hcju/isabelleps huggingface dataset:
Rm_2: shows "LINT x|M. fm x * real_cond_exp M F g x = LINT x|M. fm x * g x"
exhaust: fixes y :: "('d, 'e) xstate" obtains x1 :: "'d" where "y = Normal x1"   | x2 :: "'e" where "y = Fault x2"   | "y = Stuck"
differentiable_on_def: fixes f :: "'d \<Rightarrow> 'e"   and s :: "'d set" shows "(f differentiable_on s) = (\<forall>x\<in>s. f differentiable at x within s)"
DiffI: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
matrix_sum_add_distrib: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a mat"   and d :: "nat"   and g :: "nat \<Rightarrow> 'a mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. f k + g k) n = matrix_sum d f n + matrix_sum d g n"
eval_list_disj_Uni: fixes L :: "atomUni fmUni list"   and x :: "real" shows "evalUni (list_disj_Uni L) x = (\<exists>l\<in>set L. evalUni l x)"
prems_1: shows "i' cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
COMP_map_aux: fixes fs :: "(nat list \<Rightarrow> nat) list" assumes "\<forall>f\<in>set fs. PRIMREC f \<and> (\<exists>kf. \<forall>l. f l < ack kf (sum_list l))" shows "\<exists>k. \<forall>l. (\<Sum>f\<leftarrow>fs. f l) < ack k (sum_list l)"
valid_sharing_def: 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" shows "valid_sharing \<S> ts \<equiv> (outstanding_non_volatile_writes_unshared \<S> ts \<and> sharing_consis \<S> ts) \<and> read_only_unowned \<S> ts \<and> unowned_shared \<S> ts \<and> no_outstanding_write_to_read_only_memory \<S> ts"
eq_fold': fixes A :: "'b set" shows "Min A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> min x z)) None A)"
long_ch_by_ord_def: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set" shows "long_ch_by_ord f X \<equiv> \<exists>x\<in>X. \<exists>y\<in>X. \<exists>z\<in>X. x \<noteq> y \<and> y \<noteq> z \<and> x \<noteq> z \<and> TernaryOrdering.ordering f betw X"
solution_swap: fixes p1 :: "nat"   and n :: "nat"   and p2 :: "nat"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'b" assumes "p1 < n"   and "p2 < n" shows "solution (Fun.swap p1 p2 A) n x = solution A n x"
map2set_rel_sv: fixes Rm :: "('c \<times> 'b) set \<Rightarrow> (unit \<times> unit) set \<Rightarrow> ('a \<times> ('b \<Rightarrow> unit option)) set"   and Rk :: "('c \<times> 'b) set" assumes "single_valued (\<langle>Rk, unit_rel\<rangle>Rm)" shows "single_valued (\<langle>Rk\<rangle>map2set_rel Rm)"
prime_factorization_primorial: fixes x :: "real" shows "prime_factorization (primorial x) = mset_set {p. prime p \<and> real p \<le> x}"
right_total_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "right_total R = (\<forall>y. \<exists>x. R x y)"
fold_congs_1: fixes r :: "'a state_t_scheme"   and r' :: "'a state_t_scheme"   and v' :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and f :: "(partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool) \<Rightarrow> partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and f' :: "(partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool) \<Rightarrow> partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool" assumes "r = r'"   and "sp_impl_subj_subj r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "sp_impl_subj_subj_update f r = sp_impl_subj_subj_update f' r'"
cblinfun_extension_exists_def: fixes S :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'd" shows "cblinfun_extension_exists S \<phi> = (\<exists>B. \<forall>x\<in>S. B *\<^sub>V x = \<phi> x)"
assms_3: shows "h1 \<in> o[F](h2)"
analytic_on_def: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" shows "f analytic_on S \<equiv> \<forall>x\<in>S. \<exists>e>0. f holomorphic_on ball x e"
Sigma_Un_distrib1: fixes I :: "'a set"   and J :: "'a set"   and C :: "'a \<Rightarrow> 'b set" shows "Sigma (I \<union> J) C = Sigma I C \<union> Sigma J C"
wgraph_of_list_def: fixes l :: "('c \<times> 'd \<times> nat) list" shows "wgraph_of_list l \<equiv> THE w. \<forall>u v d. (w (u, v) = enat d) = ((u, v, d) \<in> set l)"
is_subseq_ofE: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and s' :: "nat \<Rightarrow> nat \<Rightarrow> int"   and n :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "is_subseq_of s s'" shows "\<exists>k\<ge>n. s' n = s k"
vors_2: shows "between (vertices f) ram1 ram2 = []"
sn: shows "SN (t[x::=p] \<star> k)"
OrdS: fixes v :: "i"   and x :: "\<kappa>" shows "[\<lparr>O!,x\<rparr> in v] = (\<exists>o\<^sub>1 y. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y)"
a_get_ancestors_si_locs_def: shows "get_ancestors_si_locs = Shadow_DOM.get_parent_locs \<union> get_host_locs"
min_option_def: fixes pos :: "bool"   and o1 :: "int option"   and o2 :: "int option" shows "min_option pos o1 o2 = (if le_option pos o1 o2 then o1 else o2)"
multi_drop_mem_not_eq: fixes c :: "'a"   and B :: "'a multiset" assumes "c \<in># B" shows "remove1_mset c B \<noteq> B"
abs_1: fixes abs :: "'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs one = one"
rev_nth_zero: fixes xs :: "'b list"   and x :: "'b" shows "(xs @ [x]) !. 0 = Some x"
intro: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>a x y. scale a (x + y) = scale a x + scale a y"   and "\<And>a b x. scale (a + b) x = scale a x + scale b x"   and "\<And>a b x. scale a (scale b x) = scale (a * b) x"   and "\<And>x. scale (1::'b) x = x" shows "vector_space scale"
finsert_not_fempty: fixes a :: "'a"   and A :: "'a fset" shows "finsert a A \<noteq> {||}"
rel_compp: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_fun (=) (R OO S) = rel_fun (=) R OO rel_fun (=) S"
prems_2: shows "IF x t1 t2 \<noteq> Trueif"
BExpr_low_def: fixes b :: "BExpr" shows "BExpr_low b = (\<forall>s t \<beta>. s  \<approx>\<^sub>\<beta> t \<longrightarrow> evalB b s = evalB b t)"
le_fin_range: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" shows "set w\<^sub>1 \<subseteq> set w\<^sub>2"
prodrg_pOp: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" assumes "\<forall>k\<in>I. Ring (A k)" shows "(\<plusminus>\<^bsub>r\<Pi>\<^bsub>I\<^esub> A\<^esub>) = prod_pOp I A"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "m \<le> n" shows "sum g {m..n} = g m + sum g {Suc m..n}"
domain_subset_carrier: fixes c :: "('b, 'c) chart"   and charts :: "('b, 'c) chart set" assumes "c \<in> charts" shows "domain c \<subseteq> manifold.carrier charts"
in_set_insert: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in>. xs" shows "List.insert x xs = xs"
dom_m_def: fixes N :: "('c, 'd) fmap" shows "dom_m N = mset_fset (fmdom N)"
fr_Send2: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n" shows "nSend k t = nReturn k t"
eq_obj_rng_of_frac_axioms: shows "eq_obj_rng_of_frac R S"
measurable_compose_Pair1: fixes x :: "'b"   and M1 :: "'b measure"   and g :: "'b \<times> 'c \<Rightarrow> 'd"   and M2 :: "'c measure"   and L :: "'d measure" assumes "x \<in> space M1"   and "g \<in> M1 \<Otimes>\<^sub>M M2 \<rightarrow>\<^sub>M L" shows "(\<lambda>y. g (x, y)) \<in> M2 \<rightarrow>\<^sub>M L"
simps_9: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" shows "Dom \<^bold>\<a>\<^bold>[t, u, v\<^bold>] = (Dom t \<^bold>\<star> Dom u) \<^bold>\<star> Dom v"
rep_eq: fixes x :: "'b ttree"   and xa :: "'b ttree" shows "Rep_ttree (x \<oplus>\<oplus> xa) = Rep_ttree x \<union> Rep_ttree xa"
filter_a2_map_def: fixes I :: "\<I>"   and ts :: "nat"   and tp :: "nat"   and a2_map :: "(nat, ('a option list, nat + nat) mapping) mapping" shows "filter_a2_map I ts tp a2_map = {xs. \<exists>tp'\<le>tp. case Mapping.lookup a2_map tp' of None \<Rightarrow> False | Some m \<Rightarrow> case Mapping.lookup m xs of None \<Rightarrow> False | Some tstp \<Rightarrow> ts_tp_lt' ts tp tstp}"
crel_vs_elim: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and v :: "'e"   and v\<^sub>T :: "(heap, 'f) state"   and M :: "heap" assumes "state_dp_consistency.crel_vs R v v\<^sub>T"   and "state_dp_consistency.cmem M"   and "P M" obtains v' :: "'f" and M' :: "heap" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "state_dp_consistency.cmem M'" and "P M'"
p_fun_mult_var: fixes x :: "'a"   and y :: "'a" assumes "x\<^sup>\<smile> ; x \<le> 1'" shows "(x \<cdot> y)\<^sup>\<smile> ; (x \<cdot> y) \<le> 1'"
simps_2: fixes s :: "'b set"   and t :: "'b set"   and ts :: "'b set list" shows "intersection_list_opt s (t # ts) = s \<inter> t # intersection_list_opt (s - t) ts"
eq: fixes A :: "'a set"   and B :: "'a set" assumes "infsetsum'_converges f A"   and "\<And>x. x \<in> A - B \<union> (B - A) \<Longrightarrow> f x = (0::'b)" shows "infsetsum' f A = infsetsum' f B"
simps_13: fixes a :: "floatarith" shows "max_Var_floatarith (Sqrt a) = max_Var_floatarith a"
union_diff2: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) = sum g (A - B) + sum g (B - A) + sum g (A \<inter> B)"
dord_pm_lin: fixes ord :: "('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> ('a \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a \<Rightarrow>\<^sub>0 'c" assumes "ord s t \<or> ord t s" shows "dord_pm ord s t \<or> dord_pm ord t s"
ide_char': fixes a :: "'d" shows "D.ide a = (D.arr a \<and> (D.dom a = a \<or> D.cod a = a))"
eval_Runit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = \<r>[\<lbrace>Cod t\<rbrace>] \<cdot> (\<lbrace>t\<rbrace> \<otimes> \<I>)"
set_inline2_lift_spmf1: fixes p :: "'f \<Rightarrow> 'k \<Rightarrow> ('i \<times> 'f) spmf"   and callee :: "'e \<Rightarrow> 'l \<Rightarrow> ('j \<times> 'e, 'k, 'i) gpv"   and gpv :: "('d, 'l, 'j) gpv"   and s :: "'e"   and s' :: "'f" shows "set_spmf (inline2 (\<lambda>s x. lift_spmf (p s x)) callee gpv s s') \<subseteq> range Inl"
IH: fixes z :: "complex" assumes "z \<in> S" shows "(deriv ^^ n) (\<lambda>w. - f w) z = - (deriv ^^ n) f z"
assms_1: shows "I \<subseteq> U\<^sub>2"
intro: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. le x x" shows "mono_setup_loc le"
sorted_list_of_set_empty: shows "ordered_powerprod_lin.sorted_list_of_set {} = []"
Abs_ordinal_cases2: fixes x :: "ordinal" obtains z :: "ord0" where "x = Abs_ordinal (ord0rel `` {z})"
inv_on_f_range: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "y \<in> f ` A" shows "inv_on f A y \<in> A"
integrable_Beta: fixes a :: "real"   and b :: "real" assumes "0 < a"   and "0 < b" shows "set_integrable lborel {0..1} (\<lambda>t. t powr (a - 1) * (1 - t) powr (b - 1))"
PB: fixes x :: "'a" shows "proper_it (it_b x) (it_b' x)"
\<zeta>_tms_len_ss_tms_wt_t_le: fixes A :: "'v pvar \<Rightarrow> zhmultiset" assumes "legal_zpassign A" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta> (length ss))) * eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t)) \<le> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (coef_hd \<zeta>' (length ss'))) * eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t))"
get_def: fixes h :: "heap"   and a :: "'d array" shows "Array.get h a = map from_nat (arrays h TYPEREP('d) (addr_of_array a))"
ogets_NF_wp: fixes P :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" shows "ovalidNF (\<lambda>s. P (f s) s) (ogets f) P"
SOME_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(SOME i. i \<in> Basis) \<in> Basis"
ww'_1: shows "w \<le> - 1 \<and> w * exp w = x"
echelon_form_of_column_k_det_iarrays_def: fixes A' :: "'b \<times> 'b iarray iarray \<times> nat \<times> ('b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b)"   and k :: "nat" shows "echelon_form_of_column_k_det_iarrays A' k = (let (det_P, A, i, bezout) = A' in if i \<noteq> nrows_iarray A \<and> A !! i !! k = (0::'b) \<and> \<not> vector_all_zero_from_index (i + 1, column_iarray k A) then (- (1::'b) * det_P, echelon_form_of_column_k_iarrays (A, i, bezout) k) else (det_P, echelon_form_of_column_k_iarrays (A, i, bezout) k))"
mem_simps_9: fixes A :: "'f"   and C :: "'f set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
lookup_tail: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x \<Rightarrow>\<^sub>0 nat" shows "lookup (i.punit.tail p) v = (if lex_pm_strict v (i.lpp p) then lookup p v else (0::'b))"
chamber_system_image: shows "f \<turnstile> domain.\<C> = codomain.\<C> \<inter> (f \<turnstile> X)"
element_ord_generates_cyclic: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "finite_group G"   and "a \<in> carrier G"   and "group.ord G a = order G" shows "Finite_And_Cyclic_Groups.cyclic_group G a"
stateSpace_def: fixes ss :: "('d, 'e) fmap set"   and vs :: "'d set" shows "stateSpace ss vs \<equiv> \<forall>s. s \<in> ss \<longrightarrow> fmdom' s = vs"
coprime_abs_right_iff: fixes k :: "int"   and l :: "int" shows "coprime k \<bar>l\<bar> = coprime k l"
face_cycle_succ_def: shows "GM.face_cycle_succ \<equiv> edge_succ GM \<circ> edge_rev GM"
gg: shows "snd (adjuster_wit (f \<bullet> g / n_g # mus_acc) f (tl gs)) \<in> carrier_vec n"
execn_Normal_elim_cases_5: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Guard f g c,Normal s\<rangle> =n\<Rightarrow> t" obtains "s \<in> g" and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"   | "t = Fault f" and "s \<notin> g"
ide_char: fixes a :: "'a2" shows "C2.ide a = (C2.arr a \<and> C2.dom a = a \<and> C2.cod a = a)"
dist_triangle_half_r: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and y :: "'b"   and x1 :: "'b"   and e :: "real"   and x2 :: "'b" assumes "class.metric_space dist uniformity open"   and "dist y x1 < e / 2"   and "dist y x2 < e / 2" shows "dist x1 x2 < e"
cases: fixes a1 :: "binding list"   and a2 :: "POPLmarkRecord.type"   and a3 :: "POPLmarkRecord.type" assumes "a1 \<turnstile> a2 <: a3" obtains \<Gamma> :: "binding list" and S :: "POPLmarkRecord.type" where "a1 = \<Gamma>__" and "a2 = S" and "a3 = Top" and "\<Gamma>__ \<turnstile>\<^sub>w\<^sub>f" and "\<Gamma>__ \<turnstile>\<^sub>w\<^sub>f S"   | \<Gamma> :: "binding list" and i :: "nat" where "a1 = \<Gamma>'__" and "a2 = TVar i__" and "a3 = TVar i__" and "\<Gamma>'__ \<turnstile>\<^sub>w\<^sub>f" and "\<Gamma>'__ \<turnstile>\<^sub>w\<^sub>f TVar i__"   | \<Gamma> :: "binding list" and i :: "nat" and U :: "POPLmarkRecord.type" and T :: "POPLmarkRecord.type" where "a1 = \<Gamma>''__" and "a2 = TVar i" and "a3 = T" and "\<Gamma>''__\<langle>i\<rangle> = \<lfloor>TVarB U\<rfloor>" and "\<Gamma>''__ \<turnstile> \<up>\<^sub>\<tau> (Suc i) 0 U <: T"   | \<Gamma> :: "binding list" and T\<^sub>1 :: "POPLmarkRecord.type" and S\<^sub>1 :: "POPLmarkRecord.type" and S\<^sub>2 :: "POPLmarkRecord.type" and T\<^sub>2 :: "POPLmarkRecord.type" where "a1 = \<Gamma>'''__" and "a2 = S\<^sub>1__ \<rightarrow> S\<^sub>2__" and "a3 = T\<^sub>1__ \<rightarrow> T\<^sub>2__" and "\<Gamma>'''__ \<turnstile> T\<^sub>1__ <: S\<^sub>1__" and "\<Gamma>'''__ \<turnstile> S\<^sub>2__ <: T\<^sub>2__"   | \<Gamma> :: "binding list" and T\<^sub>1 :: "POPLmarkRecord.type" and S\<^sub>1 :: "POPLmarkRecord.type" and S\<^sub>2 :: "POPLmarkRecord.type" and T\<^sub>2 :: "POPLmarkRecord.type" where "a1 = \<Gamma>''''__" and "a2 = (\<forall><:S\<^sub>1. S\<^sub>2)" and "a3 = (\<forall><:T\<^sub>1. T\<^sub>2)" and "\<Gamma>''''__ \<turnstile> T\<^sub>1 <: S\<^sub>1" and "TVarB T\<^sub>1 \<Colon> \<Gamma>''''__ \<turnstile> S\<^sub>2 <: T\<^sub>2"   | \<Gamma> :: "binding list" and fs :: "(char list \<times> POPLmarkRecord.type) list" and fs' :: "(char list \<times> POPLmarkRecord.type) list" where "a1 = \<Gamma>" and "a2 = RcdT fs" and "a3 = RcdT fs'" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f RcdT fs" and "unique fs'" and "\<forall>(l, T)\<in>set fs'. \<exists>S. (l, S) \<in> set fs \<and> \<Gamma> \<turnstile> S <: T"
unit_char': fixes a :: "'b" assumes "B.obj a" shows "FoH.unit a = F (H.unit a) \<cdot>\<^sub>D \<tau>.F.unit (H.map\<^sub>0 a)"
a1: shows "real_of_float (mid (compute_bound_tm prec a a tg)) \<in>\<^sub>r compute_bound_tm prec I a tg"
phis'_codem_correct: fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val" assumes "uninst_code.phi_equiv_mapping (p_g g s1_phis) g nodes_of_phis (uninst_code.ssa.phiNodes_of (p_g g s1_phis) g)"   and "next \<in> Mapping.keys (p_g g s1_phis g)" shows "uninst_code.phis'_codem (p_g g s1_phis) g next (uninst_code.substitution_code (p_g g s1_phis) g next) nodes_of_phis = uninst_code.phis'_code (p_g g s1_phis) g next"
not_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
arr_Rel_def: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T = [T\<lparr>ArrVal\<rparr>, T\<lparr>ArrDom\<rparr>, T\<lparr>ArrCod\<rparr>]\<^sub>\<circ>"
ttl_def: fixes tllist :: "('f, 'g) tllist" shows "ttl tllist = (case tllist of TNil b \<Rightarrow> TNil b | TCons x21 x22 \<Rightarrow> x22)"
Frct_def: fixes p :: "int \<times> int" shows "Frct p = Fract (fst p) (snd p)"
borel_measurable_indicator_iff: fixes A :: "'c set"   and M :: "'c measure" shows "(indicator A \<in> borel_measurable M) = (A \<inter> space M \<in> sets M)"
M2: fixes p :: "'a"   and q :: "'a"   and r :: "'a"   and s :: "'a" assumes "p \<parallel> q"   and "r \<parallel> s" shows "p \<parallel> s \<oplus> ((\<exists>t. p \<parallel> t \<and> t \<parallel> s) \<oplus> (\<exists>t. r \<parallel> t \<and> t \<parallel> q))"
components_are_iso: fixes a :: "'a" assumes "A.ide a" shows "A.iso (\<phi>'.map a)"
cosh_real_strict_mono: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x < y" shows "cosh x < cosh y"
bisimilar_simulation_step: fixes P :: "('state, 'effect) L_state"   and Q :: "('state, 'effect) L_state"   and \<alpha> :: "('act, 'effect) L_action"   and P' :: "('state, 'effect) L_state" assumes "P \<sim>\<cdot>\<^sub>L Q"   and "bn \<alpha> \<sharp>* Q"   and "P \<rightarrow>\<^sub>L \<langle>\<alpha>,P'\<rangle>" obtains Q' :: "('state, 'effect) L_state" where "Q \<rightarrow>\<^sub>L \<langle>\<alpha>,Q'\<rangle>" and "P' \<sim>\<cdot>\<^sub>L Q'"
mod_eqE: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo b c" obtains d :: "'c" where "b = plus a (times c d)"
eqp: shows "(\<Prod>k\<in>Basis. m (axis_index k)) = prod m UNIV"
leg1_composite: shows "\<mu>\<nu>.leg1 = \<mu>.leg1 \<cdot> \<p>\<^sub>1[\<mu>.leg0, \<nu>.leg1]"
SplitGuards_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and c :: "('e, 'd, 'f) com"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c\<^sub>1 Q,A"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c\<^sub>2 UNIV,UNIV"   and "(c\<^sub>1 \<inter>\<^sub>g c\<^sub>2) = Some c" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
undefined_is_map_on_empty: fixes f :: "'c set \<Rightarrow> 'd set" assumes "f = (\<lambda>x. undefined)" shows "Set_Theory.map f {} {}"
id_comp: fixes g :: "'b \<Rightarrow> 'c" shows "id \<circ> g = g"
perm_ty: fixes pi :: "(name \<times> name) list"   and T :: "ty" shows "pi \<bullet> T = T"
t: shows "t = \<langle>l, x, r\<rangle>"
fsubstp_fsubst_eq: shows "fsubstp = (\<lambda>x. x \<in> fsubst)"
single_add: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "single k (a + b) = single k a + single k b"
transp_on_mulex_on: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b multiset set" shows "transp_on (mulex_on P A) B"
simps_1: fixes a :: "('a, 'b) Combinators"   and x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators" shows "member a (x \<oplus> xs) = (member a x \<or> member a xs)"
assms: shows "aEvalUni (NeqUni (a, b, c)) x \<noteq> aEvalUni (NeqUni (a, b, c)) y \<and> x < y"
dist_eq: fixes A :: "(real, 2) vec"   and B :: "(real, 2) vec" shows "(dist A B)\<^sup>2 = (A $ 1 - B $ 1)\<^sup>2 + (A $ 2 - B $ 2)\<^sup>2"
fin_CANDS: fixes M :: "trm"   and x :: "name"   and B :: "ty" assumes "\<not> fin M x"   and "x:M \<in> \<parallel>(B)\<parallel>" shows "x:M \<in> AXIOMSn B \<or> x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)"
lp_geq_lp_from_successor: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and G :: "'a list"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and vtx2 :: "'a"   and vtx1 :: "'a"   and w :: "'a \<Rightarrow> nat" assumes "geq_arg f"   and "geq_arg g"   and "\<forall>vtx. ListMem vtx G \<longrightarrow> \<not> R vtx vtx"   and "R vtx2 vtx1"   and "ListMem vtx1 G"   and "top_sorted_abs R G" shows "f (w vtx2) (wlp R w g f vtx1 G) \<le> wlp R w g f vtx2 G"
assms_2: shows "\<S>\<bullet>\<C> \<turnstile> [e] : ts _> ts'"
gt_proper_sub: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary t"   and "proper_sub s t" shows "t >\<^sub>t s"