hcju/isabelleps huggingface dataset:
echelon_form_of_det_iarrays_def: fixes A :: "'c iarray iarray"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c" shows "echelon_form_of_det_iarrays A bezout = echelon_form_of_upt_k_det_iarrays (1::'c, A) (ncols_iarray A - 1) bezout"
hermitian_def: fixes A :: "'b mat" shows "hermitian A = (adjoint A = A)"
three_SPRAY_ge4: fixes x :: "'a" assumes "three_SPRAY x" shows "\<exists>Q1\<in>\<P>. \<exists>Q2\<in>\<P>. \<exists>Q3\<in>\<P>. \<exists>Q4\<in>\<P>. Q1 \<noteq> Q2 \<and> Q1 \<noteq> Q3 \<and> Q1 \<noteq> Q4 \<and> Q2 \<noteq> Q3 \<and> Q2 \<noteq> Q4 \<and> Q3 \<noteq> Q4"
DiffD1: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A - B" shows "c \<in> A"
encode_all_operator_effects_def: fixes \<Pi> :: "'b strips_problem"   and ops :: "'b strips_operator list"   and t :: "nat" shows "encode_all_operator_effects \<Pi> ops t \<equiv> let l = List.product [0..<t] ops in foldr (\<^bold>\<and>) (map (\<lambda>(t, op). encode_operator_effect \<Pi> t op) l) (\<^bold>\<not> \<bottom>)"
compatible_add: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
1: shows "[\<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<^bold>\<not>(\<^bold>\<exists>F. \<lbrace>x\<^sup>P,F\<rbrace>) in v]"
collapse: fixes tree :: "'b tree" shows "Node (rank tree) (tree.root tree) (children tree) = tree"
exp_suminf_prodinf_real: fixes f :: "nat \<Rightarrow> real" assumes "\<And>n. 0 \<le> f n"   and "abs_convergent_prod (\<lambda>n. exp (f n))" shows "(\<Prod>i. exp (f i)) = exp (suminf f)"
congruence_rule_def: fixes t :: "'b"   and l :: "'b" shows "congruence_rule t l = transl_rule (A_Cmp (A_Cmp (A_Lbl t) (A_Lbl l)) (A_Lbl t) \<sqsubseteq> A_Lbl l)"
coboundedI1: fixes a :: "'c DBMEntry"   and c :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<preceq> c" shows "linordered_monoid.min a b \<preceq> c"
val_ArrLen_IntgT: fixes l :: "Location"   and s :: "Store" assumes "isArrLenLoc l" shows "\<tau> s@@l = IntgT"
gf: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> S" shows "g (f x) = x"
map_FGl_def: fixes l1 :: "'aj \<Rightarrow> 'aw"   and l2 :: "'ak \<Rightarrow> 'ax"   and l3 :: "'aq \<Rightarrow> 'bc"   and co1 :: "'al \<Rightarrow> 'ay"   and co2 :: "'am \<Rightarrow> 'az"   and co3 :: "'ar \<Rightarrow> 'bd"   and co4 :: "'as \<Rightarrow> 'be"   and contra1 :: "'ba \<Rightarrow> 'an"   and contra2 :: "'bb \<Rightarrow> 'ao"   and contra3 :: "'bf \<Rightarrow> 'at"   and contra4 :: "'bg \<Rightarrow> 'au" shows "map_FGl l1 l2 l3 co1 co2 co3 co4 contra1 contra2 contra3 contra4 = map_F (map_G l1 l2 co1 co2 contra1 contra2) l1 l3 co1 co3 co4 contra1 contra3 contra4"
one_closed: shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
seq_simps_1: shows "seq\<cdot>\<bottom> = \<bottom>"
intro: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and b :: "'ah" assumes "category A"   and "category B"   and "constant_functor_axioms B b" shows "constant_functor A B b"
distinct_1: shows "SayIKnow \<noteq> SayNothing"
rec_ldec_def: shows "rec_ldec = CN rec_predecessor [CN rec_pdec1 [rec_swap (rec_iter rec_pdec2)]]"
not_iRelease: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(\<not> (P t'. t' \<R> t I. Q t)) = ((\<diamond> t I. \<not> Q t) \<and> (\<box> t I. P t \<longrightarrow> (\<diamond> t I \<down>\<le> t. \<not> Q t)))"
COCOUNT_eq: fixes A :: "real set" shows "(A \<in> sets COCOUNT) = (countable A \<or> countable (UNIV - A))"
subgroupE_1: fixes H :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set set" assumes "subgroup H (relative_homology_group (int n) (nsphere n) (equator n))" shows "H \<subseteq> carrier (relative_homology_group (int n) (nsphere n) (equator n))"
simps_2: fixes f :: "(val \<times> val) fset"   and f' :: "(val \<times> val) fset" shows "val.Fun f \<sqsubseteq> val.Fun f' = (fset f \<subseteq> fset f')"
the_elem_image_unique: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "A \<noteq> {}"   and "\<And>y. y \<in> A \<Longrightarrow> f y = f x" shows "the_elem (f ` A) = f x"
erlang_distributed_sum: fixes I :: "'c set"   and l :: "real"   and X :: "'c \<Rightarrow> 'a \<Rightarrow> real"   and k :: "'c \<Rightarrow> nat" assumes "finite I"   and "I \<noteq> {}"   and "0 < l"   and "\<And>i. i \<in> I \<Longrightarrow> distributed M lborel (X i) (\<lambda>x. ennreal (erlang_density (k i) l x))"   and "indep_vars (\<lambda>i. borel) X I" shows "distributed M lborel (\<lambda>x. \<Sum>i\<in>I. X i x) (\<lambda>x. ennreal (erlang_density ((\<Sum>i\<in>I. Suc (k i)) - 1) l x))"
inject_2: fixes x21 :: "char list"   and x22 :: "expr"   and y21 :: "char list"   and y22 :: "expr" shows "(Cast x21 x22 = Cast y21 y22) = (x21 = y21 \<and> x22 = y22)"
f_plus_minus: fixes f :: "'b \<Rightarrow> real"   and x :: "'b" shows "f x = pp f x - np f x"
Step1_4_7H: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> unchanged (e p, c p, r p, m p, rmhist!p) \<longrightarrow> unchanged (rtrner memCh!p, S1 rmhist p, S2 rmhist p, S3 rmhist p, S4 rmhist p, S5 rmhist p, S6 rmhist p)"
extr_insert_IK_eq: fixes bad :: "agent set"   and X :: "msg"   and IK :: "msg set"   and CH :: "chan set" shows "extr bad (insert X IK) CH = insert X (extr bad IK CH)"
mon_s_def: fixes fg :: "('g, 'h, 'i, 'f, 'j) flowgraph_rec_scheme"   and s :: "'g list" shows "mon_s fg s \<equiv> \<Union> {mon_n fg n |n. n \<in> set s}"
less_eq_rat_def: fixes x :: "rat"   and y :: "rat" shows "(x \<le> y) = (x < y \<or> x = y)"
1: shows "satC \<xi> (map Neg (map2 (Pr \<circ> Guard) (arOf f) (map singl (GE.getTvars (arOf f))))) = (\<not> list_all2 (eintP \<circ> Guard) (arOf f) (map singl al))"
hyps: fixes cs :: "'edge list" shows "upd_rev_cs cs (a # as') = a # upd_rev_cs cs as'"
liouville_lambda_power_even: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "even m" shows "liouville_lambda (n ^ m) = (1::'b)"
long_border_bordered: fixes w :: "'b list"   and x :: "'b list" assumes "\<^bold>|w\<^bold>| < \<^bold>|x\<^bold>| + \<^bold>|x\<^bold>|"   and "x \<le>b w" shows "w\<^sup><\<inverse>x \<inverse>\<^sup>>x \<le>b x"
mod_subset_cmul: fixes a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and p :: "nat"   and X :: "'c topology"   and k :: "int" assumes "(a, b) \<in> mod_subset p X" shows "(frag_cmul k a, frag_cmul k b) \<in> mod_subset p X"
TC_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "TC R a b \<equiv> \<forall>P. (\<forall>x y. R x y \<longrightarrow> P x y) \<and> (\<forall>x y z. P x y \<and> P y z \<longrightarrow> P x z) \<longrightarrow> P a b"
netmap_add_disjoint: fixes s1 :: "'d net_state"   and s2 :: "'d net_state"   and \<sigma> :: "nat \<Rightarrow> 'd" assumes "\<forall>i\<in>net_ips s1 \<union> net_ips s2. the ((netmap s1 ++ netmap s2) i) = \<sigma> i"   and "net_ips s1 \<inter> net_ips s2 = {}" shows "\<forall>i\<in>net_ips s1. the (netmap s1 i) = \<sigma> i"
ntrs_stack_decomp: fixes u :: "'a"   and s :: "'a list"   and r :: "'a list"   and c :: "'a list multiset"   and ww :: "('b, 'c) label list list"   and s' :: "'a list"   and c' :: "'a list multiset"   and fg :: "('a, 'b, 'c, 'd, 'e) flowgraph_rec_scheme" assumes "((u # s @ r, c), ww, s', c') \<in> trcl (ntrs fg)" obtains v :: "'a" and rr :: "'a list" where "s' = v # rr @ r" and "((u # s, c), ww, v # rr, c') \<in> trcl (ntrs fg)"
assms_6: shows "mod_arr = arr'(m - 1 := Min_mset (arr_mset arr l\<^sub>0 m))"
simps_6: shows "\<infinity>\<^sub>2 +\<^sub>2 [0]\<^sub>2 = \<infinity>\<^sub>2"
t0_le: shows "t 0 \<le> t0 + e1'"
r_def: shows "r = run\<^sub>t (delta \<Sigma>) (initial \<phi>) w"
DC_def: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "a \<Sqinter> b \<equiv> \<lambda>ab Q s. min (a ab Q s) (b ab Q s)"
fmdom'_notI: fixes m :: "('b, 'a) fmap"   and x :: "'b" assumes "m $$ x = None" shows "x \<notin> fmdom' m"
REFL: fixes r :: "('d \<times> 'd) set" assumes "wo_rel r" shows "Refl r"
subdegree_power_ge: fixes f :: "'a fps"   and n :: "nat" assumes "f ^ n \<noteq> 0" shows "n * subdegree f \<le> subdegree (f ^ n)"
preserves_seq: fixes g :: "'a2"   and f :: "'a2" assumes "A2.seq g f" shows "B2.seq (F2 g) (F2 f)"
cast_inversions_3: fixes x :: "('b, 'c, 'a) unboxed"   and b :: "'a" assumes "cast_Ubx2 x = Some b" shows "x = OpUbx2 b"
nat_induct_non_zero: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "0 < n"   and "P 1"   and "\<And>n. \<lbrakk>0 < n; P n\<rbrakk> \<Longrightarrow> P (Suc n)" shows "P n"
vertical_chop_length_stable: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u--w \<longrightarrow> \<parallel>ext v\<parallel> = \<parallel>ext u\<parallel> \<and> \<parallel>ext v\<parallel> = \<parallel>ext w\<parallel>"
post_anno_asize: fixes C :: "'a acom" shows "post C = anno C (length (annos C) - 1)"
infplays_eq: fixes p :: "'c list"   and s1 :: "'c list \<Rightarrow> 'c"   and s2 :: "'c list \<Rightarrow> 'c" assumes "\<And>p'. prefix p p' \<Longrightarrow> augment_list s1 p' = augment_list s2 p'" shows "strategy_progression s1 p = strategy_progression s2 p"
inject: fixes x2 :: "'b"   and y2 :: "'b" shows "(Some x2 = Some y2) = (x2 = y2)"
cases: fixes x :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec \<times> (('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" obtains P :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" and A :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" and Q :: "(('b, 2) Finite_Cartesian_Product.vec, 2) Finite_Cartesian_Product.vec" where "x = (P, A, Q)"
binary_product_cone_axioms: shows "binary_product_cone (\<cdot>) f0 f1 p0 p1"
idx_build_is_index: fixes t :: "'a dlist"   and f :: "'a \<Rightarrow> 'b" assumes "ls.invar t" shows "hll_idx.is_index f (ls.\<alpha> t) (hll_idx.idx_build f t)"
sgnx_able_sgnx: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "f sgnx_able F" shows "(f has_sgnx sgnx f F) F"
gcd_poly_code: fixes p :: "'b poly"   and q :: "'b poly" shows "gcd p q = gcd_poly_code p q"
pushforward_def: fixes T :: "('a, 'c) partial_object_scheme"   and f :: "'d \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'd" shows "pushforward T f g \<equiv> compose (carrier T) f g"
mat_mult_index: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'c list list"   and nc :: "nat"   and m2 :: "'c list list"   and i :: "nat"   and j :: "nat"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI ze pl ti nr m1 m2 ! j ! i = scalar_prodI ze pl ti (row m1 i) (col m2 j)"
trans: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set"   and R :: "('procS, 'procT) Proc" assumes "P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q"   and "Q \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> R" shows "P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> R"
Call: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and statT :: "ref_ty"   and ps :: "expr list"   and pTs :: "ty list"   and mn :: "mname"   and statDeclT :: "ref_ty"   and m :: "mhead"   and pTs' :: "ty list" assumes "E,dt\<Turnstile>e\<Colon>-RefT statT"   and "E,dt\<Turnstile>ps\<Colon>\<doteq>pTs"   and "max_spec (prg E) (cls E) statT \<lparr>name = mn, parTs = pTs\<rparr> = {((statDeclT, m), pTs')}" shows "E,dt\<Turnstile>{cls E,statT,invmode m e}e\<cdot>mn( {pTs'}ps)\<Colon>-resTy m"
simps_1: fixes a :: "'a" shows "hideDupsRec a [] = []"
greaterThanAtMost_empty_iff2: fixes k :: "'a"   and l :: "'a" shows "({} = {k<..l}) = (\<not> k < l)"
simps_1: fixes p :: "fm"   and q :: "fm" shows "d_\<beta> (And p q) = (\<lambda>k. d_\<beta> p k \<and> d_\<beta> q k)"
vertices_restrict: fixes G :: "('e, 'd) labeled_graph" shows "vertices (restrict G) = vertices G"
share_append_Ghost\<^sub>s\<^sub>b': fixes sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}" shows "share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ [Ghost\<^sub>s\<^sub>b A L R W])) \<S> = share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<S>"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<sqinter> b) \<sqinter> c = a \<sqinter> (b \<sqinter> c)"
af_ltl_continuation_suffix: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl"   and i :: "nat" shows "w \<Turnstile> \<phi> = suffix i w \<Turnstile> af \<phi> (prefix i w)"
hyps_1: shows "TVarB X S \<in> set \<Gamma>"
state_update_vs_allInstances_generic_empty: fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a" shows "mk \<lparr>heap = Map.empty, assocs = A\<rparr> \<Turnstile> OclAllInstances_generic pre_post Type \<doteq> Set{}"
ord_def: fixes x :: "nat" shows "local.ord x \<equiv> SOME d. \<forall>n. (x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>) = (d dvd n)"
SN_partI: fixes x :: "'b"   and r :: "('b \<times> 'b) set" assumes "\<And>y. (x, y) \<in> r \<Longrightarrow> y \<in> SN_part r" shows "x \<in> SN_part r"
simps: fixes as :: "'b atom set"   and lb :: "nat \<Rightarrow> 'b option"   and ub :: "nat \<Rightarrow> 'b option" shows "as \<doteq> (lb, ub) = (\<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub))"
simps: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "combine_tm (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps =1 (let (c\<^sub>0, c\<^sub>1) = if dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in filter_tm (\<lambda>p. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps \<bind> find_closest_pair_tm (c\<^sub>0, c\<^sub>1))"
subst_lit: fixes \<sigma> :: "'f \<Rightarrow> 'g"   and v :: "'e" shows "\<sigma> \<dagger> \<guillemotleft>v\<guillemotright> = \<guillemotleft>v\<guillemotright>"
pmf_None_eq_weight_spmf: fixes p :: "'a spmf" shows "pmf p None = 1 - weight_spmf p"
replicate_zero_coeff: fixes n :: "nat" shows "local.coeff (replicate n \<zero>) = (\<lambda>_. \<zero>)"
simps_1: fixes P :: "'c \<Rightarrow> bool" shows "filter_option P None = None"
gnodes_nodes: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "gnodes A w V \<subseteq> UNIV \<times> nodes A"
smult_0_right: fixes a :: "'a" shows "Polynomial.smult a 0 = 0"
comp_assoc: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and h :: "'a \<Rightarrow> 'c" shows "f \<circ> g \<circ> h = (f \<circ>\<circ>\<circ> (\<circ>)) g h"
LocalSecretsComposition_neg_loc_s: fixes secret :: "Secrets"   and P :: "specID"   and ch :: "chanID" assumes "sKS secret \<notin> LocalSecrets P"   and "exprChannel ch (sE secret)"   and "sKS secret \<notin> specKeysSecrets P" shows "ch \<notin> loc P"
assms_3: fixes y :: "'a" assumes "x = \<lbrakk>Final y\<rbrakk>" shows "p"
\<delta>_prod_finite: fixes \<delta>1 :: "('d, 'e) ta_rule set"   and \<delta>2 :: "('f, 'e) ta_rule set" assumes "finite \<delta>1"   and "finite \<delta>2" shows "finite (\<delta>_prod \<delta>1 \<delta>2)"
distinct_1: fixes x1 :: "'d"   and x2 :: "'e" shows "Inl x1 \<noteq> Inr x2"
bij_betw_ord_of: fixes n :: "nat" shows "bij_betw ord_of {0..<n} (hfset (ord_of n))"
null_uni_uniq_3: fixes v :: "i" shows "[\<^bold>\<exists>y. y\<^sup>P \<^bold>= (\<^bold>\<iota>x. Null (x\<^sup>P)) in v]"
F_into_set: shows "Functor F : AA \<longrightarrow> Set"
ceuclidean_cinner: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and x :: "'c"   and y :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "cinner x y = (\<Sum>b\<in>CBasis. cinner x b * cnj (cinner y b))"
poly_Cons_eq: fixes c :: "'a"   and cs :: "'a list"   and d :: "'a"   and ds :: "'a list" shows "(poly (c # cs) = poly (d # ds)) = (c = d \<and> poly cs = poly ds)"
cfif: shows "f\<inverse> O f \<subseteq> e \<union> f\<inverse> \<union> f"
get_root_node_pure: fixes get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_get_root_node get_root_node get_parent" shows "pure (get_root_node ptr) h"
pointbase_is_continuous: fixes f :: "'c set \<Rightarrow> 'd set" shows "continuous (pointbase f)"
msg_fresh_def: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" shows "msg_fresh \<sigma> m \<equiv> case m of Rreq hopsc x xa xb xc oipc osnc sipc \<Rightarrow> 1 \<le> osnc \<and> (sipc \<noteq> oipc \<longrightarrow> oipc \<in> kD (rt (\<sigma> sipc)) \<and> osnc \<le> nsqn (rt (\<sigma> sipc)) oipc \<and> (nsqn (rt (\<sigma> sipc)) oipc = osnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) oipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) oipc) = Aodv_Basic.inv)) | Rrep hopsc dipc dsnc x sipc \<Rightarrow> 1 \<le> dsnc \<and> (sipc \<noteq> dipc \<longrightarrow> dipc \<in> kD (rt (\<sigma> sipc)) \<and> dsnc \<le> nsqn (rt (\<sigma> sipc)) dipc \<and> (nsqn (rt (\<sigma> sipc)) dipc = dsnc \<longrightarrow> the (dhops (rt (\<sigma> sipc)) dipc) \<le> hopsc \<or> the (flag (rt (\<sigma> sipc)) dipc) = Aodv_Basic.inv)) | Rerr destsc sipc \<Rightarrow> \<forall>ripc\<in>dom destsc. ripc \<in> kD (rt (\<sigma> sipc)) \<and> the (destsc ripc) - 1 \<le> nsqn (rt (\<sigma> sipc)) ripc | _ \<Rightarrow> True"
x_readable: shows "x \<in> \<C> \<or> (\<forall>i<length cms\<^sub>1. x \<notin> snd (cms\<^sub>1 ! i) AsmNoReadOrWrite)"
pi: shows "(\<lambda>z. f (- z)) contour_integrable_on \<gamma>"
gcd: shows "int (gcd (m - m') (order \<G>)) = 1"
lincomb_not_depend: fixes A :: "rat vec set"   and N :: "rat vec set"   and a :: "rat vec \<Rightarrow> rat" assumes "finite A"   and "A \<subseteq> N"   and "LinearCombinations.submodule class_ring N gs.V" shows "gs.lincomb a A = module.lincomb (gs.vs N) a A"
The_opt_eq_some: fixes p :: "'a \<Rightarrow> bool"   and x :: "'a" shows "(The_opt p = Some x) = (p x \<and> (\<forall>y. p y \<longrightarrow> x = y))"
max_less_iff_conj: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "(signed.max x y <s z) = (x <s z \<and> y <s z)"
same_face: shows "G.face_cycle_set rev.a' = G.face_cycle_set a'"
eq_refl: fixes x :: "'b list"   and y :: "'b list" assumes "x = y" shows "subseq x y"
w_cmpser_is_d_gchain: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "w_cmpser G n f" shows "d_gchain G n f"
true_clss_of_interp_iff_equal: fixes J :: "'b set"   and I :: "'b set" shows "(J \<TTurnstile>s clss_of_interp I) = (J = I)"
N_sg_HN: fixes H :: "'a set"   and N :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> N" shows "\<natural>H \<diamondop> N \<guillemotright> N "
pFresh: shows "presFreshAll (rec MOD) (recAbs MOD) MOD"
alist_and_negation_type_to_match_expr_f_matches: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and C :: "'d \<Rightarrow> 'b"   and spts :: "'d negation_type list"   and a :: "action"   and p :: "'c" shows "matches \<gamma> (alist_and (NegPos_map C spts)) a p = (\<forall>m\<in>set spts. matches \<gamma> (negation_type_to_match_expr_f C m) a p)"
DAcc_DSources: fixes X :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "(X \<in> DAcc i C) = (C \<in> DSources i X)"
ker_def: fixes f :: "'a \<Rightarrow> 'b" shows "ker f = {a. f a = (0::'b)}"
le_iff_add: fixes a :: "'q multiset"   and b :: "'q multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
ChainsD: fixes x :: "'a"   and C :: "'a set"   and r :: "('a \<times> 'a) set"   and y :: "'a" assumes "x \<in> C"   and "C \<in> Chains r"   and "y \<in> C" shows "(x, y) \<in> r \<or> (y, x) \<in> r"
extend_set_INT_distrib: fixes B :: "'e \<Rightarrow> 'a set"   and A :: "'e set" shows "extend_set h (\<Inter> (B ` A)) = (\<Inter>x\<in>A. extend_set h (B x))"
unstream_filterS: fixes a :: "('b, 'c) Stream"   and p :: "'b \<rightarrow> tr" assumes "a \<noteq> \<bottom>" shows "unstream\<cdot>(filterS\<cdot>p\<cdot>a) = filterL\<cdot>p\<cdot>(unstream\<cdot>a)"
vint_minus_def: shows "vint_minus = (\<lambda>x\<in>\<^sub>\<circ>\<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>. (int_of_vint (x\<lparr>0\<^sub>\<nat>\<rparr>) - int_of_vint (x\<lparr>1\<^sub>\<nat>\<rparr>))\<^sub>\<int>)"
mono: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset"   and d :: "'c multiset" assumes "a \<subseteq># c"   and "b \<subseteq># d" shows "a \<inter># b \<subseteq># c \<inter># d"
tiny_digraph_axioms: shows "tiny_digraph \<alpha> \<CC>"
1: shows "int (proots_count (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]) {x. 0 < x}) \<le> changes (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) \<and> even (changes (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) - int (proots_count (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]) {x. 0 < x}))"
1: shows "code_rate enc X - \<H>(X) = (\<Sum>i\<in>L. fi i * real (cw_len i)) + (\<Sum>i\<in>L. fi i * log b (fi i))"
fin_string_length: fixes L :: "char list"   and k :: "nat" assumes "length L \<le> k"   and "length L \<le> MAX_STR" shows "List.member (fin_string_upto k) (Abs_fin_string L)"
build_first: fixes w :: "nat \<Rightarrow> 'a" shows "w 0 ## w [Suc 0..] = w"
eq0: fixes n :: "nat" shows "mat_extension dims vars1 (testN n) = tensor_P (testN n) (1\<^sub>m K)"
rat_times_code: fixes p :: "rat"   and q :: "rat" shows "quotient_of (p * q) = (let (a, c) = quotient_of p; (b, d) = quotient_of q in Rat.normalize (a * b, c * d))"
infdist_mono: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A \<subseteq> B"   and "A \<noteq> {}" shows "infdist x B \<le> infdist x A"