" shows "strict_mono (\<lambda>x. x + n)"
arg_elim: fixes f :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "f x"   and "x = y" shows "f y"
m'_def: shows "m' = drop_perp (Rep_hyp2 c') l'"
discr_Par: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis c1"   and "PL_Indis.discr tval aval indis c2" shows "PL_Indis.discr tval aval indis (c1 | c2)"
inf_prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<sqinter> ys \<equiv> xs \<down> inf_prefix_length xs ys"
unat_of_bl_128_16_less_helper: fixes b :: "16 word" shows "unat (of_bl (to_bl b)) < 2 ^ 16"
add_im_sheaf_def: shows "imS.add_im_sheaf \<equiv> \<lambda>V. +\<^bsub>identity S \<^sup>\<inverse> S V\<^esub>"
D_def: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list" shows "D r \<tau> \<sigma> \<sigma>' \<tau>' = ((r|\<sigma> @ \<tau>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r \<and> (r|\<tau> @ \<sigma>'|, r|\<tau>| + r|\<sigma>|) \<in> mul_eq r)"
map_poly_idom_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "map_poly_idom_hom hom \<equiv> idom_hom hom"
add_one_2_stop: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_add_one2 m lm (b, Oc # list)" shows "wprepare_stop m lm (tl b, hd b # Oc # list)"
img_point_elem_set: fixes x :: "'s"   and a :: "'s" assumes "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>" shows "img x \<in> local.set a"
sets_restrict_UNIV: fixes M :: "'b measure" shows "sets (restrict_space M UNIV) = sets M"
qSwapAll_involutive: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and A :: "('d, 'e, 'f, 'g, 'h) qAbs" shows "X #[[x \<and> y]]_zs #[[x \<and> y]]_zs = X \<and> A $[[x \<and> y]]_zs $[[x \<and> y]]_zs = A"
ideal_prod_in_carrier: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "I \<cdot> J \<subseteq> carrier R"
simps_2: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b" shows "f ^^ Suc n = f \<circ> f ^^ n"
invar_def: fixes t :: "(('a \<times> 'b) \<times> color) tree" shows "M.invar t \<equiv> rbt t \<and> sorted1 (Tree2.inorder t)"
valid_valid': fixes l :: "color list" assumes "valid l" shows "valid' l"
length_deleteAround: fixes g :: "graph"   and v :: "nat"   and ps :: "(nat \<times> nat) list" shows "|deleteAround g v ps| \<le> |ps|"
IH1: assumes "simple_labels \<Gamma>"   and "wellformed \<Gamma>"   and "\<exists>pn. p1 \<in> subterms (\<Gamma> pn)"   and "l \<in> labels \<Gamma> p1" shows "\<forall>p'\<in>sterms \<Gamma> p1. l \<in> labels \<Gamma> p'"
simps_4: fixes v :: "'b"   and va :: "'b"   and xs :: "('b, 'c) Combinators list" shows "singleCombinators (DenyAllFromTo v va # xs) = singleCombinators xs"
option2list_def: fixes n :: "'c option" shows "option2list n \<equiv> case n of None \<Rightarrow> [] | Some s \<Rightarrow> [s]"
cases: fixes x :: "tm \<times> tm" obtains x :: "name" and p :: "tm" and q :: "tm" and v :: "name" and y :: "name" and x' :: "name" where "atom x \<sharp> (p, q, v, y, x')" and "atom x' \<sharp> (p, q, v, y)" and "atom y \<sharp> (p, q, v)" and "atom v \<sharp> (p, q)" and "xa__ = (p, q)"
ex_Inf: fixes A :: "'b set" shows "\<exists>inf. is_Inf A inf"
unique_monic_hensel_factorization: fixes C :: "int poly"   and Fs :: "int poly multiset"   and n :: "nat" assumes "unique_factorization_m C (1, Fs)"   and "monic C"   and "square_free_m C"   and "n \<noteq> 0" shows "\<exists>Gs. poly_mod.unique_factorization_m (p ^ n) C (1, Gs)"
least_mapp_least_map_eq: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "least_mapp f (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> least_map f S)"
row_length_Nil: shows "rat_poly.row_length [] = 0"
closed_adh: fixes T :: "'b set set"   and A :: "'b set" assumes "topology T"   and "A \<subseteq> carrier\<^bsub>T\<^esub>" shows "(A closed\<^bsub>T\<^esub>) = (\<forall>x\<in>carrier\<^bsub>T\<^esub>. x adh\<^bsub>T\<^esub> A \<longrightarrow> x \<in> A)"
complex_of_real_Re: fixes k :: "complex" assumes "is_real k" shows "cor (Re k) = k"
tagged_division_ofD_6: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_division_of i" shows "\<Union> {K. \<exists>x. (x, K) \<in> s} = i"
trcl_uncons: fixes c :: "'c"   and a :: "'d"   and w :: "'d list"   and c' :: "'c"   and t :: "('c \<times> 'd \<times> 'c) set" assumes "(c, a # w, c') \<in> trcl t" shows "\<exists>ch. (c, a, ch) \<in> t \<and> (ch, w, c') \<in> trcl t"
get_tag_name_locs_impl: shows "get_tag_name_locs = CD.a_get_tag_name_locs"
append1_eq_conv: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list"   and y :: "'b" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
exhaust_sel: fixes Direction :: "('a, 'b) Direction" obtains "Direction = Direction (Direction.lt Direction) (Direction.LBI Direction) (Direction.UBI Direction) (Direction.LB Direction) (Direction.UB Direction) (Direction.LI Direction) (Direction.UI Direction) (Direction.UBI_upd Direction) (Direction.LE Direction) (Direction.GE Direction) (Direction.le_rat Direction)"
hequiv_names: fixes H :: "(('c, 'd) fm list \<times> 'd) set"   and i :: "'d"   and j :: "'d" assumes "hequiv H i j" shows "i \<in> names H"
cases: fixes x :: "(char list, char list, char list list \<times> t) namespace \<times> t" obtains tenvT :: "(char list, char list, char list list \<times> t) namespace" and tv :: "char list" where "x = (tenvT__, Tvar tv)"   | tenvT :: "(char list, char list, char list list \<times> t) namespace" and ts :: "t list" and tn :: "tctor" where "x = (tenvTa__, Tapp ts tn)"   | tenvT :: "(char list, char list, char list list \<times> t) namespace" and n :: "nat" where "x = (tenvT, Tvar_db n)"
p_t_stop: shows "other.p other.t_stop = other.p_max"
all_orientable_characterisations_6: shows "(\<forall>x. orientable_1 x) = (\<forall>x. orientable_7 x)"
\<phi>_in_terms_of_\<eta>o: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = G f \<cdot>\<^sub>D \<eta>o y"
unary_wt_sym_0_gt: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> 'c"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and f :: "'b"   and g :: "'b" assumes "kbo_std_basis ground_heads_var gt_sym arity_sym arity_var wt_sym \<epsilon> \<delta> extf"   and "arity_sym f = 1"   and "wt_sym f = (0::'c)" shows "gt_sym f g \<or> g = f"
dist_triangle_lt: fixes x :: "'a"   and z :: "'a"   and y :: "'a"   and e :: "real" assumes "dist x z + dist y z < e" shows "dist x y < e"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_left_moving_1_B ires rs (l, r) = (\<exists>ml mr rn. l = Bk \<up> ml @ Oc # Oc # ires \<and> r = Bk \<up> mr @ Oc \<up> Suc rs @ Bk \<up> rn \<and> Suc 0 < ml + mr \<and> 0 < mr)"
topological_tendstoI: fixes "open" :: "'e set \<Rightarrow> bool"   and l :: "'e"   and f :: "'f \<Rightarrow> 'e"   and F :: "'f filter" assumes "class.topological_space open"   and "\<And>S. \<lbrakk>open S; l \<in> S\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F x in F. f x \<in> S" shows "filterlim f (topological_space.nhds open l) F"
le_t_stop_iff: fixes a :: "real"   and x :: "real"   and v :: "real" assumes "a \<noteq> 0" shows "(x \<le> movement.t_stop a v) = (if a < 0 then 0 \<le> movement.p' a v x else movement.p' a v x \<le> 0)"
assms_2: shows "\<turnstile> F \<longrightarrow> G"
execl_sub_aux: fixes Q :: "instr list \<Rightarrow> int \<times> (char list \<Rightarrow> int) \<times> int list \<Rightarrow> bool"   and P' :: "instr list"   and pc :: "int"   and s :: "char list \<Rightarrow> int"   and stk :: "int list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and P :: "instr list"   and F :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> (int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and m :: "nat"   and n :: "nat"   and cfs' :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "\<And>m n. \<forall>k\<in>{m..<n}. Q P' (((pc, s, stk) # cfs) ! k) \<Longrightarrow> P' \<Turnstile> map (off P) (case m of 0 \<Rightarrow> (pc, s, stk) # take n cfs | Suc m \<Rightarrow> F cfs m n)"   and "\<forall>k\<in>{m..<n + m + length cfs'}. Q P' ((cfs' @ (pc, s, stk) # cfs) ! (k - m))" shows "P' \<Turnstile> (pc - size P, s, stk) # map (off P) (take n cfs)"
factors_as_mset_factors: fixes x :: "'a"   and fs :: "'a list" assumes "x \<noteq> (0::'a)"   and "x \<noteq> (1::'a)" shows "comm_semiring_1_class.factors fs x = mset_factors (mset fs) x"
take_lemma: fixes x :: "CValue"   and y :: "CValue" assumes "\<And>n. \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>x = \<psi>\<^sup>E\<^bsub>n\<^esub>\<cdot>y" shows "x = y"
mem_existence_ivl_iv_defined_2: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "x0 \<in> X"
gin: fixes n :: "nat" shows "from_nat_into \<G> n \<in> \<G>"
pr_naturality_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and g :: "'b"   and b :: "'b"   and d :: "'b"   and f :: "'b"   and a :: "'b"   and c :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "partial_magma.arr C g"   and "partial_magma.dom C g = b"   and "partial_magma.cod C g = d"   and "partial_magma.arr C f"   and "partial_magma.dom C f = a"   and "partial_magma.cod C f = c" shows "C (pr1 c d) (elementary_category_with_binary_products.prod C pr0 pr1 f g) = C f (pr1 a b)"
finite_lprefix_nitpick_simps_2: fixes xs :: "'b llist" shows "Coinductive_List.finite_lprefix <> xs = True"
RF_essential: fixes \<Gamma> :: "('c, 'd) web_scheme"   and S :: "'c set" shows "RF\<^bsub>\<Gamma>\<^esub> (\<E>\<^bsub>\<Gamma>\<^esub> S) = RF\<^bsub>\<Gamma>\<^esub> S"
set_plus_mono: fixes C :: "'b set"   and D :: "'b set"   and a :: "'b" assumes "C \<subseteq> D" shows "a +o C \<subseteq> a +o D"
Pi_pmf_insert': fixes A :: "'a set"   and x :: "'a"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf" assumes "finite A"   and "x \<notin> A" shows "Pi_pmf (insert x A) dflt p = p x \<bind> (\<lambda>y. Pi_pmf A dflt p \<bind> (\<lambda>f. return_pmf (f(x := y))))"
fpairsD_2: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> fpairs r" shows "\<exists>a b. x = [a, b]\<^sub>\<circ>"
perm_empty: fixes xs :: "'d list" shows "([] <~~> xs) = (xs = [])"
starT: fixes x :: "'b" assumes "x \<in> T" shows "open_segment b x \<subseteq> rel_interior T"
simps: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "v"   and a5 :: "(pat \<times> exp0) list"   and a6 :: "v"   and a7 :: "'b state \<times> (v, v) result" shows "evaluate_match a1 a2 a3 a4 a5 a6 a7 = ((\<exists>ck env v1 err_v s. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = v1 \<and> a5 = [] \<and> a6 = err_v \<and> a7 = (s, Rerr (Rraise err_v))) \<or> (\<exists>ck env env' v1 p pes e bv err_v s. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = v1 \<and> a5 = (p, e) # pes \<and> a6 = err_v \<and> a7 = bv \<and> allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s) p v1 [] = Match env' \<and> evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e bv) \<or> (\<exists>ck env v1 p e pes bv s err_v. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = v1 \<and> a5 = (p, e) # pes \<and> a6 = err_v \<and> a7 = bv \<and> allDistinct (pat_bindings p []) \<and> pmatch (c env) (refs s) p v1 [] = No_match \<and> evaluate_match ck env s v1 pes err_v bv) \<or> (\<exists>ck env v1 p e pes s err_v. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = v1 \<and> a5 = (p, e) # pes \<and> a6 = err_v \<and> a7 = (s, Rerr (Rabort Rtype_error)) \<and> pmatch (c env) (refs s) p v1 [] = Match_type_error) \<or> (\<exists>ck env v1 p e pes s err_v. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = v1 \<and> a5 = (p, e) # pes \<and> a6 = err_v \<and> a7 = (s, Rerr (Rabort Rtype_error)) \<and> \<not> allDistinct (pat_bindings p [])))"
ws_a: shows "ws a = action_translation (J (Suc n)) (justifying_ws (J (Suc n)) a'')"
Node: shows "t = \<langle>ls, x, rs\<rangle>"
assms_3: fixes a' :: "real"   and x :: "real" assumes "a \<le> a'"   and "a' \<le> x" shows "f integrable_on {a'..x}"
assms_2: fixes l :: "nat list" shows "g l + sum_list l < ack kg (sum_list l)"
in_Reals_norm: fixes z :: "complex" assumes "z \<in> \<real>" shows "cmod z = \<bar>Re z\<bar>"
m_ivl_anti_mono: fixes y :: "ivl"   and x :: "ivl" assumes "y \<le> x" shows "m_ivl x \<le> m_ivl y"
iIN_Suc_insert_conv: fixes n :: "nat"   and d :: "nat" shows "insert (Suc (n + d)) [n\<dots>,d] = [n\<dots>,Suc d]"
fixing_ide_gives_functor_1: fixes a1 :: "'a" assumes "ide a1" shows "functor (\<cdot>) (\<cdot>) (\<lambda>f2. local.Prod (a1, f2))"
And: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and \<psi> :: "('msaux, 'muaux) mformula"   and \<psi>' :: "Formula.formula"   and pos :: "bool"   and \<chi> :: "Formula.formula"   and buf :: "event_data option list set list \<times> event_data option list set list" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "wf_mformula \<sigma> j P V n R \<psi> \<psi>'"   and "if pos then \<chi> = formula.And \<phi>' \<psi>' else \<chi> = formula.And \<phi>' (formula.Neg \<psi>') \<and> fv \<psi>' \<subseteq> fv \<phi>'"   and "wf_mbuf2' \<sigma> P V j n R \<phi>' \<psi>' buf" shows "wf_mformula \<sigma> j P V n R (MAnd (fv \<phi>') \<phi> pos (fv \<psi>') \<psi> buf) \<chi>"
foldl_un_empty_eq: fixes i :: "'a set"   and ww :: "'a set list" shows "foldl (\<union>) i ww = i \<union> foldl (\<union>) {} ww"
coboundedI2: fixes b :: "'e"   and c :: "'e"   and a :: "'e" assumes "b \<le> c" shows "inf a b \<le> c"
subU: shows "cball (0::'a) \<delta> \<subseteq> U"
distinct_7: fixes x2 :: "('a, 'b, 'c, 'd) Tree"   and x31 :: "'d"   and x32 :: "'b" shows "tNot x2 \<noteq> tPred x31 x32"
Eps_Opt_eq_None: fixes P :: "'a \<Rightarrow> bool" shows "(Eps_Opt P = None) = (\<not> Ex P)"
dual_le: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (y ^ o \<le> x ^ o)"
case_distrib: fixes h :: "'f \<Rightarrow> 'e"   and f1 :: "'g \<Rightarrow> 'f"   and f2 :: "'h \<Rightarrow> 'f"   and sum :: "'g + 'h" shows "h (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (case sum of Inl x \<Rightarrow> h (f1 x) | Inr x \<Rightarrow> h (f2 x))"
subst_id_on_type_scheme_list: fixes A :: "type_scheme list"   and S :: "nat \<Rightarrow> typ" assumes "\<forall>x\<in>free_tv A. S x = TVar x" shows "$ S A = A"
summable_powser_split_head: fixes f :: "nat \<Rightarrow> 'a"   and z :: "'a" shows "summable (\<lambda>n. f (Suc n) * z ^ n) = summable (\<lambda>n. f n * z ^ n)"
simps_2: fixes X :: "'b list set"   and r :: "'b rexp" shows "lang_trm (Trn X r) = X \<cdot> lang r"
Refl_Field_Restr: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" assumes "Refl r" shows "Field (Restr r A) = Field r \<inter> A"
notin_range_Some: fixes x :: "'h option" shows "(x \<notin> range Some) = (x = None)"
preal_mult_def: fixes r :: "preal"   and s :: "preal" shows "r * s \<equiv> Abs_preal (mult_set (Rep_preal r) (Rep_preal s))"