hcju/isabelleps huggingface dataset:
le_sup_iff: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "less_eq_prod (sup_prod x y) z = (less_eq_prod x z \<and> less_eq_prod y z)"
assms_2: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length_muaux args aux) \<phi>) rel1"
listrel_iff_nth: fixes xs :: "'a list"   and ys :: "'b list"   and r :: "('a \<times> 'b) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>n<length xs. (xs ! n, ys ! n) \<in> r))"
obs_intra_singleton_element: fixes m :: "'node"   and n :: "'node"   and S :: "'node SDG_node set" assumes "m \<in> obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>" shows "obs_intra n \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {m}"
add_node_def: fixes v :: "'a"   and G :: "'a graph" shows "add_node v G \<equiv> \<lparr>nodes = {v} \<union> nodes G, edges = edges G\<rparr>"
shadow_root_different_delete_get: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and shadow_root_ptr' :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "shadow_root_ptr \<noteq> shadow_root_ptr'"   and "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr' h = Some h'" shows "get shadow_root_ptr h' = get shadow_root_ptr h"
list_all_iff: fixes P :: "'d \<Rightarrow> bool"   and x :: "'d list" shows "list_all P x = Ball (set x) P"
cong_refl: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b b a"
bind_cong: fixes x :: "'m option"   and y :: "'m option"   and f :: "'m \<Rightarrow> 'n option"   and g :: "'m \<Rightarrow> 'n option" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "Option.bind x f = Option.bind y g"
hyps: fixes a :: "'a mod_ring" shows "a ^ CARD('a) ^ n = a"
enough_flushs_nth_update: fixes 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 \<D> :: "bool"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "enough_flushs ts"   and "i < length ts"   and "\<not> \<D> \<longrightarrow> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "enough_flushs (ts[i := (p, is, \<theta>, sb, \<D>, \<O>, \<R>)])"
init_fin_silent_moveD: fixes t :: "'t"   and s :: "status"   and x :: "'x1"   and m :: "'m1"   and s' :: "status"   and x' :: "'x1"   and m' :: "'m1" assumes "\<tau>trsys.silent_move (r1.init_fin t) r1.init_fin_\<tau>move ((s, x), m) ((s', x'), m')" shows "r1.silent_move t (x, m) (x', m') \<and> s = s' \<and> s' = Running"
aadd_minus_less: fixes b :: "ant"   and a :: "ant" assumes "b \<noteq> - \<infinity> \<and> b \<noteq> \<infinity>"   and "a < 0" shows "a + b < b"
minim_def: fixes A :: "'a set" shows "minim A \<equiv> THE b. isMinim A b"
rtrancl_refl: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k" shows "r\<^sup>*\<^sup>* a a"
degree_zero_imp_empty_roots: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)"   and "degree p = 0" shows "roots p = {#}"
gim: shows "g ` (\<Union> \<F> - C) \<subseteq> T"
assms_2: shows "set_pmf p' \<subseteq> set_pmf p"
ind: fixes i :: "nat"   and c :: "nat" assumes "i < |ls|"   and "distinct ls"   and "last ls \<noteq> u"   and "ls ! i = u"   and "u \<in> set ls" shows "nextElem ls c u = ls ! Suc i"
rel_restrict_alt_def: fixes R :: "('a \<times> 'a) set"   and A :: "'a set" shows "rel_restrict R A = Restr R (- A)"
eval_Tensor: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Arr t"   and "Arr u" shows "\<lbrace>t \<^bold>\<otimes> u\<rbrace> = \<lbrace>t\<rbrace> \<otimes> \<lbrace>u\<rbrace>"
diseq: fixes y :: "fin_string"   and x :: "char" assumes "y \<in> set (fin_string_upto j)" shows "fin_string_empty \<noteq> fin_string_cons x y"
eq1: shows "(bw_diff ^^ j) (\<lambda>z. \<Sum>i = 1..j. z + (int i - int (aa i) - 1) gchoose i + (z + (int i - int (bb i) - 1) gchoose i)) = (\<lambda>_. if j = 0 then 0 else 2)"
elims: fixes x :: "('b word \<times> 'b word) list"   and y :: "'b wordinterval" assumes "l2wi x = y" obtains "x = []" and "y = Empty_WordInterval"   | s :: "'b word" and e :: "'b word" where "x = [(s__, e__)]" and "y = WordInterval s__ e__"   | s :: "'b word" and e :: "'b word" and v :: "'b word \<times> 'b word" and va :: "('b word \<times> 'b word) list" where "x = (s, e) # v # va" and "y = RangeUnion (WordInterval s e) (l2wi (v # va))"
asym_Sup_of_chain: fixes C :: "('c \<times> 'c) set set" assumes "set_chain C"   and "\<And>R. R \<in> C \<Longrightarrow> asym R" shows "asym (\<Union> C)"
cases: fixes x :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) \<times> (nat \<times> ('sf, 'sc, 'sz) step) list" obtains D :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" and S :: "(nat \<times> ('sf, 'sc, 'sz) step) list" where "x = (D, S)"
well_shaped_map_tree: fixes t :: "'b tree"   and f :: "'b \<Rightarrow> 'c" assumes "well_shaped t" shows "well_shaped (map_tree f t)"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c"   and f :: "'c"   and \<rho> :: "'c"   and g :: "'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "partial_magma.ide V r"   and "partial_magma.ide V f"   and "partial_magma.in_hom V \<rho> g (H r f)" shows "tabulation_data_axioms V H r \<rho> f g"
minus_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c" shows "pol_coeff S (fst c, \<lambda>j. -\<^sub>a\<^bsub>S\<^esub> snd c j)"
conj_assoc: fixes P :: "'b upred"   and Q :: "'b upred"   and S :: "'b upred" shows "((P \<and> Q) \<and> S) = (P \<and> Q \<and> S)"
Compl_set_less_eq_Compl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b set" assumes "class.linorder less_eq less" shows "ord.set_less_eq less_eq (- A) (- B) = ord.set_less_eq less_eq B A"
finite_vars_of_cnf: fixes cnf :: "lit uprod set" assumes "finite cnf" shows "finite (vars_of_cnf cnf)"
prime_factorization_primepow: fixes n :: "'a" assumes "primepow n" shows "prime_factorization n = replicate_mset (multiplicity (aprimedivisor n) n) (aprimedivisor n)"
bit_signed_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and w :: "'c word"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "bit (ring_1.signed one plus zero minus w) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit w (min (LENGTH('c) - Suc 0) n))"
reindex_bij_betw: fixes h :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "bij_betw h S T" shows "(\<Sum>x\<in>S. g (h x)) = sum g T"
intros_5: fixes \<S> :: "s_context"   and ts :: "t list"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and n :: "nat"   and \<C> :: "t_context" assumes "\<S>\<bullet>Some ts \<tturnstile>_ i vs;es : ts"   and "length ts = n" shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vs es] : [] _> ts"
simps_16: fixes M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fa Pi M slp = slp_of_fa_cnst Pi Pi M slp"
B_1: shows "(l, s) = dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (l, s')"
tauStepChainPar2: fixes Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and \<Psi>\<^sub>P :: "'b"   and P :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<Longrightarrow>\<^sub>\<tau> Q'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* Q" shows "\<Psi> \<rhd> P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P \<parallel> Q'"
map_index_len: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and ls :: "'b list"   and n :: "nat" shows "length (map_index f ls n) = length ls"
lincomb_union: fixes U :: "'a vec set"   and U' :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "U \<subseteq> carrier_vec n"   and "U' \<subseteq> carrier_vec n"   and "U \<inter> U' = {}"   and "finite U"   and "finite U'"   and "a \<in> U \<union> U' \<rightarrow> UNIV" shows "lincomb a (U \<union> U') = lincomb a U + lincomb a U'"
CompDefinedI: fixes f :: "'g"   and C :: "('h, 'g, 'i) Category_scheme"   and g :: "'g" assumes "f \<in> mor\<^bsub>C\<^esub>"   and "g \<in> mor\<^bsub>C\<^esub>"   and "cod\<^bsub>C\<^esub> f = dom\<^bsub>C\<^esub> g" shows "f \<approx>>\<^bsub>C\<^esub> g"
finsum_insert: fixes F :: "'e set"   and a :: "'e"   and f :: "'e \<Rightarrow> 'c" assumes "finite F"   and "a \<notin> F"   and "f \<in> F \<rightarrow> carrier V"   and "f a \<in> carrier V" shows "finsum V f (insert a F) = f a \<oplus>\<^bsub>V\<^esub> finsum V f F"
assms: shows "a \<in> soldAllocations'' N \<Omega>"
add_idemp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a" assumes "class.idemp_monoid_add plus zero" shows "plus x x = x"
iff_4k: fixes r :: "real"   and k :: "nat"   and m :: "nat"   and n :: "nat"   and m' :: "nat"   and n' :: "nat" assumes "r = real k"   and "odd k" shows "((4 * real m + r) / (2 * 2 ^ n) = (4 * real m' + r) / (2 * 2 ^ n')) = (m = m' \<and> n = n')"
induces_monoidal_category: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "free_monoidal_category C" shows "monoidal_category (free_monoidal_category.comp C) (elementary_monoidal_category.T (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C)) (elementary_monoidal_category.\<alpha> (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C)) (elementary_monoidal_category.\<iota> (free_monoidal_category.unity\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C))"
l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_remove_child_wf2\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs heap_is_wellformed parent_child_rel"
is_dghmD_6: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF>\<lparr>HomDom\<rparr> = \<AA>"
ccApprox_mono: fixes t :: "Vars.var ttree"   and t' :: "Vars.var ttree" assumes "paths t \<subseteq> paths t'" shows "ccApprox t \<sqsubseteq> ccApprox t'"
l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"
dis: shows "\<not> disjnt (affine hull T) (rel_interior U)"
subformulas\<^sub>\<mu>_list_length: fixes \<phi> :: "'a ltln" shows "length (subformulas\<^sub>\<mu>_list \<phi>) = card (subformulas\<^sub>\<mu> \<phi>)"
cases: fixes a1 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and a2 :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list"   and a3 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "a1 \<Rightarrow>\<^sup>*\<^bsub>a2\<^esub> a3" obtains final :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" where "a1 = final__" and "a2 = []" and "a3 = final__"   | conf' :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" and T :: "((Vars.var \<times> exp) list \<times> exp \<times> stack_elem list) list" and final :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" and conf :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" where "a1 = conf" and "a2 = conf' # T" and "a3 = final" and "conf' \<Rightarrow>\<^sup>*\<^bsub>T\<^esub> final" and "conf \<Rightarrow> conf'"
assms_2: shows "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>addr a,s\<^sub>1\<rangle>"
that: fixes L :: "V set list" assumes "\<Union> (list.set L) = D"   and "\<And>X. X \<in> list.set L \<Longrightarrow> indecomposable (tp X)"   and "\<And>M. \<lbrakk>M \<subseteq> D; \<And>X. X \<in> list.set L \<Longrightarrow> tp X \<le> tp (M \<inter> X)\<rbrakk> \<Longrightarrow> tp M = tp D" shows "thesis"
unreachable_from_init: fixes s :: "Store"   and l :: "Location"   and T :: "Javatype" shows "\<not> s\<turnstile> l reachable_from init T"
guardedFrameStatEq: fixes P :: "('a, 'b, 'c) psi" assumes "guarded P" shows "extractFrame P \<simeq>\<^sub>F \<bottom>\<^sub>F"
ide_def: fixes a :: "'a" shows "C.ide a \<equiv> a \<cdot>\<^sub>C a \<noteq> C.null \<and> (\<forall>f. (f \<cdot>\<^sub>C a \<noteq> C.null \<longrightarrow> f \<cdot>\<^sub>C a = f) \<and> (a \<cdot>\<^sub>C f \<noteq> C.null \<longrightarrow> a \<cdot>\<^sub>C f = f))"
h_def: shows "h = (\<lambda>x. if x \<in> A then f x else if x \<in> B - A then g x else x)"
simps: fixes p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list" shows "combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps = (let (c\<^sub>0, c\<^sub>1) = if dist (case p\<^sub>0\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>L of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case p\<^sub>0\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1\<^sub>R of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) then (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in Let (filter (\<lambda>p. dist (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) ps) (find_closest_pair (c\<^sub>0, c\<^sub>1)))"
outs_gpv_\<I>_full: shows "outs_gpv \<I>_full = outs'_gpv"
disc_1: fixes x1 :: "('c, 'd) hd" shows "is_Hd (Hd x1)"
negative_cycle_dest_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and n :: "nat" assumes "\<not> cycle_free M n" shows "\<exists>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<and> len M i i xs < \<one>"
less_eq_top_is_Top: fixes x :: "'b top" assumes "top.Top \<le> x" shows "x = top.Top"
foldl_error: fixes g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'a + 'b"   and n :: "'c"   and e :: "'a"   and xs :: "'d list" shows "snd (foldl (\<lambda>(i, m) x. (g x i, m \<bind> (\<lambda>_. f x i))) (n, Inl e) xs) = Inl e"
trans_D: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" assumes "P \<sqsubseteq>\<^sub>D Q"   and "Q \<sqsubseteq>\<^sub>D S" shows "P \<sqsubseteq>\<^sub>D S"
simps: fixes a1 :: "('d, 'e) ta_rule set"   and a2 :: "'e tree"   and a3 :: "'d" shows "accs_laz a1 a2 a3 = (\<exists>q f qs \<delta> ts. a1 = \<delta> \<and> a2 = NODE f ts \<and> a3 = q \<and> q \<rightarrow> f qs \<in> \<delta> \<and> list_all_zip (accs_laz \<delta>) ts qs)"
UU_def: shows "UU \<equiv> {(a, b). a < b}"
Dom_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Dom (Inv t) = Cod t"
bisim1Val2': fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and n :: "nat"   and v :: "'addr val"   and xs :: "'addr val list" shows "P,e,n,h \<turnstile>' (Val v, xs) \<leftrightarrow> ([v], xs, length (compE2 e), None)"
dvd_imp_size_le: fixes a :: "'a"   and b :: "'a" assumes "a dvd b"   and "b \<noteq> (0::'a)" shows "euclidean_size a \<le> euclidean_size b"
prems_1: shows "\<forall>j<b. (x # xs) ! j < a"
senior_states_cases_subset: fixes p :: "'b"   and n :: "nat"   and q :: "'b" obtains "senior_states p n \<subset> senior_states q n"   | "senior_states p n = senior_states q n"   | "senior_states q n \<subset> senior_states p n"
dom_def: fixes m :: "'g \<Rightarrow> 'h option" shows "dom m = {a. m a \<noteq> None}"
ih_24: shows "InvariantVarsM (getM state') F0 Vbl"
cscc_outer_invarI: fixes it :: "'v set"   and D :: "'v set"   and l :: "'v set list" assumes "outer_invar it D"   and "outer_invar it D \<Longrightarrow> cscc_invar_ext G l D" shows "cscc_outer_invar it (l, D)"
word_clz_max: fixes w :: "'b word" shows "word_clz w \<le> size w"
ray_meets_line_def: fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and betw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and ra :: "'a"   and rb :: "'a"   and la :: "'a"   and lb :: "'a" assumes "TarskiAbsolute cong betw" shows "TarskiAbsolute.ray_meets_line betw ra rb la lb = (\<exists>x. TarskiAbsolute.on_ray betw x ra rb \<and> TarskiAbsolute.on_line betw x la lb)"
steps: shows "\<Gamma>\<turnstile> (r, s) \<rightarrow>\<^sup>+ (r', s')"
subtopology_UNIV: fixes U :: "'b topology" shows "subtopology U UNIV = U"
zero_backward_terminating_path_root: fixes r :: "'a" assumes "point r" shows "backward_terminating_path_root r 0"
assms: shows "vpairs r \<subseteq>\<^sub>\<circ> x \<times>\<^sub>\<circ> y"
walk_2: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow> w" shows "walk [v, w]"
emeasure_completion: fixes S :: "'c set"   and M :: "'c measure" assumes "S \<in> sets (completion M)" shows "emeasure (completion M) S = emeasure M (main_part M S)"
sums_mono_reindex: fixes g :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "strict_mono g"   and "\<And>n. n \<notin> range g \<Longrightarrow> f n = (0::'a)" shows "(\<lambda>n. f (g n)) sums c = f sums c"
SV_inv2_invariant: shows "reach sv_TS \<subseteq> Vinv2"
key_order_of_nat_term_order_def: fixes to :: "'a nat_term_order" shows "key_order_of_nat_term_order to = Abs_key_order (nat_term_compare to)"
length_bv_not: fixes w :: "bit list" shows "length (bv_not w) = length w"
rep_eq: shows "generator list_generator = (list_has_next, list_next)"
assms_3: shows "\<not> ps (S t i) p \<noteq> None"
call_option_def: fixes T :: "nat"   and K :: "real" shows "call_option T K = (\<lambda>w. max (prices Mkt stk T w - K) 0)"
less_eq_vec_def: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "(x \<le> y) = (\<forall>i. x $ i \<le> y $ i)"
\<phi>: shows "\<guillemotleft>\<phi> : a \<rightarrow> a'\<guillemotright> \<and> local.map \<phi> = Cop_S.MkArr (\<Phi>'.Y a) (\<Phi>'.Y a') \<Phi>\<Psi>'.map"
xt1_5: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x"   and "x \<le> y" shows "x = y"
restrict_out: fixes x :: "'c"   and A :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "x \<notin> A" shows "(m |` A) x = None"
tAA: shows "trace (A * A) = trace (P * B * B * adjoint P)"
ins: shows "finprod R f (insert a A) = f a \<otimes> finprod R f A"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "I.satC \<xi> \<equiv> list_ex (I.satL \<xi>)"
chain_def: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'b" assumes "class.po below" shows "po.chain below Y = (\<forall>i. below (Y i) (Y (Suc i)))"
last_el_def: shows "last_el = val (int CARD('a) - 1)"
grid_onedim_split: fixes b :: "(nat \<times> int) list"   and ds :: "nat set"   and d :: "nat" shows "grid b (ds \<union> {d}) = grid b ds \<union> grid (child b left d) (ds \<union> {d}) \<union> grid (child b right d) (ds \<union> {d})"
X_def: shows "X = monom (monom (1::'a) 1) 0"
mu_hoare_r: fixes R :: "('a \<times> 'a) set"   and F :: "'b hrel \<Rightarrow> 'b hrel"   and p :: "'b upred"   and e :: "('a, 'b) uexpr"   and q :: "'b upred" assumes "wf R"   and "Monotonic F"   and "\<And>st P. \<lbrace>p \<and> (e, \<guillemotleft>st\<guillemotright>)\<^sub>u \<in>\<^sub>u \<guillemotleft>R\<guillemotright>\<rbrace> P \<lbrace>q\<rbrace>\<^sub>u \<Longrightarrow> \<lbrace>p \<and> e =\<^sub>u \<guillemotleft>st\<guillemotright>\<rbrace> F P \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<rbrace> \<mu> F \<lbrace>q\<rbrace>\<^sub>u"
gpath_subset: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) list"   and v :: "nat \<times> 'b"   and U :: "(nat \<times> 'b) set" assumes "gpath A w V r v"   and "set (gstates r v) \<subseteq> U" shows "gpath A w U r v"
bracket_app: fixes i :: "nat"   and s :: "dB"   and s' :: "dB" assumes "term_bracket i s = Some s'" shows "s' \<degree> Var i \<leftrightarrow> s"
rep_eq: fixes x :: "mini_alg" shows "ma_base x = snd (snd (Rep_mini_alg x))"
assms_1: shows "p \<prec>\<^sub>p monomial c v"
covector_restrict_comp_conv: fixes x :: "'a"   and y :: "'a" assumes "covector x" shows "y \<sqinter> x \<le> y * x\<^sup>T"
invariance_of_domain_subspaces: fixes U :: "'a set"   and S :: "'a set"   and V :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "openin (top_of_set U) S"   and "subspace U"   and "subspace V"   and "dim V \<le> dim U"   and "continuous_on S f"   and "f ` S \<subseteq> V"   and "inj_on f S" shows "openin (top_of_set V) (f ` S)"
in_filtermap_iff: fixes f' :: "'a \<Rightarrow> 'b"   and h' :: "'c \<Rightarrow> 'a"   and F' :: "'c filter"   and g' :: "'a \<Rightarrow> 'b" shows "(f' \<in> O[filtermap h' F'](g')) = ((\<lambda>x. f' (h' x)) \<in> O[F'](\<lambda>x. g' (h' x)))"
inp1: fixes s :: "real" assumes "0 \<le> s"   and "s \<le> t" shows "mk_v I (OVar vid1) (sol 0, bb) (sol s) \<in> Contexts I pid1 UNIV"
pure_permute_id: fixes p :: "perm" shows "permute p = (\<lambda>x. x)"
lattice_lemma: fixes T :: "'b set"   and X :: "'b set"   and C :: "'b set set"   and B :: "'b set"   and act :: "('b \<times> 'b) set"   and F :: "'b program"   and r :: "'b set" assumes "T \<inter> X \<in> C"   and "B \<subseteq> X"   and "act \<in> Acts F"   and "lattice C"   and "T \<in> C"   and "B \<in> C"   and "closed F T B C" shows "T \<inter> (wp act X \<inter> awp F (X \<union> cl C (T \<inter> r)) \<union> X) \<in> C"
completeTA_lang: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option" assumes "det_tree_automaton TA A" shows "ta_lang (det_tree_automaton.completeTA TA A) = ta_lang TA"