hcju/isabelleps huggingface dataset:
valid_subdiv_imp_valid_one_chain: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "chain_subdiv_chain one_chain1 subdiv"   and "boundary_chain one_chain1"   and "boundary_chain subdiv"   and "\<forall>(k, \<gamma>)\<in>subdiv. valid_path \<gamma>" shows "\<forall>(k, \<gamma>)\<in>one_chain1. valid_path \<gamma>"
assms_1: shows "select_return_top cts c1 c2 = TopType ctm"
chine_simps_1: shows "C.arr H\<mu>H\<nu>\<pi>.chine"
an1: shows "[a ^ (n - 1) = 1] (mod n)"
isin_set_inorder: fixes t :: "'a btree"   and x :: "'a" assumes "sorted_less (inorder t)" shows "isin t x = (x \<in> set (inorder t))"
Rats_cases: fixes q :: "'a" assumes "q \<in> \<rat>" obtains r :: "rat" where "q = of_rat r"
set_elem_nth: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "\<exists>m<length xs. xs ! m = x"
assms: shows "\<forall>v. freedb e v \<longrightarrow> closed (\<Gamma> v)"
assms_1: shows "t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
inf_def: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y = min x y"
sup_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
drop_map: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "drop n (map f xs) = map f (drop n xs)"
disjnt_Union1: fixes \<A> :: "'b set set"   and B :: "'b set" shows "disjnt (\<Union> \<A>) B = (\<forall>A\<in>\<A>. disjnt A B)"
elims: fixes x :: "('b \<times> nat) option"   and y :: "('b \<times> nat) option" assumes "unlock_lock x = y" obtains "x = None" and "y = None"   | t :: "'b" and n :: "nat" where "x = \<lfloor>(t, n)\<rfloor>" and "y = (case n of 0 \<Rightarrow> None | Suc n' \<Rightarrow> \<lfloor>(t, n')\<rfloor>)"
subgroup_adjoin_induct: fixes H0 :: "'a set"   and P :: "('a, 'b) monoid_scheme \<Rightarrow> bool" assumes "subgroup H0 G"   and "P (G\<lparr>carrier := H0\<rparr>)"   and "\<And>H a. \<lbrakk>subgroup H G; H0 \<subseteq> H; a \<in> carrier G - H; P (G\<lparr>carrier := H\<rparr>)\<rbrakk> \<Longrightarrow> P (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "P G"
gvcon': shows "continuous_on (cbox (0, 0) (1, 1)) (\<lambda>z. vector_derivative g (at (fst z)))"
is_interval_connected: fixes S :: "'b set" assumes "is_interval S" shows "connected S"
design_iso_block_intersect_num_eq: fixes b1 :: "'a set"   and b2 :: "'a set" assumes "b1 \<in># \<B>"   and "b2 \<in># \<B>" shows "b1 |\<inter>| b2 = \<pi> ` b1 |\<inter>| \<pi> ` b2"
end_point_iff1: fixes x :: "'a" assumes "path x" shows "is_point (end_points x) = (\<not> no_end_points x)"
Cons_4: fixes k' :: "'a" assumes "k' \<in> fst ` set xs" shows "lt k k'"
Loebian_imp_Noetherian: fixes x :: "'a" assumes "Loebian x" shows "Noetherian x"
of_nat_power_less_of_nat_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less (semiring_1.of_nat one plus zero x) (power.power one times (semiring_1.of_nat one plus zero b) w) = (x < b ^ w)"
scaling_linear: fixes f :: "'a \<Rightarrow> 'a"   and c :: "real" assumes "f (0::'a) = (0::'a)"   and "\<forall>x y. dist (f x) (f y) = c * dist x y" shows "linear f"
conv: shows "fds_abs_converges (fds_norm (f * g)) s"
finite_borel_measure'_def: fixes M :: "'a measure" shows "finite_borel_measure' M \<equiv> finite_measure M \<and> finite_borel_measure'_axioms M"
lfinite_eq_range_llist_of: fixes xs :: "'a llist" shows "lfinite xs = (xs \<in> range llist_of)"
sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<le> x + y"
le_divide_eq_1_pos: 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 a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq one (divide b a) = less_eq a b"
ce_set_pair_by_minsky: fixes A :: "nat set"   and B :: "nat set" assumes "A \<in> ce_sets"   and "B \<in> ce_sets" obtains M :: "(nat \<times> (nat, nat) Op) set" where "finite M" and "deterministic M" and "0 \<notin> fst ` M" and "Suc 0 \<notin> fst ` M" and "\<And>x vs. \<lbrakk>vs 0 = x; vs 1 = 0; x \<in> A \<union> B\<rbrakk> \<Longrightarrow> \<exists>vs'. ((2, vs), 0, vs') \<in> (step M)\<^sup>* \<or> ((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*" and "\<And>x vs vs'. \<lbrakk>vs 0 = x; vs 1 = 0; ((2, vs), 0, vs') \<in> (step M)\<^sup>*\<rbrakk> \<Longrightarrow> x \<in> A" and "\<And>x vs vs'. \<lbrakk>vs 0 = x; vs 1 = 0; ((2, vs), Suc 0, vs') \<in> (step M)\<^sup>*\<rbrakk> \<Longrightarrow> x \<in> B"
\<psi>\<psi>: shows "alw \<psi> ys1"
map_iterator_genord_key_filter_correct: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option"   and R :: "'e \<times> 'f \<Rightarrow> 'e \<times> 'f \<Rightarrow> bool"   and P :: "'e \<Rightarrow> bool" assumes "map_iterator_genord it m R" shows "map_iterator_genord (map_iterator_key_filter P it) (m |` {k. P k}) R"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray"   and i :: "nat"   and fs :: "int vec list" shows "LLL_impl_inv (f, mu, ds) i fs = (list_repr i f (map ((!) fs) [0..<m]) \<and> d_repr ds fs \<and> mu_repr mu fs)"
Abs_freeRes_inverse: fixes y :: "freeRes_Rep" assumes "y \<in> Collect freeRes_Rep_set" shows "Rep_freeRes (Abs_freeRes y) = y"
countable_iff_lepoll: fixes A :: "'a set" shows "countable A = (A \<lesssim> UNIV)"
diff2: shows "k-smooth_on (apply_chart c3 ` (V \<inter> W)) (apply_chart c2 \<circ> inv_chart c3)"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and R :: "('b \<times> 'b) set" shows "((a1, a2) \<in> Omega_Algebra_Models.omega R) = (\<exists>x y z. a1 = x \<and> a2 = z \<and> (x, y) \<in> R \<and> (y, z) \<in> Omega_Algebra_Models.omega R)"
constant_function_def: fixes S :: "'a set"   and a :: "'b" shows "constant_function S a = (\<lambda>x\<in>S. a)"
aprimedivisor_of_prime: fixes p :: "'a" assumes "prime p" shows "aprimedivisor p = p"
lem: fixes m :: "nat"   and n :: "nat" assumes "m - Suc 0 < n" shows "m < Suc n"
assms: shows "((\<sigma>, NodeS i p R), a, s') \<in> onode_sos S"
gt_lbound_def: fixes c :: "'a"   and b :: "'a option" shows "c >\<^sub>l\<^sub>b b = \<rhd>\<^sub>l\<^sub>b (<) c b"
tllist_ord_ttlI: fixes xs :: "('d, 'b) tllist"   and ys :: "('d, 'b) tllist" assumes "local.tllist_ord xs ys" shows "local.tllist_ord (ttl xs) (ttl ys)"
induct: fixes x :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and H :: "chan set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> extr bad IK H"   and "\<And>M. M \<in> IK \<Longrightarrow> P M"   and "\<And>c A B M. \<lbrakk>Chan c A B M \<in> H; c = insec \<or> c = auth \<or> A \<in> bad \<or> B \<in> bad\<rbrakk> \<Longrightarrow> P M" shows "P x"
cball_diff_sphere: fixes a :: "'b"   and r :: "real" shows "cball a r - sphere a r = ball a r"
fst_divmod: fixes m :: "num"   and n :: "num" shows "fst (divmod m n) = numeral m div numeral n"
submonoid_axioms: shows "submonoid S R (\<cdot>) \<one>"
trans_inv_image: fixes r :: "('g \<times> 'g) set"   and f :: "'h \<Rightarrow> 'g" assumes "trans r" shows "trans (inv_image r f)"
relcomppI: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and b :: "'b"   and s :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'c" assumes "r a b"   and "s b c" shows "(r OO s) a c"
SUP_const: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and f :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Sup ((\<lambda>i. f) ` A) = f"
finite_orbit: fixes s :: "'e"   and f :: "'e \<Rightarrow> 'e" assumes "s \<in> orbit f s" shows "finite (orbit f s)"
assms: shows "ones \<equiv> Abs_fps (\<lambda>n. 1::'a)"
W_lcoset_bij_betw_singletons: fixes S :: "'b set"   and w :: "'b" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>" shows "bij_betw ((+o) w) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) 0) (dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w})"
RepFun_cong: fixes A :: "hf"   and B :: "hf"   and f :: "hf \<Rightarrow> hf"   and g :: "hf \<Rightarrow> hf" assumes "A = B"   and "\<And>x. x \<^bold>\<in> B \<Longrightarrow> f x = g x" shows "RepFun A f = RepFun B g"
Ctxt_valid_def: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "( \<Turnstile> G ) = (\<forall>A. A \<in> G \<longrightarrow>  \<Turnstile> Call : A )"
assms_1: shows "\<Union> (deps ` M) \<subseteq> I ` insert_messages M"
add_tan_eq: fixes x :: "'a"   and y :: "'a" assumes "cos x \<noteq> (0::'a)"   and "cos y \<noteq> (0::'a)" shows "tan x + tan y = sin (x + y) / (cos x * cos y)"
assms: fixes cs :: "(Term_Class.term \<times> sterm) list" assumes "\<And>pat t. (pat, t) \<in> set cs \<Longrightarrow> P t" shows "P (Sabs cs)"
fmempty_lookup: fixes x :: "'d" shows "{$$} $$ x = None"
len: shows "e_length (block i t) = Suc (t + 1)"
x: shows "x = fst (split_n_m_x (dim_col A) (dim_row A) X)"
surj_pair: fixes p :: "'k \<times> 'l" shows "\<exists>x y. p = (x, y)"
simps: fixes e1 :: "'a"   and a1 :: "'b"   and r1 :: "nat"   and ts1 :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and e2 :: "'a"   and a2 :: "'b"   and r2 :: "nat"   and ts2 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.link (BinomialHeapStruc.Node e1 a1 r1 ts1) (BinomialHeapStruc.Node e2 a2 r2 ts2) = (if a1 \<le> a2 then BinomialHeapStruc.Node e1 a1 (Suc r1) (BinomialHeapStruc.Node e2 a2 r2 ts2 # ts1) else BinomialHeapStruc.Node e2 a2 (Suc r2) (BinomialHeapStruc.Node e1 a1 r1 ts1 # ts2))"
reaches_step_1: fixes x :: "'a"   and y :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state"   and z :: "'a" assumes "(x reaches y) s"   and "(y points_to z) s" shows "(x reaches z) s"
assms_2: fixes x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>" shows "thesis"
assms_1: shows "p = {l}groupcast(fips, fmsg) . qq"
sum_arith_progression: fixes f :: "nat \<Rightarrow> 'b"   and N :: "nat"   and a :: "nat" shows "(\<Sum>r<N. \<Sum>i<a. f (i * N + r)) = sum f {..<a * N}"
aezfun_left_minus: fixes a :: "('a, 'b) aezfun" shows "- a + a = 0"
pow_commutes_list: fixes u :: "'b list"   and n :: "nat" shows "u \<^sup>@ n \<cdot> u = u \<cdot> u \<^sup>@ n"
strict_weak_trans: fixes a :: "'alt"   and b :: "'alt"   and c :: "'alt" assumes "a \<prec>[R i] b"   and "R i b c" shows "a \<prec>[R i] c"
differentiable_norm_at: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "norm differentiable at a"
assms: shows "r \<in> p_e modRules2 T S"
connected_components_of_eq_empty: fixes X :: "'b topology" shows "(connected_components_of X = {}) = (topspace X = {})"
2: shows "f |\<subseteq>| HOLZF.Domain f |\<times>| HOLZF.Range f"
seq_mult_simp: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "t \<in> carrier Zp\<^bsup>\<omega>\<^esup>" shows "(s \<otimes>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> t) k = s k \<otimes> t k"
scale_ms_aux_def: fixes c :: "real"   and xs :: "('b \<times> real) msllist" shows "scale_ms_aux c xs = scale_shift_ms_aux (const_expansion c, 0) xs"
index_mat_four_block_2: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and D :: "'a mat" shows "dim_row (four_block_mat A B C D) = dim_row A + dim_row D"
uminus: fixes x :: "'a" shows "f (- x) = - f x"
trivial_outside: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and v :: "'b" assumes "ChamberComplexEndomorphism X f"   and "v \<notin> \<Union> X" shows "f v = v"
ring_simprules_4: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus>\<^bsub>R\<^esub> y \<in> carrier R"
context_conjI: fixes P :: "bool"   and Q :: "bool" assumes "P"   and "P \<Longrightarrow> Q" shows "P \<and> Q"
preserves_ide: fixes a :: "'d" assumes "D.ide a" shows "D.ide (\<I>\<^sub>D \<otimes>\<^sub>D a)"
Star_Star: fixes r :: "'a language" shows "Star (Star r) = Star r"
spmf_try_spmf: fixes p :: "'h spmf"   and q :: "'h spmf"   and x :: "'h" shows "spmf (TRY p ELSE q) x = spmf p x + pmf p None * spmf q x"
piecewise_continuous_onI: fixes I :: "'c set"   and a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd"   and l :: "'c \<Rightarrow> 'd"   and u :: "'c \<Rightarrow> 'd" assumes "finite I"   and "continuous_on ({a..b} - I) f"   and "\<And>i. \<lbrakk>i \<in> I; a < i; i \<le> b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> l i) (at_left i)"   and "\<And>i. \<lbrakk>i \<in> I; a \<le> i; i < b\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> u i) (at_right i)" shows "piecewise_continuous_on a b I f"
simps_1: fixes x1 :: "'l"   and x2 :: "'m"   and y1 :: "'l"   and y2 :: "'m" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
combine_append_replicate: fixes Us :: "'a list"   and Ks :: "'a list"   and n :: "nat" assumes "set Us \<subseteq> carrier R" shows "combine (Ks @ replicate n \<zero>) Us = combine Ks Us"
subst_domain_subst_compose: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "subst_domain (\<sigma> \<circ>\<^sub>s \<tau>) = subst_domain \<sigma> - {x. \<exists>y. \<sigma> x = Var y \<and> \<tau> y = Var x} \<union> (subst_domain \<tau> - subst_domain \<sigma>)"
LIM_compose2: fixes f :: "'a \<Rightarrow> 'b"   and b :: "'b"   and a :: "'a"   and g :: "'b \<Rightarrow> 'c"   and c :: "'c" assumes "f \<midarrow>a\<rightarrow> b"   and "g \<midarrow>b\<rightarrow> c"   and "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b" shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
app_vrrestrictionD_2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>r\<^sub>\<circ> A" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
contradiction': fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list"   and Q_hit :: "'c list"   and Q_hit_pre :: "'c list"   and Q_hit_post :: "'c list" assumes "ProofStepInduct_y_neq_new_last G v0 v1 paths P_new sep_size P_k_pre y P_k_post"   and "Q_hit \<in> ProofStepInduct_NonTrivial.Q G v0 v1 paths P_new sep_size"   and "y \<in> set Q_hit"   and "Q_hit = Q_hit_pre @ y # Q_hit_post" shows "False"
ring: shows "ring (RDirProd_list (map (\<lambda>i. R Quot I (Suc i)) [0..<Suc n]))"
c_d: shows "(c, d) \<in> thetaZOSeqD"
sumhr_add_lbound_zero: fixes m :: "nat star"   and k :: "nat star"   and f :: "nat \<Rightarrow> real" shows "sumhr (m + k, k, f) = 0"
set_allConsec: fixes \<Delta> :: "'a"   and \<Delta>s' :: "'a list"   and \<Delta>s :: "'a list"   and \<Delta>1 :: "'a" assumes "\<Delta> \<in> set \<Delta>s'"   and "\<Delta>s = \<Delta>s' ## \<Delta>1" shows "\<exists>\<Delta>2. (\<Delta>, \<Delta>2) \<in> allConsec \<Delta>s"
assms_4: shows "(\<Sum>x\<in>X. of_int (\<beta> x) * exp x) = 0"
pure_universal_release_simp: fixes \<nu> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "pure_universal \<nu>" shows "w \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<nu> = w \<Turnstile>\<^sub>n \<nu>"
r_inv_ex: fixes x :: "'a" shows "\<exists>y. x ** y = one"
eventually_uniformity_metric: fixes P :: "'a \<times> 'a \<Rightarrow> bool" shows "eventually P uniformity = (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))"
viable_market_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market" assumes "disc_filtr_prob_space M F" shows "disc_filtr_prob_space.viable_market M F Mkt = (\<forall>p. stock_portfolio Mkt p \<longrightarrow> \<not> disc_filtr_prob_space.arbitrage_process M F Mkt p)"
real_div_down_def: fixes p :: "nat"   and i :: "int"   and j :: "int" shows "real_div_down p i j = truncate_down (Suc p) (real_of_int i / real_of_int j)"
unbounded: fixes B :: "real" shows "\<forall>\<^sub>F x in at_infinity. B \<le> cmod (f x)"
g_def: shows "g = Cn 4 r_snoc [recf.Id 4 1, app]"
ps_get_imp_def: fixes psi :: "'a Heap.array"   and u :: "nat" shows "ps_get_imp psi u \<equiv> Array.nth psi u"
set_times_plus_distrib: fixes a :: "'c"   and b :: "'c"   and C :: "'c set" shows "a *o (b +o C) = a * b +o a *o C"
key2_not_currk: fixes s :: "state"   and r :: "room"   and g :: "guest"   and g' :: "guest"   and k :: "key" assumes "s \<in> reach"   and "owns s r = \<lfloor>g\<rfloor>"   and "g \<noteq> g'" shows "(k, currk s r) \<notin> cards s g'"
splitNlist_correct: fixes p :: "'d \<Rightarrow> bool"   and i :: "'d"   and nl :: "('f, 'd) Node list"   and l :: "('f, 'd) Node list"   and n :: "('f, 'd) Node"   and r :: "('f, 'd) Node list" assumes "\<And>a b. p a \<Longrightarrow> p (a + b)"   and "\<not> p i"   and "p (i + gmnl nl)"   and "splitNlist p i nl = (l, n, r)" shows "\<not> p (i + gmnl l) \<and> p (i + gmnl l + gmn n) \<and> nl = l @ n # r"
angles_6: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "angle b1 c1 a1 = angle a2 c2 b2"
FL_valid_Not: fixes P :: "'state"   and x :: "('b, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> FL_Formula.Not x = (\<not> P \<Turnstile> x)"
holomorphic_fds_eval: fixes A :: "complex set"   and f :: "complex fds" assumes "A \<subseteq> {z. conv_abscissa f < ereal (Re z)}" shows "eval_fds f holomorphic_on A"
algebraicE: fixes x :: "'a" assumes "algebraic x" obtains p :: "'a poly" where "\<And>i. poly.coeff p i \<in> \<int>" and "p \<noteq> 0" and "poly p x = (0::'a)"
reduced_bdt_checkers_def: fixes ifex_of :: "'b \<Rightarrow> 'c ifex"   and val :: "'b \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> bool" shows "reduced_bdt_checkers ifex_of val \<equiv> (\<forall>b s. val_ifex (ifex_of b) s = val b s) \<and> (\<forall>b. reduced (ifex_of b) {})"
dom_eq_empty_conv: fixes f :: "'c \<Rightarrow> 'd option" shows "(dom f = {}) = (f = Map.empty)"
length_nth_simps_4: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat" shows "(x # xs) ! Suc n = xs ! n"
linear_injective_isomorphism: fixes f :: "('c, 'd) vec \<Rightarrow> ('c, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "\<exists>f'. Vector_Spaces.linear (*s) (*s) f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
TryRed: fixes 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 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"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s\<rangle> \<rightarrow> \<langle>try e' catch(C V) e\<^sub>2,s'\<rangle>"
lm134: fixes X :: "'a set"   and f :: "('a \<times> 'b) set" assumes "finite X" shows "MiscTools.sum' f X = MiscTools.sum' f (X \<inter> Domain f)"
neutral': fixes \<sigma> :: "'b \<Rightarrow> 'b"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "\<sigma> permutes alts"   and "is_pref_profile R"   and "a \<in> alts" shows "pmf (sds (permute_profile \<sigma> R)) (\<sigma> a) = pmf (sds R) a"
intro: fixes weak_gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and default :: "'a"   and mono :: "'a \<Rightarrow> bool" assumes "\<And>xys. \<forall>x y. (x, y) \<in> set xys \<longrightarrow> weak_gt x y \<Longrightarrow> \<exists>gt bound. mono_matrix_carrier gt default bound mono \<and> (\<forall>x y. (x, y) \<in> set xys \<longrightarrow> gt x y)" shows "weak_complexity_linear_poly_order_carrier weak_gt default mono"
omega_unfold_eq: fixes x :: "'a" shows "x \<cdot> x\<^sup>\<omega> = x\<^sup>\<omega>"
Diff_subset_conv: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(A - B \<subseteq> C) = (A \<subseteq> B \<union> C)"
eq1_2: shows "y1 \<bullet> y1 = x \<bullet> y1"
ro_cond_def: fixes p :: "bool Heap" shows "ro_cond p \<equiv> \<forall>h. success p h \<longrightarrow> snd (the (execute p h)) = h"
representation_basis: fixes basis :: "'b set"   and b :: "'b" assumes "vs1.independent basis"   and "b \<in> basis" shows "vs1.representation basis b = (\<lambda>v. if v = b then 1::'a else (0::'a))"
sub_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "sub_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; rs1 = get_operand_w5 (op_list ! 1); rd = get_operand_w5 (op_list ! 3) in do operand2 \<leftarrow> gets (get_operand2 op_list); curr_win \<leftarrow> get_curr_win (); rs1_val \<leftarrow> gets (user_reg_val curr_win rs1); psr_val \<leftarrow> gets (cpu_reg_val PSR); c_val \<leftarrow> gets (\<lambda>s. get_icc_C psr_val); result \<leftarrow> gets (\<lambda>s. if instr_name = arith_type SUB \<or> instr_name = arith_type SUBcc then rs1_val - operand2 else rs1_val - operand2 - ucast c_val); rd_val \<leftarrow> gets (user_reg_val curr_win rd); new_rd_val \<leftarrow> gets (\<lambda>s. if rd \<noteq> 0 then result else rd_val); write_reg new_rd_val curr_win rd; sub_instr_sub1 instr_name result rs1_val operand2 od"
cond_helper2: fixes p :: "'a"   and y :: "'a"   and x :: "'a"   and z :: "'a"   and w :: "'a" assumes "test p"   and "pres y (!p)" shows "!p \<cdot> (p \<cdot> x + !p \<cdot> y)\<^sup>\<dagger> \<cdot> (p \<cdot> z + !p \<cdot> w) = !p \<cdot> y\<^sup>\<dagger> \<cdot> w"
cos_0_iff_canon: fixes \<phi> :: "real" assumes "cos \<phi> = 0"   and "- pi < \<phi>"   and "\<phi> \<le> pi" shows "\<phi> = pi / 2 \<or> \<phi> = - pi / 2"
not_extf_gt_nil_singleton_if_\<delta>_eq_\<epsilon>: fixes s :: "('s, 'v) tm"   and f :: "'s" assumes "wary s"   and "\<delta> = \<epsilon>" shows "\<not> extf f (>\<^sub>t) [] [s]"
simple_rotations_bij: fixes r :: "Vertex \<Rightarrow> Vertex" assumes "r \<in> simple_rotations" shows "bij r"
rep_eq: fixes x :: "'b"   and xa :: "'c"   and xb :: "('b, 'c) mapping" shows "rep (update x xa xb) = rep xb(x \<mapsto> xa)"
Cd_domain: fixes d :: "'b"   and X :: "'a set" shows "Bossy.Cd d (X \<inter> Field (BPd d)) = Bossy.Cd d X"