hcju/isabelleps huggingface dataset:
linorder_lattice_inf_le1: fixes x :: "'a"   and y :: "'a" shows "inf x y \<le> x"
pw_conc_inres: fixes R :: "('a \<times> 'b) set"   and S' :: "'b nres" shows "inres (\<Down> R S') = (\<lambda>s. nofail S' \<longrightarrow> (\<exists>s'. (s, s') \<in> R \<and> inres S' s'))"
nat_of_ipv6addr_ipv6addr_of_nat_mod: fixes n :: "nat" shows "nat_of_ipv6addr (ipv6addr_of_nat n) = n mod 2 ^ 128"
Par_transT_simp: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b" shows "\<not> PL.transT tval aval (c1 | c2, s) s'"
DOWN_mapsto: shows "SC.DOWN \<in> SC.Univ \<rightarrow> UNIV"
card_distinct: fixes xs :: "'a list" assumes "card (set xs) = length xs" shows "distinct xs"
n_mult_def: fixes R :: "('c, 'd) ring_scheme"   and f :: "'e \<Rightarrow> 'c" assumes "UP_cring R" shows "UP_cring.n_mult R f = (\<lambda>n. [n] \<cdot>\<^bsub>R\<^esub> f n)"
filter_is_Write\<^sub>s\<^sub>b_empty: fixes xs :: "'b memref list" assumes "filter is_Write\<^sub>s\<^sub>b xs = []" shows "\<forall>r\<in>set xs. is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r"
sums_add: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and g :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "f sums a"   and "g sums b" shows "(\<lambda>n. f n + g n) sums (a + b)"
IH_1: assumes "isin x1_ x" shows "size x1_ = Suc (size (delete x x1_))"
extractable_partition: fixes K :: "msg set"   and Bad :: "agent set"   and NI :: "msg set"   and I :: "msg set" assumes "Keys_bad K Bad"   and "NI \<subseteq> payload"   and "I \<subseteq> valid" shows "extractable K I \<subseteq> {implInsec A B M |A B M. M \<in> extr Bad NI (local.abs I)} \<union> {implConfid A B M |A B M. (A, B) \<in> UNIV \<and> M \<in> extr Bad NI (local.abs I)} \<union> {implAuth A B M |A B M. M \<in> extr Bad NI (local.abs I)} \<union> {implSecure A B M |A B M. (A, B) \<in> UNIV \<and> M \<in> extr Bad NI (local.abs I)}"
ASSp: fixes x :: "char list"   and e :: "aexp"   and s :: "char list \<Rightarrow> int" shows "\<down>\<^sub>t (x ::= e, s) = Suc 0"
cv_hom_meet: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> 1\<^sub>\<pi> \<sqinter> y \<cdot> 1\<^sub>\<pi>) \<parallel> U = x \<cdot> 1\<^sub>\<pi> \<parallel> U \<sqinter> y \<cdot> 1\<^sub>\<pi> \<parallel> U"
is_image_is_image_spec: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and P' :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B" shows "Tarski_neutral_dimensionless.Reflect Bet Cong P' P A B = Tarski_neutral_dimensionless.ReflectL Bet Cong P' P A B"
point_measurable_def: fixes M :: "'c measure"   and S :: "'d set"   and f :: "'c \<Rightarrow> 'd" shows "point_measurable M S f \<equiv> f ` space M \<subseteq> S \<and> (\<forall>r\<in>range f \<inter> S. f -` {r} \<inter> space M \<in> sets M)"
map_id_on_def: fixes xs :: "'c set" shows "map_id_on xs \<equiv> \<lambda>x. if x \<in> xs then Some x else None"
comp_def: fixes g :: "'d"   and f :: "'d" shows "Dop.comp g f \<equiv> f \<cdot>\<^sub>D g"
simps_1: fixes uu :: "'c monom \<times> 'd" shows "monom_mult_poly uu [] = []"
binomial_r_part_sum: fixes m :: "nat" shows "sum ((choose) (2 * m + 1)) {..m} = 2 ^ (2 * m)"
fs_carrier: fixes n :: "nat"   and fs :: "int vec list" assumes "fs_int_indpt n fs" shows "set fs \<subseteq> carrier_vec n"
drop_Suc: fixes n :: "nat"   and xs :: "'g list" shows "drop (Suc n) xs = drop n (tl xs)"
le_minus_iff: 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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq a (uminus b) = less_eq b (uminus a)"
sub_composition_closed: fixes a :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set" assumes "a \<in> \<ww>"   and "b \<in> \<ww>" shows "add_local_ring_at a b \<in> \<ww>"
swapcols_carrier: fixes l :: "nat"   and k :: "nat"   and A :: "'b mat"   and n :: "nat"   and m :: "nat" shows "(swapcols l k A \<in> carrier_mat n m) = (A \<in> carrier_mat n m)"
total_preorder_on_axioms_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "total_preorder_on_axioms carrier le \<equiv> \<forall>x y. x \<in> carrier \<longrightarrow> y \<in> carrier \<longrightarrow> le x y \<or> le y x"
op_type_alt_simps_6: fixes op :: "string_ternop + collection_ternop"   and k :: "call_kind"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho> :: "'a OCL_Types.type"   and \<upsilon> :: "'a OCL_Types.type" shows "ternop_type op k \<tau> \<sigma> \<rho> \<upsilon> = ((\<exists>opa. op = Inl opa \<and> k = DotCall \<and> string_ternop_type opa \<tau> \<sigma> \<rho> \<upsilon>) \<or> (\<exists>opa. op = Inr opa \<and> k = ArrowCall \<and> collection_ternop_type opa \<tau> \<sigma> \<rho> \<upsilon>))"
hyps_2: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate" assumes "(a2 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile>a2 \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
is_subset_MST_def: fixes w :: "'b set \<Rightarrow> nat"   and g :: "'b ugraph"   and A :: "('b \<times> 'b) set" shows "is_subset_MST w g A \<equiv> \<exists>t. is_MST w g t \<and> A \<subseteq> edges t"
diff_power_eq_sum: fixes x :: "'a"   and n :: "nat"   and y :: "'a" shows "x ^ Suc n - y ^ Suc n = (x - y) * (\<Sum>p<Suc n. x ^ p * y ^ (n - p))"
assms: shows "ipv6_unparsed_compressed_to_preferred (ipv6_preferred_to_compressed ip) = Some ip'"
set_vwalk_arcs_rev: fixes p :: "'a list" shows "set (vwalk_arcs (rev p)) = {(v, u). (u, v) \<in> set (vwalk_arcs p)}"
assms_1: shows "\<forall>nid. honest nid \<longrightarrow> prefix sbc (bc (\<sigma>\<^bsub>nid\<^esub>t \<langle>nid \<rightarrow> t\<rangle>\<^bsub>n\<^sub>s\<^esub>))"
f: fixes b :: "'b" assumes "b \<in> B" shows "f (the_inv_into A f b) = b"
set_prefixes_eq: fixes xs :: "'a list" shows "set (prefixes xs) = {ys. prefix ys xs}"
cross7_mult_left: fixes c :: "real"   and x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(c *\<^sub>R x) \<times>\<^sub>7 y = c *\<^sub>R x \<times>\<^sub>7 y"
set_iterator_emp_def: fixes c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and \<sigma>0 :: "'a" shows "set_iterator_emp c f \<sigma>0 = \<sigma>0"
open_halfspace_gt: fixes b :: "real"   and a :: "'b" shows "open {x. b < a \<bullet> x}"
graph_plan_lemma_4: fixes as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list"   and P :: "('c, 'd) fmap \<times> ('c, 'd) fmap \<Rightarrow> bool"   and vs :: "'c set"   and s :: "('c, 'd) fmap"   and s' :: "('c, 'd) fmap" assumes "\<forall>a. ListMem a as \<and> P a \<longrightarrow> fmdom' (snd a) \<inter> vs = {}"   and "sat_precond_as s as"   and "sat_precond_as s' (filter (\<lambda>a. \<not> P a) as)"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "fmrestrict_set vs (exec_plan s as) = fmrestrict_set vs (exec_plan s' (filter (\<lambda>a. \<not> P a) as))"
upper_circle_map: shows "{x. 1 / x \<in> (\<lambda>x. x + 1) ` {x. Im x < sqrt 3 * Re x}} = upper_circle_01"
conf_Addr: fixes G :: "prog"   and s :: "st"   and a :: "loc"   and T :: "ty" shows "G,s\<turnstile>Addr a\<Colon>\<preceq>T = (\<exists>obj. heap s a = Some obj \<and> G\<turnstile>obj_ty obj\<preceq>T)"
n_eq_Suc_sub1_conv: fixes n :: "nat" shows "(n = Suc (n - Suc 0)) = (n \<noteq> 0)"
inter_scheme_strict_sorted: fixes l :: "nat"   and U :: "nat list set" assumes "Form l U"   and "0 < l" shows "strict_sorted (inter_scheme l U)"
leg0_commutes: shows "trg.cod.leg0 \<cdot> trg.chine = trg.leg0"
set_of_ivl_def: fixes x :: "'b \<times> 'b" shows "set_of_ivl x = {fst x..snd x}"
runiq_def: fixes R :: "('a \<times> 'b) set" shows "runiq R = (\<forall>X. trivial X \<longrightarrow> trivial (R `` X))"
simps_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and p :: "'b point" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.point_mult divide times one plus minus uminus a 0 p = Infinity"
cases: fixes r :: "l2_state" obtains ik :: "msg set" and secret :: "msg set" and progress :: "fid_t \<Rightarrow> Runs.var set option" and signalsInit :: "signal \<Rightarrow> nat" and signalsResp :: "signal \<Rightarrow> nat" and signalsInit2 :: "signal \<Rightarrow> nat" and signalsResp2 :: "signal \<Rightarrow> nat" and chan :: "chan set" and bad :: "agent set" where "r = \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, chan = chan, bad = bad\<rparr>"
Semi: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and d :: "com"   and R :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<turnstile> {P} c {Q}"   and "C \<turnstile> {Q} d {R}" shows "C \<turnstile> {P} c; d {R}"
Loop_def: shows "Loop \<equiv> {(X, mrk). finite (- mrk) \<and> finite X \<and> X \<subseteq> mrk \<and> mrk \<subseteq> reach root \<and> reach root \<inter> - mrk \<subseteq> path X mrk}"
rep_eq: fixes x :: "float"   and xa :: "float" shows "dist x xa = dist (real_of_float x) (real_of_float xa)"
sorted_less_rev_set_unique: fixes xs :: "'b list"   and ys :: "'b list" assumes "Sorted_Less.sorted (rev xs)"   and "Sorted_Less.sorted (rev ys)"   and "set xs = set ys" shows "xs = ys"
mono: fixes callee :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf"   and x :: "('a, 'b, 'c) gpv \<times> 'd" shows "spmf.mono_body (\<lambda>exec_gpv. case x of (c, s) \<Rightarrow> the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y)))"
span_same: fixes S :: "'a vec set" assumes "S \<subseteq> mat_kernel A" shows "KA.span S = MN.span S"
inj_onD: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e"   and y :: "'e" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
invariant3p_correct_jvm_state_mexecT: fixes \<Phi> :: "String.literal \<Rightarrow> String.literal \<Rightarrow> (ty list \<times> ty err list) option list" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P" shows "invariant3p (mexecT P) (correct_jvm_state \<Phi>)"
sourcesD_3: fixes a :: "'a"   and \<mu> :: "'a" assumes "a \<in> sources \<mu>" shows "\<mu> \<star> a \<noteq> null"
map_concat: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list list" shows "map f (concat xs) = concat (map (map f) xs)"
prems_3: shows "IF x t1 t2 \<noteq> Falseif"
simps_15: fixes x11 :: "'c option"   and x12 :: "'d" shows "set1_list_op (Insert x11 x12) = set_option x11"
subgame_VV: fixes p :: "Player"   and V' :: "'a set" shows "(if p = Even then V0\<^bsub>G'.subgame V'\<^esub> else V\<^bsub>G'.subgame V'\<^esub> - V0\<^bsub>G'.subgame V'\<^esub>) = V' \<inter> G'.VV p"
heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_is_l_heap_is_wellformed: shows "l_heap_is_wellformed ShadowRootClass.type_wf ShadowRootClass.known_ptr heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes get_disconnected_nodes"
is_closed_seq_conv_subseq: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "convergent_subseq s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"
pdevs_val_of_list_of_pdevs: fixes e :: "nat \<Rightarrow> real"   and I :: "real set"   and X :: "'a pdevs" assumes "e \<in> UNIV \<rightarrow> I"   and "0 \<in> I" obtains e' :: "nat \<Rightarrow> real" where "pdevs_val e (pdevs_of_list (map snd (list_of_pdevs X))) = pdevs_val e' X" and "e' \<in> UNIV \<rightarrow> I"
enat_sub_add_same: fixes n :: "enat"   and m :: "enat" assumes "n \<le> m" shows "m = m - n + n"
cases: fixes x :: "nat list" obtains "x = []"   | a :: "nat" and as :: "nat list" where "x = a # as"
HPow_mono: fixes x :: "hf"   and y :: "hf" assumes "x \<le> y" shows "HPow x \<le> HPow y"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'g list" assumes "i < n" shows "take n xs ! i = xs ! i"
f_in_S: fixes x :: "'b" assumes "x \<in> S" shows "f x \<in> S"
mod_relH: fixes as :: "nat set"   and h :: "heap"   and h' :: "heap"   and P :: "assn" assumes "relH as h h'" shows "((h, as) \<Turnstile> P) = ((h', as) \<Turnstile> P)"
stutter_equiv_element_right: fixes \<sigma> :: "nat \<Rightarrow> 'f"   and \<tau> :: "nat \<Rightarrow> 'f"   and m :: "nat" assumes "\<sigma> \<approx> \<tau>" obtains n :: "nat" where "\<sigma> m = \<tau> n" and "\<forall>j<n. \<exists>k<m. \<sigma> k = \<tau> j"
inf_llist_lnth: fixes xs :: "'a llist" assumes "\<not> lfinite xs" shows "inf_llist (($) xs) = xs"
assms_2: shows "lock_thread_ok (locks s) (thr s)"
fup1: fixes f :: "'b \<rightarrow> 'a" shows "fup\<cdot>f\<cdot>\<bottom> = \<bottom>"
countable_infiniteE': fixes A :: "'b set" assumes "countable A"   and "infinite A" obtains g :: "nat \<Rightarrow> 'b" where "bij_betw g UNIV A"
subst_noop_1: fixes e :: "exp"   and y :: "Vars.var" shows "e[y::=y] = e"
pochhammer_eq_0_iff: fixes a :: "'b"   and n :: "nat" shows "(pochhammer a n = (0::'b)) = (\<exists>k<n. a = - of_nat k)"
intros_5: fixes \<psi> :: "'a ltln"   and \<chi> :: "'a ltln"   and \<phi> :: "'a ltln" assumes "\<psi> \<turnstile>\<^sub>s \<chi>" shows "\<phi> and\<^sub>n \<psi> \<turnstile>\<^sub>s \<chi>"
scalar_product: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v \<bullet> Mod_Type_Connect.from_hma\<^sub>v w = scalar_product v w"
fps_power_nth: fixes a :: "'a fps"   and m :: "nat"   and n :: "nat" shows "fps_nth (a ^ m) n = (if m = 0 then fps_nth 1 n else \<Sum>v\<in>natpermute n m. \<Prod>j = 0..m - 1. fps_nth a (v ! j))"
Dynkin_subset: fixes N :: "'a set set" assumes "N \<subseteq> {D \<in> events. prob (X \<inter> D) = prob X * prob D}" shows "Dynkin (space M) N \<subseteq> {D \<in> events. prob (X \<inter> D) = prob X * prob D}"
dom_def: fixes f :: "'b" shows "C'.dom f = (if C'.domains f \<noteq> {} then SOME a. a \<in> C'.domains f else C'.null)"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_after_move (as, lm) (s, l, r) ires = (\<exists>rn m lm1 lm2. lm = lm1 @ m # lm2 \<and> (if lm1 = [] then l = Oc \<up> Suc m @ Bk # Bk # ires else l = Oc \<up> Suc m @ Bk # <rev lm1> @ Bk # Bk # ires) \<and> r = <lm2> @ Bk \<up> rn)"
Fs_dim_3: shows "mat_of_rows n (map gso [0..<length fs]) \<in> carrier_mat (length fs) n"
y: shows "y \<in> prefixes X \<and> is_sub_seq a a' y"
rcs_subset: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and a :: "'c" assumes "Group G"   and "G \<guillemotright> H "   and "a \<in> carrier G" shows "H \<bullet>\<^bsub>G\<^esub> a \<subseteq> carrier G"
induct: fixes x :: "'a"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> Wellfounded.acc r"   and "\<And>x. \<lbrakk>\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r; \<And>y. (y, x) \<in> r \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P x"
list_tail_coinc: fixes n1 :: "'a"   and r1 :: "'a list"   and n2 :: "'a"   and r2 :: "'a list" assumes "n1 # r1 = n2 # r2" shows "n1 = n2 \<and> r1 = r2"
adv: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> (\<I>_common_core \<oplus>\<^sub>\<I> \<I>_common_rest)) \<turnstile>g \<A> \<surd>" shows "advantage \<A> (obsf_resource (sim |\<^sub>= 1\<^sub>C |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
elims: fixes x :: "'a extN"   and xa :: "'a extN"   and y :: "'a extN" assumes "x + xa = y" obtains "xa__ = extN.Bot" and "xa = extN.Bot" and "ya__ = extN.N 0"   | uu_ :: "'a" where "xa__ = extN.Bot" and "xa = extN.Val uu_" and "ya__ = extN.N 1"   | y :: "nat" where "xa__ = extN.Bot" and "xa = extN.N yb__" and "ya__ = extN.N yb__"   | "xa__ = extN.Bot" and "xa = extN.Top" and "ya__ = extN.N 1"   | uv_ :: "'a" where "xa__ = extN.Val uv_" and "xa = extN.Bot" and "ya__ = extN.N 1"   | uw_ :: "'a" and ux_ :: "'a" where "xa__ = extN.Val uw_" and "xa = extN.Val ux_" and "ya__ = extN.N 2"   | uy_ :: "'a" and y :: "nat" where "xa__ = extN.Val uy_" and "xa = extN.N yc__" and "ya__ = extN.N (yc__ + 1)"   | uz_ :: "'a" where "xa__ = extN.Val uz_" and "xa = extN.Top" and "ya__ = extN.N 2"   | x :: "nat" where "xa__ = extN.N xb__" and "xa = extN.Bot" and "ya__ = extN.N xb__"   | x :: "nat" and va_ :: "'a" where "xa__ = extN.N xc__" and "xa = extN.Val va_" and "ya__ = extN.N (xc__ + 1)"   | x :: "nat" and y :: "nat" where "xa__ = extN.N xd__" and "xa = extN.N yd__" and "ya__ = extN.N (xd__ + yd__)"   | x :: "nat" where "xa__ = extN.N x" and "xa = extN.Top" and "ya__ = extN.N (x + 1)"   | "xa__ = extN.Top" and "xa = extN.Bot" and "ya__ = extN.N 1"   | vb_ :: "'a" where "xa__ = extN.Top" and "xa = extN.Val vb_" and "ya__ = extN.N 2"   | y :: "nat" where "xa__ = extN.Top" and "xa = extN.N y" and "ya__ = extN.N (y + 1)"   | "xa__ = extN.Top" and "xa = extN.Top" and "ya__ = extN.N 2"
\<nu>_stable_def: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<nu>_stable \<phi> w = (\<F>\<G> \<phi> w = \<G> \<phi> w)"
ffold_set_union: fixes x :: "'m"   and A :: "'m fset"   and B :: "'m fset" shows "ffold funionM x (A |\<union>| B) = ffold funionM (ffold funionM x A) B"
suc: shows "\<pi> (Suc k) = \<pi>' (Suc k')"
eq_fold: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and top :: "'a"   and bot :: "'a"   and normalize :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "bounded_quasi_semilattice_set f top bot normalize" shows "bounded_quasi_semilattice_set.F f top bot A = (if finite A then Finite_Set.fold f top A else bot)"
hide_set_empty: fixes P :: "'a process" shows "P \ {} = P"
hyps: fixes b :: "'a node ref option" shows "<os_list xs b> os_rem x b <os_list (removeAll x xs)>\<^sub>t"
not_call_simps_8: fixes l :: "'d"   and fdata :: "'a \<Rightarrow> nat"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_call ({l}deliver(fdata) . p)"
x_in: shows "(x1, x2, x3) \<in> edges (rem_unPath xs G)"
nat_pow_one: fixes n :: "nat" shows "(1::'a) ^ n = (1::'a)"
de': shows "[B,d',C]\<^bsub>T\<^esub> = [B,e',C]\<^bsub>T\<^esub>"
degen_acc_run_complete_aux1: fixes m :: "('Q, 'more) igb_graph_rec_scheme \<Rightarrow> 'b"   and r :: "nat \<Rightarrow> 'Q \<times> nat"   and i :: "nat"   and n :: "nat"   and q :: "'Q" assumes "num_acc \<noteq> 0"   and "graph_defs.is_run (degeneralize_ext m) r"   and "\<exists>j\<ge>i. n \<in> acc (fst (r j))"   and "r i = (q, n)" shows "\<exists>j\<ge>i. \<exists>q'. r j = (q', n) \<and> n \<in> acc q'"
tangent_field_def: fixes c :: "real \<Rightarrow> 'a"   and x :: "real" shows "tangent_field c x = restrict0 diff_fun_space (\<lambda>f. frechet_derivative (f \<circ> c) (at x) 1)"
convert_Some: fixes t1 :: "t"   and v :: "v"   and t2 :: "t"   and sx :: "sx option"   and v' :: "v" assumes "types_agree t1 v"   and "cvt t2 sx v = Some v'" shows "\<lparr>[$C v, $Cvtop t2 Convert t1 sx]\<rparr> \<leadsto> \<lparr>[$C v']\<rparr>"
subspace_is_vs: fixes W :: "rat vec set" assumes "VectorSpace.subspace class_ring W gs.V" shows "vectorspace class_ring (gs.vs W)"
inline_lift_spmf1: fixes p :: "'e \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'e) spmf"   and gpv :: "('d, 'h, 'i) gpv"   and s :: "'e" shows "inline (\<lambda>s c. lift_spmf (p s c)) gpv s = lift_spmf (map_spmf projl (inline1 (\<lambda>s c. lift_spmf (p s c)) gpv s))"
partCompat_partGen: fixes theta :: "('a \<times> 'a) set"   and P :: "'b set set"   and f :: "'b \<Rightarrow> 'a" assumes "sym theta"   and "trans theta"   and "partCompat P theta f" shows "partCompat (partGen P) theta f"
cmp_simps'_5: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "D.cod (\<Phi> (\<mu>, \<nu>)) = F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)"
comp_cod_arr: fixes f :: "'a"   and b :: "'a" assumes "S.arr f"   and "S.cod f = b" shows "S.comp b f = f"
Abs_SkewBinomialHeap_inject: fixes x :: "unit + ('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem"   and y :: "unit + ('a, 'b) BsSkewBinomialHeapStruc.BsSkewElem" assumes "x \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}"   and "y \<in> {q. BsSkewBinomialHeapStruc.bs_invar q}" shows "(Abs_SkewBinomialHeap x = Abs_SkewBinomialHeap y) = (x = y)"
Var_diff_Op: fixes xs :: "'varSort"   and x :: "'var"   and delta :: "'f"   and inp :: "'d \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qAbs set option" shows "Var xs x \<noteq> Op delta inp binp"
WTrtCond: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and e\<^sub>1 :: "char list exp"   and T\<^sub>1 :: "ty"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty"   and T :: "ty" assumes "P,E,h \<turnstile> e : Boolean"   and "P,E,h \<turnstile> e\<^sub>1 : T\<^sub>1"   and "P,E,h \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<or> P \<turnstile> T\<^sub>2 \<le> T\<^sub>1"   and "P \<turnstile> T\<^sub>1 \<le> T\<^sub>2 \<longrightarrow> T = T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T\<^sub>1 \<longrightarrow> T = T\<^sub>1" shows "P,E,h \<turnstile> if (e) e\<^sub>1 else e\<^sub>2 : T"
simps_1: fixes s :: "'a" shows "lst_conv (s, []) = s"
assms: shows "check \<C> (es @ [e]) ts \<noteq> Bot"
divideR_right: fixes r :: "real"   and y :: "'b"   and x :: "'b" assumes "r \<noteq> 0" shows "(y = x /\<^sub>R r) = (r *\<^sub>R y = x)"
Memop: fixes i :: "nat"   and ts :: "('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'sb \<times> 'dirty \<times> 'owns \<times> 'rels) list"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'sb"   and \<D> :: "'dirty"   and \<O> :: "'owns"   and \<R> :: "'rels"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'shared"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'sb"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "'dirty"   and \<O>' :: "'owns"   and \<R>' :: "'rels"   and \<S>' :: "'shared" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>m (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "(ts, m, \<S>) \<Rightarrow> (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')], m', \<S>')"
purge_Cons_notIn: fixes u :: "'a"   and G :: "'a set"   and c :: "'b"   and ucl :: "('a \<times> 'b) list" assumes "u \<in> G" shows "GM_sec_model.purge G ((u, c) # ucl) = GM_sec_model.purge G ucl"
h3help: shows "list_constr (a # I) (length qs) \<longrightarrow> a < length qs"
plus_perm_def: fixes p :: "'b perm"   and q :: "'b perm" shows "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"