hcju/isabelleps huggingface dataset:
in_hom_def: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>1 b\<guillemotright> \<equiv> a \<in> C1.domains f \<and> b \<in> C1.codomains f"
augment_edge_cf_def: fixes cf :: "nat \<times> nat \<Rightarrow> 'b" shows "augment_edge_cf cf \<equiv> \<lambda>(u, v) \<Delta>. cf((u, v) := cf (u, v) - \<Delta>, (v, u) := cf (v, u) + \<Delta>)"
assms_2: shows "atom i' \<sharp> (A, i, B)"
simps_3: fixes \<delta> :: "int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and p\<^sub>0 :: "int \<times> int"   and v :: "int \<times> int"   and va :: "(int \<times> int) list" shows "find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va) = (let (\<delta>', p\<^sub>1) = find_closest_bf_code p\<^sub>0 (take 7 (v # va)) in if \<delta> \<le> \<delta>' then find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1) (v # va) else find_closest_pair_code (\<delta>', p\<^sub>0, p\<^sub>1) (v # va))"
call_no_result: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and chain :: "char list"   and rs :: "'c rule list"   and rrs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma> chain = Some rs"   and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z Undecided"   and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m (Call chain) # rrs \<Rightarrow>\<^sub>z t"
free_basis_def: fixes G :: "'b list set" shows "\<BB>\<^sub>F G \<equiv> \<BB> \<langle>G\<rangle>\<^sub>F"
distinct_32: fixes name' :: "name"   and pi' :: "pi"   and name1 :: "name"   and name2 :: "name"   and pi :: "pi" shows "<\<nu>name'>pi' \<noteq> name1{name2}.pi"
leg0_uniquely_isomorphic_2: shows "\<exists>!\<theta>. \<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> r\<^sub>0\<guillemotright>"
inter_add_right1: fixes x :: "'a"   and N :: "'a multiset"   and M :: "'a multiset" assumes "x \<notin># N" shows "N \<inter># add_mset x M = N \<inter># M"
simps_2: fixes b :: "'d" shows "lsumr (Inr (Inl b)) = Inl (Inr b)"
e_nth_append_zeros_big: fixes b :: "nat"   and i :: "nat"   and z :: "nat" assumes "e_length b \<le> i" shows "e_nth (e_append_zeros b z) i = 0"
oghoare_catch_valid: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P\<^sub>1 :: "'e set"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and Q :: "'e set"   and P\<^sub>2 :: "'e set"   and c\<^sub>2 :: "('e, 'd, 'f) com"   and A :: "'e set" assumes "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P\<^sub>1 c\<^sub>1 Q, P\<^sub>2"   and "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P\<^sub>2 c\<^sub>2 Q, A" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P\<^sub>1 Catch c\<^sub>1 c\<^sub>2 Q, A"
finterD1: fixes c :: "'e"   and A :: "'e fset"   and B :: "'e fset" assumes "c |\<in>| A |\<inter>| B" shows "c |\<in>| A"
preserves_reflects_arr: fixes f :: "'a \<times> 'a" shows "arr (\<sigma> f) = CC.arr f"
vsv_ex1: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "\<exists>!b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
brw_\<alpha>_def: shows "brw_\<alpha> = (\<lambda>(Q, W, rcm). (dom Q, W, rcm))"
alg_mk_auto_def: fixes aOps :: "('e, 'd, 'c list) MapOps"   and tOps :: "('f, 'd \<times> 'b, 'd) MapOps"   and simInit :: "'b \<Rightarrow> 'd"   and k_dfs :: "('e, 'f) AlgState" shows "alg_mk_auto aOps tOps simInit k_dfs \<equiv> \<lparr>pInit = simInit, pTrans = \<lambda>obs ec. the (MapOps.lookup tOps (aTrans k_dfs) (ec, obs)), pAct = \<lambda>ec. the (MapOps.lookup aOps (aActs k_dfs) ec)\<rparr>"
le_fin: fixes y :: "eint" assumes "y \<in> A" shows "eint m \<le> y \<and> y \<le> eint n"
distinct_11: fixes x3 :: "fmla"   and x41 :: "name"   and x42 :: "fmla" shows "neg x3 \<noteq> exi x41 x42"
intros: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b" assumes "r a b" shows "r\<inverse>\<inverse> b a"
surjective: fixes r :: "'a s_scheme" shows "r = \<lparr>inst = inst r, funcs = s.funcs r, tab = s.tab r, mem = s.mem r, globs = s.globs r, \<dots> = s.more r\<rparr>"
ide_iff_IDE: fixes a :: "'c local.term set" shows "ide a = IDE a"
len_gt_0: fixes len_of :: "'b itself \<Rightarrow> nat" assumes "class.len len_of" shows "0 < len_of TYPE('b)"
refine: fixes c :: "nat \<times> nat \<Rightarrow> int"   and s :: "nat"   and t :: "nat"   and N :: "nat" assumes "Edka_Impl c s t N" shows "(uncurry2 (augment_imp N), uncurry2 (PR_CONST (Network.augment_cf_impl c))) \<in> (asmtx_assn N int_assn)\<^sup>d *\<^sub>a (list_assn (nat_assn \<times>\<^sub>a nat_assn))\<^sup>k *\<^sub>a int_assn\<^sup>k \<rightarrow>\<^sub>a asmtx_assn N int_assn"
rnotinf: shows "r \<notin> A\<^sup>\<omega>"
assms: fixes e :: "exp0"   and s' :: "'a state"   and env :: "v sem_env" assumes "e \<in> set es"   and "clock s' \<le> clock s" shows "\<exists>s'' r. evaluate True env s' e (s'', r)"
comp: fixes f :: "nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1'"   and "g \<in> PrimRec1'" shows "(\<lambda>x. f (g x)) \<in> PrimRec1'"
invar_abs: shows "\<alpha> ` wa_invar WAC \<subseteq> wa_invar WAA"
prefixeq_butlast: fixes xs :: "'b list" shows "butlast xs \<le>p xs"
stutstep_def: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "stutstep s n \<equiv> s n = s (Suc n)"
has_derivative_in_compose: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and g :: "'c \<Rightarrow> 'd"   and g' :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x) within f ` s)" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
disjnt_empty2: fixes A :: "'c set" shows "disjnt A {}"
minus_mat_seq_def: fixes A :: "complex mat"   and X :: "nat \<Rightarrow> complex mat" shows "minus_mat_seq A X = (\<lambda>n. A - X n)"
simps: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and \<sigma> :: "'e \<Rightarrow> ('d, 'e, 'f) val option"   and l :: "'e" shows "\<R>\<^sub>S \<alpha> \<beta> \<sigma> l = \<sigma> (inv \<beta> l) \<bind> (\<lambda>v. Some (\<R>\<^sub>V \<alpha> \<beta> v))"
elements_mat_def: fixes A :: "'a mat" shows "elements_mat A = set (concat (map (\<lambda>i. map (\<lambda>j. A $$ (i, j)) [0..<dim_col A]) [0..<dim_row A]))"
max_Var_floatarith_fold_const_fa: fixes fa :: "floatarith" shows "max_Var_floatarith (fold_const_fa fa) \<le> max_Var_floatarith fa"
xivp_3: shows "Y\<^sub>1 \<in> U \<rightarrow> S"
map_mp: fixes m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option"   and k :: "'a"   and v :: "'b" assumes "m \<subseteq>\<^sub>m m'"   and "m k = Some v" shows "m' k = Some v"
is_nth_power_Suc_0: fixes n :: "nat" shows "is_nth_power n (Suc 0)"
two_composable_arrows_of_spans_axioms: shows "two_composable_arrows_of_spans (\<cdot>) prj0 prj1 \<pi> \<rho>"
neg_numeral_le_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (uminus (numeral.numeral one plus n)) zero"
orthogonal_complement_antimono: fixes B :: "'a set"   and A :: "'a set" assumes "B \<subseteq> A" shows "orthogonal_complement A \<subseteq> orthogonal_complement B"
finite_lists_length_eq: fixes A :: "'c set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs \<subseteq> A \<and> length xs = n}"
span_eq: fixes S :: "'b set"   and T :: "'b set" shows "(ideal S = ideal T) = (S \<subseteq> ideal T \<and> T \<subseteq> ideal S)"
entt_fr_refl: fixes F :: "assn"   and F' :: "assn"   and A :: "assn" assumes "F \<Longrightarrow>\<^sub>t F'" shows "F * A \<Longrightarrow>\<^sub>t F' * A"
in_arcsD2: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b" assumes "pair_wf_digraph G"   and "(u, v) \<in> parcs G" shows "v \<in> pverts G"
simps_1: fixes f :: "'a fps" shows "fps_nth_deriv 0 f = f"
norm_Pair2: fixes x :: "'c" shows "norm (x, 0::'d) = norm x"
lspasl_c_der: fixes h1 :: "'b"   and h2 :: "'b"   and h0 :: "'b"   and h3 :: "'b" assumes "h1,h2\<triangleright>h0"   and "h1,h3\<triangleright>h0" shows "(h1,h2\<triangleright>h0) \<and> h2 = h3"
f_def: shows "f \<equiv> \<lambda>x. if x \<in> {0..c} then x powr a else exp (- x / 2)"
rsp: shows "rel_fun (eq_onp overrider) (eq_onp overrider) (\<lambda>F x y. F y x) (\<lambda>F x y. F y x)"
splus_eq_splus: fixes t :: "'a"   and u :: "'t" shows "t \<oplus> u = nat_term_class.splus (term_of_pair (t, the_min)) u"
ide_dom: fixes f :: "'c" assumes "C.arr f" shows "C.ide (C.dom f)"
post_del_def: fixes t :: "('b \<times> nat) tree"   and t' :: "('c \<times> nat) tree" shows "post_del t t' \<equiv> invar t' \<and> (lvl t' = lvl t \<or> lvl t' + 1 = lvl t) \<and> (lvl t' = lvl t \<and> sngl t \<longrightarrow> sngl t')"
add_less_mono2: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j" shows "k + i < k + j"
has_derivative_iff_norm: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and x :: "'d"   and s :: "'d set" shows "(f has_derivative f') (at x within s) = (bounded_linear f' \<and> ((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
generate_into_G: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> comm.additive.generate (R \<inter> H)" shows "a \<in> R"
simps_5: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and vs :: "(variable \<times> typ) list"   and ns :: "variable set"   and Hs :: "Core.term list"   and t :: "Core.term"   and p :: "proofterm" shows "replay' thy vs ns Hs (AbsP t p) = (let t' = subst_bvs (map (\<lambda>(x, y). Fv x y) vs) t; rep = replay' thy vs ns (t' # Hs) p in if typ_of t' = Some propT \<and> term_ok thy t' then map_option ((\<longmapsto>) t') rep else None)"
q2: shows "Suc (i div s * s + i mod s) = i div s * s + Suc (i mod s)"
assms_3: shows "Arg2pi w \<le> Arg2pi z"
add_left_cancel: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.cancel_semigroup_add plus" shows "(plus a b = plus a c) = (b = c)"
pair_resTy_def: fixes p :: "'a \<times> 'b" shows "resTy p = resTy (snd p)"
\<tau>_def: fixes i :: "nat" shows "\<tau> i = (\<Sum>l\<leftarrow>[3..<Suc i]. (n (l - 2) + n i) * (n (l - 1) + n i))"
mat_delete_dim_1: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "dim_row (mat_delete A i j) = dim_row A - 1"
onode_bcast: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d"   and m :: "'e"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd) \<times> 'e seq_action \<times> (nat \<Rightarrow> 'c) \<times> 'd) set"   and i :: "nat"   and R :: "nat set" assumes "((\<sigma>, s), broadcast m, \<sigma>', s') \<in> S" shows "((\<sigma>, NodeS i s R), R:*cast(m), \<sigma>', NodeS i s' R) \<in> onode_sos S"
Suc_num_args: fixes t :: "('a, 'b) tm" assumes "is_App t" shows "Suc (num_args (fun t)) = num_args t"
wagner_1_var: fixes x :: "'a" shows "(x\<^sup>\<star> \<cdot> x)\<^sup>\<omega> = x\<^sup>\<omega>"
Vr_n_val_Vr: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "carrier (Vr K v) = carrier (Vr K (n_val K v))"
simps_12: fixes e1 :: "'a aexp"   and e2 :: "'a aexp"   and e1' :: "'a aexp"   and e2' :: "'a aexp" shows "local.less_gexp_aux (Gt e1 e2) (Eq e1' e2') = False"
poly_pinfty_gt_lc: fixes p :: "real poly" assumes "0 < lead_coeff p" shows "\<exists>n. \<forall>x\<ge>n. lead_coeff p \<le> poly p x"
C_below_C: fixes i :: "nat"   and j :: "nat" shows "(C\<^bsup>i\<^esup> \<sqsubseteq> C\<^bsup>j\<^esup>) = (i \<le> j)"
reconstruct_share: fixes S1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> unit \<Rightarrow> 'a spmf"   and R1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'a spmf"   and S2_14_OT :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'b spmf"   and R2_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'b spmf"   and protocol_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and adv_14_OT :: "real"   and x :: "bool" assumes "gmw_base S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT" shows "share x \<bind> reconstruct = return_spmf x"
no_recording_cs_if_not_snapshotted: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and cid :: "nat"   and q :: "'a" assumes "trace init t final"   and "\<not> ps (S t i) p \<noteq> None"   and "channel cid = Some (q, p)" shows "cs (S t i) cid = cs init cid"
wffp_imp_appendR: fixes sl1 :: "('St, 'U, 'C) state list"   and sl2 :: "('St, 'U, 'C) state list" assumes "wffp (sl1 @ sl2)"   and "sl2 \<noteq> []" shows "wffp sl2"
plus_of_int: fixes x :: "'a"   and n :: "int" shows "f (x + of_int n) = f x"
all_conv_all_lookup: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('b, 'c) mapping_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Mapping2.all P t = (\<forall>k v. RBT_Mapping2.lookup t k = Some v \<longrightarrow> P k v)"
boundedE: fixes b :: "'d"   and c :: "'d"   and a :: "'d" assumes "sup b c \<le> a" obtains "b \<le> a" and "c \<le> a"
trans_subset_SN: fixes R :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "trans R"   and "R \<subseteq> r \<union> s"   and "SN r"   and "SN s" shows "SN R"
measurable_subprob_algebra_density: fixes N :: "'a measure"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'b measure" assumes "sigma_finite_measure N"   and "space N \<noteq> {}"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M N)"   and "\<And>x. x \<in> space M \<Longrightarrow> integral\<^sup>N N (f x) \<le> 1" shows "(\<lambda>x. density N (f x)) \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
proj_def: fixes n :: "'b"   and S :: "('b strand_label \<times> 'c) list" shows "proj n S \<equiv> filter (\<lambda>s. is_LabelN n s \<or> is_LabelS s) S"
of_nat_sum_mset: fixes A :: "nat multiset" shows "of_nat (\<Sum>\<^sub># A) = \<Sum>\<^sub># (image_mset of_nat A)"
simps_1: fixes mag :: "'c"   and dim :: "(int, 'd) dimvec"   and more :: "'e" shows "mag \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = mag"
exec_move_CallI1: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and M :: "String.literal"   and es :: "'addr expr1 list" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e\<bullet>M(es)) h s ta h' s'"
A2_def: shows "A2 = mat_of_cols (dim_row A) (map (col A) [1..<dim_col A])"
inject_3: fixes x31 :: "char list"   and x32 :: "char list list"   and y31 :: "char list"   and y32 :: "char list list" shows "(StepC2 x31 x32 = StepC2 y31 y32) = (x31 = y31 \<and> x32 = y32)"
local_guarantee_preservation: fixes i :: "nat" assumes "i < length cms" shows "preserves_local_guarantee_compliance (\<R>_rel (cms ! i))"
hyps: fixes bs :: "'a list"   and m :: "nat" assumes "boundslt (length bs) p"   and "bound m p"   and "m < length bs" shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
lset_conv_lnth: fixes xs :: "'c llist" shows "lset xs = {lnth xs n |n. enat n < llength xs}"
funionM_return_empty2: fixes x :: "'m" shows "funionM x (return {||}) = x"
LAssRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>V:=e,s,b\<rangle> \<rightarrow> \<langle>V:=e',s',b'\<rangle>"
image_mset_Diff: fixes B :: "'a multiset"   and A :: "'a multiset"   and f :: "'a \<Rightarrow> 'b" assumes "B \<subseteq># A" shows "image_mset f (A - B) = image_mset f A - image_mset f B"
linear_0: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f" shows "f 0 = 0"
polyadd_0_2: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "0\<^sub>p +\<^sub>p p = p"
real_ivl_eqs_3: fixes r :: "real" assumes "0 < r" shows "ball (r / 2) (r / 2) = {0<--<r}"
rep_eq: fixes x :: "'c"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmdrop x xa) = map_drop x (fmlookup xa)"
ord_recp_total: fixes T :: "hf"   and G :: "hf \<Rightarrow> hf"   and H :: "hf \<Rightarrow> hf"   and x :: "hf" shows "\<exists>y. ord_recp T G H x y"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "R.isomorphic a a'" obtains f :: "'a" where "R.in_hom f a a' \<and> R.iso f"
plus_inf: fixes f :: "atom fm"   and xs :: "real list" assumes "nqfree f" shows "\<exists>x. \<forall>y\<ge>x. R.I (inf\<^sub>+ f) xs = R.I f (y # xs)"
IH_2: fixes T :: "real tensor" assumes "T \<in> set\<^sub>v (tensors_from_net m2)" shows "dims T = input_sizes m2"
preserves_arr: fixes f :: "('a, 'a) CC.arr" assumes "CC.arr f" shows "arr (if CC.arr f then CC.Map f else null)"
disjoint_subset: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "disjoint A B"   and "A \<subseteq> B \<union> C" shows "A \<subseteq> C"
abs_eq: fixes x :: "'a \<Rightarrow> 'b" shows "pp_of_fun x = PP (Abs_poly_mapping x)"
analz_insert_freshK: fixes evs :: "event list"   and KAB :: "nat"   and K :: "nat" assumes "evs \<in> bankerb_gets"   and "KAB \<notin> range shrK" shows "(Key K \<in> analz (insert (Key KAB) (knows Spy evs))) = (K = KAB \<or> Key K \<in> analz (knows Spy evs))"
sl_def: shows "Err.sl = (\<lambda>(A, r, f). (err A, Err.le r, lift2 f))"
aencrypt_def: fixes \<alpha> :: "'grp"   and msg :: "'grp" shows "aencrypt \<alpha> msg = sample_uniform (order \<G>) \<bind> (\<lambda>y. return_spmf (\<^bold>g [^] y, \<alpha> [^] y \<otimes> msg))"
get_M_pure: fixes ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'j"   and h :: "(_) heap" shows "pure (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h"
rtc_isPath: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> E\<^sup>*" shows "\<exists>p. isPath u p v"
lsnd_\<alpha>_finite: fixes l :: "'a list" shows "finite (ListSetImpl_NotDist.lsnd_\<alpha> l)"
preserves_inv: fixes f :: "'a" assumes "local.iso f" shows "trg (local.inv f) = local.inv (trg f)"
assms_4: shows "a1 \<in> b1 \<rightarrow> carrier R"
arr_Rel_ArrCod_in_Vset: shows "F\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
ba'': fixes w :: "'a" assumes "w \<in> S" shows "a'' \<bullet> w = b - a' \<bullet> z"
map_comp: fixes g1 :: "'k \<Rightarrow> 'h"   and g2 :: "'l \<Rightarrow> 'i"   and g3 :: "'m \<Rightarrow> 'j"   and f1 :: "'n \<Rightarrow> 'k"   and f2 :: "'o \<Rightarrow> 'l"   and f3 :: "'p \<Rightarrow> 'm"   and v :: "('n, 'o, 'p) generat" shows "map_generat g1 g2 g3 (map_generat f1 f2 f3 v) = map_generat (g1 \<circ> f1) (g2 \<circ> f2) (g3 \<circ> f3) v"
irreducible_mult_unit_left: fixes a :: "'a"   and p :: "'a" assumes "is_unit a" shows "irreducible (a * p) = irreducible p"
low_proc_non0\<^sub>1_correct: fixes p :: "prog" assumes "vars p = vars\<^sub>0" shows "low_proc_non0\<^sub>1 p = low_proc_non0 p"
run_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a"   and x :: "nat \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'c" shows "run r u x y = (\<forall>i. r (u i) (u (Suc i)) (x i) (y i))"
Diff_insert2: fixes A :: "'e set"   and a :: "'e"   and B :: "'e set" shows "A - insert a B = A - {a} - B"
subspace_scale: fixes S :: "('d \<Rightarrow>\<^sub>0 'c) set"   and x :: "'d \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "phull.subspace S"   and "x \<in> S" shows "c \<cdot> x \<in> S"
suffixes_append: fixes xs :: "'b list"   and ys :: "'b list" shows "suffixes (xs @ ys) = suffixes ys @ map (\<lambda>zs. zs @ ys) (suffixes xs)"
LLL_inv_initial_state_bound: shows "LLL_bound_invariant True True 0 fs_init"
degree_pdevs_of_list_eq: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<noteq> (0::'b)" shows "degree (pdevs_of_list xs) = length xs"
card_of_Well_order: fixes A :: "'a set" shows "Well_order |A|"
tfr\<^sub>s\<^sub>t\<^sub>p_list_all_alt_def: fixes S :: "('fun, 'var) strand_step list" shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p S = ((\<forall>a t t'. \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t \<in> set S \<and> (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t') \<and> (\<forall>X F. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> (\<forall>x\<in>fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X. \<exists>a. \<Gamma> (Var x) = Var a) \<or> (\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<longrightarrow> T = [] \<or> (\<exists>s\<in>set T. s \<notin> Var ` set X))))"
match_simplematcher_SrcDst_2: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "'a ipt_iprange"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (Match (Dst X)) a p = (p_dst p \<in> ipt_iprange_to_set X)"
surjective_embedding_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "(embedding_map X Y f \<and> f ` topspace X = topspace Y) = homeomorphic_map X Y f"
simps_2: fixes f :: "'a vec"   and g :: "'a vec"   and ng :: "'a"   and n_gs :: "('a vec \<times> 'a) list"   and mus :: "'a list"   and g' :: "'a vec" shows "gram_schmidt.mus_adjuster f ((g, ng) # n_gs) mus g' = (let a = f \<bullet> g / ng in gram_schmidt.mus_adjuster f n_gs (a # mus) (- a \<cdot>\<^sub>v g + g'))"
lst_of_sp_is_vert: fixes g :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es :: "'c edge list"   and v2 :: "'c" assumes "Graph.subpath g v1 es v2" shows "v2 \<in> Graph.vertices g"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_child_nodes to_tree_order" shows "l_to_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs to_tree_order"
delete2_ssubseqs: fixes as :: "'b list"   and bs :: "'b list"   and cs :: "'b list"   and a :: "'b"   and b :: "'b" shows "as @ bs @ cs \<in> ssubseqs (as @ [a] @ bs @ [b] @ cs)"
hyps_8: shows "(fst (shift1 (shift_list (h, U) x qs, B - {(h, U)})) = []) = (fst (shift_list (h, U) x qs, B - {(h, U)}) = [])"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat, 'b) fmap" shows "fmlookup (higherPowers\<^sub>f x xa xb) = (\<lambda>xc. fmlookup xb (inc_above x xa xc))"