hcju/isabelleps huggingface dataset:
simps: fixes x :: "('a \<aproc> \<times> 'a \<aproc> set list \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a + call \<times> (nat \<Rightarrow> 'a option) \<times> ((nat \<times> char list) \<times> 'a \<Rightarrow> 'a \<aproc> set) \<times> 'a) discr" shows "abs_g\<cdot>x = (case undiscr x of Inl (PC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> {} | Inl (PP (Plus c), [], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa], bb) \<Rightarrow> \<bottom> | Inl (PP (Plus c), [x, xa, cnts], ve, b) \<Rightarrow> let b' = \<anb> b c; \<beta> = [c \<mapsto> b] in {((c, \<beta>), cont) |cont. cont \<in> cnts} | Inl (PP (Plus c), x # xa # cnts # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts], bb) \<Rightarrow> \<bottom> | Inl (PP (prim.If ct cf), [x, cntts, cntfs], ve, b) \<Rightarrow> (let b' = \<anb> b ct; \<beta> = [ct \<mapsto> b] in {((ct, \<beta>), cnt) |cnt. cnt \<in> cntts}) \<union> (let b' = \<anb> b cf; \<beta> = [cf \<mapsto> b] in {((cf, \<beta>), cnt) |cnt. cnt \<in> cntfs}) | Inl (PP (prim.If ct cf), x # cntts # cntfs # ac # lista, bb) \<Rightarrow> \<bottom> | Inl (AStop, [], bb) \<Rightarrow> \<bottom> | Inl (AStop, [x], xa, xb) \<Rightarrow> {} | Inl (AStop, x # a # lista, bb) \<Rightarrow> \<bottom> | Inr (App lab f vs, \<beta>, ve, b) \<Rightarrow> let fs = \<aA> f \<beta> ve; as = map (\<lambda>v. \<aA> v \<beta> ve) vs; b' = \<anb> b lab in {((lab, \<beta>), f') |f'. f' \<in> fs} | Inr (call.Let lab ls c', \<beta>, ve, b) \<Rightarrow> {})"
set_times_mono3: fixes a :: "'c"   and C :: "'c set"   and D :: "'c set" assumes "a \<in> C" shows "a *o D \<subseteq> C * D"
fls_base_factor_mult: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_base_factor (f * g) = fls_base_factor f * fls_base_factor g"
add_map_sq_matrix: fixes f :: "'e \<Rightarrow> 'c"   and A :: "('e, 'd) sq_matrix"   and g :: "'e \<Rightarrow> 'c" shows "map_sq_matrix f A + map_sq_matrix g A = map_sq_matrix (\<lambda>x. f x + g x) A"
assms_3: shows "Mapping.lookup (phis g) x = Some vs"
newton_interpolation_poly_int_None: fixes xs_ys :: "(int \<times> int) list"   and q :: "int poly" assumes "distinct (map fst xs_ys)"   and "newton_interpolation_poly_int xs_ys = None"   and "\<And>x y. (x, y) \<in> set xs_ys \<Longrightarrow> poly q x = y"   and "degree q < length xs_ys" shows "False"
concat_append: fixes xs :: "'b list list"   and ys :: "'b list list" shows "concat (xs @ ys) = concat xs @ concat ys"
Example2_lemma2_aux2: fixes j :: "nat"   and s :: "nat"   and b :: "nat \<Rightarrow> 'c"   and t :: "'c" assumes "j \<le> s" shows "sum (b(s := t)) {0..<j} = sum b {0..<j}"
order_trans: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
trms\<^sub>l\<^sub>s\<^sub>t_append: fixes A :: "('c \<times> ('a, 'b) strand_step) list"   and B :: "('c \<times> ('a, 'b) strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>t (A @ B) = trms\<^sub>l\<^sub>s\<^sub>t A \<union> trms\<^sub>l\<^sub>s\<^sub>t B"
assms_1: shows "u \<turnstile>\<^bsub>v,n\<^esub> M"
adj_dual: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "f \<stileturn> g = \<partial>\<^sub>F g \<stileturn> \<partial>\<^sub>F f"
rtrancl_while_Some: fixes p :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b list"   and x :: "'b"   and ws :: "'b list"   and Z :: "'b set" assumes "rtrancl_while p f x = Some (ws, Z)" shows "if ws = [] then Z = {(x, y). y \<in> set (f x)}\<^sup>* `` {x} \<and> (\<forall>z\<in>Z. p z) else \<not> p (hd ws) \<and> hd ws \<in> {(x, y). y \<in> set (f x)}\<^sup>* `` {x}"
dtor_corec_o_map: fixes g :: "'g \<Rightarrow> 'f"   and h :: "'e \<Rightarrow> ('g, 'g coF + 'e) coF0" shows "map_coF g \<circ> dtor_corec_coF h = dtor_corec_coF (map_pre_coF g (map_sum (map_coF g) id) \<circ> h)"
assms: shows "b \<notin>\<^sub>\<circ> A"
intro: fixes G_invar :: "'a \<Rightarrow> bool"   and g :: "'a" assumes "G_invar g" shows "Dijkstra_Impl_axioms G_invar g"
keys_simps_2: fixes c :: "color"   and l :: "('b, 'c) rbt"   and k :: "'b"   and v :: "'c"   and r :: "('b, 'c) rbt" shows "RBT_Impl.keys (Branch c l k v r) = RBT_Impl.keys l @ k # RBT_Impl.keys r"
multiplicity_prime_power: fixes p :: "'a"   and n :: "nat" assumes "prime_elem p" shows "multiplicity p (p ^ n) = n"
other: fixes l' :: "'b" assumes "l \<noteq> l'"   and "\<not> rstep l s i"   and "rstep l' s i"   and "dist_step l (Suc i) < dist_step l i" shows "thesis"
fmlookup_dom'_iff: fixes x :: "'c"   and m :: "('c, 'd) fmap" shows "(x \<in> fmdom' m) = (\<exists>a. m $$ x = Some a)"
sdrop_while_SCons: fixes P :: "'d \<Rightarrow> bool"   and a :: "'d"   and s :: "'d stream" shows "sdrop_while P (a ## s) = (if P a then sdrop_while P s else a ## s)"
v_le_p: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "v cfg \<le> p (state cfg)"
elims: fixes x :: "bit list"   and xa :: "nat"   and xb :: "nat"   and y :: "bit list" assumes "select x xa xb = y" obtains i :: "nat" and n :: "nat" where "x = []" and "xa = i__" and "xb = n__" and "y = (if i__ \<le> 0 then selecthelp [] n__ else select [] (i__ - 1) (n__ - 1))"   | xc :: "bit" and l :: "bit list" and i :: "nat" and n :: "nat" where "x = xc # l" and "xa = i" and "xb = n" and "y = (if i \<le> 0 then selecthelp (xc # l) n else select l (i - 1) (n - 1))"
ContractsWithBossyConstants_axioms: shows "ContractsWithBossyConstants Xd Xh Pd Ch d1h1 d1h2 d1h3 d2h1 d2h2 d2h3 d3h1 d3h2 d3h3 ds"
Expr2ZFType0: fixes e :: "(ZF, ZF) Expression" assumes "ZFType (Expr2ZF e) = 0" shows "e = Vx"
01: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "0 \<le> g x \<and> g x \<le> 1"
domA_fv_pure: fixes \<Gamma> :: "('a \<times> 'b) list" shows "fv \<Gamma> = domA \<Gamma>"
use_implies_allDef: fixes g :: "'g"   and m :: "'node"   and v :: "'var \<times> 'node \<times> Def"   and m' :: "'node"   and ms :: "'node list" assumes "lookupDef g m (var g v) = v"   and "m \<in> set (\<alpha>n g)"   and "var g v \<in> uses g m'"   and "g \<turnstile> m-ms\<rightarrow>m'"   and "\<forall>x\<in>set (tl ms). var g v \<notin> defs g x" shows "v \<in> braun_ssa.allDefs g (defNode v)"
prj_simps_3: shows "C.cod trg_f.prj\<^sub>1 = trg.apex"
M_e: shows "(\<lambda>\<sigma>. expr_sem \<sigma> e) \<in> M \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
collapse_def: fixes v :: "'a"   and PDPE :: "'a set list \<times> 'a set \<times> ('a \<times> 'a) set" shows "collapse v PDPE \<equiv> let (p, D, pE) = PDPE; i = idx_of p v; p = collapse_aux p i in (p, D, pE)"
nprv_impI: fixes \<phi> :: "'fmla"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "nprv (insert \<phi> F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F (imp \<phi> \<psi>)"
dropWhile_eq_Nil_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(dropWhile P xs = \<epsilon>) = (\<forall>x\<in>set xs. P x)"
convex_supp_sum: fixes S :: "'c set"   and u :: "'d \<Rightarrow> real"   and I :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "convex S"   and "supp_sum u I = 1"   and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)" shows "supp_sum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
sorted_toList: fixes xs :: "'c odlist" shows "sorted (toList xs)"
length_fusion_def: fixes g :: "('a, 'b) generator"   and s :: "'b" shows "length_fusion g s = length (list.unfoldr g s)"
poly_differentiable: fixes p :: "real poly"   and x :: "real" shows "poly p differentiable at x"
append_assoc: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "append (append xs ys) zs = append xs (append ys zs)"
primes_psi_def: shows "primes_psi = sum_upto mangoldt"
finite_has_minimal2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and a :: "'a" assumes "class.order less_eq less"   and "finite A"   and "a \<in> A" shows "\<exists>m\<in>A. less_eq m a \<and> (\<forall>b\<in>A. less_eq b m \<longrightarrow> m = b)"
P_mod_mult_const: fixes I :: "'a set"   and J :: "'a set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "ideal S I"   and "ideal S J"   and "pol_coeff S (n, f)"   and "P_mod R S X I (polyn_expr R X n (n, f))"   and "pol_coeff S (0, g)"   and "P_mod R S X J (polyn_expr R X 0 (0, g))" shows "P_mod R S X (I \<diamondsuit>\<^sub>r\<^bsub>S\<^esub> J) (polyn_expr R X n (n, f) \<cdot>\<^sub>r polyn_expr R X 0 (0, g))"
img_insert: shows "range_vars (\<sigma>1(v := t)) = range_vars \<sigma>1 \<union> fv t"
prime_power_exp_nat: fixes p :: "nat"   and n :: "nat"   and x :: "nat"   and k :: "nat" assumes "normalization_semidom_class.prime p"   and "n \<noteq> 0"   and "x ^ n = p ^ k" shows "\<exists>i. x = p ^ i"
allConsec_set: fixes \<Delta>1 :: "'a"   and \<Delta>2 :: "'a"   and \<Delta>s :: "'a list" assumes "(\<Delta>1, \<Delta>2) \<in> allConsec \<Delta>s" shows "\<Delta>1 \<in> set \<Delta>s \<and> \<Delta>2 \<in> set \<Delta>s"
elims: fixes x :: "'a"   and xa :: "('a \<times> 'b) list"   and y :: "('a \<times> 'b) list" assumes "AList.delete_aux x xa = y" obtains k :: "'a" where "x = k__" and "xa = []" and "y = []"   | k :: "'a" and k' :: "'a" and v :: "'b" and xs :: "('a \<times> 'b) list" where "x = k" and "xa = (k', v) # xs" and "y = (if k = k' then xs else (k', v) # AList.delete_aux k xs)"
Rep_nonzero_inject: fixes x :: "'b nonzero"   and y :: "'b nonzero" shows "(Rep_nonzero x = Rep_nonzero y) = (x = y)"
eq4: fixes s :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> keys p" shows "lookup (monomial c t + p) s = lookup p s"
LT29: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" assumes "|~ P \<longrightarrow> \<circle>P \<or> \<circle>Q"   and "|~ P \<and> Unchanged v \<longrightarrow> \<circle>P" shows "\<turnstile> P \<longrightarrow> \<box>P \<or> \<diamond>Q"
block_Suc_2: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_1 b k" shows "block i (Suc t) = e_snoc (e_update b k 1) 0"
F_lemma: fixes m :: "nat"   and r :: "nat"   and t :: "nat" assumes "rec_exec rec_halt [m, r] = t" shows "rec_exec rec_F [m, r] = valu (rght (conf m r t))"
zerofun_imp_empty_supp: shows "supp 0 = {}"
independent_split_3: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "R.independent K (Us @ Vs)" shows "R.Span K Us \<inter> R.Span K Vs = {\<zero>}"
obj_eval_Obj: fixes t :: "'a E.term" assumes "E.Obj t" shows "B.obj (EVAL t)"
image_minus_const_atLeastAtMost': fixes d :: "'b"   and a :: "'b"   and b :: "'b" shows "(\<lambda>t. t - d) ` {a..b} = {a - d..b - d}"
cases: fixes a1 :: "('c \<times> ('d, 'e) cconstraint \<times> 'f \<times> 'd list \<times> 'c) set \<times> ('c \<Rightarrow> ('d, 'e) cconstraint)"   and a2 :: "'d set"   and a3 :: "'d \<Rightarrow> 'e" assumes "valid_abstraction a1 a2 a3" obtains A :: "('c \<times> ('d, 'e) cconstraint \<times> 'f \<times> 'd list \<times> 'c) set \<times> ('c \<Rightarrow> ('d, 'e) cconstraint)" and k :: "'d \<Rightarrow> 'e" and X :: "'d set" where "a1 = A" and "a2 = X" and "a3 = k" and "\<forall>(x, m)\<in>clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>" and "collect_clkvt (trans_of A) \<subseteq> X" and "finite X"
fps_of_ratfps_divide: fixes f :: "'b ratfps"   and g :: "'b ratfps" shows "fps_of_ratfps (f / g) = fps_of_ratfps f / fps_of_ratfps g"
abs_fun_eq': fixes a :: "'b"   and x :: "'a"   and b :: "'b"   and y :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
Induction_axP_sf: fixes p :: "tm" shows "Sigma_fm (Induction_axP p)"
bdd_above_Icc: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq (ord.atLeastAtMost less_eq a b)"
SReal_divide_numeral: fixes r :: "real star"   and w :: "num" assumes "r \<in> \<real>" shows "r / numeral w \<in> \<real>"
inducts: fixes P :: "'d frame \<Rightarrow> bool"   and frame :: "'d frame" assumes "\<And>assertion. P (\<langle>\<epsilon>, assertion\<rangle>)"   and "\<And>name frame. P frame \<Longrightarrow> P (\<lparr>\<nu>name\<rparr>frame)" shows "P frame"
distinct_21: fixes x3 :: "('b, 'c) fm"   and x5 :: "('b, 'c) fm" shows "(\<^bold>\<not> x3) \<noteq> (\<^bold>\<diamond> x5)"
k'_1: shows "i' icd\<^bsup>\<pi>\<^esup>\<rightarrow> k'"
eq_iff_3: fixes tm1 :: "tm"   and tm2 :: "tm"   and tm1a :: "tm"   and tm2a :: "tm" shows "(Eats tm1 tm2 = Eats tm1a tm2a) = (tm1 = tm1a \<and> tm2 = tm2a)"
lex_def: fixes p :: "real \<times> real"   and q :: "real \<times> real" shows "Counterclockwise_2D_Arbitrary.lex p q = (fst p < fst q \<or> fst p = fst q \<and> snd p < snd q \<or> p = q)"
ldec_iter: fixes z :: "nat"   and n :: "nat" shows "ldec z n = pdec1 (Iter pdec2 n z) - 1"
float_eqI: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "sign x = sign y"   and "fraction x = fraction y"   and "IEEE.exponent x = IEEE.exponent y" shows "x = y"
simps: fixes u :: "'a Vector"   and v :: "'a Vector" shows "u \<ominus> v = \<lparr>tdir = tdir u - tdir v, xdir = xdir u - xdir v, ydir = ydir u - ydir v, zdir = zdir u - zdir v\<rparr>"
h3: shows "\<forall>i F. freeIn var F \<longrightarrow> freeIn var (g F i)"
assms_2: shows "has_least_fixpoint (f \<circ> g)"
naturality: fixes f :: "'a" assumes "\<sigma>.A.arr f" shows "B (\<sigma> (\<sigma>.A.cod f)) (F f) = B (G f) (\<sigma> (\<sigma>.A.dom f))"
SC_Resolution: fixes \<Gamma> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> {#}" shows "\<Union> (cnf ` nnf ` set_mset \<Gamma>) \<turnstile> \<box>"
converseD: fixes a :: "'c"   and b :: "'d"   and r :: "('d \<times> 'c) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
l14: fixes x :: "'a"   and y :: "'a" shows "- (x \<sqinter> - - y) \<le> - (x \<sqinter> y)"
offs_pred_zero_cons: fixes ns :: "nat list"   and ub :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" assumes "offs_pred ns ub (x # xs) index key mi ma"   and "index key x (length ns) mi ma = i"   and "i < length ns"   and "offs_num (length ns) (x # xs) index key mi ma 0 = 0" shows "offs_next ns ub (x # xs) index key mi ma 0 \<le> offs_next (ns[i := Suc (ns ! i)]) ub xs index key mi ma 0"
invP: shows "A \<Turnstile> (S, U \<rightarrow>) P"
antisym: fixes b :: "'c multiset"   and a :: "'c multiset" assumes "b \<subseteq># a"   and "a \<subseteq># b" shows "a = b"
finite_Inf_Sup: fixes A :: "'a set set" shows "\<Sqinter> (\<Squnion> ` A) \<le> \<Squnion> (\<Sqinter> ` {f ` A |f. \<forall>Y\<in>A. f Y \<in> Y})"
Confidentiality_lemma: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "nat"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Number Ts, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace>\<rbrace>) \<in> set evs"   and "Key authK \<notin> analz (knows Spy evs)"   and "servK \<in> symKeys"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> kerbIV_gets"   and "Key servK \<in> analz (knows Spy evs)" shows "expiredSK Ts evs"
foldli_transform: fixes iti :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S0 :: "'d set"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "set_iterator_genord iti S0 R" shows "\<exists>l0. distinct l0 \<and> S0 = set l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0"
phase_rel_invD: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state" assumes "phase_rel_inv s" shows "(\<forall>m. sys_ghost_hs_in_sync m s, sys_ghost_hs_phase s, gc_phase s, sys_phase s, filter is_mw_Phase (sys_mem_store_buffers gc s)) \<in> phase_rel"
simp_el_hull': fixes b :: "'b list"   and G :: "'b list set" assumes " b \<in>B \<langle>G\<rangle> " shows " b \<in>B G "
mult_imp_div_pos_less: 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 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 y :: "'b"   and x :: "'b"   and z :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero y"   and "less x (times z y)" shows "less (divide x y) z"
hom_F_C2_Powerset: fixes X :: "'c set" shows "\<exists>f. bij_betw f (Pow X) (homr \<F>\<^bsub>X\<^esub> C2)"
TESL_interp_commute: fixes \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi>\<^sub>1 @ \<Phi>\<^sub>2 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk>\<lbrakk> \<Phi>\<^sub>2 @ \<Phi>\<^sub>1 \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
u: shows "a * p - e * N * b * q = (p\<^sup>2 + N * q\<^sup>2) * u"
n0: fixes m :: "nat" assumes "n0 \<le> m" shows "(f m)\<^sub>F n \<in> (\<lambda>k. (k)\<^sub>F n) ` K n0"
implem_LtK_aux: fixes x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and K :: "ltkey" shows "implem (Chan x A B M) \<noteq> LtK K"
A: shows "A \<in> sets (COCOUNT \<Otimes>\<^sub>M M)"
simps_2: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview (Suc n) = {t \<leadsto> envTrans eact aact (tLast t) |t eact aact. t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n \<and> eact \<in> set (envAction (tLast t)) \<and> (\<forall>a. aact a \<in> set (JKBP.jAction jkbp (PreEnvironmentJView.mkM envVal jview (PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n)) t a))}"
Ana_var: fixes x :: "'var" shows "Ana (Var x) = ([], [])"
simps_1: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list" shows "subtract_list_sorted (x # xs) (y # ys) = (if x = y then subtract_list_sorted xs (y # ys) else if x < y then x # subtract_list_sorted xs (y # ys) else subtract_list_sorted (x # xs) ys)"
mset_lran: fixes a :: "int \<Rightarrow> 'a"   and l :: "int"   and h :: "int" shows "mset (lran a l h) = arr_mset a l h"
token_run_intial_state: fixes x :: "nat" shows "\<UU>.token_run x x = Abs (Unf\<^sub>G \<phi>)"
terminal_path_from_def: fixes c :: "'form"   and "is" :: "nat list" shows "terminal_path_from c is = map (edge_at c) (rev (prefixes is))"
vec_eq_norm_smult_normalized: fixes v :: "complex vec" shows "v = vec_norm v \<cdot>\<^sub>v vec_normalize v"
widen_Class: fixes P :: "'b prog"   and T :: "ty"   and C :: "String.literal" shows "P \<turnstile> T \<le> Class C = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C) \<or> C = Object \<and> (\<exists>A. T = A\<lfloor>\<rceil>))"
indep_in_subI_subset: fixes \<E>' :: "'a set"   and \<E> :: "'a set"   and X :: "'a set" assumes "\<E>' \<subseteq> \<E>"   and "X_insert.indep_in \<E>' X" shows "X_insert.indep_in \<E> X"
\<tau>\<mu>\<nu>: shows "arr \<tau> \<and> arr \<mu> \<and> arr \<nu>"
idx_intro: fixes i :: "nat" assumes "i < length s" shows "idx s \<lbrakk>I (s ! i)\<rbrakk> = Inr (Suc i)"
product_run_None: fixes \<iota>\<^sub>m :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and \<delta>\<^sub>m :: "'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'a"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "\<iota>\<^sub>m k = None" shows "run (\<Delta>\<^sub>\<times> \<delta>\<^sub>m) \<iota>\<^sub>m w i k = None"
simps_4: fixes bs :: "int list"   and a :: "Cooper.num" shows "Inum bs (Neg a) = - Inum bs a"
pigeon_cols_1: shows "r1 \<in> {..<k + 1}"
mat_of_cblinfun_compose: fixes F :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c"   and G :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "mat_of_cblinfun (F o\<^sub>C\<^sub>L G) = mat_of_cblinfun F * mat_of_cblinfun G"
cert_okD3: fixes c :: "'c list"   and n :: "nat"   and T :: "'c"   and B :: "'c"   and A :: "'c set"   and pc :: "nat" assumes "cert_ok c n T B A"   and "B \<in> A"   and "pc < n" shows "c ! Suc pc \<in> A"
order_iff: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b \<le>s a) = (a = signed.max a b)"
WhileRedsThrow: fixes b :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 c :: "char list exp" assumes "P \<turnstile> \<langle>b,s\<rangle> \<rightarrow>* \<langle>throw e,s'\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<rangle> \<rightarrow>* \<langle>throw e,s'\<rangle>"
pow_mono: fixes b :: "'a"   and a :: "'a"   and n :: "nat" assumes "b \<le> a"   and "(0::'a) \<le> b" shows "b ^ n \<le> a ^ n \<and> (0::'a) \<le> b ^ n"
hom_mset_factors: fixes F :: "'b poly multiset"   and p :: "'b poly" assumes "mset_factors F p" shows "mset_factors (image_mset poly_lift F) (poly_lift p)"
solves_ode_subset: fixes x :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set"   and S :: "real set" assumes "(x solves_ode f) T X"   and "S \<subseteq> T" shows "(x solves_ode f) S X"
assms_3: fixes K :: "'a set" assumes "subgroup K G"   and "H \<subseteq> K" shows "E \<subseteq> K"
assms_5: shows "mix_pmf \<alpha> p r \<succeq>[\<R>] mix_pmf \<alpha> q r"
le_less_trans: fixes x :: "'d word"   and y :: "'d word"   and z :: "'d word" assumes "x \<le>s y"   and "y <s z" shows "x <s z"
t_twice_correct: fixes ires :: "cell list"   and rs :: "nat"   and n :: "nat" shows "\<exists>stp ln rn. steps0 (Suc 0, Bk # Bk # ires, Oc \<up> Suc rs @ Bk \<up> n) (tm_of abc_twice @ shift (mopup (Suc 0)) (length (tm_of abc_twice) div 2)) stp = (0, Bk \<up> ln @ Bk # Bk # ires, Oc \<up> Suc (2 * rs) @ Bk \<up> rn)"
locks_a_conv: fixes ls :: "'g lock_actions"   and ntsjswss :: "('h, 'i, 'j) new_thread_action list \<times> 'h conditional_action list \<times> ('h, 'k) wait_set_action list \<times> 'h interrupt_action list \<times> 'l list" shows "\<lbrace>(ls, ntsjswss)\<rbrace>\<^bsub>l\<^esub> = ls"
Suc_le_eq: fixes m :: "nat"   and n :: "nat" shows "(Suc m \<le> n) = (m < n)"
takeWhile_eq_take_P_nth: fixes n :: "nat"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<And>i. \<lbrakk>i < n; i < length xs\<rbrakk> \<Longrightarrow> P (xs ! i)"   and "n < length xs \<Longrightarrow> \<not> P (xs ! n)" shows "takeWhile P xs = take n xs"
is_real_rot_to_x_axis: fixes z :: "complex" assumes "z \<noteq> 0" shows "is_real (cis (- arg z) * z)"
inverts_components: fixes a :: "'a" assumes "ide a" shows "inverse_arrows (\<ll> a) (\<ll>' a)"