hcju/isabelleps huggingface dataset:
elims_2: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'a list"   and resOf :: "'b \<Rightarrow> 'a"   and parOf :: "'c \<Rightarrow> 'a list"   and x :: "('b, 'c) lit" assumes "Signature TYPE('a) wtFsym wtPsym"   and "Signature.wtL wtFsym wtPsym arOf resOf parOf x" obtains a :: "('b, 'c) atm" where "x = Pos a__" and "Signature.wtA wtFsym wtPsym arOf resOf parOf a__"   | a :: "('b, 'c) atm" where "x = Neg a" and "Signature.wtA wtFsym wtPsym arOf resOf parOf a"
store_mem_exists: fixes i :: "nat"   and \<S> :: "s_context"   and s :: "s" assumes "i < length (s_inst \<S>)"   and "store_typing s \<S>" shows "Option.is_none (memory (s_inst \<S> ! i)) = Option.is_none (inst.mem (inst s ! i))"
Suc_fMax_pred_fimage: fixes p :: "nat"   and P :: "nat fset" assumes "Suc p |\<in>| P"   and "0 |\<notin>| P" shows "Suc (fMax ((\<lambda>x. x - Suc 0) |`| P)) = fMax P"
hd_conv_nth: fixes xs :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "hd xs = xs ! 0"
generated_ringE: fixes a :: "'a set" assumes "a \<in> M.generated_ring" obtains C :: "'a set set" where "finite C" and "disjoint C" and "C \<subseteq> sigma_sets \<Omega> M" and "a = \<Union> C"
MPair_imp_eq_right: fixes X :: "msg"   and Y :: "msg"   and X' :: "msg"   and Y' :: "msg" assumes "MPair X Y = MPair X' Y'" shows "Y = Y'"
norm_le: fixes x :: "'c"   and y :: "'d" shows "(norm x \<le> norm y) = (inner x x \<le> inner y y)"
two_64: shows "2 ^ LENGTH(64) = 18446744073709551616"
defined_by_str: fixes s :: "'b list"   and X :: "'b list set"   and A :: "'b list set" assumes "s \<in> X"   and "X \<in> UNIV // \<approx>A" shows "X = \<approx>A `` {s}"
zero_le_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero one"
tmmul: fixes vs :: "'b list"   and bs :: "'b list"   and t :: "tm"   and i :: "poly" shows "Itm vs bs (tmmul t i) = Itm vs bs (tm.Mul i t)"
proots_rectangle_code1: fixes p :: "complex poly"   and lb :: "complex"   and ub :: "complex" shows "proots_rectangle p lb ub = (if Re lb < Re ub \<and> Im lb < Im ub then if p \<noteq> 0 then if no_proots_line p lb (Complex (Re ub) (Im lb)) \<and> no_proots_line p (Complex (Re ub) (Im lb)) ub \<and> no_proots_line p ub (Complex (Re lb) (Im ub)) \<and> no_proots_line p (Complex (Re lb) (Im ub)) lb then let p1 = p \<circ>\<^sub>p [:lb, Complex (Re ub - Re lb) 0:]; pR1 = map_poly Re p1; pI1 = map_poly Im p1; gc1 = gcd pR1 pI1; p2 = p \<circ>\<^sub>p [:Complex (Re ub) (Im lb), Complex 0 (Im ub - Im lb):]; pR2 = map_poly Re p2; pI2 = map_poly Im p2; gc2 = gcd pR2 pI2; p3 = p \<circ>\<^sub>p [:ub, Complex (Re lb - Re ub) 0:]; pR3 = map_poly Re p3; pI3 = map_poly Im p3; gc3 = gcd pR3 pI3; p4 = p \<circ>\<^sub>p [:Complex (Re lb) (Im ub), Complex 0 (Im lb - Im ub):]; pR4 = map_poly Re p4; pI4 = map_poly Im p4; gc4 = gcd pR4 pI4 in nat (- (changes_alt_itv_smods 0 1 (pR1 div gc1) (pI1 div gc1) + changes_alt_itv_smods 0 1 (pR2 div gc2) (pI2 div gc2) + changes_alt_itv_smods 0 1 (pR3 div gc3) (pI3 div gc3) + changes_alt_itv_smods 0 1 (pR4 div gc4) (pI4 div gc4)) div 4) else Code.abort STR ''proots_rectangle fails when there is a root on the border.'' (\<lambda>_. proots_rectangle p lb ub) else Code.abort STR ''proots_rectangle fails when p=0.'' (\<lambda>_. proots_rectangle p lb ub) else 0)"
MI_pred_MI: fixes intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "MModel intT intF intP" shows "MI_pred MI"
cexp_cong1: fixes p1 :: "('d \<times> 'd) set"   and r1 :: "('e \<times> 'e) set"   and q :: "('f \<times> 'f) set" assumes "p1 =o r1"   and "Card_order q" shows "p1 ^c q =o r1 ^c q"
mop_closed: shows "mop A \<in> carrier A \<rightarrow> carrier A"
all_not_in_conv: fixes A :: "'d set" shows "(\<forall>x. x \<notin> A) = (A = {})"
c1_1: shows "conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1''')) \<in> conv ars"
prems_2: shows "bdt (Node lt a rt) var = Some bdt1"
element_type_det: fixes \<tau> :: "'b OCL_Types.type"   and \<sigma>\<^sub>1 :: "'b OCL_Types.type"   and \<sigma>\<^sub>2 :: "'b OCL_Types.type" assumes "element_type \<tau> \<sigma>\<^sub>1"   and "element_type \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
qr: fixes n :: "nat" assumes "0 < n" shows "MPoly_Type.monom (monomial (Suc 0) x') 1 ^ n = MPoly_Type.monom (monomial (Suc 0) x') 1 + qr n * ((MPoly_Type.monom (monomial (Suc 0) x') 1)\<^sup>2 - MPoly_Type.monom (monomial (Suc 0) x') 1)"
finite_Collect_conjI: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
limit_ordinal_not_0: shows "\<not> limit_ordinal 0"
ex_inj: assumes "class.countable TYPE('b)" shows "\<exists>to_nat. inj to_nat"
isRefV_simps_5: fixes C :: "CTypeId"   and a :: "nat" shows "isRefV (objV C a) = True"
subset_subst_pairs_diff_exists': fixes D :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set"   and \<I> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and d :: "('a, 'b) Term.term \<times> ('a, 'b) Term.term" assumes "finite D" shows "\<exists>Di\<subseteq>D. Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I> \<subseteq> {d \<cdot>\<^sub>p \<I>} \<and> d \<cdot>\<^sub>p \<I> \<notin> D - Di \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<I>"
assms_1: fixes x :: "real"   and y :: "real" assumes "b \<le> x"   and "x \<le> y" shows "f y \<le> f x"
internalize_ordLeq: fixes r' :: "('a \<times> 'a) set"   and r :: "('b \<times> 'b) set" shows "(r' \<le>o r) = (\<exists>p. Field p \<subseteq> Field r \<and> r' =o p \<and> p \<le>o r)"
neutral: fixes A :: "'k set"   and g :: "'k \<Rightarrow> 'l" assumes "\<forall>x\<in>A. g x = (0::'l)" shows "sum g A = (0::'l)"
arr_dom_iff_arr: fixes f :: "'a2" shows "A2.arr (A2.dom f) = A2.arr f"
rt_fresher_mapupd: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "dip \<in> kD rt"   and "the (rt dip) \<sqsubseteq> r" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> rt(dip \<mapsto> r)"
erH_rHom_cf: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and s :: "'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "s \<in> carrier S" shows "erH R S X A B Y h s = h s"
rSym: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> X" shows "(Q, P) \<in> bisim O X O bisim"
comp: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and g :: "'d \<Rightarrow> 'b" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_cbilinear (\<lambda>x y. cblinfun_apply (f x) (g y))"
quantale_homset_iff: fixes f :: "'b \<Rightarrow> 'c" shows "(f \<in> quantale_homset) = (comp_pres f \<and> Sup_pres f)"
prems_1: shows "n -a' # as'\<rightarrow>* n'"
intros_3: fixes b :: "'a"   and \<alpha> :: "'b"   and a :: "'a"   and ars :: "('a \<times> 'b \<times> 'a) set"   and ss :: "(bool \<times> 'b \<times> 'a) list" assumes "(b, \<alpha>, a) \<in> ars"   and "(b, ss) \<in> conv ars" shows "(a, (False, \<alpha>, b) # ss) \<in> conv ars"
sigma_sets_induct_disjoint: fixes G :: "'b set set"   and \<Omega> :: "'b set"   and A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "Int_stable G"   and "G \<subseteq> Pow \<Omega>"   and "A \<in> sigma_sets \<Omega> G"   and "\<And>A. A \<in> G \<Longrightarrow> P A"   and "P {}"   and "\<And>A. \<lbrakk>A \<in> sigma_sets \<Omega> G; P A\<rbrakk> \<Longrightarrow> P (\<Omega> - A)"   and "\<And>A. \<lbrakk>disjoint_family A; range A \<subseteq> sigma_sets \<Omega> G; \<And>i. P (A i)\<rbrakk> \<Longrightarrow> P (\<Union> (range A))" shows "P A"
in_affected: fixes a :: "'a"   and f :: "'a perm" shows "(a \<in> affected f) = (f \<langle>$\<rangle> a \<noteq> a)"
rel_envTI: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and M :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" assumes "(R ===> M) f g" shows "Monomorphic_Monad.rel_envT R M (EnvT f) (EnvT g)"
ln_less_zero: fixes x :: "real" assumes "0 < x"   and "x < 1" shows "real_ln x < 0"
inj_term_list_def: fixes es :: "'a list" shows "\<langle>es\<rangle> = In3 (map expr_of es)"
finite_induct: fixes F :: "'k set"   and P :: "'k set \<Rightarrow> bool" assumes "finite F"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P F"
filter_le_avg_def: fixes s :: "(nat, unit) HashMap.hashmap" shows "filter_le_avg s \<equiv> let a = avg s in hs.iterate s (\<lambda>x s. if x \<le> a then my_ins x s else s) (my_empty ())"
span_closed: fixes U :: "'a vec set"   and v :: "'a vec" assumes "U \<subseteq> carrier_vec n"   and "v \<in> local.span U" shows "v \<in> carrier_vec n"
continuous_on_closest_point: fixes S :: "'a set"   and t :: "'a set" assumes "convex S"   and "closed S"   and "S \<noteq> {}" shows "continuous_on t (closest_point S)"
"1_1": fixes x :: "'a poly" assumes "k \<noteq> 0"   and "x = root_unity k"   and "x dvd p"   and "p \<noteq> 0" shows "decompose_prod_root_unity_main (map_poly hom (p div x)) k = map_prod id (map_poly hom) (decompose_prod_root_unity_main (p div x) k)"
jordan_matrix_poly_bound: fixes n_as :: "(nat \<times> 'a) list"   and N :: "nat" assumes "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; 0 < n\<rbrakk> \<Longrightarrow> norm a \<le> 1"   and "\<And>n a. \<lbrakk>(n, a) \<in> set n_as; norm a = 1\<rbrakk> \<Longrightarrow> n \<le> N" shows "\<exists>c1. \<forall>k. local.norm_bound (jordan_matrix n_as ^\<^sub>m k) (c1 + real k ^ (N - 1))"
Entry_dominates: fixes g :: "'g"   and n :: "'node" assumes "invar g"   and "n \<in> set (\<alpha>n g)" shows "old.dominates g (Entry g) n"
X: shows "X \<in> sets (count_space A \<Otimes>\<^sub>M count_space B)"
mirror_half_segments_of_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "e \<in> UNIV \<rightarrow> {- 1<..<1}"   and "length (half_segments_of_aform X) \<noteq> 1" shows "list_all (\<lambda>seg. ccw' (fst seg) (snd seg) (aform_val e X)) (map (pairself (mirror_point (fst X))) (half_segments_of_aform X))"
IH: fixes H :: "real fps" shows "(fps_XD' G ^^ n) H = (\<Sum>m\<le>n. S n m * G ^ m * (fps_deriv ^^ m) H)"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'b" assumes "A.ide a"   and "A.ide a'"   and "\<guillemotleft>g : F a' \<rightarrow>\<^sub>B F a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<rightarrow>\<^sub>A a\<guillemotright> \<and> F f = g"
insert_eq_iff: fixes a :: "'b"   and A :: "'b set"   and b :: "'b"   and B :: "'b set" assumes "a \<notin> A"   and "b \<notin> B" shows "(Set.insert a A = Set.insert b B) = (if a = b then A = B else \<exists>C. A = Set.insert b C \<and> b \<notin> C \<and> B = Set.insert a C \<and> a \<notin> C)"
lunit_commutes_with_L: fixes f :: "'a" assumes "ide f" shows "\<l>[L f] = L \<l>[f]"
prems_2: shows "\<forall>x y. x \<le> g x y \<and> y \<le> g x y"
Un_Int_eq_4: fixes T :: "'a set"   and S :: "'a set" shows "T \<inter> (S \<union> T) = T"
comp_eq_elim: fixes a :: "'d \<Rightarrow> 'c"   and b :: "'b \<Rightarrow> 'd"   and c :: "'e \<Rightarrow> 'c"   and d :: "'b \<Rightarrow> 'e" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
antisym: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm b a"   and "lex_pm a b" shows "a = b"
discr_Atm: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and atm :: "'c" assumes "PL_Indis indis" shows "PL_Indis.discr tval aval indis (Atm atm) = PL_Indis.presAtm aval indis atm"
add: shows "\<exists>i j. N_Chop(lan v,i,j) \<and> |i| = x \<and> |j| = y"
natural_transformation_axioms: shows "natural_transformation (\<cdot>) (\<cdot>) (\<lambda>f. f \<otimes> \<I>) local.map \<rho>"
degree_eq_length_coeffs: fixes p :: "'a poly" shows "degree p = length (coeffs p) - 1"
value_distinct_1: fixes cs :: "(Term_Class.term \<times> sterm) list"   and name :: "name"   and ts :: "sterm list" shows "Sabs cs \<noteq> name $$ ts"
constrains_empty: fixes F :: "'b program"   and B :: "'b set" shows "F \<in> {} co B"
disjunctive_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask"   and "\<And>n. \<not> bit a n \<or> \<not> bit b n" shows "plus a b = or a b"
cmp_simps_5: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi>\<^sub>F (f, g)) = F (f \<star>\<^sub>C g)"
nZ': shows "pderiv (charpoly A) \<noteq> 0"
complex_of_real_cmod: fixes x :: "complex" assumes "0 \<le> x" shows "complex_of_real (cmod x) = x"
SPEC_cons_rule: fixes m :: "'c nres"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'c \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>"   and "\<And>x. \<Phi> x \<Longrightarrow> \<Psi> x" shows "m \<le> SPEC \<Psi>"
assms_1: shows "\<not> fic (M[a\<turnstile>c>b]) c"
euclid_ext_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus" shows "euclidean_ring_gcd.euclid_ext_aux divide minus zero times one modulo unit_factor normalize one zero zero one a b = (euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b, gcd a b)"
qPsubst_preserves_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and zs :: "'varSort"   and z :: "'var" assumes "qGood X"   and "qGoodEnv rho"   and "qFresh zs z X"   and "qFreshEnv zs z rho" shows "qFresh zs z X #[[rho]]"
closure_act_1a: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<A>\<^bold>\<A>(\<^bold>\<A>\<phi> \<^bold>\<equiv> \<phi>) in v]"
tauChainInduct: fixes x1 :: "('a, 'b, 'c) psi"   and x2 :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> x1 \<Longrightarrow>\<^sup>^\<^sub>\<tau> x2"   and "\<And>a. P a a"   and "\<And>a b c. \<lbrakk>\<Psi> \<rhd> a \<Longrightarrow>\<^sup>^\<^sub>\<tau> b; P a b; \<Psi> \<rhd> b \<longmapsto> \<tau> \<prec> c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
ta_rstates_def: fixes TA :: "('a, 'b, 'c) tree_automaton_rec_scheme" shows "ta_rstates TA = ta_initial TA \<union> \<delta>_states (ta_rules TA)"
offset_suc: fixes k :: "nat"   and x :: "'a"   and xs :: "'a list"   and n :: "nat" assumes "offset k x xs \<noteq> None" shows "offset (Suc n) x xs = Some (Suc (the (offset n x xs)))"
R_def: shows "R \<equiv> \<lambda>n. EM_remainder (2 * N + 1) (\<lambda>x. - fact (2 * N + 2) / x ^ (2 * N + 3)) (int n)"
distinct_1: fixes a :: "'e"   and b' :: "'f" shows "Inl a \<noteq> Inr b'"
1: shows "\<Theta>,{} \<turnstile> subst_typ' [((Var (String.Literal True True True False False True False STR ''a'', 0), full_sort), \<tau>), ((Var (String.Literal True True True False False True False STR ''b'', 0), full_sort), \<tau>')] eq_combination_ax"
ordertype_singleton: fixes r :: "('a \<times> 'a) set"   and x :: "'a" assumes "wf r" shows "ordertype {x} r = 1"
simps_1: fixes n :: "nat" shows "zhmset_of_enat (enat n) = of_nat n"
matches_ex_entry_def: fixes P :: "'a prog"   and C :: "char list"   and pc :: "nat"   and xcp :: "ex_entry" shows "matches_ex_entry P C pc xcp \<equiv> let (s, e, C', h, d) = xcp in s \<le> pc \<and> pc < e \<and> P \<turnstile> C \<preceq>\<^sup>* C'"
simps_16: fixes x2 :: "'b" shows "set_literal (x2\<inverse>) = {x2}"
leB: fixes n :: "nat" shows "Sigma_Algebra.measure lebesgue (S n) \<le> B"
properPath_strong_coind: fixes phi :: "'a stream \<Rightarrow> 'a treeFI \<Rightarrow> bool"   and as :: "'a stream"   and tr :: "'a treeFI" assumes "phi as tr"   and "\<And>as tr. phi as tr \<Longrightarrow> shd as = lab tr"   and "\<And>as tr. phi as tr \<Longrightarrow> \<exists>tr'\<in>set (sub tr). phi (stl as) tr' \<or> properPath (stl as) tr'" shows "properPath as tr"
arrow: shows "D.ide (Go x) \<and> \<guillemotleft>\<epsilon>o x : F (Go x) \<rightarrow>\<^sub>C x\<guillemotright>"
to_partial_def: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and f :: "'d \<Rightarrow> 'e" assumes "sifum_security_init dma \<C>_vars eval" shows "to_partial f = (\<lambda>x. Some (f x))"
octo_inverse_cnj: fixes x :: "octo" shows "inverse x = Octonions.cnj x /\<^sub>R (norm x)\<^sup>2"
idom: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<cdot>\<^sub>r b = \<zero>" shows "a = \<zero> \<or> b = \<zero>"
selection_closed_id: fixes uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "x = inf (uminus (uminus x)) x"
fmeasurable_cbox: fixes a :: "'b"   and b :: "'b" shows "cbox a b \<in> fmeasurable lborel"
game_0_def: fixes \<A> :: "(bool, '\<beta>, '\<gamma>) gpv" shows "prf'.game_0 \<A> = key_seed_gen \<bind> (\<lambda>key. exec_gpv (prf'.prf_oracle key) \<A> () \<bind> (\<lambda>(b, uu_). return_spmf b))"
strongly_strategyproof_profile_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" shows "RD.strongly_strategyproof_profile R i Ri' = SD (R i) (RD (R(i := Ri'))) (RD R)"
bs_env_closed: fixes \<rho> :: "(nat \<times> bval) list"   and \<rho>' :: "(nat \<times> exp) list" assumes "bs_env \<rho> \<rho>'" shows "closed_env \<rho>'"
preserves_cod: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "cod (L f) = L (cod f)"
simps_7: fixes \<E> :: "('a, 'b, 'c) cntxt"   and e1 :: "('a, 'b, 'c) expr"   and e :: "('a, 'b, 'c) expr" shows "(AssignL\<^sub>\<E> \<E> e1) [e] = Assign (\<E> [e]) e1"
corec2_def: fixes s1 :: "'f \<Rightarrow> ('e, 'e JF1 + 'f, 'e JF2 + 'd) F1"   and s2 :: "'d \<Rightarrow> ('e, 'e JF1 + 'f, 'e JF2 + 'd) F2" shows "corec2 s1 s2 = unfold2 (case_sum (F1map id Inl Inl \<circ> dtor1) s1) (case_sum (F2map id Inl Inl \<circ> dtor2) s2) \<circ> Inr"
leg1_simps_5: shows "local.dom f\<^sub>0gh\<^sub>1.p\<^sub>1 = f\<^sub>0gh\<^sub>1.p\<^sub>1"
simps_1: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_part_1 var a b d (Eq p) = (let deg = MPoly_Type.degree p var in \<Sum>i = 0..<deg + 1. isolate_variable_sparse p var i * (a + b * Var var) ^ i * d ^ (deg - i))"
assms_1: shows "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"
partition_filter2: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "snd (partition P xs) = filter (Not \<circ> P) xs"
fsubset_antisym: fixes A :: "'c fset"   and B :: "'c fset" assumes "A |\<subseteq>| B"   and "B |\<subseteq>| A" shows "A = B"
is_SDG_path_slice1: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node"   and n'' :: "'node SDG_node" assumes "n is-ns\<rightarrow>\<^sub>d* n'"   and "n' \<in> sum_SDG_slice1 n''" shows "n \<in> sum_SDG_slice1 n''"
pset_pnorm_alt_id: fixes s :: "'a rexp"   and r :: "'a rexp" assumes "s \<in> pset r" shows "pnorm_alt s = s"
inj_std_dual: assumes "subspace S"   and "finite B" shows "inj_on local.std_dual S"
finite_Label: fixes S :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" assumes "finite S" shows "finite (Label S)"
simps_2: fixes x :: "'g"   and xs :: "'g list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
Spy_see_shrK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> yahalom" shows "(Key (shrK A) \<in> parts (knows Spy evs)) = (A \<in> bad)"
infinite_remove: fixes S :: "'g set"   and a :: "'g" assumes "infinite S" shows "infinite (S - {a})"
pmf_of_list_remove_zeros_1: fixes xs :: "('b \<times> real) list" assumes "pmf_of_list_wf xs" shows "pmf_of_list_wf (filter (\<lambda>z. snd z \<noteq> 0) xs)"
a_monoid: fixes G :: "('c, 'd) ring_scheme" assumes "abelian_monoid G" shows "Group.monoid (add_monoid G)"
trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "(b::'b) \<le> (a::'b)"   and "(c::'b) \<le> (b::'b)" shows "(c::'b) \<le> (a::'b)"
clock_commonKnowledge_relation_image_aux: fixes Y :: "'s odlist"   and as :: "'a list" shows "(\<Union>x\<in>set as. \<Union>a\<in>toSet Y. \<Union>aa\<in>toSet Y \<inter> rel_ext (\<lambda>s''. envObs x a = envObs x s''). {(a, aa)}) = Restr (\<Union>a\<in>set as. rel_ext (\<lambda>(s, s'). envObs a s = envObs a s')) (toSet Y)"
blank_def: fixes P :: "'b prog"   and C :: "char list" shows "blank P C \<equiv> (C, init_fields (ifields P C))"
intro: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and I :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>" assumes "\<And>s x y s'. \<lbrakk>(y, s') \<in> set_spmf (callee s x); I s; x \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> I s'"   and "\<And>s. I s \<Longrightarrow> \<I> \<turnstile>c callee s \<surd>" shows "callee_invariant_on callee I \<I>"
relhoare_conseq_right: fixes pre :: "('e \<times> 'f) set"   and Ra :: "('e \<times> 'e) set"   and Rc :: "('f \<times> 'f) set"   and post' :: "('e \<times> 'f) set"   and post :: "('e \<times> 'f) set" assumes "{pre} Ra, Rc {> post'}"   and "post' \<subseteq> post" shows "{pre} Ra, Rc {> post}"
sqrt_add_le_add_sqrt: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "sqrt (x + y) \<le> sqrt x + sqrt y"
nth_append_length: fixes xs :: "'e list"   and x :: "'e"   and ys :: "'e list" shows "(xs @ x # ys) ! length xs = x"
initial_arrows_exist: fixes x :: "'d" assumes "D.ide x" shows "\<exists>y e. initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G x y e"
rep_list_plus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and s :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (r + s) = rep_list r + rep_list s"