hcju/isabelleps huggingface dataset:
scale_zero_right: fixes a :: "'a" shows "a *s 0 = 0"
pbernpoly_eq_bernpoly: fixes x :: "real"   and n :: "nat" assumes "x \<in> {0..<1}" shows "pbernpoly n x = bernpoly n x"
pj_surj_to: fixes R :: "('d, 'e) Ring_scheme"   and J :: "'d set"   and X :: "'d set" assumes "Ring R"   and "ideal R J"   and "X \<in> carrier (R /\<^sub>r J)" shows "\<exists>r\<in>carrier R. pj R J r = X"
fields_table_SomeD': fixes G :: "prog"   and C :: "qtname"   and P :: "fspec \<Rightarrow> Decl.field \<Rightarrow> bool"   and fn :: "fspec"   and T :: "ty" assumes "fields_table G C P fn = Some T" shows "\<exists>f. (fn, f) \<in> set (DeclConcepts.fields G C) \<and> type f = T"
runit_commutes_with_R: fixes f :: "'a" assumes "ide f" shows "\<r>[R f] = R \<r>[f]"
leg0_in_hom_2: shows "\<guillemotleft>p\<^sub>0 : p\<^sub>0 \<Rightarrow> p\<^sub>0\<guillemotright>"
simps_23: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Call_indirect i) ts = (if table \<C> \<noteq> None \<and> i < length (types_t \<C>) then case types_t \<C> ! i of tn _> tm \<Rightarrow> type_update ts (to_ct_list (tn @ [T_i32])) (Type tm) else Bot)"
bind_pause: fixes out :: "'o"   and c :: "'i \<Rightarrow> 'm"   and f :: "'a \<times> 'w list \<Rightarrow> 'm" shows "bind (pause out c) f = pause out (\<lambda>i. bind (c i) f)"
SCND: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "set_mset \<Gamma> \<union> \<^bold>\<not> ` set_mset \<Delta> \<turnstile> \<bottom>"
Infinitesimal_HComplex: fixes x :: "real star"   and y :: "real star" assumes "x \<in> Infinitesimal"   and "y \<in> Infinitesimal" shows "HComplex x y \<in> Infinitesimal"
bv_msb_def: fixes w :: "bit list" shows "bv_msb w = (if w = [] then \<zero> else hd w)"
prems_2: fixes i :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> i = Some f'" shows "dfree f'"
g_bounds2: obtains c4 :: "real" where "\<And>x i. \<lbrakk>x\<^sub>1 \<le> x; i < k\<rbrakk> \<Longrightarrow> g_approx i x \<le> c4 * g x" and "0 < c4"
nth_insert_nth: fixes i :: "nat"   and xs :: "'b list"   and j :: "nat"   and x :: "'b" assumes "i \<le> length xs"   and "j \<le> length xs" shows "insert_nth j x xs ! i = (if i = j then x else if i < j then xs ! i else xs ! (i - 1))"
swap_cancel: fixes x :: "'sz"   and y :: "'sz"   and z :: "'sz" shows "swap x y (swap x y z) = z"
shift_left_inj: fixes xs :: "'b list"   and s1 :: "'b stream"   and s2 :: "'b stream" shows "(xs @- s1 = xs @- s2) = (s1 = s2)"
lappend_iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b llist" shows "iterates f x @@ xs = iterates f x"
UNC_I: fixes A :: "('b \<times> 'b) set" assumes "\<And>a b. \<lbrakk>a \<in> NF A; b \<in> NF A; (a, b) \<in> A\<^sup>\<leftrightarrow>\<^sup>*\<rbrakk> \<Longrightarrow> a = b" shows "UNC A"
negate_eq1: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "- x = - 1 \<cdot> x"
induct: fixes P :: "'b match_expr \<Rightarrow> bool"   and a0 :: "'b match_expr" assumes "P MatchAny"   and "\<And>uu_. P (Match uu_)"   and "P (MatchNot MatchAny)"   and "\<And>uv_. P (MatchNot (Match uv_))"   and "\<And>m. P m \<Longrightarrow> P (MatchNot (MatchNot m))"   and "\<And>m1 m2. \<lbrakk>P (MatchNot m1); P (MatchNot m2)\<rbrakk> \<Longrightarrow> P (MatchNot (MatchAnd m1 m2))"   and "\<And>m1 m2. \<lbrakk>P m1; P m2\<rbrakk> \<Longrightarrow> P (MatchAnd m1 m2)" shows "P a0"
max_absorb2: fixes x :: "'c"   and y :: "'c" assumes "x \<le> y" shows "max x y = y"
opposite_dir_8: fixes a :: "'a"   and b :: "'a option" shows "\<rhd>\<^sub>u\<^sub>b (\<lambda>x y. y < x) a b = \<lhd>\<^sub>l\<^sub>b (<) a b"
hyps: fixes v :: "'v" shows "is_path v' (rev_path xs) v = is_path v xs v'"
k: shows "(r|\<sigma>| -s r \<down>l \<beta>, {#\<alpha>#}) \<in> mul_eq r"
cancel_append_same2: fixes xs :: "'d list"   and ys :: "'d list"   and zs :: "'d list" assumes "cancel xs ys" shows "cancel (xs @ zs) (ys @ zs)"
2: shows "g = ZFfun (|dom|f) (|cod|f) ((|@|) g)"
bij_betw_quotient_domain_and_range_permutation_eq: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // domain_and_range_permutation A B = {{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B}}"
distinct_sorted_difference: fixes xs :: "'c list"   and ys :: "'c list" assumes "distinct xs"   and "distinct ys"   and "sorted xs"   and "sorted ys" shows "distinct (List_local.difference xs ys) \<and> sorted (List_local.difference xs ys)"
distinct_map: fixes r :: "Vertex \<Rightarrow> Vertex"   and a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex"   and f :: "Vertex" assumes "r \<in> complex_rotations"   and "distinct [a, b, c, d]"   and "\<forall>e\<in>{a, b, c}. r e \<noteq> f" shows "r d = f"
foldl_spmf_append: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and p :: "'c spmf"   and xs :: "'d list"   and ys :: "'d list" shows "foldl_spmf f p (xs @ ys) = foldl_spmf f (foldl_spmf f p xs) ys"
locally_respects: shows "\<forall>a s u. \<not> ifp (current s) u \<and> kprecondition s a \<longrightarrow> vpeq u s (kstep s a)"
Rep_S_inverse: fixes x :: "('b, 'd, 'e, 'f, 'g, 'h, 'i) S" shows "Abs_S (Rep_S x) = x"
nat_helper2: fixes xs :: "bit list" shows "nat_to_bv (2 ^ length xs + bv_to_nat xs) = \<one> # xs"
elims: fixes x :: "'b nel"   and y :: "'b list" assumes "n2l x = y" obtains xa :: "'b" where "x = NE_One xa__" and "y = [xa__]"   | xa :: "'b" and xs :: "'b nel" where "x = NE_Cons xa xs" and "y = xa # n2l xs"
recurrence_time_flip_sign_2: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and T :: "real"   and x :: "'b" assumes "auto_ll_on_open f X"   and "T \<in> auto_ll_on_open.existence_ivl0 f X x"   and "auto_ll_on_open.flow0 f X x T = x" shows "auto_ll_on_open.flow0 f X x (- T) = x"
factorization_lattice_diag_nonzero: fixes d :: "nat"   and u :: "'a poly"   and k :: "'a"   and i :: "nat" assumes "d = degree u"   and "d \<le> n"   and "u \<noteq> 0"   and "k \<noteq> (0::'a)"   and "i < n" shows "factorization_lattice u (n - d) k ! i $ i \<noteq> (0::'a)"
\<Phi>_hps1: fixes h :: "'b heap" shows "\<Phi>_hps [h] = \<Phi> h"
is_aligned_no_overflow'': fixes x :: "'b word"   and n :: "nat" assumes "is_aligned x n"   and "x + 2 ^ n \<noteq> 0" shows "x \<le> x + 2 ^ n"
prems: shows "x \<in> fv\<^sub>s\<^sub>s\<^sub>t (s # S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<delta>)"
fmlookup_default_def: fixes d :: "'c"   and m :: "('d, 'c) fmap"   and x :: "'d" shows "fmlookup_default d m x = (case fmlookup m x of None \<Rightarrow> d | Some v \<Rightarrow> v)"
execn_mono: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate"   and m :: "nat" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "n \<le> m" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =m\<Rightarrow> t"
arc_injective: fixes x :: "'a" assumes "arc x" shows "injective x"
prob_some: fixes x :: "'a" assumes "x \<in> s" shows "Sigma_Algebra.measure M {x} = Sigma_Algebra.measure M {SOME y. y \<in> s}"
proper_subterm_red_def: fixes t :: "'a trm"   and S :: "'a eclause set"   and \<sigma> :: "('a \<times> 'a trm) list" shows "proper_subterm_red t S \<sigma> = (\<exists>p s. subterm t p s \<and> p \<noteq> [] \<and> trm_rep (s \<lhd> \<sigma>) S \<noteq> s \<lhd> \<sigma>)"
assms_4: shows "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>"
document_ptr_kinds_def: fixes heap :: "(_) heap" shows "document_ptr_kinds heap = the |`| cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| ffilter is_document_ptr_kind (object_ptr_kinds heap)"
indep_setsI: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'a set set" assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> events"   and "\<And>A J. \<lbrakk>J \<noteq> {}; J \<subseteq> I; finite J; \<forall>j\<in>J. A j \<in> F j\<rbrakk> \<Longrightarrow> prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))" shows "indep_sets F I"
11070: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- - x \<sqinter> y) \<squnion> - x \<sqinter> - z = - (- - x \<sqinter> y)"
extend_map_affine_to_sphere1: fixes K :: "'a set"   and U :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and S :: "'a set"   and L :: "'a set" assumes "finite K"   and "affine U"   and "continuous_on (U - K) f"   and "f ` (U - K) \<subseteq> T"   and "\<And>C. \<lbrakk>C \<in> components (U - S); C \<inter> K \<noteq> {}\<rbrakk> \<Longrightarrow> C \<inter> L \<noteq> {}"   and "closedin (top_of_set U) S"   and "disjnt K S"   and "K \<subseteq> U" obtains g :: "'a \<Rightarrow> 'b" where "continuous_on (U - L) g" and "g ` (U - L) \<subseteq> T" and "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
loxodromic_repelling_fixed_point_2: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "Gromov_extension f (repelling_fixed_point f) = repelling_fixed_point f"
to_depressed_quartic: fixes a4 :: "'a"   and b :: "'a"   and a3 :: "'a"   and c :: "'a"   and a2 :: "'a"   and d :: "'a"   and a1 :: "'a"   and e :: "'a"   and a0 :: "'a"   and p :: "'a"   and q :: "'a"   and r :: "'a"   and x :: "'a"   and y :: "'a" assumes "a4 \<noteq> (0::'a)"   and "b = a3 / a4"   and "c = a2 / a4"   and "d = a1 / a4"   and "e = a0 / a4"   and "p = c - (3::'a) / (8::'a) * b\<^sup>2"   and "q = (b ^ 3 - (4::'a) * b * c + (8::'a) * d) / (8::'a)"   and "r = (- (3::'a) * b ^ 4 + (256::'a) * e - (64::'a) * b * d + (16::'a) * b\<^sup>2 * c) / (256::'a)"   and "x = y - b / (4::'a)" shows "(a4 * x ^ 4 + a3 * x ^ 3 + a2 * x\<^sup>2 + a1 * x + a0 = (0::'a)) = (y ^ 4 + p * y\<^sup>2 + q * y + r = (0::'a))"
collect_def: fixes F :: "('d \<Rightarrow> 'c set) set"   and x :: "'d" shows "BNF_Def.collect F x = (\<Union>f\<in>F. f x)"
IH: fixes y :: "nat" assumes "y < p"   and "prime y" shows "\<exists>c. Prime y \<in> set c \<and> valid_cert c \<and> real (length c) \<le> 6 * log 2 (real y) - 4 \<and> (\<forall>x\<in>set c. Pratt_Certificate.size_pratt x \<le> 3 * log 2 (real y))"
preimage_vimage_eq: fixes f :: "'c \<Rightarrow> 'd"   and U :: "'d set"   and U' :: "'d set"   and X :: "'c set" shows "f \<^sup>\<inverse> f -` U' U \<inter> X = f \<^sup>\<inverse> X (U \<inter> U')"
proots_within_empty: fixes p :: "'b poly" shows "proots_within p {} = {}"
simps_1: fixes a :: "('a, 'b) Node" shows "FingerTreeStruc.digitToTree (One a) = Single a"
accepting_pair\<^sub>R_simp: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and F :: "('c \<times> 'd \<times> 'c) set"   and I :: "('c \<times> 'd \<times> 'c) set"   and w :: "nat \<Rightarrow> 'd" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 (F, I) w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> F = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> I \<noteq> {}"
intP_def: fixes p :: "'psym"   and al :: "univ list" shows "intP p al \<equiv> eintP (Oldp p) al"
croot_0_1: fixes n :: "nat" shows "croot n 0 = 0"
PROJ_def: fixes i :: "nat"   and l :: "nat list" shows "PROJ i l = hd0 (drop i l)"
bst_of_list_snoc: fixes xs :: "'a list"   and y :: "'a" shows "bst_of_list (xs @ [y]) = Tree_Set.insert y (bst_of_list xs)"
zero_QuantT_def: shows "0 \<equiv> toQ \<lparr>mag = 0::'a, dim = QD('b), unit_sys = unit\<rparr>"
chine_assoc_props_1: shows "\<guillemotleft>\<nu>\<pi>\<rho>.chine_assoc : HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C H\<nu>H\<pi>\<rho>.chine\<guillemotright>"
first_bothNet_charn: fixes a :: "('b, 'c) Combinators" assumes "\<not> member DenyAll a" shows "first_bothNet a = {first_srcNet a, first_destNet a}"
Tsub: shows "T \<subseteq> {x. i \<bullet> x = 0}"
sup_inf_absorb: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<union># (x \<inter># y) = x"
even_count_lconjseq_alternating_order2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and s :: "'b"   and t :: "'b"   and k :: "nat"   and x :: "'b" assumes "class.group_add minus plus zero uminus"   and "plus s s = zero"   and "plus t t = zero"   and "power.power zero plus (plus s t) k = zero" shows "gcd_nat.greater_eq (count_list (group_add.lconjseq minus plus (alternating_list (2 * k) s t)) x) 2"
simps_3: fixes a :: "bool"   and as :: "bool list"   and b :: "bool"   and bs :: "bool list" shows "nfa_accepting' (a # as) (b # bs) = (a \<and> b \<or> nfa_accepting' as bs)"
elements_0_mat: fixes nr :: "nat"   and nc :: "nat" shows "elements_mat (0\<^sub>m nr nc) \<subseteq> {0::'a}"
simps: fixes x :: "'a"   and xs :: "'a list" shows "List.last (x # xs) = (if xs = [] then x else List.last xs)"
sample_uniform_def: fixes n :: "nat" shows "sample_uniform n = spmf_of_set {..<n}"
poly_fun_Hilbert_poly: fixes b :: "nat \<Rightarrow> nat" shows "poly_fun (Hilbert_poly b)"
assms: shows "\<xi> \<Turnstile>\<^sub>r \<phi>"
subresultant_smult_right: fixes c :: "'b"   and J :: "nat"   and f :: "'b poly"   and g :: "'b poly" assumes "c \<noteq> (0::'b)" shows "subresultant J f (Polynomial.smult c g) = Polynomial.smult (c ^ (degree f - J)) (subresultant J f g)"
Diff: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and a :: "'c set"   and b :: "'c set" assumes "ring_of_sets \<Omega> M"   and "a \<in> M"   and "b \<in> M" shows "a - b \<in> M"
A_fdia_var: fixes x :: "'a"   and y :: "'a" shows "A x y =  |x\<rangle> d y \<cdot> ad y"
unitary_def: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "unitary M = (mat_adj M *\<^sub>m\<^sub>m M = eye)"
upd_rev_cs_empty: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.upd_rev_cs kind cs [] = []" shows "cs = []"
empty_upd_none: fixes x :: "'a" shows "Map.empty(x := None) = Map.empty"
closed_Diff: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "topological_space.closed open S"   and "open T" shows "topological_space.closed open (S - T)"
fwi_invariant: fixes j :: "nat"   and n :: "nat"   and i'' :: "nat"   and i :: "nat"   and j'' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "j \<le> n"   and "i'' \<le> i"   and "j'' \<le> j" shows "fwi m n k i j i' j' \<le> fwi m n k i'' j'' i' j'"
nul_def: fixes mrexp3 :: "'b mrexp3" shows "nul mrexp3 = (case mrexp3 of One3 \<Rightarrow> True | Plus3 x41 x42 x43 x44 \<Rightarrow> x44 | Times3 x51 x52 x53 x54 \<Rightarrow> x54 | Star3 _dummy_ _dummya_ \<Rightarrow> True | _ \<Rightarrow> False)"
convex_scaling: fixes S :: "'a set"   and c :: "real" assumes "convex S" shows "convex ((*\<^sub>R) c ` S)"
Un_def: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = {x. x \<in> A \<or> x \<in> B}"
summable_pre_cos: fixes x :: "real" shows "summable (\<lambda>n. mssnth cos_series_stream n * x ^ n)"
scc_in_P: fixes scc :: "'val set"   and g :: "'g"   and P :: "'val set" assumes "scc \<in> condensation_nodes g P" shows "scc \<subseteq> P"
progress_e2: fixes \<S> :: "s_context"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list"   and s :: "s"   and k :: "nat"   and lholed :: "Lholed"   and j :: "nat" assumes "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts"   and "store_typing s \<S>"   and "Lfilled k lholed [$Br j] es" shows "j < k"
DF_DF: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i < j" shows "DF k i \<lless> DF k j"
iso_is_arr: fixes f :: "'a" assumes "B.iso f" shows "B.arr f"
prime_factorization_normalize: fixes x :: "'b" shows "prime_factorization (normalize x) = prime_factorization x"
PiE_singleton: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "f \<in> extensional A" shows "(\<Pi>\<^sub>E x\<in>A. {f x}) = {f}"
mult_numeral_1: fixes a :: "'b" shows "Numeral1 * a = a"
stalk\<rho>_eq: shows "\<forall>U V. U \<in> I \<and> V \<in> I \<longrightarrow> \<rho>1 U V = \<rho>2 U V"
collinear_translate_iff: fixes a :: "'b"   and A :: "'b set" shows "collinear ((+) a ` A) = collinear A"
preserves_reflects_arr: fixes f :: "'a" shows "S.arr (M.\<ll> f) = S.arr f"
conjugate_zero_iff: fixes conjugate :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "(conjugate a = zero) = (a = zero)"
assms_3: shows "vars1 \<union> vars2 \<union> vars3 \<subseteq> {0..<length dims}"
ordertype_VWF_eq_iff: fixes \<alpha> :: "V"   and A :: "V set" assumes "Ord \<alpha>"   and "small A" shows "(ordertype A VWF = \<alpha>) = (\<exists>f. bij_betw f A (elts \<alpha>) \<and> (\<forall>x\<in>A. \<forall>y\<in>A. (f x < f y) = ((x, y) \<in> VWF)))"
decode_plan_def: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" shows "\<Phi>\<inverse> \<Pi> \<A> t \<equiv> map (decode_plan' \<Pi> \<A>) [0..<t]"
RS_simps_2: shows "Maps.dom TfHgh.R\<^sub>0 = Maps.MkIde (src (tab\<^sub>0 f))"
sigma_finite_iff_density_finite: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (density M (\<lambda>x. ennreal (f x)))" shows "sigma_finite_measure (density (density M (\<lambda>x. ennreal (f x))) f) = (AE x in density M (\<lambda>x. ennreal (f x)). f x \<noteq> \<infinity>)"
chain: shows "(sterm_sem I \<theta> \<circ> (\<lambda>t. fst (mk_v I ODE (sol 0, b) (sol t))) has_derivative frechet I \<theta> (fst (mk_v I ODE (sol 0, b) (sol s))) \<circ> (\<lambda>t'. t' *\<^sub>R (\<chi>i. if i \<in> ODE_vars I ODE then ODE_sem I ODE (sol s) $ i else 0))) (at s within {0..t})"
gen_g_impl_rel_sv: fixes Rv :: "('a \<times> 'b set) set"   and Re :: "('c \<times> ('b \<times> 'b) set) set"   and Rv0 :: "('d \<times> 'b set) set"   and Rm :: "('e \<times> 'f) set" assumes "single_valued Rv"   and "single_valued Re"   and "single_valued Rv0"   and "single_valued Rm" shows "single_valued (\<langle>Rm, Rv, Re, Rv0\<rangle>gen_g_impl_rel_ext)"
m_o_def: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and opt :: "'a st option"   and X :: "char list set" assumes "Measure1 m h" shows "Measure1.m_o m h opt X = (case opt of None \<Rightarrow> h * card X + 1 | Some S \<Rightarrow> Measure1.m_s m S X)"
prod_into_DSTEP2: fixes S :: "'c"   and a :: "'d"   and G :: "('c \<times> ('c, 'd) RHS) list"   and L :: "('c + 'd) list"   and R :: "('c + 'd) list" assumes "(S, Leaf a) \<in> set G" shows "L \<cdot> [Inl S] \<cdot> R -G\<rightarrow> L \<cdot> [Inr a] \<cdot> R"
x_fresh_Y: shows "igFresh (asIMOD SEM) xs x Y"
lists_length_Suc_eq: fixes A :: "'a set"   and n :: "nat" shows "{xs. set xs \<subseteq> A \<and> length xs = Suc n} = (\<lambda>(xs, n). n # xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
part_emp: fixes J :: "'a set"   and P :: "'a set set" shows "part J (insert {} P) = part J P"
INF_lower2: fixes i :: "'b"   and A :: "'b set"   and u :: "'c"   and f :: "'b \<Rightarrow> 'c" assumes "i \<in> A"   and "u \<le> f i" shows "u \<le> \<Squnion> (f ` A)"
update_tableau_id: fixes update :: "nat \<Rightarrow> 'a \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and s :: "('b, 'a) state"   and x :: "nat"   and c :: "'a" assumes "Update update"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<T> (update x c s) = \<T> s"
pmdl_closed_spoly: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and F :: "('t \<Rightarrow>\<^sub>0 'd) set"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "p \<in> pmdl F"   and "q \<in> pmdl F" shows "spoly p q \<in> pmdl F"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs"   and "l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs" shows "l_set_mode_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs get_shadow_root get_shadow_root_locs"
smooth_on_scaleR: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. f x *\<^sub>R g x)"
can_assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set"   and c :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b"   and "partial_magma.ide (free_monoidal_category.comp C) c" shows "free_monoidal_category.can C (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C a b c)"
apply_operations_def: fixes es :: "('a \<times> ('a, 'b) operation) event list" shows "apply_operations es \<equiv> hb.apply_operations (node_deliver_messages es) (\<lambda>x. {})"
intros_8: fixes \<Gamma> :: "binding list"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "\<Gamma> \<turnstile> fs [:] fTs" shows "\<Gamma> \<turnstile> Rcd fs : RcdT fTs"
nat_mult_eq_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "0 < k" shows "(k * m = k * n) = (m = n)"
abc_step_red2: fixes s :: "nat"   and lm :: "nat list"   and p :: "abc_inst list"   and n :: "nat" shows "abc_steps_l (s, lm) p (Suc n) = (let (as', am') = abc_steps_l (s, lm) p n in abc_step_l (as', am') (abc_fetch as' p))"
zip_append: fixes xs :: "'c list"   and us :: "'d list"   and ys :: "'c list"   and vs :: "'d list" assumes "length xs = length us" shows "zip (xs @ ys) (us @ vs) = zip xs us @ zip ys vs"
prems_2: fixes "is" :: "nat list" assumes "is \<lhd> a # ds" shows "lookup_base (a # ds) v is = e is"
has_derivative_scaleR_fun: fixes y :: "'c \<Rightarrow> 'd"   and y' :: "'c \<Rightarrow> 'd"   and a :: "'c"   and A :: "'c set"   and x :: "real" assumes "(y has_derivative y') (at a within A)" shows "(x *\<^sub>R y has_derivative x *\<^sub>R y') (at a within A)"
id_assn_list: fixes xs :: "'b list"   and ys :: "'b list"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> list_assn id_assn xs ys" shows "xs = ys"
preserves_arr: fixes f :: "'a" assumes "B.arr f" shows "arr (UP f)"
preserves_cod: fixes f :: "'a" assumes "arr f" shows "cod \<t>[f] = \<Omega>.map (cod f)"
Q: shows "\<forall>sb. sb \<noteq> \<bottom> \<longrightarrow> unfold\<cdot>hb\<cdot>sb \<sqsubseteq> unfold\<cdot>(appendStep\<cdot>ha\<cdot>hb\<cdot>sb0)\<cdot>(Right\<cdot>sb)"
is_map_from_is_homomorphism: fixes U :: "'a set"   and V :: "'a set" assumes "is_open U"   and "is_open V"   and "V \<subseteq> U" shows "Set_Theory.map (imS.im_sheaf_morphisms U V) (imS.im_sheaf U) (imS.im_sheaf V)"