hcju/isabelleps huggingface dataset:
pp_of_Keys_init_syzygy_list_subset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pp_of_term ` Keys (set (init_syzygy_list bs)) \<subseteq> insert (0::'a) (pp_of_term ` Keys (set bs))"
set_plus_intro2: fixes b :: "'c"   and C :: "'c set"   and a :: "'c" assumes "b \<in> C" shows "a + b \<in> a +o C"
sorted_le_x_xs: shows "linorder_class.sorted (x # xs)"
cases: fixes P1 :: "'a \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool"   and a :: "'a \<times> 'b" assumes "pred_prod P1 P2 a" obtains aa :: "'a" and b :: "'b" where "a = (aa, b)" and "P1 aa" and "P2 b"
partition_by_median_def: fixes k :: "'b"   and ps :: "(real, 'b) vec list" shows "partition_by_median k ps = (let m = axis_median k ps; (l, r) = partition (\<lambda>p. p $ k \<le> m) ps in (l, m, r))"
InterI: fixes C :: "'c set set"   and A :: "'c" assumes "\<And>X. s2p C X \<Longrightarrow> s2p X A" shows "s2p (\<Inter> C) A"
findMin_def: fixes bq :: "('c, 'd) BsSkewBinomialTree list" shows "findMin bq = val (getMinTree bq)"
ny: shows "\<forall>w. P w (\<midarrow> y) = (\<not> O w y)"
greatest_mem: fixes L :: "('a, 'b) gorder_scheme"   and l :: "'a"   and A :: "'a set" assumes "greatest L l A" shows "l \<in> A"
closer_point_lemma: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "0 < (y - x) \<bullet> (z - x)" shows "\<exists>u>0. u \<le> 1 \<and> dist (x + u *\<^sub>R (z - x)) y < dist x y"
U: shows "splits_at \<alpha> i a1 U a2 \<and> splits_at \<gamma> j b1 U a2 \<and> LeftDerivation a1 D b1"
lem: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "Ball (set xs) P" shows "Ball (set (remove1 x xs)) P = True"
collapse_2: fixes "term" :: "'b local.term" assumes "is_Prim term" shows "\<^bold>\<langle>un_Prim term\<^bold>\<rangle> = term"
FunctorCompPreserved: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme"   and f :: "'c" assumes "Functor F"   and "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## f \<in> mor\<^bsub>CatCod F\<^esub>"
ds_ds_subseteq_ds: fixes r :: "('b \<times> 'b) set"   and S :: "'b set" assumes "trans r" shows "r \<down>s (r \<down>s S) \<subseteq> r \<down>s S"
oreachable_weakenE: fixes s :: "'d \<times> 'e"   and A :: "('d \<times> 'e, 'f) automaton"   and PS :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and PU :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and QS :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and QU :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "s \<in> oreachable A PS PU"   and "\<And>s s' a. PS s s' a \<Longrightarrow> QS s s' a"   and "\<And>s s'. PU s s' \<Longrightarrow> QU s s'" shows "s \<in> oreachable A QS QU"
inconsistent_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" shows "inconsistent A b = (\<not> consistent A b)"
offs_enum_next_le: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat"   and n :: "nat"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and k :: "nat" assumes "index_less index key"   and "i < j"   and "j < n"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i \<le> offs (enum xs index key n mi ma) k ! j"
unique_connecting_path_properties_2: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "v \<leadsto>\<^bsub>G\<^esub> w \<noteq> []"
apsnd_eq_conv: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a \<times> 'c"   and g :: "'c \<Rightarrow> 'b" shows "(apsnd f x = apsnd g x) = (f (snd x) = g (snd x))"
is_weakly_dom_component_safe_def: shows "Core_DOM_Components.is_weakly_dom_component_safe \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_weakly_dom_component_safe Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
simps_4: fixes xs :: "'b list"   and ys :: "'b list" shows "proper_interval (Some xs) (Some ys) = List_Proper_Interval.proper_interval_list_aux xs ys"
project_extend_transient_D: fixes C :: "'c set"   and F :: "'a program"   and D :: "'a set" assumes "project h C (extend h F) \<in> transient (project_set h C \<inter> D)" shows "F \<in> transient (project_set h C \<inter> D)"
ffd_ffb_adj: fixes f :: "'d \<Rightarrow> 'c set" shows "fd\<^sub>\<F> f \<stileturn> bb\<^sub>\<F> f"
freess_cons: fixes x :: "'a"   and xs :: "'a list" shows "freess (x # xs) = frees x |\<union>| freess xs"
a: fixes x :: "'b list"   and a :: "nat"   and y :: "'b list"   and b :: "nat"   and z :: "'b list"   and c :: "nat" assumes "LS x a y b z c" shows "2 \<le> a"
simps: fixes m :: "nat"   and q :: "nat"   and r :: "nat" shows "newstat m q r = (if q \<noteq> 0 then Entry m (4 * (q - 1) + 2 * scan r + 1) else 0)"
substitute''_ConsE: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "substitute'' f T (x # xs) ys" obtains zs :: "'b list" and xs' :: "'b list" and ysa :: "'b list" where "ys = x # ysa" and "zs \<in> paths (f x)" and "xs' \<in> xs \<otimes> zs" and "substitute'' (f_nxt f T x) T xs' ysa"
wf_projectTable: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and n :: "nat"   and X :: "nat set \<times> 'b option list set"   and I :: "nat set" assumes "getIJ getIJ"   and "wf_atable n X" shows "wf_atable n (New_max.projectTable I X) \<and> fst (New_max.projectTable I X) = fst X \<inter> I"
induct: fixes P :: "fm \<Rightarrow> bool"   and fm :: "fm" assumes "\<And>x1 x2. P (x1 IN x2)"   and "\<And>x1 x2. P (x1 EQ x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 OR x2)"   and "\<And>x. P x \<Longrightarrow> P (Neg x)"   and "\<And>x1 x2. P x2 \<Longrightarrow> P (SyntaxN.Ex x1 x2)" shows "P fm"
extends_to_elementary_category_with_terminal_object: shows "elementary_category_with_terminal_object (\<cdot>) \<one> trm"
W_var_geD: fixes S :: "nat \<Rightarrow> typ"   and t :: "typ"   and m :: "nat"   and e :: "expr"   and A :: "type_scheme list"   and n :: "nat" assumes "Some (S, t, m) = W e A n" shows "n \<le> m"
While: fixes b :: "bexp"   and C :: "com" assumes "\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). I (s, n) \<and> lmaps_to_axpr b True s} C { (\<lambda>(s, n). I (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1}" shows "\<turnstile>\<^sub>3\<^sub>a {(\<lambda>(s, n). I (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1} WHILE b DO C { \<lambda>(s, n). I (s, n) \<and> lmaps_to_axpr b False s}"
is_regular_def: fixes s :: "nat set \<Rightarrow> (nat \<times> nat) set"   and U :: "nat set set" shows "cring0.is_regular s U \<equiv> \<forall>\<pp>. \<pp> \<in> U \<longrightarrow> (\<exists>V. cring0.is_zariski_open V \<and> V \<subseteq> U \<and> \<pp> \<in> V \<and> cring0.is_locally_frac s V)"
blocking_on_CD_on: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set"   and h :: "'f"   and X'' :: "'d set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.blocking_on Pd Ch ds X h X''" shows "X'' \<subseteq> Contracts.CD_on Pd ds (X \<union> X'')"
g_def: fixes m :: "nat \<Rightarrow>\<^sub>0 nat" shows "g m = (\<Prod>v. f v ^ poly_mapping.lookup m v)"
lbl_lbl: fixes xa :: "'a word"   and x :: "'a word" shows "length (to_bl xa) = length (to_bl x)"
elementsAppend: fixes a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" shows "elements (a @ b) = elements a @ elements b"
sum_mset_mono: fixes K :: "'b multiset"   and f :: "'b \<Rightarrow> 'c multiset"   and g :: "'b \<Rightarrow> 'c multiset" assumes "\<And>i. i \<in># K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum_mset (+) {#} (image_mset f K) \<subseteq># comm_monoid_add.sum_mset (+) {#} (image_mset g K)"
preprm_apply_composition: fixes f :: "('b \<times> 'b) list"   and g :: "('b \<times> 'b) list"   and x :: "'b" shows "preprm_apply (preprm_compose f g) x = preprm_apply f (preprm_apply g x)"
neutral: fixes A :: "'b set"   and g :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>A. g x = True" shows "bool.F g A = True"
combined_size_bound_gso_integer: fixes x :: "int" assumes "x \<in> {fs.\<mu>' i j |i j. j \<le> i \<and> i < m} \<union> {fs.\<sigma>s l i j |i j l. i < m \<and> j \<le> i \<and> l < j}"   and "m \<noteq> 0"   and "x \<noteq> 0"   and "n \<noteq> 0" shows "log 2 \<bar>real_of_int x\<bar> \<le> real (6 + 6 * m) * log 2 (real_of_int (M * int n)) + log 2 (real m) + real m"
abs_cases: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d" assumes "class.ring_1_abs_real_line plus zero minus less_eq less times abs uminus one" obtains "a = zero"   | "less zero (abs a)"
max2_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> Field r"   and "b \<in> Field r" shows "((max2 a b, c) \<in> r) = ((a, c) \<in> r \<and> (b, c) \<in> r)"
Cons: fixes xs :: "['b]"   and x :: "'b" assumes "finite_list xs" shows "finite_list (x : xs)"
exhaust_sel: fixes vertex :: "'a vertex" obtains "vertex = SOURCE"   | "vertex = vertex.SINK"   | "vertex = \<langle>vertex.inner vertex\<rangle>"
simps_2: fixes ok :: "nat"   and f' :: "'b list"   and r' :: "'b list" shows "st_inv (App ok f' r') = (ok \<le> length f' \<and> length f' < length r')"
ex_card: fixes n :: "nat"   and A :: "'b set" assumes "n \<le> card A" shows "\<exists>S\<subseteq>A. card S = n"
method_exit_inner_path: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "method_exit n"   and "n -as\<rightarrow>\<^sub>\<iota>* n'" shows "as = []"
simps_4: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and var :: "nat" shows "liftmap f (And \<phi> \<psi>) var = And (liftmap f \<phi> var) (liftmap f \<psi> var)"
altc_sample_state_def: fixes altc_sample :: "'k \<Rightarrow> ('l \<Rightarrow> 'j) \<Rightarrow> 'j"   and p :: "'k"   and f :: "'l \<Rightarrow> ('i, 'j) stateT" shows "altc_sample_state altc_sample p f = StateT (\<lambda>s. altc_sample p (\<lambda>x. run_state (f x) s))"
prems_2: shows "bvars\<^sub>s\<^sub>t (x # S) \<inter> range_vars \<theta> = {}"
theI': fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
cases: fixes a :: "nat set" assumes "a \<in> i_set_ind" obtains n :: "nat" where "a = [n__\<dots>]"   | n :: "nat" where "a = [\<dots>na__]"   | n :: "nat" and d :: "nat" where "a = [n\<dots>,d]"   | r :: "nat" and m :: "nat" where "a = [ r__, mod m__ ]"   | r :: "nat" and m :: "nat" and c :: "nat" where "a = [ r, mod m, c ]"
cbovi: shows "b O ov\<inverse> \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
hyps: fixes k :: "int" shows "of_int (ffact n k) = ffact n (of_int k)"
num_unique_vars_Fun_Cons: fixes f :: "'a"   and t :: "('a, 'b) Term.term"   and ts :: "('a, 'b) Term.term list" shows "num_unique_vars (Fun f (t # ts)) \<le> num_unique_vars t + num_unique_vars (Fun f ts)"
assms_1: shows "c \<turnstile> ev \<mapsto> d"
is_extension: shows "\<forall>f. C.arr f \<longrightarrow> F (\<F>C.inclusion_of_generators f) = V f"
pmf_of_set_Pow_conv_bernoulli: fixes A :: "'b set"   and P :: "bool" assumes "finite A" shows "map_pmf (\<lambda>b. {x \<in> A. b x}) (Pi_pmf A P (\<lambda>_. bernoulli_pmf (1 / 2))) = pmf_of_set (Pow A)"
retractionE: fixes S :: "'b set"   and T :: "'b set"   and r :: "'b \<Rightarrow> 'b" assumes "retraction S T r" obtains "T = r ` S" and "r ` S \<subseteq> S" and "continuous_on S r" and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
cnt_0_D: fixes x :: "'b"   and xs :: "'b list" assumes "cnt x xs = 0" shows "x \<notin> set xs"
length_pos_if_in_set: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "0 < length xs"
\<sigma>1_1: shows "\<sigma>1 \<in> small P \<sigma>"
elims: fixes x :: "bool"   and xa :: "real"   and xb :: "real"   and xc :: "real"   and xd :: "real"   and xe :: "nat"   and y :: "real" assumes "\<Psi> x xa xb xc xd xe = y" obtains b :: "bool" and i :: "real" and d :: "real" and x\<^sub>1 :: "real" and x\<^sub>2 :: "real" and n :: "nat" where "x = b" and "xa = i" and "xb = d" and "xc = x\<^sub>1" and "xd = x\<^sub>2" and "xe = n" and "y = (if x\<^sub>2 \<le> real n then i * (real n - x\<^sub>2) else if real n \<le> x\<^sub>1 \<and> b then d * (x\<^sub>1 - real n) else 0)"
OAlist_tc_map_val_def: fixes f :: "'d \<Rightarrow> 'f \<Rightarrow> 'e"   and xs :: "('d, 'f) oalist_tc" shows "OAlist_tc_map_val f xs = OAlist_tc (map_val_pair f (list_of_oalist_tc xs))"
simps_2: fixes v :: "int \<times> int" shows "head (C v) = C v"
mergec_single: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'm" shows "mergec (csingle x) f = f x"
mult_pos_neg2: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "b < (0::'a)" shows "b * a < (0::'a)"
disct_fct_point_measurable: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure" assumes "disc_fct f"   and "point_measurable M (space N) f" shows "f \<in> M \<rightarrow>\<^sub>M N"
q: fixes s :: "'s"   and x :: "'call" assumes "I s"   and "consider' x" shows "interaction_bounded_by consider (callee s x) q"
Collect_curry: fixes P :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> bool" shows "{x. P (curry x)} = case_prod ` {x. P x}"
ovalidNF_property_def: fixes P :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option" shows "ovalidNF_property P x = (\<lambda>s f. x s f \<noteq> None \<and> (\<forall>r. Some r = x s f \<longrightarrow> P r s))"
f_inv_into_f: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "y \<in> f ` A" shows "f (inv_into A f y) = y"
ipset_from_cidr_ipcidr_to_interval: fixes base :: "'a word"   and len :: "nat" shows "ipset_from_cidr base len = {ipcidr_to_interval_start (base, len)..ipcidr_to_interval_end (base, len)}"
rep_eq: fixes x :: "complex" shows "(*\<^sub>V) (cblinfun_scaleC_right x) = (*\<^sub>C) x"
bijI: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f"   and "surj f" shows "bij f"
f: shows "e\<lparr>a:X\<rparr> = Env f"
ev_blocks_part_def: fixes m :: "nat"   and A :: "'a mat" shows "local.ev_blocks_part m A \<equiv> \<forall>i j k. i < j \<longrightarrow> j < k \<longrightarrow> k < m \<longrightarrow> A $$ (k, k) = A $$ (i, i) \<longrightarrow> A $$ (j, j) = A $$ (i, i)"
select_convs_3: fixes empty :: "'d"   and lookup :: "'d \<Rightarrow> 'b \<Rightarrow> 'c option"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'd"   and more :: "'e" shows "update \<lparr>MapOps.empty = empty, lookup = lookup, update = update, \<dots> = more\<rparr> = update"
sub_K_validity: fixes B :: "agent"   and L :: "nat"   and K :: "nat"   and sub_K :: "msg"   and evs :: "event list"   and A :: "agent" assumes "Gets TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs"   and "sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>"   and "A \<notin> broken"   and "evs \<in> zg" shows "Says A TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs"
lookup_update_with_trie: fixes ks :: "'e list"   and f :: "'d option \<Rightarrow> 'd"   and t :: "('e, 'd) trie"   and ks' :: "'e list" shows "lookup_trie (update_with_trie ks f t) ks' = (if ks' = ks then Some (f (lookup_trie t ks')) else lookup_trie t ks')"
\<ll>_ide_simp: fixes a :: "'a" assumes "ide a" shows "MC.\<ll> a = MC.lunit a"
Sublists_singleton: fixes w :: "'a list" shows "Sublists {w} = set (sublists w)"
onll_ostep_invariantD: fixes A :: "('a \<times> ('b, 'c, 'd, 'e) seqp, 'f) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and P :: "('a \<times> 'e) \<times> 'f \<times> 'a \<times> 'e \<Rightarrow> bool"   and \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and a :: "'f"   and \<sigma>' :: "'a"   and p' :: "('b, 'c, 'd, 'e) seqp" assumes "A \<Turnstile>\<^sub>A (S, U \<rightarrow>) onll \<Gamma> P"   and "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a" shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<sigma>, l), a, \<sigma>', l')"
distinct_5: fixes x1 :: "res_crash"   and x4 :: "e list" shows "RSCrash x1 \<noteq> RSNormal x4"
inj1: shows "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>"
parts_crypt_prikey: fixes s :: "(agent \<times> msg) set"   and K :: "key"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt K (PriKey A) \<in> parts (used s)" shows "(\<exists>n. K = Auth_ShaKey n \<and> (Asset n, Crypt (Auth_ShaKey n) (PriKey A)) \<in> s) \<or> {PriKey A, Key K} \<subseteq> spied s"
c_iop_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "c_iop G H = (\<lambda>X\<in>set_rcs G H. {z. \<exists>x\<in>X. \<exists>h\<in>H. h \<cdot>\<^bsub>G\<^esub> \<rho>\<^bsub>G\<^esub> x = z})"
reds_def: shows "reds \<equiv> \<lambda>P. {(x, (xb, xa), xd, xc). redsp P x xb xa xd xc}"
axioms_2: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "strict_ordered_set A less" shows "transitive A less"
ipresIGWls_def: fixes h :: "'i \<Rightarrow> 'j"   and MOD :: "('k, 'l, 'm, 'n, 'o, 'p, 'i, 'q, 'r) model_scheme"   and MOD' :: "('s, 't, 'u, 'n, 'v, 'w, 'j, 'x, 'y) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
Abs_lst1_fcb2': fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'd \<Rightarrow> 'c"   and c :: "'d" assumes "[[atom a]]lst. x = [[atom b]]lst. y"   and "atom a \<sharp> f a x c"   and "{atom a, atom b} \<sharp>* c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f a x c = f (p \<bullet> a) (p \<bullet> x) c"   and "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> f b y c = f (p \<bullet> b) (p \<bullet> y) c" shows "f a x c = f b y c"
OUTfromChCorrect_data14: shows "OUTfromChCorrect data14"
weak_triangle: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and z :: "'a"   and y :: "'a" assumes "Turn_into_distance f" shows "f x z \<le> sqrt 2 * max (f x y) (f y z)"
adh_closure_iff: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier"   and "x \<in> carrier" shows "(x adh A) = (x \<in> closure A)"
left_unique_rel_gcI: assumes "left_unique rel_c" shows "left_unique rel_gc"
coeffs_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" assumes "(f (lead_coeff p) = (0::'b)) = (p = 0)" shows "coeffs (map_poly f p) = map f (coeffs p)"
atLeastAtMost_iff: fixes i :: "'a word"   and l :: "'a word"   and u :: "'a word" shows "(i \<in> signed.atLeastAtMost l u) = (l \<le>s i \<and> i \<le>s u)"
run_fail_nondet: fixes return :: "'f \<Rightarrow> 'd"   and empty :: "'f" shows "run_nondet (nondetM_base.fail_nondet return empty) = return empty"
dupe2_1: shows "q.eq_m (A' * D + B' * H) U2"
comm_ring_1_ow_def: fixes U :: "'a set"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" shows "comm_ring_1_ow U times one plus zero minus uminus \<equiv> comm_ring_ow U plus zero minus uminus times \<and> zero_neq_one_ow U one zero \<and> comm_monoid_mult_ow U times one"
omax_in: fixes R :: "('a \<times> 'a) set set" assumes "finite R"   and "R \<noteq> {}"   and "\<forall>r\<in>R. Well_order r" shows "omax R \<in> R"
interesting_subst2': fixes a :: "coname"   and b :: "coname"   and P :: "trm"   and N :: "trm"   and y :: "name"   and z :: "name" assumes "a \<noteq> b"   and "a \<sharp> P"   and "b \<sharp> P" shows "N{a:=y.P}{b:=y.P} = N{b:=z.Ax z a}{a:=y.P}"
similar_mat_wit_refl: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "similar_mat_wit A A (1\<^sub>m n) (1\<^sub>m n)"
below: fixes y :: "'a" assumes "y \<in> Upper L A" shows "s \<sqsubseteq> y"
coeff_Abs_poly: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>i. n < i \<Longrightarrow> f i = (0::'b)" shows "coeff (Abs_poly f) = f"
minus_mult_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = times a (uminus b)"
lstep_def: fixes x :: "accent \<times> 'b" shows "local.lstep x = (case x of (Acute, i) \<Rightarrow> (L i)\<inverse> | (Grave, i) \<Rightarrow> R i | (Macron, i) \<Rightarrow> E i)"
f: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
exhaust: fixes y :: "'c event" obtains x1 :: "'c" where "y = Broadcast x1"   | x2 :: "'c" where "y = Deliver x2"
nat_pow_Suc: fixes x :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" shows "x [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> Suc n = x [^]\<^bsub>function_ring (carrier Zp) Zp\<^esub> n \<otimes>\<^bsub>function_ring (carrier Zp) Zp\<^esub> x"
IH2: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> fst (simp r2) \<rightarrow> v" shows "s \<in> r2 \<rightarrow> snd (simp r2) v"
neg_lc_dom_quad: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "a < 0" shows "\<exists>w. \<forall>y>w. a * y\<^sup>2 + b * y + c < 0"
select_convs_1: fixes I :: "nat"   and M :: "nat"   and N :: "nat"   and S :: "nat"   and more :: "'b" shows "vars.I \<lparr>vars.I = I, M = M, N = N, S = S, \<dots> = more\<rparr> = I"
lle_double_def: fixes x :: "double"   and y :: "double" shows "lle_double x y = (lcompare_double x y \<le> 0)"
simps_1: fixes n :: "nat"   and c :: "poly" shows "tmboundslt n (CP c) = True"
Int_Un_eq_4: fixes T :: "'b set"   and S :: "'b set" shows "T \<union> S \<inter> T = T"
1: shows "inverse_functors (\<cdot>) (\<cdot>\<acute>) \<Psi> \<Phi>"
cis_cmod: fixes a :: "complex" assumes "a \<noteq> 0" shows "cis (arg a) * cor (cmod a) = a"
assms_5: shows "range_vars \<theta> \<inter> bvars\<^sub>s\<^sub>t S = {}"
neq: fixes y :: "real" assumes "y \<in> {0..1}" shows "(1 - x * y) * z \<noteq> 1"
get_owner_document_wf_is_l_get_owner_document_wf: shows "l_get_owner_document_wf Shadow_DOM.heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs get_disconnected_nodes Shadow_DOM.get_owner_document Shadow_DOM.get_parent"
map_nth: fixes xs :: "'d list" shows "map ((!) xs) [0..<length xs] = xs"
nprod_eqTr: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" shows "f \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> g \<in> {j. j \<le> n} \<rightarrow> carrier R \<and> (\<forall>j\<le>n. f j = g j) \<longrightarrow> e\<Pi>\<^bsub>R,n\<^esub> f = e\<Pi>\<^bsub>R,n\<^esub> g"
infnorm_pos_le: fixes x :: "'a" shows "0 \<le> infnorm x"
thetaIfTT_BisT: shows "thetaIfTT \<subseteq> BisT"