hcju/isabelleps huggingface dataset:
simps: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" shows "hindrance_by \<Gamma> f \<epsilon> = (\<exists>a. a \<in> A \<Gamma> \<and> a \<notin> \<E>\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f) \<and> d_OUT f a < weight \<Gamma> a \<and> \<epsilon> < weight \<Gamma> a - d_OUT f a)"
mat_leqb_eqc_satisfies1: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and x :: "nat \<Rightarrow> rat"   and i :: "nat" assumes "x \<Turnstile>\<^sub>c\<^sub>s set (mat_leqb_eqc A b c)"   and "i < dim_vec b"   and "i < dim_row A" shows "matrix_to_lpolies A ! i \<lbrace> x \<rbrace> \<le> b $ i"
language_def: shows "language \<equiv> {xs. nextl (init M) xs \<in> final M}"
multlist_closed: fixes fs :: "'a list" assumes "set fs \<subseteq> carrier G" shows "foldr (\<otimes>) fs \<one> \<in> carrier G"
semilattice_order_axioms: shows "semilattice_order min (\<le>) (<)"
distrib: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b) * c = a * c + b * c"
l1: shows "\<forall>i<length Ds. (0::'a) \<le> ll i"
normalize_rules_singleton: fixes f :: "'b match_expr \<Rightarrow> 'b match_expr list"   and m :: "'b match_expr"   and a :: "action" shows "normalize_rules f [Rule m a] = map (\<lambda>m. Rule m a) (f m)"
monad_state_envT': assumes "monad_state Monad_Overloading.return Monad_Overloading.bind get put" shows "monad_state Monad_Overloading.return Monad_Overloading.bind get put"
list_all2_refl: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "\<And>x. P x x" shows "list_all2 P xs xs"
Collect_Int_Times: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'f set"   and B :: "'g set" shows "{(x, y). R x y} \<inter> A \<times> B = {(x, y). R x y \<and> x \<in> A \<and> y \<in> B}"
card_of_ordLeq_infinite: fixes A :: "'d set"   and B :: "'e set" assumes "|A| \<le>o |B|"   and "infinite A" shows "infinite B"
finite_measure_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure" assumes "subalgebra M F"   and "finite_measure M" shows "finite_measure (restr_to_subalg M F)"
cond_override_empty1: fixes c :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd option" shows "cond_override c Map.empty t = t"
tendsto_power: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and n :: "nat" assumes "(f \<longlongrightarrow> a) F" shows "((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
rotate1_length01: fixes xs :: "'b list" assumes "length xs \<le> 1" shows "rotate1 xs = xs"
scope_simp: fixes v :: "'form \<times> nat list"   and i :: "('form, 'var) antecedent"   and v' :: "'form \<times> nat list" shows "scope' v i v' = (\<exists>c is' is. v = (c, is) \<and> v' = (c, is') \<and> c \<in> set conclusions \<and> is' \<in> (#) 0 ` it_paths (it' c) \<and> (\<exists>i. prefix (is @ [i]) is' \<and> i = in_port_at (c, is) i))"
assms_3: shows "bij_betw \<phi> S.Univ S'.Univ"
wf_\<Gamma>_def: fixes rs :: "'a rule list" shows "wf_\<Gamma> rs \<equiv> \<forall>rsg\<in>ran \<Gamma> \<union> {rs}. \<forall>r\<in>set rsg. \<forall>chain. get_action r = Call chain \<longrightarrow> \<Gamma> chain \<noteq> None"
top_unique: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and a :: "'c" assumes "class.order_top less_eq less top" shows "less_eq top a = (a = top)"
T_TApp: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1\<^sub>1 :: "POPLmarkRecord.type"   and T\<^sub>1\<^sub>2 :: "POPLmarkRecord.type"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t\<^sub>1 : (\<forall><:T\<^sub>1\<^sub>1. T\<^sub>1\<^sub>2)"   and "\<Gamma> \<turnstile> T\<^sub>2 <: T\<^sub>1\<^sub>1" shows "\<Gamma> \<turnstile> t\<^sub>1 \<bullet>\<^sub>\<tau> T\<^sub>2 : T\<^sub>1\<^sub>2[0 \<mapsto>\<^sub>\<tau> T\<^sub>2]\<^sub>\<tau>"
hm_\<alpha>'_def: fixes m :: "('a, ('b, 'c) assoc_list) RBT.rbt" shows "hm_\<alpha>' m \<equiv> \<lambda>hc. case rm.\<alpha> m hc of None \<Rightarrow> None | Some lm \<Rightarrow> Some (lm.\<alpha> lm)"
start_of_nonzero_1: fixes ly :: "nat list"   and as :: "nat" shows "0 < start_of ly as"
consistency: fixes s1 :: "'a"   and s2 :: "'a"   and s3 :: "'a"   and rs :: "('b \<times> 'c) list" assumes "s1 \<preceq> s2"   and "s2 \<preceq> s3"   and "s3 = s1 \<star> rs" shows "\<exists>rs' rs''. s2 = s1 \<star> rs' \<and> s3 = s2 \<star> rs'' \<and> set rs' \<subseteq> set rs \<and> set rs'' \<subseteq> set rs"
card_def: shows "card \<equiv> folding.F (\<lambda>_. Suc) 0"
induced_play_def: fixes \<sigma> :: "'c list \<Rightarrow> 'c" shows "induced_play \<sigma> \<equiv> ltake (2 * N) \<circ> lmap last \<circ> ltl \<circ> strategy_progression \<sigma>"
gen_shleg_poly_def: fixes k :: "nat" shows "local.gen_shleg_poly k = (pderiv ^^ k) ([:0, 1, - 1:] ^ n)"
not_in_iff: fixes x :: "'d"   and M :: "'d multiset" shows "(x \<notin># M) = (count M x = 0)"
lins2D_4: fixes j :: "nat"   and k :: "nat"   and e :: "node fset \<times> command_gadget \<times> node fset"   and w :: "node" assumes "j < length (Inr e # \<pi>)"   and "k < length (Inr e # \<pi>)"   and "(Inr e # \<pi>) ! j = Inr e"   and "(Inr e # \<pi>) ! k = Inl w"   and "w |\<in>| thd3 e" shows "j < k"
pre_weak_rel: fixes Pre :: "'a upred"   and I :: "'a upred"   and P :: "'a upred"   and Post :: "'a upred" assumes "`Pre \<Rightarrow> I`"   and "(I \<Rightarrow> Post) \<sqsubseteq> P" shows "(Pre \<Rightarrow> Post) \<sqsubseteq> P"
intensional_simps_15: fixes P :: "'a \<Rightarrow> bool" shows "P \<longrightarrow> \<not> P \<equiv> \<not> P"
galois_aux: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(x \<sqinter> y = \<delta>) = (x \<le> y\<^sup>c)"
weak_ring_morphism_is_hom: fixes f :: "('c multiset \<Rightarrow> 'a) list \<Rightarrow> 'd"   and I :: "('c multiset \<Rightarrow> 'a) list set" assumes "weak_ring_morphism f I (poly_ring L)" shows "f \<in> ring_hom (poly_ring L) (image_ring f (poly_ring L))"
last_index_less: fixes x :: "'c"   and xs :: "'c list"   and n :: "nat" assumes "x \<in> set xs"   and "length xs \<le> n" shows "last_index xs x < n"
hd_rev: shows "coeff (Poly r) (length r - Suc 0) = hd (rev r)"
unique_servKeys: fixes A :: "agent"   and K :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and K' :: "nat"   and B' :: "agent"   and Ts' :: "msg"   and X' :: "msg" assumes "Says Tgs A (Crypt K \<lbrace>Key servK, Agent B, Ts, X\<rbrace>) \<in> set evs"   and "Says Tgs A' (Crypt K' \<lbrace>Key servK, Agent B', Ts', X'\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV_gets" shows "A = A' \<and> B = B' \<and> K = K' \<and> Ts = Ts' \<and> X = X'"
assms_4: assumes "a \<in># A"   and "b \<in># B" shows "P"
P2: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "|~ \<box>F \<longrightarrow> F \<and> \<circle>F"
in_Rats_imp_algebraic: fixes x :: "'a" assumes "x \<in> \<rat>" shows "algebraic x"
poly_entire: fixes p :: "'a list"   and q :: "'a list" shows "(poly (p *** q) = poly []) = (poly p = poly [] \<or> poly q = poly [])"
assms_1: shows "(f expands_to MS (MSLCons (C, e) xs) g) (b # basis)"
hinv_def: shows "h\<^sup>-\<^sup>1 \<equiv> inv_into (UNIV // R) h"
distinct_5: fixes x1 :: "'a"   and x4 :: "'a \<times> nat" shows "CFG_node x1 \<noteq> Actual_in x4"
sum_eq: shows "integral\<^sup>N (count_space A) f = integral\<^sup>N (count_space B) g"
asset_i_owner_ii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, Crypt (Auth_ShaKey m) (PriKey A)) \<in> s"   and "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s" shows "False"
vdomain_vsubset_VUnion2: fixes r :: "V" shows "\<D>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> \<Union>\<^sub>\<circ> (\<Union>\<^sub>\<circ> r)"
simps_6: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b \<Rightarrow> 'b" shows "rec_list f1 f2 [] = f1"
h_minus_OUT_finite: fixes i :: "nat"   and x :: "'v" shows "d_OUT (h_minus i) x \<noteq> \<top>"
l3_init_def: shows "l3_init \<equiv> {\<lparr>ik = ik_init, secret = {}, progress = Map.empty, signals = \<lambda>x. 0, l3_state.bad = bad_init\<rparr>}"
integrable_product_swap_iff: fixes f :: "'b \<times> 'a \<Rightarrow> 'c" shows "integrable (lborel \<Otimes>\<^sub>M lborel) (\<lambda>(x, y). f (y, x)) = integrable (lborel \<Otimes>\<^sub>M lborel) f"
assms_2: shows "j \<in> {b \<bullet> y |y. y \<in> sat_dual A c}"
offending: fixes G :: "'a graph" shows "c_offending_flows m G = (if c_sinvar m G then {} else {{e \<in> edges G. \<not> P e}})"
intro: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> real \<times> real" assumes "R2 i j"   and "green_typeII_cube_axioms i j twoC F" shows "green_typeII_cube i j twoC F"
set_mult_inclusion: fixes H :: "'a set"   and P :: "'a set" assumes "subgroup H G"   and "P \<subseteq> carrier G"   and "H <#>\<^bsub>G\<^esub> P \<subseteq> H" shows "P \<subseteq> H"
singl_meas_if: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "f \<in> space M \<rightarrow> space N"   and "\<forall>r\<in>range f \<inter> space N. \<exists>A\<in>sets N. range f \<inter> A = {r}" shows "point_measurable (fct_gen_subalgebra M N f) (space N) f"
val_le_trans: fixes v1 :: "val"   and v2 :: "val"   and v3 :: "val" assumes "v1 \<sqsubseteq> v2"   and "v2 \<sqsubseteq> v3" shows "v1 \<sqsubseteq> v3"
pw_nres_rel_iff: fixes a :: "'b nres"   and b :: "'c nres"   and A :: "('b \<times> 'c) set" shows "((a, b) \<in> \<langle>A\<rangle>nres_rel) = (nofail (\<Down> A b) \<longrightarrow> nofail a \<and> (\<forall>x. inres a x \<longrightarrow> inres (\<Down> A b) x))"
convex_hull_insert_segments: fixes a :: "'b"   and S :: "'b set" shows "convex hull insert a S = (if S = {} then {a} else \<Union> (closed_segment a ` (convex hull S)))"
convex_segment_1: fixes a :: "'b"   and b :: "'b" shows "convex {a--b}"
strict_mono_compose: fixes r :: "'a \<Rightarrow> 'b"   and s :: "'c \<Rightarrow> 'a" assumes "strict_mono r"   and "strict_mono s" shows "strict_mono (\<lambda>x. r (s x))"
stabilizer_supergrp_P: fixes P :: "'a set"   and p :: "nat" assumes "finite (carrier (G\<lparr>carrier := conj.stabilizer P\<rparr>))"   and "P \<in> stabGroup.subgroups_of_size p" shows "subgroup P (G\<lparr>carrier := conj.stabilizer P, carrier := group_action.stabilizer (G\<lparr>carrier := conj.stabilizer P\<rparr>) (stabGroup.conjugation_action p) P\<rparr>)"
that_1: shows "(x - y) \<bullet> rot (a - b) < 0"
SAs_States_HAStates: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto" assumes "SA \<in> SAs A" shows "States SA \<subseteq> HAStates A"
f12: shows "ucast (get_CWP (cpu_reg_val PSR s1)) = fst (fst (get_curr_win () s2))"
Ang_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool" shows "A B C Ang a \<equiv> QCongA a \<and> a A B C"
New: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and a :: "nat"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and C :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and l :: "char list \<Rightarrow> val option" assumes "new_Addr h = \<lfloor>a\<rfloor>"   and "h' = h(a \<mapsto> (C, Collect (init_obj P C)))" shows "P,E \<turnstile> \<langle>new C,(h, l)\<rangle> \<Rightarrow> \<langle>ref (a, [C]),(h', l)\<rangle>"
wellformed_transaction_strand_unlabel_memberD_6: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and x :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) stateful_strand_step"   and t :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term"   and s :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" assumes "wellformed_transaction T"   and "x \<in> set (unlabel (transaction_strand T))"   and "x = insert\<langle>t,s\<rangle>" shows "x \<in> set (unlabel (transaction_updates T))"
IH: fixes S :: "(nat \<Rightarrow> real) set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int" assumes "simplicial_chain p S c" shows "singular_subdivision p c = simplicial_subdivision p c"
mresolution_def: fixes C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "mresolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma> = (C\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>) - (L\<^sub>1 \<cdot>\<^sub>l\<^sub>s \<sigma>) \<union> ((C\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>) - (L\<^sub>2 \<cdot>\<^sub>l\<^sub>s \<sigma>))"
fmsubset_eq: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap" assumes "s1 \<subseteq>\<^sub>f s2" shows "\<forall>a. a |\<in>| fmdom s1 \<longrightarrow> fmlookup s1 a = fmlookup s2 a"
U1: shows "U1 = Mp (sdiv_poly (C - D * H) q)"
simps_17: fixes Vs :: "String.literal list"   and U :: "unit"   and o' :: "'a expr"   and e :: "'a expr" shows "compE1 Vs (sync\<^bsub>U\<^esub> (o') e) = sync\<^bsub>length Vs\<^esub> (compE1 Vs o') compE1 (Vs @ [fresh_var Vs]) e"
that: fixes h :: "heap"   and h' :: "heap" assumes "\<sigma> = Some h"   and "\<sigma>' = Some h'"   and "effect c h h' r" shows "thesis"
lemma_2_5_9_a: fixes a :: "'a"   and b :: "'a" shows "a \<le> b r\<rightarrow> a"
fps_annil: fixes f :: "('c, 'd) fps" shows "0 \<cdot> f = 0"
wary_\<rho>t: shows "wary (subst \<rho> t)"
sup_absorb1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x \<sqinter> y = x"
assms: shows "\<forall>x y. x \<succeq>[Pr] y \<longrightarrow> (\<forall>\<alpha> \<beta>. \<alpha> + \<beta> = 1 \<and> 0 \<le> \<alpha> \<and> 0 \<le> \<beta> \<longrightarrow> \<alpha> *\<^sub>R x + \<beta> *\<^sub>R y \<succeq>[Pr] y)"
roots_of_2_2: fixes p :: "int poly" shows "Ball (set (roots_of_2 p)) invariant_2"
total: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "total_preorder_on carrier le"   and "x \<in> carrier"   and "y \<in> carrier" shows "le x y \<or> le y x"
IH: fixes cs' :: "(char list \<Rightarrow> int) set acom" assumes "cs' \<le> cs" shows "(step {} ^^ n) cs' \<le> cs"
List_unique: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and as :: "'b list"   and bs :: "'b list" assumes "List h x as"   and "List h x bs" shows "as = bs"
nhds_discrete: fixes x :: "'a" shows "nhds x = principal {x}"
measure_times: fixes A :: "'a \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i \<in> events i" shows "emeasure (Pi\<^sub>M I M) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))"
elims: fixes x :: "perm"   and xa :: "('a, 'b) L_state"   and y :: "('a, 'b) L_state" assumes "x \<bullet> xa = y" obtains p :: "perm" and xb :: "'b \<times> 'b fs_set \<times> 'a" where "x = p__" and "xa = AC xb__" and "y = AC (p__ \<bullet> xb__)"   | p :: "perm" and xb :: "'b fs_set \<times> 'a" where "x = p" and "xa = EF xb" and "y = EF (p \<bullet> xb)"
boundOutputFreshSet_4: fixes xvec :: "name list"   and x :: "name"   and B :: "('d, 'e, 'f) boundOutput" shows "xvec \<sharp>* \<lparr>\<nu>x\<rparr>B = xvec \<sharp>* [x].B"
Int_iff: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
simps_1: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, send\<langle>t\<rangle>) = (l, receive\<langle>t\<rangle>)"
mem_monad_self: fixes x :: "'a star" shows "x \<in> monad x"
simps_3: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c pmf"   and x :: "'d" assumes "finite A"   and "A \<noteq> {}" shows "fold_bind_random_permutation f g x A = pmf_of_set A \<bind> (\<lambda>a. fold_bind_random_permutation f g (f a x) (A - {a}))"
rotate_id: fixes n :: "nat"   and xs :: "'a list" assumes "n mod length xs = 0" shows "rotate n xs = xs"
empty_btree_def: shows "abs_split.empty_btree = Leaf"
valid_fold_combine_Leaf: fixes w :: "nat \<times> nat"   and ts :: "'a tree list"   and h :: "'a tree"   and ts' :: "'a tree list" assumes "0 < fst w"   and "ts = h # ts'"   and "Ball (set ts) local.valid"   and "adjacent w ts" shows "local.valid (fold combine ts Leaf) \<and> l (fold combine ts Leaf) = fst w \<and> r (fold combine ts Leaf) = snd w"
size_list_simp_2: fixes xs :: "'b list" shows "\<^bold>|xs\<^bold>| = (if xs = \<epsilon> then 0 else Suc \<^bold>|tl xs\<^bold>|)"
nz_is_zero: fixes x :: "nat" assumes "nz_dom x" shows "nz x = 0"
hom_group_sum: fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> (((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and A :: "'b \<Rightarrow> ('c, 'd) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> hom (A i) (homology_group p X)"   and "\<And>i. i \<in> I \<Longrightarrow> Group.group (A i)" shows "(\<lambda>x. gfinprod (homology_group p X) (\<lambda>i. f i (x i)) I) \<in> hom (sum_group I A) (homology_group p X)"
infinity_evalUni_EqUni: fixes p :: "real \<times> real \<times> real"   and x :: "real" shows "(\<exists>x. \<forall>y<x. aEvalUni (EqUni p) y) = evalUni (substNegInfinityUni (EqUni p)) x"
rtrancl_refl: fixes a :: "'e"   and r :: "('e \<times> 'e) set" shows "(a, a) \<in> r\<^sup>*"
sets_restrict_space: fixes M :: "'b measure"   and \<Omega> :: "'b set" shows "sets (restrict_space M \<Omega>) = sets.restricted_space M \<Omega>"
convergentI: fixes T :: "'b set set"   and F :: "'b set set"   and x :: "'b" assumes "F \<longlongrightarrow>\<^bsub>T\<^esub> x"   and "x \<in> carrier\<^bsub>T\<^esub>" shows "F convergent\<^bsub>T\<^esub>"
iter2: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice" assumes "finite_or_infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (finite_or_infinite_iteration.iter seq (nil::'b::refinement_lattice) (c::'b::refinement_lattice)) (finite_or_infinite_iteration.iter seq nil c) = finite_or_infinite_iteration.iter seq nil c"
finmap_eq_iff: fixes P :: "'b \<Rightarrow>\<^sub>F 'c"   and Q :: "'b \<Rightarrow>\<^sub>F 'c" shows "(P = Q) = (Fin_Map.domain P = Fin_Map.domain Q \<and> (\<forall>i\<in>Fin_Map.domain P. (P)\<^sub>F i = (Q)\<^sub>F i))"
pos: fixes F :: "'a filter"   and H :: "('a \<Rightarrow> real) set"   and h :: "'a \<Rightarrow> real" assumes "landau_function_family F H"   and "h \<in> H" shows "\<forall>\<^sub>F x in F. 0 < h x"
del_has_real_derivative: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real"   and t :: "real"   and u :: "real set" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "(safe_distance_normal.\<tau> \<delta> has_real_derivative safe_distance_normal.\<tau>' t) (at t within u)"
forall_nat_expansion: fixes n\<^sub>0 :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>n\<ge>n\<^sub>0. P n) = (P n\<^sub>0 \<and> (\<forall>n\<ge>Suc n\<^sub>0. P n))"
assms_1: shows "es @ [Deliver (i, Insert e (Some a))] prefix of j"
insortTree_ne_Nil: fixes t :: "'a tree"   and ts :: "'a tree list" shows "insortTree t ts \<noteq> []"
subset: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "lideal I R addition multiplication zero unit" shows "I \<subseteq> R"
wlp_Seq: fixes P :: "'b \<Rightarrow> real"   and a :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real" assumes "P \<tturnstile> wlp a Q"   and "Q \<tturnstile> wlp b R"   and "well_def a"   and "well_def b"   and "unitary Q"   and "unitary R" shows "P \<tturnstile> wlp (a ;; b) R"
Resolution_cnf_sound: fixes S :: "'b literal set set"   and \<A> :: "'b \<Rightarrow> bool" assumes "S \<turnstile> \<box>" shows "\<not> cnf_semantics \<A> S"
const_map_in_set: fixes x :: "'c"   and A :: "'c set"   and v :: "'d" assumes "x \<in> A" shows "const_map v A x = Some v"
all_is_def: fixes p :: "('c, 'd) fm"   and ps :: "('c, 'd) fm list"   and xs :: "nat set" shows "all_is p ps xs \<equiv> \<forall>v\<in>xs. ps !. v = Some p"
Un_insert_right: fixes A :: "'g set"   and a :: "'g"   and B :: "'g set" shows "A \<union> insert a B = insert a (A \<union> B)"
rel_gpv''D: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and gpv :: "('f, 'h, 'j) gpv"   and gpv' :: "('g, 'i, 'k) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')"
Main2: fixes x :: "'a"   and n :: "nat" shows "d / 2 * real (card (V x \<inter> {1..n})) \<le> Max {u i x |i. i < s} + birkhoff_sum (\<lambda>x. \<bar>u s x / real s\<bar>) (n + 2 * s) x + birkhoff_sum F (n + 2 * s) x + 1 / real s * (\<Sum>i<2 * s. \<bar>u (n + i) x\<bar>)"
verit_eq_simplify_8: fixes x2 :: "num"   and y2 :: "num" shows "(num.Bit0 x2 = num.Bit0 y2) = (x2 = y2)"
greaterThan_def: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a" shows "ord.greaterThan less l \<equiv> {x. less l x}"
measure_eqI: fixes M :: "'d measure"   and N :: "'d measure" assumes "sets M = sets N"   and "\<And>A. A \<in> sets M \<Longrightarrow> emeasure M A = emeasure N A" shows "M = N"
fds_right_inverse: fixes f :: "'b fds" assumes "fds_nth f 1 \<noteq> (0::'b)" shows "f * inverse f = 1"
z_signs_R: fixes signs :: "rat list"   and I :: "nat list \<times> nat list" assumes "list_all (\<lambda>i. i < length signs) (fst I)"   and "list_all (\<lambda>i. i < length signs) (snd I)"   and "list_all (\<lambda>s. s = 1 \<or> s = 0 \<or> s = - 1) signs" shows "z_R I signs = 1 \<or> z_R I signs = 0 \<or> z_R I signs = - 1"
Field_csum: fixes r :: "('a \<times> 'a) set"   and s :: "('b \<times> 'b) set" shows "Field (r +c s) = Inl ` Field r \<union> Inr ` Field s"
nonempty_meld_2: fixes ys :: "('c, 'd) bintree option list"   and xs :: "('c, 'd) bintree option list" assumes "ys \<noteq> []" shows "meld xs ys \<noteq> []"
h1: shows "p = (\<Sum>i\<le>MPoly_Type.degree p var. isolate_variable_sparse p var i * Var var ^ i)"
IVT2: fixes f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and y :: "'a"   and a :: "'b" assumes "f b \<le> y"   and "y \<le> f a"   and "a \<le> b"   and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x" shows "\<exists>x\<ge>a. x \<le> b \<and> f x = y"
InvariantImpliedLiterals_def: fixes F :: "Literal list list"   and M :: "(Literal \<times> bool) list" shows "InvariantImpliedLiterals F M \<equiv> \<forall>l. l el elements M \<longrightarrow> formulaEntailsLiteral (F @ val2form (decisionsTo l M)) l"
FES: fixes \<AA> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<BB> :: "'b formula"   and \<Theta> :: "'b formula list" assumes "\<AA> # \<Gamma> \<Rightarrow> \<BB> # \<Theta>" shows "\<Gamma> \<Rightarrow> \<AA> \<^bold>\<rightarrow> \<BB> # \<Theta>"