hcju/isabelleps huggingface dataset:
In: fixes x :: "'a Gromov_completion" assumes "x \<in> range to_Gromov_completion" shows "Gromov_extension (g \<circ> f) x = (Gromov_extension g \<circ> Gromov_extension f) x"
float_val_lt_threshold: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a"   and "1 < LENGTH('a)" shows "valof a < threshold TYPE(('a, 'b) IEEE.float)"
that: fixes a :: "real"   and b :: "real"   and c :: "real" assumes "Rep_triangle t = [a, b, c]"   and "a \<in> angles"   and "b \<in> angles"   and "c \<in> angles"   and "a + b + c = 180"   and "a \<le> b"   and "b \<le> c" shows "thesis"
simps_10: fixes t1 :: "Syntax.term"   and t2 :: "Syntax.term"   and t' :: "Syntax.term"   and x :: "Syntax.var" shows "App t1 t2[t' / x] = App (t1[t' / x]) (t2[t' / x])"
app_hunion1: fixes x :: "hf"   and g :: "hf"   and f :: "hf" assumes "x \<^bold>\<notin> hdomain g" shows "app (f \<squnion> g) x = app f x"
eq_iff: fixes a :: "'b multiset"   and b :: "'b multiset" shows "(a = b) = (a \<subseteq># b \<and> b \<subseteq># a)"
injective_mult_closed: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.injective x"   and "matrix_stone_relation_algebra.injective y" shows "matrix_stone_relation_algebra.injective (x \<odot> y)"
subset_singleton_iff: fixes X :: "'e set"   and a :: "'e" shows "(X \<subseteq> {a}) = (X = {} \<or> X = {a})"
cong_bigtheta: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "\<Omega>[F](f) = \<Omega>[F](g)"
R_lin_independent_reduce: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and n :: "'b"   and ms :: "'b list"   and ns :: "'b list" assumes "RModule R smult M"   and "n \<in> M"   and "set ms \<subseteq> M"   and "R_scalar_mult.R_lin_independent R smult (ms @ n # ns)" shows "R_scalar_mult.R_lin_independent R smult (ms @ ns)"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r"
select_convs_2: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'a" shows "parTs \<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> = parTs"
rel_refl: fixes R1a :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'e + 'f" assumes "\<And>x. R1a x x"   and "\<And>x. R2a x x" shows "rel_sum R1a R2a x x"
one_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "subgroup H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
lr_succ_def: fixes A :: "('d, 'c) nba"   and a :: "'d"   and f :: "'c \<Rightarrow> nat option" shows "lr_succ A a f \<equiv> {g. dom g = \<Union> (transition A a ` dom f) \<and> (\<forall>p\<in>dom f. \<forall>q\<in>transition A a p. the (g q) \<le> the (f p)) \<and> (\<forall>q\<in>dom g. accepting A q \<longrightarrow> even (the (g q)))}"
antisym_conv: fixes y :: "'b \<times> 'c"   and x :: "'b \<times> 'c" assumes "less_eq_prod y x" shows "less_eq_prod x y = (x = y)"
bdd: shows "bdd_above ((\<lambda>t. cmod t + (cmod t)\<^sup>2) ` ball z d)"
sees_methods_idemp: 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 C :: "char list"   and Mm :: "char list \<Rightarrow> ((ty list \<times> ty \<times> 'b) \<times> char list) option"   and M :: "char list"   and m :: "ty list \<times> ty \<times> 'b"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>Mm'. P \<turnstile> D sees_methods Mm' \<and> Mm' M = \<lfloor>(m, D)\<rfloor>"
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.codomains (g \<cdot> f) = C.codomains g"
joint_D_gchain_n0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and g :: "nat \<Rightarrow> 'a set" assumes "D_gchain G n f"   and "D_gchain G 0 g"   and "g 0 \<subset> f n" shows "D_gchain G (Suc n) (jointfun n f 0 g)"
il: shows "A \<Turnstile> (S, U \<rightarrow>) onl \<Gamma> P"
resource_def: fixes core :: "('i, 'j, 'a, 'c, 'e, 'g) core"   and core_init :: "'i"   and rest :: "('k, 'j, 'b, 'd, 'f, 'h, 'k) rest_scheme" shows "fused_resource.resource core core_init rest = RES (fused_resource.fuse core rest) (core_init, rinit rest)"
Ls_def: shows "Ls \<equiv> {L0 x y |x y. True} \<union> {L1 x y |x y. True} \<union> {L2 x y |x y. True} \<union> {L3 x y |x y. True}"
that_1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X"   and "x \<le> y" shows "dist (f x) (f y) \<le> L * dist x y"
outstanding_non_volatile_refs_owned_or_read_only_nth_update: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and \<O> :: "nat set"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "outstanding_non_volatile_refs_owned_or_read_only \<S> ts"   and "i < length ts"   and "non_volatile_owned_or_read_only False \<S> \<O> sb" shows "outstanding_non_volatile_refs_owned_or_read_only \<S> (ts[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
infsetsum_Un_disjoint: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'a set" assumes "f abs_summable_on A"   and "f abs_summable_on B"   and "A \<inter> B = {}" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B"
lfilter_def: fixes P :: "'a \<Rightarrow> bool" shows "lfilter P \<equiv> llist.fixp_fun (\<lambda>lfilter xs. case xs of []\<^sub>l \<Rightarrow> []\<^sub>l | x #\<^sub>l xs' \<Rightarrow> if P x then x #\<^sub>l lfilter xs' else lfilter xs')"
lm075: fixes a :: "('a \<times> 'b set) set" assumes "a \<in> runiqs \<inter> Pow (UNIV \<times> (UNIV - {{}}))" shows "((\<lambda>pseudo. {(x, pseudo ,,, x) |x. x \<in> Domain pseudo}) \<circ> pseudoAllocation) a = id a"
pick_swapAbs_qSwapAbs: fixes A :: "('e, 'f, 'varSort, 'var, 'g) qAbs set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "goodAbs A" shows "pick (A $[x1 \<and> x2]_xs) $= pick A $[[x1 \<and> x2]]_xs"
RecvMarker_1: shows "\<exists>q p. t ! j = RecvMarker cid q p"
path_connectedin_path_component_of: fixes X :: "'c topology"   and x :: "'c" shows "path_connectedin X (path_component_of_set X x)"
id_cblinfun_def: shows "id_cblinfun \<equiv> CBlinfun (\<lambda>x. x)"
map_upd_triv: fixes t :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and x :: "'a" assumes "t k = \<lfloor>x\<rfloor>" shows "t(k \<mapsto> x) = t"
is_dghm_op: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<AA>' :: "V"   and \<BB>' :: "V"   and \<alpha>' :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<AA>' = op_dg \<AA>"   and "\<BB>' = op_dg \<BB>"   and "\<alpha>' = \<alpha>" shows "op_dghm \<FF> : \<AA>' \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>'\<^esub> \<BB>'"
SeqSteps: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and cfg\<^sub>1 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and cfg\<^sub>2 :: "('d, 'e, 'f) com \<times> ('d, 'f) xstate"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and c\<^sub>1' :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "\<Gamma> \<turnstile> cfg\<^sub>1 \<rightarrow>\<^sup>* cfg\<^sub>2"   and "cfg\<^sub>1 = (c\<^sub>1, s)"   and "cfg\<^sub>2 = (c\<^sub>1', s')" shows "\<Gamma> \<turnstile> (Seq c\<^sub>1 c\<^sub>2, s) \<rightarrow>\<^sup>* (Seq c\<^sub>1' c\<^sub>2, s')"
prod_cases5: fixes y :: "'a \<times> 'b \<times> 'c \<times> 'd \<times> 'e" obtains a :: "'a" and b :: "'b" and c :: "'c" and d :: "'d" and e :: "'e" where "y = (a, b, c, d, e)"
simps_8: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form" shows "subc c s (Exists p) = Exists (subc c (liftt s) p)"
msubstleneg_def: fixes c :: "poly"   and t :: "tm"   and a :: "poly"   and b :: "tm" shows "msubstleneg c t a b = Le (tm.Neg (tm.Add (tm.Mul a t) (tm.Mul c b)))"
Cut_Atom_pre: fixes k :: "'b"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "Atom k, \<Gamma> \<Rightarrow> \<Delta>"   and "\<Gamma> \<Rightarrow> Atom k, \<Delta>" shows "\<Gamma> \<Rightarrow> \<Delta>"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "nat" assumes "gcd x xa = y" obtains xb :: "nat" and ya :: "nat" where "x = xb" and "xa = ya" and "y = (if ya = 0 then xb else gcd ya (xb mod ya))"
\<phi>_in_terms_of_\<eta>: fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = G f \<cdot>\<^sub>D \<phi>\<psi>.\<eta> y"
sum_mono: fixes K :: "'b set"   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 (+) {#} f K \<subseteq># comm_monoid_add.sum (+) {#} g K"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_cases_3: fixes c :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>c: t \<doteq> s\<rangle> = fv t \<union> fv s"
poly_linepath_comp: fixes p :: "'a poly"   and a :: "'a"   and b :: "'a" shows "poly p \<circ> linepath a b = poly (p \<circ>\<^sub>p [:a, b - a:]) \<circ> of_real"
content_division: fixes \<D> :: "'c set set"   and S :: "'c set" assumes "\<D> division_of S" shows "sum (Sigma_Algebra.measure lebesgue) \<D> = Sigma_Algebra.measure lebesgue S"
subgroup_generated_by_image: fixes S :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set set" assumes "S \<subseteq> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))" shows "carrier (subgroup_generated (relative_homology_group (int n) (subtopology (nsphere n) {x. x n \<le> 0}) {x. x n = 0}) (gp ` S)) = gp ` carrier (subgroup_generated (reduced_homology_group (int n - 1) (nsphere (n - Suc 0))) S)"
exhaust_sel: fixes prod :: "'a \<times> 'b" obtains "prod = (cc prod, snd prod)"
some_right_adjoint_def: fixes f :: "'a" shows "f\<^sup>* \<equiv> SOME g. adjoint_pair f g"
typing_alt_simps_12: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and v :: "String.literal"   and \<tau> :: "'a OCL_Types.type" shows "(\<Gamma> \<turnstile>\<^sub>E Var v : \<tau>) = (fmlookup \<Gamma> v = Some \<tau>)"
pinf_6: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (x \<le> t) = False"
strict_trans: fixes b :: "'e"   and a :: "'e"   and c :: "'e" assumes "b < a"   and "c < b" shows "c < a"
vconn_implies_hpar_4: fixes \<mu> :: "'b"   and f :: "'b"   and g :: "'b" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>D g\<guillemotright>" shows "trg\<^sub>D g = trg\<^sub>D f"
mset_le_decr_left2: fixes c :: "'a multiset"   and a :: "'a multiset"   and b :: "'a multiset" assumes "c + a \<subseteq># b" shows "a \<subseteq># b"
unit_simps_4: fixes a :: "'a" assumes "obj a" shows "local.dom \<i>[a] = a \<star> a"
mod_def: fixes x :: "'a mpoly"   and y :: "'a mpoly" shows "x mod y = (THE r'. \<exists>a q r. pseudo_divmod_rel a x y q r \<and> r' = smult (inverse a) r)"
inj_map_run_eq: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and r1 :: "'g \<Rightarrow> 'e"   and r2 :: "'g \<Rightarrow> 'e" assumes "inj \<alpha>"   and "\<alpha> \<circ> r1 = \<alpha> \<circ> r2" shows "r1 = r2"
closed_Collect_le: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x \<le> g x}"
inj_on_to_nat_on: fixes A :: "'f set" assumes "countable A" shows "inj_on (to_nat_on A) A"
delta_mono: fixes p :: "fm"   and d :: "int"   and d' :: "int" assumes "iszlfm p"   and "d dvd d'"   and "d_\<delta> p d" shows "d_\<delta> p d'"
unlabel_filter_eq: fixes A :: "('a \<times> 'b) list"   and B :: "('a \<times> 'b) set" assumes "\<forall>(j, p)\<in>set A \<union> B. \<forall>(k, q)\<in>set A \<union> B. p = q \<longrightarrow> j = k" shows "filter (\<lambda>d. d \<notin> snd ` B) (unlabel A) = unlabel (filter (\<lambda>d. d \<notin> B) A)"
vec_to_iarray_nth': fixes A :: "('b, 'c) vec"   and i :: "'c" shows "vec_to_iarray A !! mod_type_class.to_nat i = A $ i"
of_nat_simps_1: shows "of_nat 0 = (0::'g)"
bounded_linear_Blinfun_apply: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "blinfun_apply (Blinfun f) = f"
span_trans: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> span S"   and "y \<in> span (insert x S)" shows "y \<in> span S"
elims_3: fixes x :: "'a ext0"   and xa :: "'a ext0" assumes "\<not> x \<le> xa" obtains uv_ :: "'a" where "xa__ = ext0.Val uv_" and "xa = ext0.Bot"   | x :: "'a" and y :: "'a" where "xa__ = ext0.Val x" and "xa = ext0.Val y" and "\<not> x \<le> y"   | "xa__ = ext0.Top" and "xa = ext0.Bot"   | ux_ :: "'a" where "xa__ = ext0.Top" and "xa = ext0.Val ux_"
defAss_defNode: fixes g :: "'g"   and m :: "'node"   and v :: "'val"   and ns :: "'node list" assumes "defAss g m v"   and "v \<in> allVars g"   and "g \<turnstile> Entry g-ns\<rightarrow>m" shows "defNode g v \<in> set ns"
ID_ccompare: fixes ccompare :: "('a \<Rightarrow> 'a \<Rightarrow> order) option"   and c :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ID ccompare = Some c" shows "class.linorder (le_of_comp c) (lt_of_comp c)"
unstream_splice_trans_Right_only: fixes g :: "('e, 'f) generator"   and h :: "('e, 'g) generator"   and sh :: "'g" shows "unstream (splice_trans g h) (Right_only sh) = unstream h sh"
equiv_inject: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and B :: "'b set" assumes "equiv A R"   and "equiv B R" shows "A = B"
that: fixes A :: "'b set" assumes "finite A"   and "A \<subseteq> B"   and "p \<in> span A" shows "thesis"
UP_smult_zero: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "p \<in> carrier (UP R)" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>UP R\<^esub> p = \<zero>\<^bsub>UP R\<^esub>"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "\<And>I. ideal I R \<Longrightarrow> \<exists>A\<subseteq>carrier R. finite A \<and> I = Idl\<^bsub>R\<^esub> A" shows "noetherian_ring_axioms R"
funny_set_falg_fmask_helper: fixes c2 :: "'a set"   and fmask2 :: "'a set"   and c1 :: "'a set"   and fmask1 :: "'a set" assumes "c2 \<subseteq> fmask2" shows "(c1 = c2 \<and> fmask1 = fmask2) = (\<forall>pkt. (pkt \<inter> fmask1 = c1) = (pkt \<inter> fmask2 = c2))"
assms_1: shows "es @ [Deliver (i, Delete n)] prefix of j"
set_sel_2: fixes a :: "'c list"   and x :: "'c" assumes "a \<noteq> []"   and "x \<in> set (tl a)" shows "x \<in> set a"
is_interp_def: fixes I :: "(char \<Rightarrow> real) \<times> (char \<Rightarrow> real \<Rightarrow> real) \<times> (char \<Rightarrow> real \<Rightarrow> bool) \<times> (char \<Rightarrow> (variable \<Rightarrow> real) set \<Rightarrow> (variable \<Rightarrow> real) set)" shows "is_interp I \<equiv> case I of (x, xa, xb, G) \<Rightarrow> \<forall>a. mono (G a)"
asTerm_equal_imp_alpha: fixes qX :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and qY :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood qX"   and "asTerm qX = asTerm qY" shows "qX #= qY"
qGoodEnv_iff: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option" shows "qGoodEnv rho = ((\<forall>xs. liftAll qGood (rho xs)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|) \<and> |{X. \<exists>ys y. rho ys y = Some X}| <o |UNIV|)"
S_reps_\<S>_aux: fixes l :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "reps (l, R) \<in> S" shows "(l, R) \<in> \<S>"
nfoldli_upt_rule: fixes lb :: "nat"   and ub :: "nat"   and I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a nres"   and P :: "'a \<Rightarrow> bool" assumes "lb \<le> ub"   and "I lb \<sigma>0"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i < ub; I i \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f i \<sigma> \<le> SPEC (I (i + 1))"   and "\<And>i \<sigma>. \<lbrakk>lb \<le> i; i \<le> ub; I i \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma>. \<lbrakk>I ub \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "nfoldli [lb..<ub] c f \<sigma>0 \<le> SPEC P"
Meet_def: fixes A :: "'b set" shows "\<Sqinter>A = (THE inf. is_Inf A inf)"
execn_Fault_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "s = Fault f" shows "t = Fault f"
Greatest_def: fixes P :: "'b word \<Rightarrow> bool" shows "signed.Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le>s x))"
tllist_set_induct: fixes x :: "'a"   and xs :: "('a, 'b) tllist"   and P :: "'a \<Rightarrow> ('a, 'b) tllist \<Rightarrow> bool" assumes "x \<in> tset xs"   and "\<And>xs. \<not> is_TNil xs \<Longrightarrow> P (thd xs) xs"   and "\<And>xs y. \<lbrakk>\<not> is_TNil xs; y \<in> tset (ttl xs); P y (ttl xs)\<rbrakk> \<Longrightarrow> P y xs" shows "P x xs"
cut_ge_Diff: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" shows "(A - B) \<down>\<ge> t = A \<down>\<ge> t - B \<down>\<ge> t"
translate_rhs_def: shows "translate_rhs = snd \<circ> fthe_elem"
inj': shows "inj_on (vec \<circ> g \<circ> (\<lambda>x. x $ 1)) (vec ` S)"
Liminf_bounded: fixes C :: "'c"   and X :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F n in F. C \<le> X n" shows "C \<le> Liminf F X"
simps: fixes a :: "nat" shows "sum_of_2_squares_nat a = (\<exists>aa b. a = aa\<^sup>2 + b\<^sup>2)"
rep_eq: fixes x :: "'b"   and xa :: "'c"   and xb :: "('b, 'c) mapping" shows "mapping.rep (Mapping.update x xa xb) = mapping.rep xb(x \<mapsto> xa)"
dims_output_witness: fixes rs :: "nat list"   and j :: "nat" assumes "2 \<le> length rs"   and "\<And>r. r \<in> set rs \<Longrightarrow> 0 < r"   and "j < rs ! 0" shows "dims (tensors_from_net (witness_l rs) $ j) = replicate (2 ^ (length rs - 2)) (last rs)"
sig_red_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "sig_red sing_reg top_tail F p q" shows "punit.lt (rep_list q) \<preceq> punit.lt (rep_list p)"
path_root_def: fixes r :: "'a"   and x :: "'a" shows "path_root r x \<equiv> r ; x \<le> x\<^sup>\<star> + x\<^sup>T\<^sup>\<star> \<and> is_inj x \<and> is_p_fun x \<and> point r"
elims_1: fixes x :: "'a Line"   and xa :: "'a Line"   and y :: "bool" assumes "meets x xa = y" obtains line1 :: "'a Line" and line2 :: "'a Line" where "x = line1" and "xa = line2" and "y = (\<exists>p. inLine p line1 \<and> inLine p line2)"
hta_has_idx_sf_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_has_idx_sf H \<equiv> hta_idx_sf H \<noteq> None"
oLimit_le_oStrictLimit: fixes f :: "nat \<Rightarrow> ordinal" shows "oLimit f \<le> oStrictLimit f"
rel_F_eq: shows "rel_F (=) (=) (=) (=) (=) (=) (=) (=) (=) = (=)"
in_hom_def: fixes f :: "(bool, 'arr list) concrete_category.arr"   and a :: "(bool, 'arr list) concrete_category.arr"   and b :: "(bool, 'arr list) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright> \<equiv> a \<in> domains f \<and> b \<in> codomains f"
is_regular_sequence_snocI: fixes f :: "'c \<Rightarrow>\<^sub>0 'd"   and fs :: "('c \<Rightarrow>\<^sub>0 'd) list" assumes "\<And>q. q * f \<in> ideal (set fs) \<Longrightarrow> q \<in> ideal (set fs)"   and "is_regular_sequence fs" shows "is_regular_sequence (fs @ [f])"
cond_case_prod_eta: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<times> 'f \<Rightarrow> 'g" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
lex_triple_def: shows "Abacus.lex_triple \<equiv> less_than <*lex*> Abacus.lex_pair"
foundation21: fixes A :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and B :: "'d state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(not A \<triangleq> not B) = (A \<triangleq> B)"
Scons_UN1_y: fixes M :: "('f, 'g) Old_Datatype.node set"   and f :: "'h \<Rightarrow> ('f, 'g) Old_Datatype.node set" shows "Scons M (\<Union> (range f)) = (\<Union>x. Scons M (f x))"
subset_wens_Join: fixes A :: "'b set"   and T :: "'b set"   and F :: "'b program"   and act :: "('b \<times> 'b) set"   and B :: "'b set"   and G :: "'b program" assumes "A = T \<inter> wens F act B"   and "T - B \<subseteq> awp F T"   and "A - B \<subseteq> awp G (A \<union> B)" shows "A \<subseteq> wens (F \<squnion> G) act B"
cong: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d" shows "comm_monoid_mult.prod times one \<equiv> comm_monoid_mult.prod times one"
iT_Div_mod_partition_card: fixes I :: "nat set"   and n :: "nat"   and d :: "nat" shows "card (I \<inter> [n * d\<dots>,d - Suc 0] \<oslash> d) = (if I \<inter> [n * d\<dots>,d - Suc 0] = {} then 0 else Suc 0)"
continuous_on_final_topologyI1: fixes Y :: "'c \<Rightarrow> 'd topology"   and Z :: "'e topology"   and g :: "'f \<Rightarrow> 'e"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'f"   and X :: "'f set" assumes "\<And>i. continuous_map (Y i) Z (g \<circ> f i)"   and "\<And>i. f i \<in> topspace (Y i) \<rightarrow> X"   and "g \<in> X \<rightarrow> topspace Z" shows "continuous_map (final_topology X Y f) Z g"
env_eq_refl: fixes \<rho> :: "(nat \<times> val) list" shows "\<rho> \<approx> \<rho>"
meth: shows "P \<turnstile> C sees M, NonStatic :  Ts\<rightarrow>T = (mxs, mxl, ins, xt) in D"
not_Some_eq: fixes x :: "'a option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
map_us_def: fixes f :: "'b \<Rightarrow> 'c" shows "map_us f = us \<circ> f \<circ> IInf"
lookup_map_entry': fixes x :: "'d"   and f :: "'c \<Rightarrow> 'c"   and m :: "('d, 'c) mapping"   and y :: "'d" shows "Mapping.lookup (Mapping.map_entry x f m) y = (if x = y then map_option f (Mapping.lookup m y) else Mapping.lookup m y)"
induct: fixes x :: "event list"   and P :: "event list \<Rightarrow> bool" assumes "x \<in> ns_public"   and "P []"   and "\<And>evsf X A B. \<lbrakk>evsf \<in> ns_public; P evsf; X \<in> synth (analz (knows A evsf))\<rbrakk> \<Longrightarrow> P (Says A B X # evsf)"   and "\<And>evsr A B X. \<lbrakk>evsr \<in> ns_public; P evsr; Says A B X \<in> set evsr\<rbrakk> \<Longrightarrow> P (Gets B X # evsr)"   and "\<And>evs1 NA A B. \<lbrakk>evs1 \<in> ns_public; P evs1; Nonce NA \<notin> used evs1\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) # evs1)"   and "\<And>evs2 NB B NA A. \<lbrakk>evs2 \<in> ns_public; P evs2; Nonce NB \<notin> used evs2; Gets B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs2\<rbrakk> \<Longrightarrow> P (Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) # evs2)"   and "\<And>evs3 A B NA NB. \<lbrakk>evs3 \<in> ns_public; P evs3; Says A B (Crypt (pubK B) \<lbrace>Nonce NA, Agent A\<rbrace>) \<in> set evs3; Gets A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs3\<rbrakk> \<Longrightarrow> P (Says A B (Crypt (pubK B) (Nonce NB)) # evs3)" shows "P x"
between_def: fixes vs :: "'a list"   and ram\<^sub>1 :: "'a"   and ram\<^sub>2 :: "'a" shows "between vs ram\<^sub>1 ram\<^sub>2 \<equiv> let (pre\<^sub>1, post\<^sub>1) = splitAt ram\<^sub>1 vs in if ram\<^sub>2 \<in> set post\<^sub>1 then let (pre\<^sub>2, post\<^sub>2) = splitAt ram\<^sub>2 post\<^sub>1 in pre\<^sub>2 else let (pre\<^sub>2, post\<^sub>2) = splitAt ram\<^sub>2 pre\<^sub>1 in post\<^sub>1 @ pre\<^sub>2"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>" shows "a *\<^sub>\<rat> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vrat_mult"