hcju/isabelleps huggingface dataset:
power_even_abs: fixes 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "even n" shows "power.power one times (abs a) n = power.power one times a n"
fs: shows "((\<approx>) ===> (\<approx>)) f f'"
pmf_nonneg: fixes p :: "'c pmf"   and x :: "'c" shows "0 \<le> pmf p x"
valid_path_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "valid_path (reversepath g) = valid_path g"
fam_morphisms_are_maps: fixes X :: "'e set"   and is_open :: "'e set \<Rightarrow> bool"   and \<FF> :: "'e set \<Rightarrow> 'f set"   and \<rho> :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'f \<Rightarrow> 'f"   and b :: "'f"   and add_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str :: "'e set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str :: "'e set \<Rightarrow> 'f"   and one_str :: "'e set \<Rightarrow> 'f"   and \<FF>' :: "'e set \<Rightarrow> 'g set"   and \<rho>' :: "'e set \<Rightarrow> 'e set \<Rightarrow> 'g \<Rightarrow> 'g"   and b' :: "'g"   and add_str' :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str' :: "'e set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str' :: "'e set \<Rightarrow> 'g"   and one_str' :: "'e set \<Rightarrow> 'g"   and fam_morphisms :: "'e set \<Rightarrow> 'f \<Rightarrow> 'g"   and U :: "'e set" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U" shows "Set_Theory.map (fam_morphisms U) (\<FF> U) (\<FF>' U)"
nabla_unfold: fixes x :: "('a \<times> 'a) set" shows "rel_nabla x \<subseteq> relfdia x (rel_nabla x)"
assign_def: fixes f :: "'a \<Rightarrow> state \<Rightarrow> state"   and v :: "'a" shows "assign f v = (\<lambda>(x, s). let (x', s') = (if x = None then f v else id) (x, s) in (x', if x' = None then s' else s))"
sorted_wrt_append: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "sorted_wrt P (xs @ ys) = (sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>list.set xs. \<forall>y\<in>list.set ys. P x y))"
simps: fixes G :: "nat set \<times> nat set set"   and k :: "nat" shows "kill_short G k = (if short_cycles G k = {} then G else kill_short (G -- choose_v G k) k)"
span_insert_subset: fixes A :: "'b set"   and B :: "'b set"   and r :: "'b" assumes "span A \<subseteq> span B"   and "r \<in> span B" shows "span (insert r A) \<subseteq> span B"
trace_induct2: fixes t :: "'a Trace"   and t' :: "'b Trace"   and P :: "'a Trace \<Rightarrow> 'b Trace \<Rightarrow> bool" assumes "tLength t = tLength t'"   and "\<And>s s'. P (tInit s) (tInit s')"   and "\<And>s s' t t'. \<lbrakk>tLength t = tLength t'; P t t'\<rbrakk> \<Longrightarrow> P (t \<leadsto> s) (t' \<leadsto> s')" shows "P t t'"
lambda_zero: shows "(\<lambda>h. 0::'b) = (*) (0::'b)"
inj_on_higherPowers: fixes n :: "nat"   and i :: "nat" shows "inj_on (higherPowers n i) {x. \<forall>j. n \<le> j \<and> j < n + i \<longrightarrow> lookup x j = (0::'b)}"
simps: fixes x :: "const"   and y :: "const" shows "(x \<le> y) = (y = Any \<or> x = y)"
upper: fixes A :: "'b set"   and x :: "'b"   and a :: "'b" assumes "lub A x"   and "a \<in> A" shows "a \<le> x"
app: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "A >> A'"   and "B >> B'" shows "App A B >> App A' B'"
tendsto_zero_powrI: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> 0) F"   and "(g \<longlongrightarrow> b) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 < b" shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> 0) F"
item_rule_def: fixes item :: "item" shows "item_rule item = (case item of Item x1 x2 x3 x4 \<Rightarrow> x1)"
intros_2: fixes p :: "Player"   and v :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "path_conforms_with_strategy p (LCons v LNil) \<sigma>"
RedFAcc: fixes h :: "heap"   and a :: "nat"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list"   and v :: "val"   and P :: "(char list list \<times> char list exp) prog"   and t :: "ty"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "h a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>"   and "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>addr a\<bullet>F{D},(h, l, sh),b\<rangle> \<rightarrow> \<langle>Val v,(h, l, sh),b\<rangle>"
le_fin_empty: fixes w :: "'item list" shows "[] \<preceq>\<^sub>F w"
cond_action_ok_final_change: fixes final1 :: "'a \<Rightarrow> bool"   and s1 :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and ca :: "'c conditional_action"   and s2 :: "('g, 'c) locks \<times> (('c \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option) \<times> 'h) \<times> ('c \<Rightarrow> 'i wait_set_status option) \<times> 'c set"   and final2 :: "'f \<Rightarrow> bool"   and ln2 :: "'j \<Rightarrow>f nat" assumes "final_thread.cond_action_ok final1 s1 t ca"   and "\<And>t. (thr s1 t = None) = (thr s2 t = None)"   and "\<And>t x1. \<lbrakk>thr s1 t = \<lfloor>(x1, no_wait_locks)\<rfloor>; final1 x1; wset s1 t = None\<rbrakk> \<Longrightarrow> \<exists>x2. thr s2 t = \<lfloor>(x2, no_wait_locks)\<rfloor> \<and> final2 x2 \<and> ln2 = no_wait_locks \<and> wset s2 t = None" shows "final_thread.cond_action_ok final2 s2 t ca"
inject_1: fixes x11 :: "nat"   and x12 :: "nat"   and y11 :: "nat"   and y12 :: "nat" shows "(Pkt x11 x12 = Pkt y11 y12) = (x11 = y11 \<and> x12 = y12)"
subgroup_generated_group_carrier: fixes G :: "('b, 'c) monoid_scheme" assumes "Group.group G" shows "subgroup_generated G (carrier G) = G"
leadsETo_mono: fixes CC' :: "'d set set"   and CC :: "'d set set"   and A :: "'d set"   and B :: "'d set" assumes "CC' \<subseteq> CC" shows "A leadsTo[CC'] B \<subseteq> A leadsTo[CC] B"
IH_1: assumes "q \<in> ctermsl p1"   and "r \<in> ctermsl q" shows "r \<in> ctermsl p1"
list_eval_Throw: fixes P :: "(char list list \<times> char list exp) prog"   and x :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and vs :: "val list"   and es' :: "char list exp list" assumes "P \<turnstile> \<langle>throw x,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>map Val vs @ throw x # es',s\<rangle> [\<Rightarrow>] \<langle>map Val vs @ e' # es',s'\<rangle>"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "known_ptr = ShadowRootClass.known_ptr" shows "l_create_character_data\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr"
wf_vars_mono: fixes V :: "'a set"   and S :: "('b, 'a) strand_step list"   and W :: "'a set" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t (V \<union> W) S"
dtype1: fixes tds :: "(char list list \<times> char list \<times> (char list \<times> t list) list) list"   and new_tdecs :: "tid_or_exn set"   and mn :: "char list list"   and s :: "'b state"   and ck :: "bool"   and env :: "v sem_env"   and locs :: "locn \<times> locn" assumes "check_dup_ctors tds \<and> new_tdecs = type_defs_to_new_tdecs mn tds \<and> disjnt new_tdecs (defined_types s) \<and> allDistinct (map (\<lambda>x. case x of (tvs, tn, ctors) \<Rightarrow> tn) tds)" shows "evaluate_dec ck mn env s (Dtype locs tds) (update_defined_types (\<lambda>_. new_tdecs \<union> defined_types s) s, Rval (make_sem_env nsEmpty (build_tdefs mn tds)))"
splitAt_dist_ram12: fixes vs :: "'c list"   and a :: "'c list"   and ram1 :: "'c"   and b :: "'c list"   and ram2 :: "'c"   and c :: "'c list" assumes "distinct vs"   and "vs = a @ ram1 # b @ ram2 # c" shows "(b, c) = splitAt ram2 (snd (splitAt ram1 vs))"
content_pCons: fixes a :: "'a"   and p :: "'a poly" shows "content (pCons a p) = gcd a (content p)"
get_elements_by_class_name_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_elements_by_class_name ptr' idd \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set sc) = (ptr' \<in> set sc)"
g_rec_def: fixes c_ls :: "nat"   and key :: "nat" shows "g_rec c_ls key = (let n = c_fst key; x = c_snd key; m = c_snd n; m1 = c_fst m; m2 = c_snd m; y1 = c_fst x; x1 = c_snd x in if y1 = 0 then if c_assoc_have_key c_ls (c_pair m1 x1) = 0 then c_cons (c_pair key (c_assoc_value c_ls (c_pair m1 x1))) c_ls else c_ls else let y2 = y1 - 1 in if c_assoc_have_key c_ls (c_pair n (c_pair y2 x1)) = 0 then let t1 = c_assoc_value c_ls (c_pair n (c_pair y2 x1)); t2 = c_pair (c_pair y2 t1) x1 in if c_assoc_have_key c_ls (c_pair m2 t2) = 0 then c_cons (c_pair key (c_assoc_value c_ls (c_pair m2 t2))) c_ls else c_ls else c_ls)"
foldB: shows "foldr (\<lambda>xs. max (length xs)) xss 0 = foldr (\<lambda>x. max (length x)) (filter (\<lambda>ys. ys \<noteq> []) xss) 0"
matrix_vector_all_zero_from_index: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "'c" shows "(\<forall>m\<ge>i. A $ m $ k = (0::'b)) = vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column k A))"
sum_list_us_le: fixes y :: "nat list"   and i :: "nat" shows "sum_list (\<^bold>u y i) \<le> i + 1"
numgcdh_pos: fixes g :: "int"   and t :: "Ferrack.num" assumes "0 \<le> g" shows "0 \<le> numgcdh t g"
dominant_term_ms_aux_1: fixes basis :: "(real \<Rightarrow> real) list"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real" assumes "basis_wf basis"   and "trimmed_ms_aux xs"   and "is_expansion_aux xs f basis" shows "f \<sim>[at_top] eval_monom (dominant_term_ms_aux xs) basis"
arith_prog_disjoint: fixes a :: "int"   and a' :: "int"   and b :: "nat" assumes "[a \<noteq> a'] (mod int b)"   and "0 < b" shows "arith_prog a b \<inter> arith_prog a' b = {}"
map_ms_aux_MSLCons: fixes f :: "'b \<Rightarrow> 'a"   and C :: "'b"   and e :: "real"   and xs :: "('b \<times> real) msllist" shows "map_ms_aux f (MSLCons (C, e) xs) = MSLCons (f C, e) (map_ms_aux f xs)"
Fvars_rawpsubst_su: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm" shows "FvarsT (rawpsubstT r txs) \<subseteq> FvarsT r - snd ` set txs \<union> \<Union> {uu_. \<exists>t x. uu_ = FvarsT t \<and> (t, x) \<in> set txs}"
card_bij': fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'a" assumes "f \<in> A \<rightarrow> B"   and "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x"   and "g \<in> B \<rightarrow> A"   and "\<And>x. x \<in> B \<Longrightarrow> f (g x) = x" shows "card A = card B"
tm_wf_tcopy_1: shows "tm_wf0 tcopy_begin"
Lambert_W_nonpos_iff: fixes x :: "real" shows "(Lambert_W x \<le> 0) = (x \<le> 0)"
a_zpi: fixes z :: "int" shows "ant z + \<infinity> = \<infinity>"
exhaust: fixes y :: "'c + 'd" obtains a :: "'c" where "y = Inl a"   | b :: "'d" where "y = Inr b"
before_dist_not1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" assumes "distinct vs"   and "before vs ram1 ram2" shows "\<not> before vs ram2 ram1"
X: fixes x :: "'a"   and y :: "'a" assumes "closed_w r"   and "awalk x r y" shows "x = y"
smult_cancel: fixes c :: "'a"   and f :: "'a poly"   and g :: "'a poly" shows "(Polynomial.smult c f = Polynomial.smult c g) = (c = (0::'a) \<or> f = g)"
Min_le: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "x \<in> A" shows "signed.Min A \<le>s x"
assoc'_eq_inv_assoc: fixes f :: "'c"   and g :: "'c"   and h :: "'c" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "\<a>\<^sub>C\<^sup>-\<^sup>1[f, g, h] = C.inv \<a>\<^sub>C[f, g, h]"
image_set_system_wellformed: fixes f :: "'a \<Rightarrow> 'c" shows "incidence_system (f ` \<V>) (blocks_image \<B> f)"
algebraic_int_minus_iff: fixes x :: "'b" shows "algebraic_int (- x) = algebraic_int x"
subseteq_guards_Spec: fixes c :: "('d, 'e, 'f) com"   and r :: "('d \<times> 'd) set" assumes "c \<subseteq>\<^sub>g Spec r" shows "c = Spec r"
MATCH1_singleton_Fun: fixes ss :: "('d, 'e) Term.term list"   and ts :: "('d, 'f) Term.term list"   and f :: "'d"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term option" assumes "length ss = length ts" shows "MATCH1 ({#(Fun f ss, Fun f ts)#}, \<sigma>) (mset (zip ss ts), \<sigma>)"
prems_4: fixes k :: "nat"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "I k h"   and "i \<le> k"   and "k < j"   and "effect (f k) h h' r" shows "I (Suc k) h'"
approx_sym: fixes x :: "'b star"   and y :: "'b star" assumes "x \<approx> y" shows "y \<approx> x"
Iod_not_less_le: fixes T :: "'a set"   and a :: "'a"   and b :: "'a" assumes "T \<subseteq> carrier D"   and "a \<in> T"   and "b \<in> T"   and "Torder (Iod D T)" shows "(\<not> a \<prec>\<^bsub>Iod D T\<^esub> b) = b \<preceq>\<^bsub>Iod D T\<^esub> a"
one_call_in_path_filter_conv: fixes x :: "'b"   and p :: "'b list" shows "one_call_in_path x p = (length (filter (\<lambda>x'. x' = x) p) \<le> 1)"
AList_Mapping_update: fixes m :: "('b \<times> 'a) list"   and k :: "'b"   and v :: "'c"   and xs :: "('b \<times> 'c) list" assumes "map_of m k = None" shows "Mapping.update k v (AList_Mapping.Mapping xs) = AList_Mapping.Mapping ((k, v) # xs)"
Sign_getmsg: fixes X :: "msg"   and Y :: "agent"   and H :: "msg set" assumes "Sign X (priK Y) \<in> analz H"   and "pubK Y \<in> analz H" shows "X \<in> analz H"
gfp_exp_unitary: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> unitary (t P)" shows "unitary (gfp_exp t)"
single_step_r: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle>" shows "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto>* \<langle>l', R'\<rangle>"
extremum_unique: fixes a :: "'b word" shows "(a \<le> 0) = (a = 0)"
sum_list_0: fixes xs :: "'c list" shows "(\<Sum>x\<leftarrow>xs. (0::'a)) = (0::'a)"
lambda_system_eq: fixes f :: "'a set \<Rightarrow> ennreal" shows "lambda_system \<Omega> M f = {l \<in> M. \<forall>x\<in>M. f (x \<inter> l) + f (x - l) = f x}"
simps_2: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_WbisT (c1 ;; c2) = (SC_WbisT c1 \<and> SC_WbisT c2 \<or> noWhile (c1 ;; c2) \<and> SC_discr (c1 ;; c2) \<or> SC_siso (c1 ;; c2))"
Quotient_rep_reflp: fixes R :: "'j \<Rightarrow> 'j \<Rightarrow> bool"   and Abs :: "'j \<Rightarrow> 'k"   and Rep :: "'k \<Rightarrow> 'j"   and T :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k" assumes "Quotient R Abs Rep T" shows "R (Rep a) (Rep a)"
obj_ty_cong: fixes vs :: "vn \<Rightarrow> val option"   and obj :: "obj" shows "obj_ty (obj\<lparr>values := vs\<rparr>) = obj_ty obj"
choose_alternating_sum: fixes n :: "nat" assumes "0 < n" shows "(\<Sum>i\<le>n. (- (1::'a)) ^ i * of_nat (n choose i)) = (0::'a)"
generated_topology_empty: fixes S :: "'c set"   and B :: "'c set set" shows "generated_topology S B {}"
q1_def: shows "q1 = p - monom_mult (lookup p (t1 \<oplus> lt f1) / lc f1) t1 f1"
sum_list_eq_0_iff_nonpos: fixes xs :: "'a list" assumes "list_all (\<lambda>x. x \<le> (0::'a)) xs" shows "(sum_list xs = (0::'a)) = (\<forall>n\<in>set xs. n = (0::'a))"
that: fixes f :: "('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set \<Rightarrow> ('a \<times> 'c \<Rightarrow>\<^sub>L 'a) set" assumes "\<And>G. G \<in> Gs \<Longrightarrow> flowsto (f G) T CX G"   and "X = \<Union> (f ` Gs)" shows "thesis"
Key_image_eq: fixes x :: "key"   and A :: "key set" shows "(Key x \<in> Key ` A) = (x \<in> A)"
assms: shows "\<CC>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
preserves_reflects_arr: fixes f :: "'c" shows "K.A.arr (\<tau>'\<tau>.map f) = \<tau>.A.arr f"
Int_empty_right: fixes A :: "'b set" shows "A \<inter> {} = {}"
padd_def: fixes a :: "'a"   and p\<^sub>1 :: "'a \<times> 'a \<times> 'a"   and p\<^sub>2 :: "'a \<times> 'a \<times> 'a" shows "padd a p\<^sub>1 p\<^sub>2 = (let (x\<^sub>1, y\<^sub>1, z\<^sub>1) = p\<^sub>1; (x\<^sub>2, y\<^sub>2, z\<^sub>2) = p\<^sub>2 in if z\<^sub>1 = \<zero> then p\<^sub>2 else if z\<^sub>2 = \<zero> then p\<^sub>1 else let d\<^sub>1 = x\<^sub>2 \<otimes> z\<^sub>1; d\<^sub>2 = x\<^sub>1 \<otimes> z\<^sub>2; l = d\<^sub>1 \<ominus> d\<^sub>2; m = y\<^sub>2 \<otimes> z\<^sub>1 \<ominus> y\<^sub>1 \<otimes> z\<^sub>2 in if l = \<zero> then if m = \<zero> then pdouble a p\<^sub>1 else (\<zero>, \<zero>, \<zero>) else let h = m [^] 2 \<otimes> z\<^sub>1 \<otimes> z\<^sub>2 \<ominus> (d\<^sub>1 \<oplus> d\<^sub>2) \<otimes> l [^] 2 in (l \<otimes> h, (d\<^sub>2 \<otimes> l [^] 2 \<ominus> h) \<otimes> m \<ominus> l [^] 3 \<otimes> y\<^sub>1 \<otimes> z\<^sub>2, l [^] 3 \<otimes> z\<^sub>1 \<otimes> z\<^sub>2))"
rot_y_ivl: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> 1"   and "0 \<le> y"   and "y \<le> 1" shows "0 \<le> (x - 1 / 2) / (2 * diamond_y_gen (x_coord (rot_x x y))) + 1 / 2 \<and> (x - 1 / 2) / (2 * diamond_y_gen (x_coord (rot_x x y))) + 1 / 2 \<le> 1"
possafety_def: fixes A :: "'b set"   and P :: "'b llist set" shows "possafety A P \<equiv> \<forall>t\<in>A\<^sup>\<spadesuit>. (\<forall>r\<in>pfinpref A t. \<exists>s\<in>A\<^sup>\<infinity>. r @@ s \<in> P) \<longrightarrow> t \<in> P"
NEGc_simps_1: fixes A :: "char list" shows "\<parallel><PR A>\<parallel> = AXIOMSc (PR A) \<union> BINDINGc (PR A) (\<parallel>(PR A)\<parallel>)"
sum_diff1'_aux: fixes F :: "'b set"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'b" assumes "finite F"   and "{i \<in> I. f i \<noteq> (0::'c)} \<subseteq> F" shows "sum' f (I - {i}) = (if i \<in> I then sum' f I - f i else sum' f I)"
subterm_lit_vars: fixes u :: "'b trm"   and L :: "'b literal" assumes "u \<in> subterms_of_lit L" shows "vars_of u \<subseteq> vars_of_lit L"
J: shows "s\<^sub>0 \<Turnstile> u\<^sub>4"
enumAppend_increase_rec: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat"   and as :: "nat list"   and bs :: "nat list" assumes "ls \<in> set ((enumAppend nmax ^^ m) (enumBase nmax))"   and "as @ bs = ls" shows "\<forall>x\<in>set as. \<forall>y\<in>set bs. x \<le> y"
measurable_bind2: fixes f :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and g :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
prime_cn: fixes f :: "recf"   and k :: "nat"   and gs :: "recf list"   and m :: "nat"   and n :: "nat" assumes "primerec f k"   and "length gs = k"   and "\<forall>i<length gs. primerec (gs ! i) m"   and "m = n" shows "primerec (Cn n f gs) m"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) atm" assumes "\<not> satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "int \<xi>'__ T1 \<noteq> int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "\<not> intP p (map (int \<xi>) Tl)"
entails_trans: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N1 :: "'a set"   and N2 :: "'a set"   and N3 :: "'a set" assumes "consequence_relation Bot entails"   and "entails N1 N2"   and "entails N2 N3" shows "entails N1 N3"
WT_gpv_inline_invar: fixes gpv :: "('g, 'b, 'c) gpv"   and s :: "'a" assumes "\<I> \<turnstile>g gpv \<surd>"   and "True" shows "\<I>' \<turnstile>g inline callee gpv s \<surd>"
directed_downset_ideals: fixes X :: "'a set" shows "directed (\<Down> X) = idealp (\<Down> X)"
adjust_step_eq: fixes st :: "nat"   and l :: "cell list"   and r :: "cell list"   and ap :: "(action \<times> nat) list"   and st' :: "nat"   and l' :: "cell list"   and r' :: "cell list" assumes "step0 (st, l, r) ap = (st', l', r')"   and "tm_wf0 ap"   and "0 < st'" shows "step0 (st, l, r) (adjust0 ap) = (st', l', r')"
cv_const2: fixes x :: "nat"   and n :: "nat"   and y :: "nat" assumes "x + n \<le> y" shows "x \<le> 0 + (y - n)"
types_preserved_call_indirect_Some: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and c :: "i32"   and j :: "nat"   and ts :: "t list"   and ts' :: "t list"   and s :: "s"   and i' :: "nat"   and cl :: "cl"   and tf :: "tf"   and arb_return :: "t list option"   and arb_labs :: "t list list"   and tvs :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c, $Call_indirect j] : ts _> ts'"   and "stab s i' (nat_of_int c) = Some cl"   and "stypes s i' j = tf"   and "cl_type cl = tf"   and "store_typing s \<S>"   and "i' < length (inst s)"   and "\<C> = (s_inst \<S> ! i')\<lparr>local := local (s_inst \<S> ! i') @ tvs, label := arb_labs, return := arb_return\<rparr>" shows "\<S>\<bullet>\<C> \<turnstile> [Callcl cl] : ts _> ts'"
leD: fixes y :: "'t"   and x :: "'t" assumes "y \<preceq>\<^sub>t x" shows "\<not> x \<prec>\<^sub>t y"
sorted_sort_key: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (map f (linorder.sort_key less_eq f xs))"
permutes_univ: fixes p :: "'b \<Rightarrow> 'b" shows "(p permutes UNIV) = (\<forall>y. \<exists>!x. p x = y)"
fail_optionT_def: shows "fail = fail_option Monad_Overloading.return"
distr_def: fixes M :: "'b measure"   and N :: "'a measure"   and f :: "'b \<Rightarrow> 'a" shows "distr M N f = measure_of (space N) (sets N) (\<lambda>A. emeasure M (f -` A \<inter> space M))"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'c" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
emeasure_subprob_space_less_top: fixes na :: "nat"   and A :: "(nat \<Rightarrow>\<^sub>F 'a) set" shows "emeasure (P' na) A \<noteq> top"
vint_uminus_transfer: shows "rel_fun cr_vint cr_vint vint_uminus_app uminus"
toFull: fixes \<sigma> :: "'tp" shows "bij_betw (toFull \<sigma>) {a. intT \<sigma> a} UNIV"
neq: fixes z :: "complex" assumes "1 \<le> Re z" shows "pre_zeta 1 z * (z - 1) + 1 \<noteq> 0"
prob_compl: fixes A :: "('b \<Rightarrow> 'a) set" assumes "A \<in> P.P.events" shows "P.P.prob (space (Pi\<^sub>M UNIV (\<lambda>_. M)) - A) = 1 - P.P.prob A"
isNSCont_NSLIM: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" assumes "isNSCont f a" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a"
simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a"   and c :: "'b" shows "option_assn P (Some a) (Some c) = P a c"
typeII_two_cubes_trace_vertical_boundaries: shows "two_chain_vertical_boundary two_chain_typeII \<subseteq> \<gamma>"