hcju/isabelleps huggingface dataset:
cos_ms_aux'_def: fixes xs :: "('b \<times> real) msllist" shows "cos_ms_aux' xs = powser_ms_aux (msllist_of_msstream cos_series_stream) (times_ms_aux xs xs)"
CHECK_def: fixes \<Phi> :: "bool"   and e :: "'c" shows "CHECK \<Phi> e \<equiv> if \<Phi> then ERETURN () else THROW e"
ras_shrink_id_refine: fixes R :: "('c \<times> 'd) set" shows "(ras_shrink, id) \<in> \<langle>R\<rangle>ras_rel \<rightarrow> \<langle>R\<rangle>ras_rel"
sigma_sets_generated_ring_eq: shows "sigma_sets \<Omega> G.generated_ring = sigma_sets \<Omega> generated_ring"
le_fin_prefix_transp: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list"   and w\<^sub>3 :: "'a list" assumes "traces ind"   and "traces.le_fin ind w\<^sub>1 w\<^sub>2"   and "w\<^sub>2 \<le> w\<^sub>3" shows "traces.le_fin ind w\<^sub>1 w\<^sub>3"
zero_le_power_eq: 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 sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a n) = (even n \<or> odd n \<and> less_eq zero a)"
in_set_spmf: fixes x :: "'d"   and p :: "'d spmf" shows "(x \<in> set_spmf p) = (Some x \<in> set_pmf p)"
IH_1: assumes "vars p1 \<subseteq> {..<n}" shows "(\<lambda>f. insertion f p1) \<in> borel_measurable (lborel_f n)"
filter_shuffles_disjoint1_2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "set xs \<inter> set ys = {}"   and "zs \<in> shuffles xs ys" shows "filter (\<lambda>x. x \<notin> set xs) zs = ys"
arr_cod_iff_arr: fixes f :: "'a \<times> 'a" shows "VV.arr (VV.cod f) = VV.arr f"
face_of_polyhedron_explicit: fixes F :: "'b set set"   and S :: "'b set"   and a :: "'b set \<Rightarrow> 'b"   and b :: "'b set \<Rightarrow> real"   and C :: "'b set" assumes "finite F"   and "S = affine hull S \<inter> \<Inter> F"   and "\<And>h. h \<in> F \<Longrightarrow> a h \<noteq> (0::'b) \<and> h = {x. a h \<bullet> x \<le> b h}"   and "\<And>F'. F' \<subset> F \<Longrightarrow> S \<subset> affine hull S \<inter> \<Inter> F'"   and "C face_of S"   and "C \<noteq> {}"   and "C \<noteq> S" shows "C = \<Inter> {S \<inter> {x. a h \<bullet> x = b h} |h. h \<in> F \<and> C \<subseteq> S \<inter> {x. a h \<bullet> x = b h}}"
unambiguous_routing_code: fixes rtbl :: "('c, 'd) routing_rule_scheme list" shows "unambiguous_routing rtbl = unambiguous_routing_code rtbl"
inf_absorb1: fixes x :: "'a upred"   and y :: "'a upred" assumes "y \<sqsubseteq> x" shows "(x \<and> y) = x"
eg: shows "gr \<in> set (simple_match_to_of_match r ifs)"
simps_2: fixes t :: "'a tree" shows "left (mirror t) = mirror (right t)"
mat_carrier: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "mat nr nc f \<in> carrier_mat nr nc"
Kbase_hom1: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<forall>j\<le>n. (Kb\<^bsub>K n P\<^esub>) j \<in> carrier K - {\<zero>}"
atLeast0_atMost_Suc: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "sum g {0..Suc n} = sum g {0..n} + g (Suc n)"
preserved_def: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x" shows "preserved f h h' = (\<forall>x. h \<turnstile> f \<rightarrow>\<^sub>r x = h' \<turnstile> f \<rightarrow>\<^sub>r x)"
eval_r: shows "eval r [encode r] = eval (Cn 1 f [h]) [encode r]"
unitET_def: shows "unitET = (\<Lambda> x. ErrorT\<cdot>(return\<cdot>(Ok\<cdot>x)))"
exec: shows "execution e (\<epsilon>(r \<mapsto> (\<epsilon>, \<epsilon>, e))) s"
list_of_vec_vec_of_list: fixes a :: "'b list" shows "list_of_vec (vec_of_list a) = a"
expansion_factor_linear_apply_cltn2: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and J :: "cltn2"   and k :: "real" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S"   and "r \<in> hyp2 \<union> S"   and "is_K2_isometry J"   and "cart2_pt r = k *\<^sub>R cart2_pt p + (1 - k) *\<^sub>R cart2_pt q" shows "expansion_factor r J *\<^sub>R cart2_append1 (apply_cltn2 r J) = (k * expansion_factor p J) *\<^sub>R cart2_append1 (apply_cltn2 p J) + ((1 - k) * expansion_factor q J) *\<^sub>R cart2_append1 (apply_cltn2 q J)"
edge_ex_vertices: fixes e :: "'a set" assumes "e \<in> E" shows "\<exists>u v. u \<noteq> v \<and> e = {u, v}"
RModuleIso: shows "RModuleIso FG (\<cdot>) V (\<star>) T W"
fmpred_map: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and m :: "('a, 'c) fmap" shows "fmpred P (fmmap f m) = fmpred (\<lambda>k v. P k (f v)) m"
assms_1: fixes var :: "Vars.var" assumes "y = Var var" shows "P"
Arg_def: fixes z :: "complex" shows "Arg z \<equiv> if z = 0 then 0 else Im (Ln z)"
out_sub12_2: shows "outside (path_image c1 \<union> path_image c2) \<subseteq> outside (path_image c2 \<union> path_image c)"
for_rec2_fwi: fixes k :: "nat"   and M :: "nat \<times> nat \<Rightarrow> 'b"   and n :: "nat"   and i :: "nat"   and j :: "nat" shows "for_rec2 (\<lambda>M. fw_upd' M k) M n i j \<le> SPEC (\<lambda>M'. M' = uncurry (fwi (curry M) n k i j))"
assms: shows "adjunction_in_bicategory (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C f g \<eta> \<epsilon>"
drop_in_carrier: fixes xs :: "'a list"   and n :: "nat" assumes "in_carrier xs" shows "in_carrier (drop n xs)"
sort_ex_norm_sort: fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set" assumes "wf_subclass_loc cs"   and "sort_ex cs s" shows "sort_ex cs (normalize_sort cs s)"
f_simp: fixes X :: "'a set" shows "f X = \<Union> {f {x} |x. x \<in> X}"
dom_comp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and f :: "'a" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.dom C (C g f) = partial_magma.dom C f"
\<Delta>\<Phi>_pass1_sum_ub: fixes h :: "'b hp list" shows "\<Phi>_hps (pass\<^sub>1 h) - \<Phi>_hps h \<le> sum_ub h"
degree_monic_char_poly: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "degree (char_poly A) = n \<and> coeff (char_poly A) n = (1::'c)"
wlsInp_iff: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "wlsInp delta inp = (wlsOpS delta \<and> sameDom (arOf delta) inp \<and> liftAll2 wls (arOf delta) inp)"
map_add_restrict_dom_right: fixes m :: "'c \<Rightarrow> 'd option"   and m' :: "'c \<Rightarrow> 'd option" shows "(m ++ m') |` dom m' = m'"
assign_eval\<^sub>w: fixes x :: "var\<^sub>C"   and e :: "aexp\<^sub>C"   and mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat" shows "eval_abv\<^sub>C \<langle>x \<leftarrow> e, mds, mem\<rangle>\<^sub>C \<langle>Stop, mds, mem(x := ev\<^sub>A\<^sub>C mem e)\<rangle>\<^sub>C"
integral_pmf_of_set: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "S \<noteq> {}"   and "finite S" shows "measure_pmf.expectation (pmf_of_set S) f = sum f S / real (card S)"
neq_Nil_rev_conv: fixes l :: "'c list" shows "(l \<noteq> []) = (\<exists>xs x. l = xs @ [x])"
J_rw: shows "J = {i. i < dim_col A \<and> i \<in> J}"
elims: fixes x :: "bool"   and y :: "enat" assumes "\<up> x = y" obtains "\<not> x" and "y = \<infinity>"   | "x" and "y = 0"
s'_D_2: shows "\<forall>x\<in>{0..1} - s'. \<gamma>2 differentiable at x"
in_cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "(f \<in> \<omega>[F](h)) = (g \<in> \<omega>[F](h))"
bisimilar_def: fixes P :: "('state, 'effect) L_state"   and Q :: "('state, 'effect) L_state" shows "P \<sim>\<cdot>\<^sub>L Q \<equiv> \<exists>R. L_transform.is_bisimulation R \<and> R P Q"
tangent_not_through_K2: fixes p :: "proj2"   and q :: "proj2" assumes "p \<in> S"   and "q \<in> hyp2" shows "\<not> proj2_incident q (polar p)"
inverse_Quantity_ext_def: fixes x :: "('a, 'b, 'c) Quantity_scheme" shows "inverse x = \<lparr>mag = inverse (mag x), dim = inverse (Quantity.dim x), \<dots> = inverse (Quantity.more x)\<rparr>"
reduced_word_for_minimal: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list"   and bs :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as"   and "bs \<in> lists A"   and "monoid_add.sum_list plus zero bs = a" shows "order.greater_eq (length bs) (length as)"
action_agree_valid_in_snapshot: fixes a :: "('d, 'e) fmap \<times> ('d, 'e) fmap"   and PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" assumes "a \<in> PROB"   and "agree (fst a) s"   and "agree (snd a) s" shows "a \<in> snapshot PROB s"
select_finite: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b option" assumes "finite S" shows "finite (select f S)"
fbox_while_unfold: fixes t :: "'a"   and x :: "'a"   and p :: "'a" shows " |while t do x od] p = (d t + d p) \<cdot> (ad t +  |x]  |while t do x od] p)"
min_top2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b" assumes "class.order_top less_eq less top" shows "ord.min less_eq x top = x"
IH: assumes "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` \<Union> (set (map P S))) = \<Union> (set (map P (S \<cdot>\<^sub>s\<^sub>t \<delta>)))"
create_gba__invar: fixes \<phi> :: "'b ltlr" shows "create_gba \<phi> \<le> SPEC gba"
cring_simprules_5: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y \<in> carrier R"
\<Phi>\<Psi>: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<AA> :: "('d \<times> 'd) set"   and \<phi> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "two_set_categories_bij_betw_Univ S S' \<AA> \<phi>"   and "partial_magma.arr S f" shows "two_set_categories_bij_betw_Univ.\<Phi> S' S \<AA> (inv_into (Collect (category.terminal S)) \<phi>) (two_set_categories_bij_betw_Univ.\<Phi> S S' \<AA> \<phi> f) = f"
preserves_reflects_arr: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a" assumes "functor A B F" shows "partial_magma.arr B (F f) = partial_magma.arr A f"
simps_1: fixes bbs :: "bool list"   and bs :: "int list" shows "Ifm bbs bs T = True"
h0: shows "satisfies_properties p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_matrix (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
hyps: shows "similar_mat (k \<cdot>\<^sub>m jordan_matrix n_as) (jordan_matrix (map (\<lambda>a. case a of (n, a) \<Rightarrow> (n, k * a)) n_as))"
IH: shows "weave A (nths xs A) (nths xs (- A)) = xs"
fund_sol_is_nontriv_solution: fixes D :: "nat" assumes "pell D" shows "pell.nontriv_solution D (pell.fund_sol D)"
C_ground: shows "ground\<^sub>l\<^sub>s C"
is_zero_row_upt_ncols: fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec" shows "is_zero_row_upt_k i (ncols A) A = (\<forall>j. A $ i $ j = (0::'c))"
fua: shows "lesvars (map (\<lambda>(x, e). (x, \<lambda>s. e (S s))) upds) = lesvars upds"
Dube: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "finite X"   and "finite F"   and "F \<subseteq> P[X]"   and "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "g \<in> punit.reduced_GB F" shows "poly_deg g \<le> Dube (card X) (maxdeg F)"
rep_eq: fixes x :: "'b fset" shows "fMax x = Max (fset x)"
exec_1_d_NormalI: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and \<Sigma> :: "(('addr, 'thread_id, 'heap) jvm_thread_action \<times> 'addr option \<times> 'heap \<times> 'addr frame list) set"   and tas :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" assumes "exec_d P t \<sigma> = Normal \<Sigma>"   and "(tas, \<sigma>') \<in> \<Sigma>" shows "P,t \<turnstile> Normal \<sigma> -tas-jvmd\<rightarrow> Normal \<sigma>'"
term_onestep: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool" assumes "well_def body" shows "\<guillemotleft> \<N> G \<guillemotright> \<tturnstile> wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) (\<lambda>s. 1)"
closure_of_singleton: fixes X :: "'c topology"   and a :: "'c" assumes "t1_space X" shows "X closure_of {a} = (if a \<in> topspace X then {a} else {})"
fmpredI: fixes m :: "('a, 'b) fmap"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x y" shows "fmpred P m"
below_refl: fixes x :: "'c" shows "x \<sqsubseteq> x"
Un_fTo_Chain: fixes CC :: "'a set set" assumes "Chain fTo D CC" shows "Chain D (\<Union> CC)"
CoxeterComplex: shows "CoxeterComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
harry_step_False_end: fixes xs :: "bool list" assumes "True \<in> set xs" shows "harry_step (xs @ [False]) = harry_step xs @ [False]"
is_extensional: fixes f :: "'a" assumes "\<not> hom.arr f" shows "\<phi>.map f = hom.null"
con_comp_ipurge_ref_aux: fixes U :: "'e option set"   and xs :: "'f list"   and p :: "'g \<Rightarrow> 'f"   and X :: "'f set"   and I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e"   and q :: "'h \<Rightarrow> 'f" assumes "U \<subseteq> range Some"   and "set xs \<subseteq> range p"   and "X \<subseteq> range p" shows "ipurge_ref_aux I D (the ` U) (map (inv p) xs) (inv p ` X) = inv p ` ipurge_ref_aux (con_comp_pol I) (con_comp_map D E p q) U xs X"
SASP_to_DIMACS'_def: fixes h :: "nat"   and prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "SASP_to_DIMACS' h prob \<equiv> SASP_to_DIMACS h (rem_implicit_pres_ops prob)"
KIR_cleanup: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "KIR f x"   and "d_IN f x \<noteq> \<top>" shows "KIR (cleanup f) x"
b': shows "P \<Turnstile> b' <\<sim>> b"
contract_vacuity: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "\<phi> \<notin> K" shows "K \<inter> (K \<^bold>* .\<not> \<phi>) = K"
union: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z (M + N) = f (comm_monoid_mset.F f z M) (comm_monoid_mset.F f z N)"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a * b * c = a * (b * c)"
pseudo_divmod_impl: fixes f :: "'a poly"   and g :: "'a poly" shows "pseudo_divmod f g = map_prod poly_of_list poly_of_list (pseudo_divmod_list (coeffs f) (coeffs g))"
is_monic_setD: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_monic_set B"   and "b \<in> B"   and "b \<noteq> 0" shows "lc b = (1::'c)"
in_set_enumerate_eq: fixes p :: "nat \<times> 'f"   and n :: "nat"   and xs :: "'f list" shows "(p \<in> set (enumerate n xs)) = (n \<le> fst p \<and> fst p < length xs + n \<and> xs ! (fst p - n) = snd p)"
P0: shows "r \<in> carrier (residue_ring (p ^ k))"
H_without_scalar_prod: shows "H = mat 2 2 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if i \<noteq> j then 1 / sqrt 2 else if i = 0 then 1 / sqrt 2 else - (1 / sqrt 2)))"
eqB: shows "B = adjoint P * (P * B * adjoint P) * P"
dist_commute: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
closure_Some: fixes R0 :: "'b rexp set"   and S0 :: "'b rexp set"   and as :: "'b list" assumes "finite R0"   and "finite S0" shows "\<exists>p. pEquivalence_Checking.closure as ([(R0, S0)], []) = Some p"
gaussian_moment_even_pos: fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R (exp (- x\<^sup>2) * x ^ (2 * k))) (sqrt pi / 2 * (fact (2 * k) / (2 ^ (2 * k) * fact k)))"
1: shows "\<guillemotleft>\<iota> \<star> f : (b \<star> b) \<star> f \<Rightarrow>\<^sub>S b \<star> f\<guillemotright>"
subset_code_2: fixes A :: "'b set"   and t :: "('b, unit) RBT.rbt" shows "(A \<subseteq> Coset t) = (\<forall>y\<in>Set t. y \<notin> A)"
tick_count_f_suc: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat" assumes "dilating f sub r" shows "tick_count r c (f (Suc n)) = tick_count r c (f n) + (if hamlet (Rep_run r (f (Suc n)) c) then 1 else 0)"
fct_gen_subalgebra_space: fixes M :: "'c measure"   and N :: "'d measure"   and X :: "'c \<Rightarrow> 'd" shows "space (fct_gen_subalgebra M N X) = space M"
assms: shows "(\<Sum>n. real_of_int (b n) / real_of_int (prod a {..n})) \<in> \<rat>"
coerce_cfun': shows "COERCE('e \<rightarrow> 'f, 'g \<rightarrow> 'h) = cfun_map\<cdot>COERCE('g, 'e)\<cdot>COERCE('f, 'h)"
x_def: fixes i :: "nat" shows "x i = (if i = 0 then 1 else 0)"
bigE: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Omega>[F](g)" obtains c :: "real" where "0 < c" and "\<forall>\<^sub>F x in F. c * norm (g x) \<le> norm (f x)"
path_evalF: fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and x :: "formula" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "x \<in> set fs" shows "\<not> evalF (counterModel f) counterAssign x"
test_one: shows "at (at (1::'a)) = (1::'a)"
ASSERT_LIVE_def: shows "ASSERT_LIVE = {. \<box> \<diamond> (\<lambda>x. x (0::'a)) .}"
traces_CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_sub: fixes A :: "'b set" shows "T (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A) \<subseteq> {s. front_tickFree s \<and> set s \<subseteq> ev ` A \<union> {tick}}"
msubsteq_nb: fixes a :: "poly"   and r :: "tm"   and t :: "tm"   and s :: "tm"   and c :: "poly"   and d :: "poly" assumes "islin (Eq (CNP 0 a r))"   and "tmbound0 t"   and "tmbound0 s" shows "bound0 (msubsteq c t d s a r)"
is_none_map_option: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b option" shows "Option.is_none (map_option f x) = Option.is_none x"
assert_unsat: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "index_valid ats s"   and "\<U> (assert a s)" shows "minimal_unsat_state_core (assert a s)"
that: fixes h :: "'a \<Rightarrow> complex" assumes "continuous_on (cball a r) h"   and "\<And>z. z \<in> cball a r \<Longrightarrow> f z = exp (h z)" shows "thesis"
assms: shows "\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> ex1 r v"
simps_4: fixes f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd" shows "(case \<bottom> of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
FL_valid_eqvt: fixes P :: "'state"   and x :: "('a, 'pred, 'act, 'effect) FL_Formula.formula"   and p :: "perm" assumes "P \<Turnstile> x" shows "p \<bullet> P \<Turnstile> p \<bullet> x"
invertible: fixes u :: "'a" assumes "u \<in> H" shows "sub.invertible u"
maximal_def: fixes S :: "'a set"   and C :: "'a set set" shows "maximal S C = (\<forall>S'\<in>C. S \<subseteq> S' \<longrightarrow> S = S')"
B: shows "B \<subseteq> {0..<dim_row A} \<times> {0..<dim_col A}"
euclerian_rev: fixes v' :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v :: "'v" shows "is_Eulerian_trail v' (rev_path ps) v = is_Eulerian_trail v ps v'"
Q2_leq: shows "Q2 \<le>\<^sub>L adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * (P' + Q) * tensor_P (1\<^sub>m N) (mat_incr K)"
inject_3: fixes x4 :: "real"   and y4 :: "real" shows "(RealVal x4 = RealVal y4) = (x4 = y4)"
sumlist_def: fixes xs :: "'a list" shows "sumlist xs \<equiv> foldr (\<oplus>) xs \<zero>"
that: fixes C' :: "'b set" assumes "C' \<subseteq> C"   and "finite C'"   and "S \<subseteq> \<Union> (f ` C')" shows "thesis"
eval_induct: fixes e :: "exp"   and s :: "loc \<Rightarrow> nat"   and n :: "nat"   and s' :: "loc \<Rightarrow> nat"   and P :: "exp \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> bool" assumes "(e, s) -|-> (n, s')"   and "\<And>n s. P (N n) s n s"   and "\<And>s x. P (X x) s (s x) s"   and "\<And>e0 e1 f n0 n1 s s0 s1. \<lbrakk>(e0, s) -|-> (n0, s0); P e0 s n0 s0; (e1, s0) -|-> (n1, s1); P e1 s0 n1 s1\<rbrakk> \<Longrightarrow> P (Op f e0 e1) s (f n0 n1) s1"   and "\<And>c e n s s0 s1. \<lbrakk>(c, s) -[eval \<inter> {((e, s), n, s'). P e s n s'}]-> s0; (c, s) -[eval]-> s0; (e, s0) -|-> (n, s1); P e s0 n s1\<rbrakk> \<Longrightarrow> P (VALOF c RESULTIS e) s n s1" shows "P e s n s'"
has_locks_release_all_conv': fixes t :: "'b"   and t' :: "'b"   and l :: "('b \<times> nat) option" assumes "t \<noteq> t'" shows "has_locks (release_all l t') t = has_locks l t"
sf_set1: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "sf \<phi>[M]\<^sub>\<Sigma>\<^sub>2 \<union> sf (\<phi>[M]\<^sub>\<Pi>\<^sub>1) \<subseteq> (\<Union>\<psi>\<in>sf \<phi>. sf \<psi>[M]\<^sub>\<Sigma>\<^sub>2 \<union> sf (\<psi>[M]\<^sub>\<Pi>\<^sub>1))"
negligible_interval_1: fixes a :: "'a"   and b :: "'a" shows "negligible (cbox a b) = (box a b = {})"
adal_w': shows "(ad, al) \<in> action_loc P E w'"
C_2: fixes x :: "'b" shows "norm (g' x) \<le> norm x * C"
fun_rel_id: fixes R1 :: "('b \<times> 'b) set"   and R2 :: "('c \<times> 'c) set" assumes "R1 = Id"   and "R2 = Id" shows "R1 \<rightarrow> R2 = Id"
opt_bst_wpl2_eq_pair: fixes i :: "int"   and j :: "int" shows "opt_bst_wpl2 i j = (opt_bst2 i j, local.wpl i j (opt_bst2 i j))"
Sup_eqI: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> less_eq z y) \<Longrightarrow> less_eq x y" shows "Sup A = x"
setPair4: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "{a, b} = {c, d}"   and "a \<noteq> c" shows "a = d"
prems_4: shows "share (x # sb) \<S> a = Some False"
tensor_mat_is_assoc: fixes A :: "complex Matrix.mat"   and B :: "complex Matrix.mat"   and C :: "complex Matrix.mat" shows "A \<Otimes> (B \<Otimes> C) = A \<Otimes> B \<Otimes> C"
x_def: fixes xs :: "'b list"   and i :: "nat" shows "??.Newton_Interpolation.x xs i = xs ! i"