hcju/isabelleps huggingface dataset:
d: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> - V" shows "d \<le> dist x y"
adv_P2_def: fixes n :: "nat"   and m1 :: "bool \<times> bool"   and m2 :: "bool"   and D :: "bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf" shows "OT_12.adv_P2 n m1 m2 D = \<bar>spmf (R2 n m1 m2 \<bind> D) True - spmf (funct_OT_12 m1 m2 \<bind> (\<lambda>(o1, o2). S2 n m2 o2 \<bind> D)) True\<bar>"
equiv_subst_var: fixes v :: "nat \<Rightarrow> 'a"   and x\<^sub>j :: "nat"   and lp' :: "linear_poly"   and lp :: "linear_poly" shows "v x\<^sub>j = lp' \<lbrace> v \<rbrace> \<longrightarrow> lp \<lbrace> v \<rbrace> = subst_var x\<^sub>j lp' lp \<lbrace> v \<rbrace>"
simps_1: fixes i :: "int" shows "leaves i \<langle>\<rangle> = {(i, 0)}"
cINF_lower: fixes f :: "'e \<Rightarrow> 'd multiset"   and A :: "'e set"   and x :: "'e" assumes "subset_mset.bdd_below (f ` A)"   and "x \<in> A" shows "\<Sqinter> (f ` A) \<subseteq># f x"
changes_sign: fixes p :: "real Polynomial.poly" shows "\<forall>x y. sign p x \<noteq> sign p y \<and> x < y \<longrightarrow> (\<exists>c\<in>root_list p. x \<le> c \<and> c \<le> y)"
intro: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad return bind"   and "monad_commute_axioms bind" shows "monad_commute return bind"
is_mset_set_def: fixes X :: "'b multiset" shows "is_mset_set X \<equiv> \<forall>x\<in>#X. count X x = 1"
UP_r_distr: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "r \<otimes>\<^bsub>P\<^esub> (p \<oplus>\<^bsub>P\<^esub> q) = r \<otimes>\<^bsub>P\<^esub> p \<oplus>\<^bsub>P\<^esub> r \<otimes>\<^bsub>P\<^esub> q"
fps_zero_code: shows "0 = fps_of_ratfps 0"
prems_4: shows "(t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
c_d: shows "(c, d) \<in> thetaSSeqI"
map_rec: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map f xs = rec_list [] (\<lambda>x _. (#) (f x)) xs"
mset_add: fixes a :: "'c"   and A :: "'c multiset" assumes "a \<in># A" obtains B :: "'c multiset" where "A = add_mset a B"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "(if x \<in> S then f' x else (\<lambda>x. 0::'b)) (b1 + b2) = (if x \<in> S then f' x else (\<lambda>x. 0::'b)) b1 + (if x \<in> S then f' x else (\<lambda>x. 0::'b)) b2"
that_2: assumes "\<not> f x"   and "\<not> f (\<pi> \<bullet> x)" shows "thesis"
bhvks2: fixes t :: "nat \<Rightarrow> cnf"   and p :: "PROB"   and kId :: "'kid"   and t' :: "nat \<Rightarrow> 'KS"   and P :: "PROB set"   and q :: "PROB" assumes "t \<in> arch"   and "p = prob kId" shows "sb.eval kId t t' 0 (\<box>\<^sub>b[\<lambda>ks. sub P = ksrp ks \<and> q \<in> P \<longrightarrow> (q, p) \<in> sb]\<^sub>b)"
abstract_primitive_in_doubt_deny_Deny2: fixes \<beta> :: "'b common_primitive \<Rightarrow> ('b, 'c) tagged_packet_scheme \<Rightarrow> ternaryvalue"   and m :: "'b common_primitive match_expr"   and p :: "('b, 'c) tagged_packet_scheme"   and disc :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "primitive_matcher_generic \<beta>"   and "normalized_nnf_match m"   and "\<not> matches (\<beta>, in_doubt_deny) m Accept p" shows "\<not> matches (\<beta>, in_doubt_deny) (abstract_primitive disc m) Accept p"
IH: fixes n :: "'a itrm" assumes "n \<in> CF" shows "norm_nn n n' \<simeq> n \<diamondop> n'"
image_mset_of_subset_list: fixes \<eta> :: "'c \<Rightarrow> 'b"   and C' :: "'c multiset"   and lC :: "'b list" assumes "image_mset \<eta> C' = mset lC" shows "\<exists>qC'. map \<eta> qC' = lC \<and> mset qC' = C'"
disj_ac_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a OR b) OR c = a OR b OR c"
lead_coeff_Q': fixes f :: "nat \<Rightarrow> complex"   and n :: "nat" shows "poly.coeff (power_sum_puzzle_existence.Q' f n) n = 1"
derivativeReflexive: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and a :: "subject"   and x :: "name" assumes "Id \<subseteq> Rel" shows "derivative P P a x Rel"
HT_partial_strengthen_modset: fixes \<pi> :: "char list \<Rightarrow> com option"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_partial \<pi> P c Q" shows "HT_partial \<pi> P c (\<lambda>s\<^sub>0 s. Q s\<^sub>0 s \<and> modifies (lhsv \<pi> c) s s\<^sub>0)"
UINF_is_exists: fixes P :: "'c \<Rightarrow> 'b upred" shows "(\<Sqinter> x \<bullet> P x) = ushEx P"
ereal_divide_left_mono: fixes y :: "ereal"   and x :: "ereal"   and z :: "ereal" assumes "y \<le> x"   and "0 < z"   and "0 < x * y" shows "z / x \<le> z / y"
simps_4: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "upds (Arg x # S) = upds S"
omega_prev: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "0 \<in>\<^sub>\<circ> n" obtains k :: "V" where "n = ZFC_in_HOL.succ k"
hq: shows "(\<forall>q\<in>insert x F. q \<noteq> 0) \<longrightarrow> x \<noteq> 0"
fps_shift_times_fps_X_power: fixes n :: "nat"   and f :: "'a fps" assumes "n \<le> subdegree f" shows "fps_shift n f * fps_X ^ n = f"
pdevs_val_independent_pdevs2: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and xs :: "(real \<times> real) list" assumes "e \<in> UNIV \<rightarrow> I" shows "\<exists>e'. e' \<in> UNIV \<rightarrow> I \<and> pdevs_val e (pdevs_of_list (independent_pdevs xs)) = pdevs_val e' (pdevs_of_list xs)"
tendsto_add_ereal_nonneg: fixes x :: "ereal"   and y :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and F :: "'a filter"   and g :: "'a \<Rightarrow> ereal" assumes "x \<noteq> - \<infinity>"   and "y \<noteq> - \<infinity>"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
mirrored_step: shows "revision_step (\<alpha> r) s\<^sub>1' (\<R>\<^sub>G \<alpha> \<beta> s\<^sub>2)"
abs_summable_on_reindex_bij_betw: fixes g :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "bij_betw g A B" shows "((\<lambda>x. f (g x)) abs_summable_on A) = (f abs_summable_on B)"
hermitian_one: fixes n :: "nat" shows "hermitian (1\<^sub>m n)"
Rep_Assertion_inject: fixes x :: "'c Assertion"   and y :: "'c Assertion" shows "(Rep_Assertion x = Rep_Assertion y) = (x = y)"
left_distrib': fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<cdot> (y \<squnion> z) = x \<cdot> y \<squnion> x \<cdot> z"
lsc_sequentially: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> ereal" shows "lsc_at x0 f = (\<forall>x c. x \<longlonglongrightarrow> x0 \<and> (\<forall>n. f (x n) \<le> c) \<longrightarrow> f x0 \<le> c)"
comm_monoid_add_hom_0_axioms: shows "comm_monoid_add_hom_0 hom"
testN_dim: fixes k :: "nat" shows "testN k \<in> carrier_mat N N"
cong3_conga: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "C \<noteq> B"   and "A B C Cong3 A' B' C'" shows "A B C CongA A' B' C'"
op_find_lasso_spec_def: shows "op_find_lasso_spec \<equiv> igb_graph.find_lasso_spec"
diff_col_ex3: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "\<exists>D. A \<noteq> D \<and> B \<noteq> D \<and> C \<noteq> D \<and> Col A B D"
dom_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "H.VVV.dom f = (if H.VVV.arr f then H.VxVxV.dom f else H.VxVxV.null)"
det_field_def: fixes A :: "'a mat" shows "det_field A = det_code fst_sel_fun field_mute_fun A"
cases: fixes a1 :: "'b llist"   and a2 :: "'b llist" assumes "lprefix a1 a2" obtains xs :: "'b llist" where "a1 = LNil" and "a2 = xs__"   | xs :: "'b llist" and ys :: "'b llist" and x :: "'b" where "a1 = LCons x xs" and "a2 = LCons x ys" and "lprefix xs ys"
asymptotics1': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>1 \<le> x"   and "i < k" shows "(bs ! i * x) powr p * (1 + ln x powr (- e / 2)) \<le> (bs ! i * x + (hs ! i) x) powr p * (1 + ln (bs ! i * x + (hs ! i) x) powr (- e / 2))"
cSup_mem_eq: fixes c :: "'b"   and X :: "'b set" assumes "c \<in> X"   and "\<forall>x\<in>X. x \<le> c" shows "Sup X = c"
leastI2_order: fixes x :: "hf"   and a :: "hf"   and Q :: "hf \<Rightarrow> bool" assumes "x \<^bold>\<in> a"   and "\<And>y. y \<^bold>\<in> a \<Longrightarrow> Rep_hf x \<le> Rep_hf y"   and "\<And>z. \<lbrakk>z \<^bold>\<in> a; \<forall>y. y \<^bold>\<in> a \<longrightarrow> Rep_hf z \<le> Rep_hf y\<rbrakk> \<Longrightarrow> Q z" shows "Q (least a)"
hinsert_hdiff_if: fixes x :: "hf"   and A :: "hf"   and B :: "hf" shows "(A \<triangleleft> x) - B = (if x \<^bold>\<in> B then A - B else A - B \<triangleleft> x)"
Seed_holds_facesAt_eq: fixes p :: "nat" shows "facesAt_eq Seed\<^bsub>p\<^esub>"
B: shows "\<forall>\<^sub>F n in sequentially. Sigma_Algebra.measure M (T ((1 / 2) ^ n)) < Sigma_Algebra.measure M S + epsilon"
step: shows "\<Gamma>\<turnstile> (cs @ xs, css, s) \<rightarrow> (cs' @ xs, css, t)"
alist_of_eqI: fixes p :: "('c, 'd) pq"   and q :: "('c, 'd) pq" assumes "alist_of p = alist_of q" shows "p = q"
signed_take_bit_of_minus_1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n (uminus one) = uminus one"
simps_1: fixes x :: "'a"   and lt :: "('a, 'c) rbt"   and z :: "'a"   and v :: "'c"   and rt :: "('a, 'c) rbt"   and y :: "'a"   and s :: "'c"   and b :: "('a, 'c) rbt" shows "rbt_del_from_left x (Branch B lt z v rt) y s b = balance_left (rbt_del x (Branch B lt z v rt)) y s b"
abs_eq: fixes xa :: "'d"   and x :: "'b" shows "FL_Formula.Pred\<^sub>\<alpha> xa x = FL_Formula.abs_Tree\<^sub>\<alpha> (FL_Formula.Tree.tPred xa x)"
empty_tokens_def: fixes T :: "(symbol \<times> character list) set" shows "empty_tokens T = {t \<in> T. chars_of_token t = []}"
simps_2: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "ltl_step e (Some s) r (l, i) = (let possibilities = possible_steps e s r l i in if possibilities = {||} then (None, [], r) else let (s', t) = SOME x. x |\<in>| possibilities in (Some s', evaluate_outputs t i r, evaluate_updates t i r))"
pos_divideR_less_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and b :: "'a"   and a :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less (scaleR (inverse c) b) a = less b (scaleR c a)"
C_prod_eq_D_times_E: shows "prod id E * prod id D = prod id C"
simps_12: fixes f :: "'b"   and v :: "'b local.term"   and va :: "'b local.term" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v \<^bold>\<otimes> va = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> v \<^bold>\<otimes> va"
induct: fixes x1 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and x2 :: "com"   and x3 :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool"   and P :: "((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> com \<Rightarrow> ((char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>3\<^sub>a {x1} x2 { x3}"   and "P ($ 1) SKIP ($ 0)"   and "\<And>x a Q. P ((\<lambda>(ps, t). x \<in> dom ps \<and> vars a \<subseteq> dom ps \<and> Q (ps(x \<mapsto> paval a ps), t)) \<and>* $ 1) (x ::= a) Q"   and "\<And>Pa b c\<^sub>1 Q c\<^sub>2. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b True s} c\<^sub>1 { Q}; P (\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b True s) c\<^sub>1 Q; \<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b False s} c\<^sub>2 { Q}; P (\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b False s) c\<^sub>2 Q\<rbrakk> \<Longrightarrow> P ((\<lambda>(s, n). Pa (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1) (IF b THEN c\<^sub>1 ELSE c\<^sub>2) Q"   and "\<And>Pa C Q F. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {Pa} C { Q}; P Pa C Q\<rbrakk> \<Longrightarrow> P (Pa \<and>* F) C (Q \<and>* F)"   and "\<And>Pa C\<^sub>1 Q C\<^sub>2 R. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {Pa} C\<^sub>1 { Q}; P Pa C\<^sub>1 Q; \<turnstile>\<^sub>3\<^sub>a {Q} C\<^sub>2 { R}; P Q C\<^sub>2 R\<rbrakk> \<Longrightarrow> P Pa (C\<^sub>1;; C\<^sub>2) R"   and "\<And>Pa b C. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b True s} C { (\<lambda>(s, n). Pa (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1}; P (\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b True s) C ((\<lambda>(s, n). Pa (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1)\<rbrakk> \<Longrightarrow> P ((\<lambda>(s, n). Pa (s, n) \<and> vars b \<subseteq> dom s) \<and>* $ 1) (WHILE b DO C) (\<lambda>(s, n). Pa (s, n) \<and> lmaps_to_axpr b False s)"   and "\<And>Pa c Q P' Q'. \<lbrakk>\<turnstile>\<^sub>3\<^sub>a {Pa} c { Q}; P Pa c Q; \<And>s n. P' (s, n) \<Longrightarrow> Pa (s, n); \<And>s n. Q (s, n) \<Longrightarrow> Q' (s, n)\<rbrakk> \<Longrightarrow> P P' c Q'" shows "P x1 x2 x3"
length_drop: fixes n :: "nat"   and xs :: "'d list" shows "length (xs \<up> n) = length xs - n"
when_False: fixes X :: "'c \<Rightarrow> (unit \<times> 'c) set \<times> bool" shows "when False X = return ()"
NDtrans: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F"   and "F \<triangleright> \<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> G"
inject_3: fixes x3 :: "ty"   and x1 :: "name"   and x2 :: "trm"   and y3 :: "ty"   and y1 :: "name"   and y2 :: "trm" shows "((\<lambda>x1:x3. x2) = (\<lambda>y1:y3. y2)) = (x3 = y3 \<and> [x1].x2 = [y1].y2)"
clw_rel_br: fixes a :: "'a \<Rightarrow> 'c set"   and I :: "'a \<Rightarrow> bool" shows "clw_rel (br a I) = br (\<lambda>xs. \<Union> (a ` set xs)) (\<lambda>xs. Ball (set xs) I)"
my_U_is_defined_2: fixes p :: "'a pmf" assumes "p \<in> local.\<P>" shows "p \<approx>[\<R>] mix_pmf (local.my_U p) local.some_best local.some_worst"
ntrp_mon_env_w_no_ctx: fixes s :: "'n list"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list el_step list"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), w, s', c') \<in> trcl (ntrp fg)" shows "mon_env fg w \<inter> mon_s fg s = {}"
ereal_incseq_approx: fixes a :: "ereal"   and b :: "ereal" assumes "a < b" obtains X :: "nat \<Rightarrow> real" where "incseq X" and "\<And>i. a < ereal (X i)" and "\<And>i. ereal (X i) < b" and "(\<lambda>x. ereal (X x)) \<longlonglongrightarrow> b"
UN_ball_bex_simps_1: fixes A :: "'a set set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>\<Union> A. P x) = (\<forall>y\<in>A. \<forall>x\<in>y. P x)"
wide_subdigraphI: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<And>a. a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr> \<Longrightarrow> a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>"
echelon_form_of_upt_k_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "echelon_form_of_upt_k A k bezout = fst (foldl (echelon_form_of_column_k bezout) (A, 0) [0..<Suc k])"
inf_sup_ord_4: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq y (sup x y)"
subpath_refl: fixes a :: "real"   and g :: "real \<Rightarrow> 'a" shows "subpath a a g = linepath (g a) (g a)"
assms_2: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"
no_icd_pd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and k :: "nat" assumes "is_path \<pi>"   and "\<forall>l\<ge>n. \<not> k icd\<^bsup>\<pi>\<^esup>\<rightarrow> l"   and "n \<le> k" shows "\<pi> k pd\<rightarrow> \<pi> n"
set_mset_eq_empty_iff: fixes M :: "'a multiset" shows "(set_mset M = {}) = (M = {#})"
fds_eqI: fixes f :: "'b fds"   and g :: "'b fds" assumes "\<And>n. 0 < n \<Longrightarrow> fds_nth f n = fds_nth g n" shows "f = g"
cases: fixes a1 :: "32 Word.word"   and a2 :: "real" assumes "a1 \<equiv>\<^sub>E a2" obtains r :: "real" where "a1 = POS_INF" and "a2 = ra__" and "real_of_int (sint POS_INF) \<le> ra__"   | r :: "real" where "a1 = NEG_INF" and "a2 = r" and "r \<le> real_of_int (sint NEG_INF)"   | w :: "32 Word.word" where "a1 = w" and "a2 = real_of_int (sint w)" and "real_of_int (sint w) < real_of_int (sint POS_INF)" and "real_of_int (sint NEG_INF) < real_of_int (sint w)"
isomTr0: fixes F :: "('e, 'f) Group_scheme" assumes "Group F" shows "F \<cong> F"
csa1: shows "set (characterize_consistent_signs_at_roots_copr p qs1) \<subseteq> set signs1"
A: fixes n :: "nat" assumes "1 \<le> n" shows "exp (Ln (1 + z / of_nat n)) = 1 + z / of_nat n"
block_design_multiple: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and n :: "nat" assumes "block_design point_set block_collection u_block_size"   and "0 < n" shows "block_design point_set (repeat_mset n block_collection) u_block_size"
subdprodr: fixes x :: "nat list"   and y :: "nat list" assumes "(x, y) \<in> local.Minimal_Solutions"   and "(x, y) \<notin> Special_Solutions" shows "subdprodr y"
monadic_nfoldli_refine_aux: fixes \<Gamma> :: "assn"   and Rs :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and c :: "'c \<Rightarrow> bool Heap"   and c' :: "'d \<Rightarrow> bool nres"   and Rl :: "'f \<Rightarrow> 'e \<Rightarrow> assn"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c Heap"   and f' :: "'f \<Rightarrow> 'd \<Rightarrow> 'd nres"   and l' :: "'f list"   and l :: "'e list"   and s' :: "'d"   and s :: "'c" assumes "\<And>s s'. hn_refine (\<Gamma> * hn_ctxt Rs s' s) (c s) (\<Gamma> * hn_ctxt Rs s' s) bool_assn (c' s')"   and "\<And>x x' s s'. hn_refine (\<Gamma> * hn_ctxt Rl x' x * hn_ctxt Rs s' s) (f x s) (\<Gamma> * hn_invalid Rl x' x * hn_invalid Rs s' s) Rs (f' x' s')" shows "hn_refine (\<Gamma> * hn_ctxt (list_assn Rl) l' l * hn_ctxt Rs s' s) (imp_nfoldli l c f s) (\<Gamma> * hn_invalid (list_assn Rl) l' l * hn_invalid Rs s' s) Rs (monadic_nfoldli l' c' f' s')"
b1d_ub: shows "- Q $$ (0, 1) * (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q) $$ (0, 0) = u * b"
ocU_is_expected_utility_bernoulli: fixes outcomes :: "'a set"   and \<R> :: "('a pmf \<times> 'a pmf) set" assumes "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "finite outcomes"   and "outcomes \<noteq> {}"   and "continuous_vnm (\<P> outcomes) \<R>" shows "\<forall>x\<in>\<P> outcomes. \<forall>y\<in>\<P> outcomes. x \<succeq>[\<R>] y = (measure_pmf.expectation y (ocU outcomes \<R>) \<le> measure_pmf.expectation x (ocU outcomes \<R>))"
rel_mono_strong0: fixes R1 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and R2 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and x :: "('r, 't, 'v) gpv"   and y :: "('s, 'u, 'v) gpv"   and R1a :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and R2a :: "'t \<Rightarrow> 'u \<Rightarrow> bool" assumes "rel_gpv R1 R2 x y"   and "\<forall>z1\<in>results'_gpv x. \<forall>y1\<in>results'_gpv y. R1 z1 y1 \<longrightarrow> R1a z1 y1"   and "\<forall>z2\<in>outs'_gpv x. \<forall>y2\<in>outs'_gpv y. R2 z2 y2 \<longrightarrow> R2a z2 y2" shows "rel_gpv R1a R2a x y"
distributed_def: fixes M :: "'a measure"   and N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" shows "distributed M N X f = (distr M N X = density N f \<and> f \<in> borel_measurable N \<and> X \<in> M \<rightarrow>\<^sub>M N)"
shift_eq: fixes u :: "'b list"   and v :: "'b stream"   and w :: "'b stream" shows "(u @- v = w) = (u = stake (length u) w \<and> v = sdrop (length u) w)"
br_invar_def: fixes \<delta> :: "('a, 'b) ta_rule set" shows "br_invar \<delta> \<equiv> {(Q, W). W \<subseteq> Q \<and> Q \<subseteq> b_accessible \<delta> \<and> bacc_step \<delta> (Q - W) \<subseteq> Q \<and> finite Q}"
posd: shows "has_bochner_integral M (\<lambda>w. fpos w - fneg w) ((\<Sum>w\<in>range (pseudo_proj_True n). prob (pseudo_proj_True n -` {w} \<inter> space M) * fpos w) - (\<Sum>w\<in>range (pseudo_proj_True n). prob (pseudo_proj_True n -` {w} \<inter> space M) * fneg w))"
fst_abss: fixes st :: "'s \<times> ('c \<Rightarrow> real)" shows "fst (abss st) = fst st"
assms_2: shows "a \<in>\<^sub>\<circ> f i"
abs_conv_abscissa_cong: fixes f :: "'b fds"   and g :: "'b fds" assumes "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n" shows "abs_conv_abscissa f = abs_conv_abscissa g"
Box_def: fixes \<alpha> :: "('e, 'f, 'g) hp"   and P :: "('e, 'f, 'g) formula" shows "([[\<alpha>]]P) = ! (\<langle> \<alpha> \<rangle> ! P)"
homeomorphic_simply_connected_eq: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T" shows "simply_connected S = simply_connected T"
witness_weights_def: fixes rs :: "nat list"   and y :: "nat"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y" shows "deep_model_correct_params_y.witness_weights shared_weights rs = extract_weights shared_weights (witness_l rs)"
assms_2: shows "wf\<^sub>e\<^sub>s\<^sub>t V A"
case_eq_if: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'g \<Rightarrow> 'd"   and generat :: "('e, 'f, 'g) generat" shows "(case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = (if is_Pure generat then f1 (generat.result generat) else f2 (generat.output generat) (continuation generat))"
composites_are_arrows_2: shows "hseq (g \<star> f) g"
dc1: shows "density_context V V' \<Gamma> (\<lambda>\<sigma>. \<delta> \<sigma> * (if P \<sigma> then 1 else 0))"
valid_null_ref_def: fixes r :: "'d option" shows "valid_null_ref r \<equiv> case r of None \<Rightarrow> \<lambda>s. True | Some r' \<Rightarrow> obj_at (\<lambda>s. True) r'"
isSPath_sg_incoming: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat"   and u1 :: "nat"   and v1 :: "nat"   and u2 :: "nat" assumes "isSimplePath u p v"   and "(u1, v1) \<in> set p"   and "u1 \<noteq> u2" shows "(u2, v1) \<notin> set p"
first_in_tree_order_impl: shows "first_in_tree_order = local.a_first_in_tree_order"
Limsup_filtermap_ge: fixes F :: "'d filter"   and g :: "'e \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'e" shows "Limsup F (\<lambda>x. g (f x)) \<le> Limsup (filtermap f F) g"
Suc_neq_Zero: fixes m :: "nat"   and R :: "bool" assumes "Suc m = 0" shows "R"
assms_2: shows "\<forall>f\<in>\<Pi>\<^sub>E a\<in>A. generate G {a}. finprod G f A = \<one> \<longrightarrow> (\<forall>a\<in>A. f a = \<one>)"