hcju/isabelleps huggingface dataset:
listset_permE: fixes ys :: "'b list"   and xs :: "'b set list"   and f :: "nat \<Rightarrow> nat"   and xs' :: "'b set list" assumes "ys \<in> listset xs"   and "bij_betw f {..<length xs} {..<length xs'}"   and "\<And>i. i < length xs \<Longrightarrow> xs' ! i = xs ! f i" obtains ys' :: "'b list" where "ys' \<in> listset xs'" and "length ys' = length ys" and "\<And>i. i < length ys \<Longrightarrow> ys' ! i = ys ! f i"
Nml_VcompNml_1: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Nml (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u)"
well_valued0_take_r: fixes t :: "'b tree"   and as :: "'b list" assumes "well_shaped t"   and "well_valued0 as t" shows "well_valued0 (take (r t) as) t"
character_principal: fixes G :: "('c, 'd) monoid_scheme" assumes "finite_comm_group G" shows "character G (principal_char G)"
d: shows "d = d div 2 + (d + 1) div 2"
aligned_mask_range_cases: fixes p :: "'b word"   and n :: "nat"   and p' :: "'b word"   and n' :: "nat" assumes "is_aligned p n"   and "is_aligned p' n'" shows "mask_range p n \<inter> mask_range p' n' = {} \<or> mask_range p n \<subseteq> mask_range p' n' \<or> mask_range p' n' \<subseteq> mask_range p n"
plus_mult_axioms: shows "plus_mult id (*) zer (+) inver"
append_vec_eq: fixes v :: "'b vec"   and n :: "nat"   and v' :: "'b vec"   and w :: "'b vec"   and w' :: "'b vec" assumes "v \<in> carrier_vec n"   and "v' \<in> carrier_vec n" shows "(v @\<^sub>v w = v' @\<^sub>v w') = (v = v' \<and> w = w')"
assms_2: shows "\<forall>m\<in>M. homogeneous\<^sub>l\<^sub>s\<^sub>t m A Sec \<and> m \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t A)"
dist_triangle_half_l: fixes x1 :: "'a"   and y :: "'a"   and e :: "real"   and x2 :: "'a" assumes "dist x1 y < e / 2"   and "dist x2 y < e / 2" shows "dist x1 x2 < e"
associatedD2: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and b :: "'b" assumes "monoid_cancel G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "\<exists>u\<in>Units G. a = b \<otimes>\<^bsub>G\<^esub> u"
implies_isotone_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leadsto> y \<le> x \<leadsto> (y \<squnion> z)"
gcd_closed: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "gcd_condition_monoid G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "somegcd G a b \<in> carrier G"
expand: fixes stateT :: "('d, 'e) stateT"   and stateT' :: "('d, 'e) stateT" assumes "run_state stateT = run_state stateT'" shows "stateT = stateT'"
coclop_adj: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "f \<stileturn> g" shows "coclop (f \<circ> g)"
32: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (y \<squnion> x) = y \<squnion> x"
max_x_impl'_def: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "max_x_impl' a b y i = (if i < length a \<and> big_d' a b y i \<noteq> [] then Min (set (big_d' a b y i)) else Max (set b))"
measurable_def: fixes A :: "'e measure"   and B :: "'f measure" shows "A \<rightarrow>\<^sub>M B = {f \<in> space A \<rightarrow> space B. \<forall>y\<in>sets B. f -` y \<inter> space A \<in> sets A}"
le_add_same_cancel2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># b + a) = ({#} \<subseteq># b)"
Pair_inject: fixes a :: "'r"   and b :: "'s"   and a' :: "'r"   and b' :: "'s" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
cSUP_le_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd multiset"   and u :: "'d multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (f ` A)" shows "(Sup (f ` A) \<subseteq># u) = (\<forall>x\<in>A. f x \<subseteq># u)"
set_keys_to_list: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "set (keys_to_list p) = keys p"
zero_less_double_add_iff_zero_less_single_add: fixes a :: "'a" shows "((0::'a) < a + a) = ((0::'a) < a)"
label_4: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis"   and "x \<in> cbox (0::'a) One \<and> label x i = 0" shows "x \<bullet> i \<le> f x \<bullet> i"
findMin_def: fixes q :: "('c, 'd) SkewBinomialHeap" shows "findMin q \<equiv> BsSkewBinomialHeapStruc.bs_findMin (Rep_SkewBinomialHeap q)"
simps_8: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and vc :: "'b"   and vd :: "'b"   and l :: "'b set list" shows "smaller (AllowPortFromTo v va vb) (DenyAllFromTo vc vd) l = (AllowPortFromTo v va vb = DenyAllFromTo vc vd \<or> (if bothNet (AllowPortFromTo v va vb) = bothNet (DenyAllFromTo vc vd) then case DenyAllFromTo vc vd of DenyAllFromTo a b \<Rightarrow> AllowPortFromTo v va vb = DenyAllFromTo b a | _ \<Rightarrow> True else position (bothNet (AllowPortFromTo v va vb)) l \<le> position (bothNet (DenyAllFromTo vc vd)) l))"
distinct_adj_remdups_adj: fixes xs :: "'a list" shows "distinct_adj (remdups_adj xs)"
add_mset_inter_add_mset: fixes a :: "'a"   and A :: "'a multiset"   and B :: "'a multiset" shows "add_mset a A \<inter># add_mset a B = add_mset a (A \<inter># B)"
sum_Pinfty: fixes f :: "'a \<Rightarrow> ereal"   and P :: "'a set" shows "(sum f P = \<infinity>) = (finite P \<and> (\<exists>i\<in>P. f i = \<infinity>))"
antisym_subset: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "antisymp r"   and "\<And>a a'. s a a' \<Longrightarrow> r a a'" shows "antisymp s"
assms_3: shows "(d, t) \<rightarrow>*t t'"
subspace_def: fixes S :: "'b set" shows "m1.subspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. s1 c x \<in> S))"
widen_trans: fixes P :: "'b prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "subtype P S U"   and "subtype P U T" shows "subtype P S T"
signed_take_bit_eq_iff_take_bit_eq: fixes n :: "nat"   and a :: "'a"   and b :: "'a" shows "(signed_take_bit n a = signed_take_bit n b) = (take_bit (Suc n) a = take_bit (Suc n) b)"
smult_vec_length: fixes x :: "real"   and v :: "complex vec" assumes "0 \<le> x" shows "\<parallel>complex_of_real x \<cdot>\<^sub>v v\<parallel> = x * \<parallel>v\<parallel>"
fcard_mono: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B" shows "fcard A \<le> fcard B"
coeff_sign_changes_reduce_root: fixes a :: "'a"   and p :: "'a poly" assumes "(0::'a) < a" shows "sign_changes (coeffs (reduce_root a p)) = sign_changes (coeffs p)"
bdd_above_bot: fixes bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.order_bot bot less_eq less" shows "preorder.bdd_below less_eq A"
ts_inv_into_ts_ok_Ex: fixes Q :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool"   and I :: "'g \<Rightarrow> 'f option"   and ts :: "'g \<Rightarrow> ('h \<times> 'j \<Rightarrow>f nat) option"   and m :: "'i" assumes "ts_inv Q I ts m" shows "ts_ok (\<lambda>t x m. \<exists>i. Q i t x m) ts m"
ex_eq: shows "IArray.exists (\<lambda>i. matrix_to_iarray A !! i !! mod_type_class.to_nat j = (1::'a) \<and> mod_type_class.to_nat j = least_non_zero_position_of_vector (row_iarray i (matrix_to_iarray A))) (IArray [0..<nrows_iarray (matrix_to_iarray A)])"
mono_strict_invE: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less"   and "order.mono less_eq f"   and "f x < f y" obtains "less x y"
connected_punctured_ball: fixes a :: "'a"   and r :: "real" assumes "2 \<le> DIM('a)" shows "connected (ball a r - {a})"
set_card: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.ide S a" shows "|set_category_data.set S img a| <o \<AA>"
is_hashmap'_prec: fixes m :: "'c \<Rightarrow> 'd option"   and l :: "('c \<times> 'd) list list"   and ht :: "('c, 'd) hashtable"   and F1 :: "assn"   and m' :: "'c \<Rightarrow> 'd option"   and l' :: "('c \<times> 'd) list list"   and F2 :: "assn"   and h :: "heap \<times> nat set" shows "\<forall>s s'. h \<Turnstile> is_hashmap' m l ht * F1 \<and>\<^sub>A is_hashmap' m' l' ht * F2 \<longrightarrow> l = l' \<and> m = m'"
prefixes_Suc_length: fixes s :: "nat \<Rightarrow> nat option"   and j :: "nat"   and t :: "nat" assumes "r1_bc s" shows "e_length (r1_bc.prefixes s j (Suc t)) = Suc (e_length (r1_bc.prefixes s j t) + the (eval (r1_bc.r_find_k s) [r1_bc.prefixes s j t]))"
match_ex_table_SomeD: fixes P :: "'a prog"   and C :: "char list"   and pc :: "nat"   and xt :: "ex_table"   and pc' :: "nat"   and d' :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d')\<rfloor>" shows "\<exists>(f, t, D, h, d)\<in>set xt. matches_ex_entry P C pc (f, t, D, h, d) \<and> h = pc' \<and> d = d'"
undirected_backflows: fixes G :: "'b graph" shows "undirected G = \<lparr>nodes = nodes G, edges = edges G \<union> backflows (edges G)\<rparr>"
simps_2: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'i, 'j) new_thread_action"   and tas :: "('f, 'i, 'j) new_thread_action list" shows "thread_oks ts (ta # tas) = (thread_ok ts ta \<and> thread_oks (redT_updT' ts ta) tas)"
Interpretation_def: fixes I :: "('a, 'b, 'c, 'd) Interpretation" shows "Interpretation I \<equiv> (Category iC\<^bsub>I\<^esub> \<and> Signature iS\<^bsub>I\<^esub>) \<and> (\<forall>A. A \<in> Ty\<^bsub>iS\<^bsub>I\<^esub>\<^esub> \<longrightarrow> Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub> \<in> obj\<^bsub>iC\<^bsub>I\<^esub>\<^esub>) \<and> (\<forall>f A B. f \<in> Sig iS\<^bsub>I\<^esub> : A \<rightarrow> B \<longrightarrow> Fn\<lbrakk>f\<rbrakk>\<^bsub>I\<^esub> maps\<^bsub>iC\<^bsub>I\<^esub>\<^esub> Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub> to Ty\<lbrakk>B\<rbrakk>\<^bsub>I\<^esub>)"
dependent_zero: fixes A :: "'b set" assumes "(0::'b) \<in> A" shows "dependent A"
d\<^sub>0_inject: fixes x :: "\<o>"   and y :: "\<o>" assumes "d\<^sub>0 x = d\<^sub>0 y" shows "x = y"
openin_rel_interior: fixes S :: "'b set" shows "openin (top_of_set (affine hull S)) (rel_interior S)"
1: shows "disjoint [k] ks \<and> disjoint [m] ms"
fold_delta_a_init_a_mrexps: fixes w :: "'b list"   and s :: "'b rexp" shows "fold delta_a w (init_a s) \<in> UNIV \<times> mrexps s"
Collect_case_prod: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
implem_LtK: fixes implem :: "chan \<Rightarrow> msg"   and x :: "chan"   and K :: "ltkey" assumes "semivalid_implem implem" shows "implem x \<noteq> LtK K"
emeasure_Diff_null_set: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M"   and "A \<in> sets M" shows "emeasure M (A - B) = emeasure M A"
take_bit_eq_mask: fixes n :: "nat"   and a :: "'a" shows "take_bit n a = a AND mask n"
measurable_countable_negligible_Union: fixes S :: "nat \<Rightarrow> 'c set"   and B :: "real" assumes "\<And>n. S n \<in> lmeasurable"   and "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"   and "\<And>n. (\<Sum>k\<le>n. Sigma_Algebra.measure lebesgue (S k)) \<le> B" shows "\<Union> (range S) \<in> lmeasurable"
union_is_single: fixes M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" shows "(M + N = \<LM> a  \<RM>) = (M = \<LM> a  \<RM> \<and> N = \<Empt> \<or> M = \<Empt> \<and> N = \<LM> a  \<RM>)"
int_lt_neq_Suc_lt: fixes a :: "real"   and b :: "real" assumes "a \<in> \<int>"   and "b \<in> \<int>"   and "a < b"   and "a + 1 \<noteq> b" shows "a + 1 < b"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "inj_semiring_hom hom"   and "bijective hom" shows "semiring_isom hom"
Cond: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and b :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> b) c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P \<inter> - b) c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c\<^sub>1 c\<^sub>2 Q,A"
cl_inv_def: fixes gp :: "'a gState_scheme \<times> pState" shows "cl_inv gp = (case gp of (g, p) \<Rightarrow> length (pState.channels p) \<le> length (gState.channels g))"
wls_Op_inj: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option"   and delta' :: "'opSym"   and inp' :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp' :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp"   and "wlsInp delta' inp'"   and "wlsBinp delta' binp'" shows "(Op delta inp binp = Op delta' inp' binp') = (delta = delta' \<and> inp = inp' \<and> binp = binp')"
cut_greater_finite: fixes I :: "'a set"   and t :: "'a" assumes "finite I" shows "finite (I \<down>> t)"
ord_p_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "ord_p p q \<equiv> ord_strict_p p q \<or> p = q"
pick_2: fixes c :: "nat" assumes "c \<in> C" shows "j \<le> pick c"
bvars\<^sub>s\<^sub>s\<^sub>t_Cons: fixes a :: "('d, 'c) stateful_strand_step"   and A :: "('d, 'c) stateful_strand_step list" shows "bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a) \<union> bvars\<^sub>s\<^sub>s\<^sub>t A"
take_bit_of_1_eq_0_iff: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "(take_bit n one = zero) = (n = 0)"
common_prefix_find: fixes z :: "'a"   and xs :: "'a list"   and ys :: "'a list" assumes "z \<in> set (common_prefix xs ys)" shows "find_first z xs = find_first z ys"
sympI: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
simps_3: fixes prog :: "cmd"   and c :: "cmd" shows "labels_nodes prog (_Exit_) c = False"
typeof_unboxed_eq_const_3: fixes x :: "('a, 'b, 'c) unboxed" shows "(typeof x = Some Ubx2) = (\<exists>b. x = OpUbx2 b)"
Union: fixes a :: "nat \<Rightarrow> 'b set"   and A :: "'b set set" assumes "\<And>i. a i \<in> \<sigma>_algebra A" shows "\<Union> (range a) \<in> \<sigma>_algebra A"
prod_ord_pairD: fixes xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list"   and P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'a" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "prod_ord_pair P xs ys"   and "k \<in> fst ` set xs \<union> fst ` set ys" shows "P k (lookup_pair xs k) (lookup_pair ys k)"
order_equiv_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<sqsubseteq> x"   and "y \<simeq> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
fresh_set_fresh_forall: fixes y :: "'d"   and xs :: "'a list" shows "atom y \<sharp> xs = (\<forall>x\<in>set xs. atom y \<sharp> x)"
disjnt_def: fixes A :: "'c set"   and B :: "'c set" shows "disjnt A B = (A \<inter> B = {})"
comm_append_are_replicate: fixes xs :: "'g list"   and ys :: "'g list" assumes "xs @ ys = ys @ xs" shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
c4: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> y \<cdot> 1\<^sub>\<pi> = x \<cdot> (y \<cdot> 1\<^sub>\<pi>)"
cr_option_le_spmf_def: fixes x :: "'b option"   and p :: "'b spmf" shows "cr_option_le_spmf x p = ord_spmf (=) (return_pmf x) p"
sset_streams: fixes s :: "'a stream"   and A :: "'a set" assumes "i.R s \<subseteq> A" shows "s \<in> streams A"
foldl_append: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and a :: "'d"   and xs :: "'e list"   and ys :: "'e list" shows "foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"
SUP_mono: 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 A :: "'d set"   and B :: "'e set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. less_eq (f n) (g m)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
IH: fixes rs\<^sub>1 :: "'a rule list"   and rs\<^sub>2 :: "'a rule list"   and t' :: "state" assumes "rs = rs\<^sub>1 @ rs\<^sub>2"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t'" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t'\<rangle> \<Rightarrow> t"
\<Sigma>_protocols_to_commitments_axioms_def: fixes init :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and response :: "'c \<Rightarrow> 'b \<Rightarrow> 'e \<Rightarrow> 'f spmf"   and check :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and Rel :: "('a \<times> 'b) set"   and S_raw :: "'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'f) spmf"   and \<A>ss :: "'a \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'f \<Rightarrow> 'b spmf"   and challenge_space :: "'e set"   and valid_pub :: "'a set"   and G :: "('a \<times> 'b) spmf" shows "\<Sigma>_protocols_to_commitments_axioms init response check Rel S_raw \<A>ss challenge_space valid_pub G \<equiv> (Sigma_Protocols.\<Sigma>_protocols_base.\<Sigma>_protocol init response check Rel S_raw \<A>ss challenge_space valid_pub \<and> (\<forall>h w. (h, w) \<in> set_spmf G \<longrightarrow> (h, w) \<in> Rel)) \<and> lossless_spmf G \<and> (\<forall>h w. lossless_spmf (init h w)) \<and> (\<forall>r w e. lossless_spmf (response r w e))"
us1: fixes u :: "'a mod_ring poly" assumes "u \<in> set us1" shows "monic u \<and> irreducible u"
B: fixes n :: "nat" shows "dist basepoint ((inv f ^^ n) c) \<le> e"
order_refl: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" assumes "class.preorder less_eq less" shows "less_eq x x"
right_idem: fixes a :: "'c multiset"   and b :: "'c multiset" shows "a \<union># b \<union># b = a \<union># b"
discrete_compact_finite_iff: fixes S :: "'c set" shows "(discrete S \<and> compact S) = finite S"
bi_uniqueDl: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d"   and z :: "'c" assumes "bi_unique A"   and "A x y"   and "A z y" shows "x = z"
rel_interior_projection: fixes S :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b set"   and y :: "'a"   and z :: "'b" assumes "convex S"   and "f = (\<lambda>y. {z. (y, z) \<in> S})" shows "((y, z) \<in> rel_interior S) = (y \<in> rel_interior {y. f y \<noteq> {}} \<and> z \<in> rel_interior (f y))"
nat_pow_eone: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "[1] \<cdot>\<^bsub>P\<^esub> x = x"
step_alpha: fixes A :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and l :: "'d"   and Z :: "('c \<Rightarrow> real) set"   and l' :: "'d"   and Z' :: "('c \<Rightarrow> real) set" assumes "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto> \<langle>l', Z'\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha> \<langle>l', Closure\<^sub>\<alpha> Z'\<rangle>"
propos_subset: fixes \<phi> :: "'c ltl" shows "propos \<phi> \<subseteq> nested_propos \<phi>"
inf_sup_ord_1: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and x :: "'b::type"   and y :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) (y::'b::type)) x"
butterfly_def: fixes s :: "'d"   and t :: "'c" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
iconfs_map_throw: fixes sh :: "sheap"   and vs :: "val list"   and e :: "'a exp"   and es' :: "'a exp list" assumes "iconfs sh (map Val vs @ throw e # es')" shows "iconf sh e"
map_replicate: fixes f :: "'l \<Rightarrow> 'k"   and n :: "nat"   and x :: "'l" shows "map f (replicate n x) = replicate n (f x)"
numeral_code_3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit1 n) = (let m = numeral.numeral one plus n in plus (plus m m) one)"
transcendental_i_pi: shows "\<not> algebraic (\<i> * complex_of_real pi)"
expansion_def: fixes A :: "'a set"   and \<phi> :: "'a" shows "A \<oplus> \<phi> \<equiv> Cn (A \<union> {\<phi>})"
axioms_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "symmetry_functor A1 A2" shows "category A2"
Rep_uint32_inverse: fixes x :: "uint32" shows "Abs_uint32 (Rep_uint32 x) = x"
vprop: shows "(SOME U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r}) \<in> sets (stream_space borel) \<and> (SOME U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r}) \<inter> range X = {r}"
inR1: shows "s \<otimes> g1 \<in> stabilizer G \<phi> x #> r1"
elims: fixes x :: "'b routing_rule list"   and xa :: "'b word"   and y :: "'b routing_action" assumes "routing_table_semantics x xa = y" obtains uu_ :: "'b word" where "x = []" and "xa = uu_" and "y = routing_action undefined"   | r :: "'b routing_rule" and rs :: "'b routing_rule list" and p :: "'b word" where "x = r # rs" and "xa = p" and "y = (if prefix_match_semantics (routing_match r) p then routing_action r else routing_table_semantics rs p)"
brw_algo_def: fixes Qi :: "'a set"   and \<delta> :: "('a, 'b) ta_rule set" shows "brw_algo Qi \<delta> \<equiv> \<lparr>wa_cond = brw_cond Qi, wa_step = brw_step \<delta>, wa_initial = brw_initial \<delta>, wa_invar = brw_invar \<delta>\<rparr>"
s_def: shows "s = (\<lambda>i. sum_mat (A i))"
normalize_set_of_vec_def: fixes X :: "'b set" shows "normalize_set_of_vec X = Projections.normalize ` X"
fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_Cons: fixes s :: "('b, 'a) Term.term"   and t :: "('c, 'a) Term.term"   and F :: "(('b, 'a) Term.term \<times> ('c, 'a) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s ((s, t) # F) = fv s \<union> fv t \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F"
ys_split: shows "ys = prefix @ x # suffix \<and> concurrent_set x suffix"
T_def: fixes x :: "'a" shows "T x \<equiv> t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)"
lang_is_union_of_finals: fixes A :: "'b list set" shows "A = \<Union> (finals A)"