hcju/isabelleps huggingface dataset:
tendstoD: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "filterlim f (topological_space.nhds open l) F"   and "0 < e" shows "\<forall>\<^sub>F x in F. dist (f x) l < e"
simps: fixes a :: "rat"   and b :: "rat" shows "qdsnd (QDelta a b) = b"
partition_on_intersect_on_elements: fixes A :: "'a set"   and C :: "'a set"   and P :: "'a set set" assumes "partition_on (A \<union> C) P"   and "\<forall>X\<in>P. \<exists>x. x \<in> X \<inter> C" shows "partition_on C ((\<lambda>X. X \<inter> C) ` P)"
open_PInfty: fixes A :: "ereal set" assumes "open A"   and "\<infinity> \<in> A" shows "\<exists>x. {ereal x<..} \<subseteq> A"
oStrictLimit_lub: fixes f :: "nat \<Rightarrow> ordinal"   and x :: "ordinal" assumes "\<forall>n. f n < x" shows "oStrictLimit f \<le> x"
evaluate_net_Conv_all1: fixes m :: "real mat convnet"   and input :: "real vec list"   and i :: "nat"   and nr :: "nat" assumes "valid_net' m"   and "input_sizes m = map dim_vec input"   and "i < nr" shows "evaluate_net (Conv (all1_matrix nr (output_size' m)) m) input $ i = sum_list (list_of_vec (evaluate_net m input))"
compliant_stateful_ACS_only_state_violations_union': fixes \<T> :: "'b stateful_policy"   and G :: "'b graph"   and M :: "'b SecurityInvariant_configured list" assumes "stateful_policy_compliance \<T> G M" shows "\<Union> (get_offending_flows M (stateful_policy_to_network_graph \<T>)) \<subseteq> backflows (flows_state \<T>)"
in_keys_minusI1: fixes t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<in> keys p"   and "t \<notin> keys q" shows "t \<in> keys (p - q)"
flush_step_preserves_separated_tmps: fixes i :: "nat"   and ts :: "((stmt \<times> nat) \<times> instr list \<times> (nat \<Rightarrow> 'a option) \<times> (stmt \<times> nat) memref list \<times> 'b \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "stmt \<times> nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and m' :: "nat \<Rightarrow> nat"   and sb' :: "(stmt \<times> nat) memref list"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m', sb', \<O>', \<R>', \<S>')"   and "separated_tmps ts" shows "separated_tmps (ts[i := (p, is, \<theta>, sb', \<D>, \<O>', \<R>')])"
stutter_invariantI: fixes \<phi> :: "'b pltl" assumes "\<And>\<sigma> \<tau>. \<lbrakk>\<sigma> \<approx> \<tau>; \<sigma> \<Turnstile>\<^sub>p \<phi>\<rbrakk> \<Longrightarrow> \<tau> \<Turnstile>\<^sub>p \<phi>" shows "stutter_invariant \<phi>"
Min_in: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'g set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
compile: shows "rec_ci (Pr n f g) = (ap, arity, fp)"
prime_elem_dvd_cases: fixes p :: "'a"   and k :: "'a"   and m :: "'a"   and n :: "'a" assumes "p * k dvd m * n"   and "prime_elem p" shows "(\<exists>x. k dvd x * n \<and> m = p * x) \<or> (\<exists>y. k dvd m * y \<and> n = p * y)"
limit_mat_def: fixes X :: "nat \<Rightarrow> complex mat"   and A :: "complex mat"   and m :: "nat" shows "limit_mat X A m = (\<forall>n. X n \<in> carrier_mat m m \<and> A \<in> carrier_mat m m \<and> (\<forall>i<m. \<forall>j<m. (\<lambda>n. X n $$ (i, j)) \<longlonglongrightarrow> A $$ (i, j)))"
targetnodes_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and xs :: "'a list" assumes "CFG sourcenode targetnode valid_edge Entry" shows "CFG.targetnodes targetnode xs \<equiv> map targetnode xs"
ptensor_encode2_encode1: shows "partial_state.encode1 st2.dims0 st2.vars2' = partial_state.encode2 st2.dims0 st2.vars1'"
simps_1: fixes x1a :: "'b"   and y1a :: "'b" shows "(\<^bold>\<langle>x1a\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>y1a\<^bold>\<rangle>\<^sub>0) = (x1a = y1a)"
Rep_proj2_inverse: fixes x :: "proj2" shows "Abs_proj2 (Rep_proj2 x) = x"
mem_case_prodI2: fixes p :: "'c \<times> 'd"   and z :: "'e"   and c :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set" assumes "\<And>a b. p = (a, b) \<Longrightarrow> z \<in> c a b" shows "z \<in> (case p of (a, b) \<Rightarrow> c a b)"
lepoll_def: fixes A :: "'c set"   and B :: "'d set" shows "A \<lesssim> B \<equiv> \<exists>f. inj_on f A \<and> f ` A \<subseteq> B"
Gauss_Jordan_column_k_PA_def: fixes A' :: "(('d, 'e) vec, 'e) vec \<times> nat \<times> (('d, 'f) vec, 'e) vec"   and k :: "nat" shows "Gauss_Jordan_column_k_PA A' k = (let P = fst A'; i = fst (snd A'); A = snd (snd A'); from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if (\<forall>m\<ge>from_nat_i. A $ m $ from_nat_k = (0::'d)) \<or> i = nrows A then (P, i, A) else let Gauss = Gauss_Jordan_in_ij_PA (P, A) from_nat_i from_nat_k in (fst Gauss, i + 1, snd Gauss))"
abs_of_neg: fixes abs :: "'a \<Rightarrow> 'a"   and 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 a :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less"   and "less a zero" shows "abs a = uminus a"
inc_f: fixes x :: "'a" shows "incseq (\<lambda>i. f i x)"
hseq_leg\<^sub>0: shows "hseq (tab\<^sub>0 (local.dom \<tau>)) \<mu>.chine"
hs_dfs_dwa_def: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap"   and post :: "'a \<Rightarrow> 'a dlist" shows "hs_dfs_dwa \<Sigma>i post \<equiv> \<lparr>dwa_cond = hs_dfs_cond, dwa_step = hs_dfs_step post, dwa_initial = hs_dfs_initial \<Sigma>i, dwa_invar = hs_dfs_invar \<Sigma>i post\<rparr>"
inf_OrdSum_def: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
space_bind_empty: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b measure" assumes "space M = {}" shows "space (M \<bind> f) = {}"
inv_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree" assumes "invh l"   and "invh r"   and "bheight l = bheight r"   and "invc l"   and "invc r" shows "invh (join l r) \<and> bheight (join l r) = bheight l \<and> invc2 (join l r) \<and> (color l = Black \<and> color r = Black \<longrightarrow> invc (join l r))"
circ_left_top: fixes x :: "'a" shows "surjective (x\<^sup>\<star>)"
finsum_rdistr: fixes A :: "'b set"   and a :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "finite A" shows "a * sum f A = (\<Sum>i\<in>A. a * f i)"
ide_is_iso: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.iso a"
product_mset_def: fixes A :: "'b multiset"   and B :: "'c multiset" shows "product_mset A B = A \<times># B"
npZero_sub: fixes R :: "('c, 'd) Ring_scheme"   and n :: "nat" assumes "Ring R" shows "0 < n \<longrightarrow> \<zero>\<^bsub>R\<^esub>^\<^bsup>R n\<^esup> = \<zero>\<^bsub>R\<^esub>"
flip_at_simps_2: fixes a :: "'b"   and b :: "'b" shows "(a \<leftrightarrow> b) \<bullet> b = a"
simps_2: fixes x :: "brick"   and y :: "brick list" shows "codomain_block (x # y) = codomain x + codomain_block y"
dom_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "B.VVV.dom f = (if B.VVV.domains f \<noteq> {} then SOME a. a \<in> B.VVV.domains f else B.VVV.null)"
pdom_res_empty: fixes f :: "('a, 'b) pfun" shows "{} \<lhd>\<^sub>p f = {}\<^sub>p"
bisim_lang_eq: fixes as :: "'a list"   and ps :: "('a rexp \<times> 'a rexp) set"   and r :: "'a rexp"   and s :: "'a rexp" assumes "is_bisimulation as ps"   and "(r, s) \<in> ps" shows "lang r = lang s"
projection_kernel_is_orthogonal_complement: fixes M :: "'b set" assumes "closed_csubspace M" shows "projection M -` {0::'b} = orthogonal_complement M"
subcs_map: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and G :: "('a, 'b) form list" shows "subcs c s G = map (subc c s) G"
plus_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "plus_hom hom \<equiv> \<forall>x y. hom (x + y) = hom x + hom y"
h0: fixes i :: "nat"   and j :: "nat" assumes "i < length subsets"   and "j < length signs" shows "map (\<lambda>index_list. map (z index_list) signs) subsets ! i ! j = z (subsets ! i) (signs ! j)"
Z_def: shows "Z \<equiv> Abs_bset {A, C}"
monoid_mult_isom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "monoid_mult_isom hom \<equiv> inj_monoid_mult_hom hom \<and> bijective hom"
simps_1: fixes k :: "nat" shows "into_WW k [] = []"
flat_ord_def: fixes b :: "'e"   and x :: "'e"   and y :: "'e" shows "flat_ord b x y = (x = b \<or> x = y)"
lincomb_insert2: fixes S :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "finite S"   and "S \<subseteq> carrier_vec n"   and "True"   and "v \<notin> S"   and "v \<in> carrier_vec n" shows "lincomb a (insert v S) = a v \<cdot>\<^sub>v v + lincomb a S"
b_in_bs: fixes k :: "nat"   and as :: "real list"   and bs :: "real list"   and i :: "nat" assumes "akra_bazzi_params k as bs"   and "i < k" shows "bs ! i \<in> set bs"
v: fixes i :: "nat" shows "vertices (S i) \<subseteq> vertices C"
add_mset_lt_left_lt: fixes a :: "'a"   and b :: "'a"   and A :: "'a multiset" assumes "a < b" shows "add_mset a A < add_mset b A"
subcls1D: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "C \<noteq> Object \<and> (\<exists>fs ms. class P C = \<lfloor>(D, fs, ms)\<rfloor>)"
out__acute: fixes B :: "'p"   and A :: "'p"   and C :: "'p" assumes "B Out A C" shows "Acute A B C"
dim_image_le: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and S :: "('a, 'b) vec set" assumes "Vector_Spaces.linear (*s) (*s) f" shows "vec.dim (f ` S) \<le> vec.dim S"
ereal_distrib_left: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<noteq> \<infinity> \<or> b \<noteq> - \<infinity>"   and "a \<noteq> - \<infinity> \<or> b \<noteq> \<infinity>"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
lphil0_phil: fixes e :: "dining_event"   and s :: "nat" assumes "e \<in> Tr\<^sub>l\<^sub>p s" shows "phil e = 0"
sup_closed: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L" shows "\<Squnion>\<^bsub>L\<^esub>A \<in> carrier L"
simps_1: fixes \<alpha> :: "game"   and \<phi> :: "fml" shows "Diamondo (Agame \<alpha>) (Afml \<phi>) = Afml (\<langle> \<alpha> \<rangle> \<phi>)"
oracle_decrypt_def: fixes k :: "'key"   and L :: "'cipher set"   and c :: "'cipher" shows "oracle_decrypt k L c = return_spmf (if c \<in> L then None else decrypt k c, L)"
r2f_Sup_pres: shows "Sup_pres \<F>"
induct: fixes P :: "poly \<Rightarrow> poly \<Rightarrow> bool"   and a0 :: "poly"   and a1 :: "poly" assumes "\<And>c c'. P (C c) (C c')"   and "\<And>c c' n' p'. \<lbrakk>P (C c) c'; P (C c) p'\<rbrakk> \<Longrightarrow> P (C c) (CN c' n' p')"   and "\<And>c n p c'. \<lbrakk>P c (C c'); P p (C c')\<rbrakk> \<Longrightarrow> P (CN c n p) (C c')"   and "\<And>c n p c' n' p'. \<lbrakk>P c (CN c' n' p'); P p (CN c' n' p'); P (CN c n p) c'; P (CN c n p) p'; P (CN c n p) c'; P (CN c n p) p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (Bound v) b"   and "\<And>v va b. P (Add v va) b"   and "\<And>v va b. P (Sub v va) b"   and "\<And>v va b. P (Mul v va) b"   and "\<And>v b. P (Neg v) b"   and "\<And>v va b. P (Pw v va) b"   and "\<And>v va vb vc. P (CN v va vb) (Bound vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Add vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Sub vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Mul vc vd)"   and "\<And>v va vb vc. P (CN v va vb) (Neg vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Pw vc vd)"   and "\<And>a v. P a (Bound v)"   and "\<And>a v va. P a (Add v va)"   and "\<And>a v va. P a (Sub v va)"   and "\<And>a v va. P a (Mul v va)"   and "\<And>a v. P a (Neg v)"   and "\<And>a v va. P a (Pw v va)"   and "\<And>vc v va vb. P (Bound vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Add vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Sub vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Mul vc vd) (CN v va vb)"   and "\<And>vc v va vb. P (Neg vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Pw vc vd) (CN v va vb)" shows "P a0 a1"
lift_add: fixes a :: "'a"   and b :: "'a" shows "lift (a + b) = lift a + lift b"
Ks'_def: shows "Ks' = (if length Ks \<le> length Us then Ks @ replicate (length Us - length Ks) \<zero> else take (length Us) Ks)"
order_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "order r A"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
insert_somewhere: fixes ref :: "'b option"   and r :: "'b"   and list :: "'b list"   and oid :: "'b" assumes "ref = None \<or> ref = Some r \<and> r \<in> set list" shows "\<exists>xs ys. list = xs @ ys \<and> insert_spec list (oid, ref) = xs @ oid # ys"
bidi_1: fixes a :: "'b" assumes "a \<notin> arcs G" shows "edge_rev M a = a"
img_mkPoint_1: fixes a :: "'s" assumes "S.ide a" shows "S.mkPoint a \<in> S.set a \<rightarrow> S.hom S.unity a"
\<psi>_img_\<phi>_img: fixes A :: "'t set" assumes "A \<subseteq> S'.Univ" shows "INV.\<psi> ` \<psi> ` A = A"
Guard_is_action: fixes g :: "'d \<Rightarrow> bool"   and P :: "('c, 'd) action" shows "R (g \<circ> alpha_rp.more \<circ> fst \<longrightarrow> \<not> action.relation_of P\<^sup>f\<^sub>f \<turnstile> g \<circ> alpha_rp.more \<circ> fst \<and> action.relation_of P\<^sup>t\<^sub>f \<or> \<not> g \<circ> alpha_rp.more \<circ> fst \<and> (\<lambda>(A, A'). tr A' = tr A \<and> wait A')) \<in> {p. is_CSP_process p}"
gtr_empty_conf1: fixes w :: "'d list"   and c' :: "'c multiset"   and S :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'c \<times> 'c multiset) set" shows "(({#}, w, c') \<in> trcl (gtr S)) = (w = [] \<and> c' = {#})"
Nil: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and ys :: "('x \<Rightarrow>\<^sub>0 nat) list" shows "ordered_powerprod_lin.lexordp [] (y # ys)"
fimage_vunion_right: fixes r :: "V"   and A :: "V"   and B :: "V" shows "r `\<^sub>\<bullet> (A \<union>\<^sub>\<circ> B) = r `\<^sub>\<bullet> A \<union>\<^sub>\<circ> r `\<^sub>\<bullet> B"
echelon_form_of_euclidean_def: fixes A :: "(('b, 'c) vec, 'd) vec" shows "echelon_form_of_euclidean A = echelon_form_of A euclid_ext2"
IH: assumes "n = lookupDefNode g m v"   and "n \<notin> set (tl ns)"   and "x \<in> set (tl ns)" shows "var g v' \<noteq> v"
distinct_5: fixes x4 :: "int" shows "Unit \<noteq> Intg x4"
bind_distrib_sup2: fixes m :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres"   and g :: "'d \<Rightarrow> 'c nres" shows "m \<bind> (\<lambda>x. sup (f x) (g x)) = sup (m \<bind> f) (m \<bind> g)"
assms_1: shows "seq \<pp>\<^sub>0[a, b] h"
simps: fixes a1 :: "('exp, 'id) MWLsCom"   and a2 :: "'id \<Rightarrow> 'val"   and a3 :: "('exp, 'id) MWLsCom list"   and a4 :: "('exp, 'id) MWLsCom option"   and a5 :: "'id \<Rightarrow> 'val" shows "SRabbr a1 a2 a3 a4 a5 = ((\<exists>\<iota> m. a1 = skip\<^bsub>\<iota>\<^esub> \<and> a2 = m \<and> a3 = [] \<and> a4 = None \<and> a5 = m) \<or> (\<exists>e m v x \<iota>. a1 = x:=\<^bsub>\<iota>\<^esub> e \<and> a2 = m \<and> a3 = [] \<and> a4 = None \<and> a5 = m(x := v) \<and> E e m = v) \<or> (\<exists>c1 m \<alpha> m' c2. a1 = c1;c2 \<and> a2 = m \<and> a3 = \<alpha> \<and> a4 = Some c2 \<and> a5 = m' \<and> SRabbr c1 m \<alpha> None m') \<or> (\<exists>c1 m \<alpha> c1' m' c2. a1 = c1;c2 \<and> a2 = m \<and> a3 = \<alpha> \<and> a4 = Some (c1';c2) \<and> a5 = m' \<and> SRabbr c1 m \<alpha> (Some c1') m') \<or> (\<exists>b m \<iota> c1 c2. a1 = if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi \<and> a2 = m \<and> a3 = [] \<and> a4 = Some c1 \<and> a5 = m \<and> BMap (E b m) = True) \<or> (\<exists>b m \<iota> c1 c2. a1 = if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi \<and> a2 = m \<and> a3 = [] \<and> a4 = Some c2 \<and> a5 = m \<and> BMap (E b m) = False) \<or> (\<exists>b m \<iota> c. a1 = while\<^bsub>\<iota>\<^esub> b do c od \<and> a2 = m \<and> a3 = [] \<and> a4 = Some (c;while\<^bsub>\<iota>\<^esub> b do c od) \<and> a5 = m \<and> BMap (E b m) = True) \<or> (\<exists>b m \<iota> c. a1 = while\<^bsub>\<iota>\<^esub> b do c od \<and> a2 = m \<and> a3 = [] \<and> a4 = None \<and> a5 = m \<and> BMap (E b m) = False) \<or> (\<exists>\<iota> V m. a1 = spawn\<^bsub>\<iota>\<^esub> V \<and> a2 = m \<and> a3 = V \<and> a4 = None \<and> a5 = m))"
a_notin_segment: fixes a :: "'a" shows "a \<notin> segment D a"
order_step: fixes As :: "'a tensor list"   and ds :: "nat list" assumes "\<And>A. A \<in> set As \<Longrightarrow> P A"   and "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds" shows "P (subtensor_combine ds As)"
iso_fiter_ext: fixes f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "mono f" shows "mono (\<lambda>x. y \<sqinter> f x)"
element_power_order_eq_1': fixes x :: "'a mod_ring poly" assumes "x \<in> carrier (mult_of R)" shows "x [^]\<^bsub>mult_of R\<^esub> CARD('a) ^ degree f = x"
oconf_new: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)"   and a :: "nat"   and x :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" assumes "P,h \<turnstile> obj \<surd>"   and "h a = None" shows "P,h(a \<mapsto> x) \<turnstile> obj \<surd>"
exp_bound_lemma: fixes z :: "'a" assumes "norm z \<le> 1 / 2" shows "norm (exp z) \<le> 1 + 2 * norm z"
homotopic_join_subpaths1: fixes g :: "real \<Rightarrow> 'b"   and s :: "'b set"   and u :: "real"   and v :: "real"   and w :: "real" assumes "path g"   and "path_image g \<subseteq> s"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}"   and "u \<le> v"   and "v \<le> w" shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
Id_O_R: fixes R :: "('c \<times> 'd) set" shows "Id O R = R"
power_Suc: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" shows "power.power one times a (Suc n) = times a (power.power one times a n)"
in_own_span: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and S :: "'i set" assumes "module R M"   and "S \<subseteq> carrier M" shows "S \<subseteq> LinearCombinations.module.span R M S"
simps_1: fixes vT :: "variable \<times> typ"   and n :: "nat"   and v :: "variable"   and T :: "typ" shows "bind_fv2 vT n (Fv v T) = (if vT = (v, T) then Bv n else Fv v T)"
\<delta>\<^sub>0_def_2: shows "\<delta>\<^sub>0' = dist_code p p\<^sub>0"
f_bounded: fixes x :: "'b" assumes "x \<in> space N" shows "\<bar>f x\<bar> \<le> B"
atLeastatMost_empty_iff2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "({} = subset_mset.atLeastAtMost a b) = (\<not> a \<subseteq># b)"
ruleSet: fixes R' :: "('a sequent list \<times> 'a sequent) set"   and R :: "('a sequent list \<times> 'a sequent) set"   and Ps :: "'a sequent list"   and C :: "'a sequent" assumes "R' \<subseteq> upRules"   and "R = Ax \<union> R'"   and "(Ps, C) \<in> R*" shows "\<exists>S r. extendRule S r = (Ps, C) \<and> (r \<in> R' \<or> r \<in> Ax)"
numeral_monom: fixes n :: "num" shows "numeral n = monom (numeral n) 0"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot>\<^sub>B f = n \<and> f \<cdot>\<^sub>B n = n"
brn_gt_0: fixes c :: "('test, 'atom, 'choice) cmd" assumes "proper c" shows "0 < brn c"
abs_eq: fixes xb :: "'b \<Rightarrow> 'b ttree"   and xa :: "'b set"   and x :: "'b ttree" shows "substitute xb xa x = Abs_ttree (Collect (substitute' xb xa x))"
assms: fixes C :: "'a set set" assumes "Ball C open"   and "s \<subseteq> \<Union> C" shows "\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union> C'"
encode_problem_parallel_correct_vi: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat"   and k :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t"   and "k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) (\<Phi>\<inverse> \<Pi> \<A> t))" shows "trace_parallel_plan_strips (\<Pi>\<^sub>I) (\<Phi>\<inverse> \<Pi> \<A> t) ! k = \<Phi>\<^sub>S\<inverse> \<Pi> \<A> k"
S: fixes k :: "nat" shows "real k \<le> S k \<bullet> (1::'a) \<and> eval_fds h (S k) = (0::'a)"
periodic_orbit_recurrence_times_closed: fixes x :: "'a" assumes "periodic_orbit x" shows "closed {T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}"
le_filter_def: fixes F :: "'b filter"   and F' :: "'b filter" shows "(F \<le> F') = (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
ctor2_diff: fixes x :: "('d, 'd IF1, 'd IF2) F2"   and y :: "('d, 'd IF1, 'd IF2) F2" shows "(ctor2 x = ctor2 y) = (x = y)"
not_Null_eq: fixes x :: "'c ref" shows "(x \<noteq> Null) = (\<exists>y. x = Ref y)"
word_arith_nat_mult: fixes a :: "'b word"   and b :: "'b word" shows "a * b = word_of_nat (unat a * unat b)"
plus_in_intervalI: fixes x :: "'a"   and X :: "'a interval"   and y :: "'a"   and Y :: "'a interval" assumes "x \<in>\<^sub>i X"   and "y \<in>\<^sub>i Y" shows "x + y \<in>\<^sub>i X + Y"
card_le_if_inj_on_rel: fixes B :: "'b set"   and A :: "'c set"   and r :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "finite B"   and "\<And>a. a \<in> A \<Longrightarrow> \<exists>b. b \<in> B \<and> r a b"   and "\<And>a1 a2 b. \<lbrakk>a1 \<in> A; a2 \<in> A; b \<in> B; r a1 b; r a2 b\<rbrakk> \<Longrightarrow> a1 = a2" shows "card A \<le> card B"
elims_1: fixes x :: "('b, 'c) Combinators list"   and xa :: "'b set list"   and y :: "bool" assumes "FWNormalisationCore.sorted x xa = y" obtains l :: "'b set list" where "x = []" and "xa = l__" and "y"   | xb :: "('b, 'c) Combinators" and l :: "'b set list" where "x = [xb__]" and "xa = la__" and "y"   | xb :: "('b, 'c) Combinators" and ya :: "('b, 'c) Combinators" and zs :: "('b, 'c) Combinators list" and l :: "'b set list" where "x = xb # ya # zs" and "xa = l" and "y = (smaller xb ya l \<and> FWNormalisationCore.sorted (ya # zs) l)"
hb_trans: fixes m1 :: "'msgid \<times> 'op"   and m2 :: "'msgid \<times> 'op"   and m3 :: "'msgid \<times> 'op" assumes "hb m1 m2"   and "hb m2 m3" shows "hb m1 m3"
Iagree_ODE: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set"   and a :: "'f" assumes "Iagree I J V"   and "Inr (Inr a) \<in> V" shows "ODEs I a = ODEs J a"
equator_def: shows "equator \<equiv> \<lambda>n. {x. x n = 0}"
fps_cos_odd: fixes c :: "'a" shows "fps_cos (- c) = fps_cos c"
sync_rules_7: fixes d :: "'c \<Rightarrow> 'b"   and a :: "'c"   and C :: "'b set"   and c :: "'b"   and Q :: "'b process"   and P :: "'b process" assumes "d a \<notin> C"   and "c \<in> C" shows "(c \<rightarrow> Q \<lbrakk>C\<rbrakk> d`!`a \<rightarrow> P) = (d`!`a \<rightarrow> (c \<rightarrow> Q \<lbrakk>C\<rbrakk> P))"
stern_brocot_iterate_aux_def: shows "stern_brocot_iterate_aux \<equiv> tree_iterate (\<lambda>s. s \<otimes> LL) (\<lambda>s. s \<otimes> UR)"
left_Crypt: fixes K :: "nat"   and X :: "msg" shows "left (Crypt K X) = left X"
add_right_cancel: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y = x + z) = (y = z)"