hcju/isabelleps huggingface dataset:
pbd_points_index1: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and ps :: "'b set" assumes "PBD point_set block_collection sizes"   and "ps \<subseteq> point_set"   and "card ps = 2" shows "block_collection index ps = 1"
admissible_image: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "'a set \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'a" assumes "partial_function_definitions le lub"   and "ccpo.admissible lub le (P \<circ> g)"   and "\<And>x y. f x = f y \<Longrightarrow> x = y"   and "\<And>x. f (g x) = x" shows "ccpo.admissible (img_lub f g lub) (img_ord f le) P"
f'_def: shows "f' = (\<lambda>z t. - (poch' z * complex_of_real (t + a) powr (- 1 - 2 * of_nat n - z) - Ln (complex_of_real (t + a)) * complex_of_real (t + a) powr (- 1 - 2 * of_nat n - z) * pochhammer z (Suc (2 * n))))"
i_expand_i_drop_mult: fixes n :: "nat"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" shows "f \<odot> k \<Up> (n * k) = f \<Up> n \<odot> k"
iso_incl_trans: fixes A :: "'c ring"   and B :: "'c ring"   and C :: "'c ring" assumes "A \<lesssim> B"   and "B \<lesssim> C" shows "A \<lesssim> C"
measure_measure_spmf_conv_measure_pmf: fixes p :: "'c spmf"   and A :: "'c set" shows "Sigma_Algebra.measure (measure_spmf p) A = measure_pmf.prob p (Some ` A)"
lift_def: fixes h :: "'g \<Rightarrow> 'f"   and inp :: "'e \<Rightarrow> 'g option" shows "lift h inp \<equiv> \<lambda>i. case inp i of None \<Rightarrow> None | Some v \<Rightarrow> Some (h v)"
andfold_MatchExp_not_disc_mapMatch: fixes disc :: "'d \<Rightarrow> bool"   and C :: "'c \<Rightarrow> 'd"   and ls :: "'c list" assumes "\<forall>a. \<not> disc (C a)" shows "\<not> has_disc disc (andfold_MatchExp (map (Match \<circ> C) ls))"
lnull_def: fixes llist :: "'c llist" shows "lnull llist = (llist = LNil)"
mask_Suc_0: shows "mask (Suc 0) = (1::'b)"
a: shows "weak_unit a \<and> ide a \<and> S.ide a"
fi_map_def: fixes x :: "'v" shows "fi_map x = (if x \<in> f ` V then Some (fi x) else None)"
G_UN: shows "Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle> = (\<lambda>g. BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S) `` {g})) ` FreeGroup S"
adm_POR_base_lf_rep: fixes r :: "(Three \<Rightarrow> ValD) admS dual \<times> (Three \<Rightarrow> ValD) admS" shows "adm (\<lambda>x. x \<in> POR_base_lf_rep r)"
local_ring_def: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "local_ring R addition multiplication zero unit \<equiv> ring R addition multiplication zero unit \<and> local_ring_axioms R addition multiplication zero unit"
aadd_0_l: fixes x :: "ant" shows "0 + x = x"
uniqify_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list" shows "map bsmapt (uniqify q) = SkewBinomialHeapStruc.uniqify (map bsmapt q)"
ide_hcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "C.ide \<nu>"   and "C.ide \<mu>"   and "src\<^sub>C \<nu> = trg\<^sub>C \<mu>" shows "C.ide (\<nu> \<star>\<^sub>C \<mu>)"
\<phi>: shows "wf_mformula \<sigma> j n R \<phi> \<phi>''"
bisim1BlockNone: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and V :: "nat"   and T :: "ty" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,{V:T=None; e},h \<turnstile> ({V:T=None; e'}, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
gp_k_eq_gp_x: shows "swap heap (parent l x) x (parent l (parent l k)) = heap (parent l (parent l x))"
C_msort_le: fixes xs :: "'c list"   and k :: "nat" assumes "length xs = 2 ^ k" shows "C_msort xs \<le> k * 2 ^ k"
\<sigma>_via_\<mu>': fixes n :: "nat"   and fs :: "'a vec list"   and l :: "nat"   and i :: "nat"   and j :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs" shows "gram_schmidt_fs_lin_indpt.\<sigma> n fs (Suc l) i j = (if l = 0 then gram_schmidt_fs_lin_indpt.\<mu>' n fs i 0 * gram_schmidt_fs_lin_indpt.\<mu>' n fs j 0 else (gram_schmidt_fs_lin_indpt.\<mu>' n fs l l * gram_schmidt_fs_lin_indpt.\<sigma> n fs l i j + gram_schmidt_fs_lin_indpt.\<mu>' n fs i l * gram_schmidt_fs_lin_indpt.\<mu>' n fs j l) / gram_schmidt_fs_lin_indpt.\<mu>' n fs (l - 1) (l - 1))"
collapse: fixes prog :: "('a, 'b, 'c) prog" shows "Prog (the_prog prog) = prog"
beta_coerce: fixes x :: "'f" shows "COERCE('f, 'e)\<cdot>x = PRJ('e)\<cdot>(EMB('f)\<cdot>x)"
finite_char_subset: fixes C :: "'b set set" assumes "subset_closed C" shows "C \<subseteq> mk_finite_char C"
fls_compose_power_X_intpow: fixes d :: "nat"   and n :: "int" assumes "0 < d" shows "fls_compose_power (fls_X_intpow n) d = fls_X_intpow (int d * n)"
mset_set_eq_iff: fixes A :: "'d set"   and B :: "'d set" assumes "finite A"   and "finite B" shows "(mset_set A = mset_set B) = (A = B)"
eval_finalId: fixes e :: "char list exp"   and P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap" assumes "final e" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e,s\<rangle>"
inverse_nonpositive_iff_nonpositive: 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" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) zero = less_eq a zero"
tree_iterate_fusion: fixes h :: "'a \<Rightarrow> 'b"   and l :: "'a \<Rightarrow> 'a"   and l' :: "'b \<Rightarrow> 'b"   and r :: "'a \<Rightarrow> 'a"   and r' :: "'b \<Rightarrow> 'b"   and x :: "'a" assumes "h \<circ> l = l' \<circ> h"   and "h \<circ> r = r' \<circ> h" shows "map_tree h (tree_iterate l r x) = tree_iterate l' r' (h x)"
dyn_accessible_from_static_declC: fixes G :: "prog"   and accC :: "qtname"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m in C dyn_accessible_from accC"   and "is_static m" shows "G \<turnstile> m in declclass m dyn_accessible_from accC"
hyps: assumes "distinct (map fst ps)" shows "distinct (map fst (delete_aux k ps))"
summable_def: fixes f :: "nat \<Rightarrow> 'a" shows "summable f = (\<exists>s. f sums s)"
domain_eq: shows "Domainp (pcr_rbt (=) (=)) = is_rbt"
polyn_ring_X_nonzero: shows "X \<noteq> \<zero>"
word_and_1: fixes n :: "'b word" shows "n AND 1 = (if n !! 0 then 1 else 0)"
measurable_If_restrict_space_iff: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "{x \<in> space M. P x} \<in> sets M" shows "((\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M N) = (f \<in> restrict_space M {x. P x} \<rightarrow>\<^sub>M N \<and> g \<in> restrict_space M {x. \<not> P x} \<rightarrow>\<^sub>M N)"
assms_1: shows "t \<turnstile> s1 \<approx> s2"
B_def: shows "\<B> = Abs (Abs (Abs (Var 2 \<degree> (Var 1 \<degree> Var 0))))"
a: shows "Vx : A \<turnstile> d\<equiv>e : B \<in> Axioms.Theory T"
power4_eq_xxxx: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'d" assumes "class.monoid_mult one times" shows "power.power one times x 4 = times (times (times x x) x) x"
unx_none_simps_2: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (free name) = None"
intros: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, e, S)"   and "\<And>x. x \<in> thunks \<Gamma> \<Longrightarrow> ae x = up\<cdot>0" shows "consistent (ae, a, as) (\<Gamma>, e, S)"
cases: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" obtains "x = DenyAll"   | xa :: "(int \<times> int \<times> protocol) set set" and y :: "(int \<times> int \<times> protocol) set set" where "x = DenyAllFromTo xa__ y__"   | xa :: "(int \<times> int \<times> protocol) set set" and y :: "(int \<times> int \<times> protocol) set set" and p :: "protocol \<times> int" where "x = AllowPortFromTo xaa__ ya__ p"   | xa :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" and y :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" where "x = xa \<oplus> y"
3: shows "strip (local.\<gamma>\<^sub>c C) = c"
real_vector_eq_affinity: fixes m :: "real"   and y :: "'c"   and x :: "'c"   and c :: "'c" assumes "m \<noteq> 0" shows "(y = m *\<^sub>R x + c) = (y /\<^sub>R m - c /\<^sub>R m = x)"
eval_cases_4: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains "e' = Val v" and "s' = s"
Base: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language" assumes "R x y" shows "regular_cong R x y"
tt_eqI: fixes v :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys p"   and "\<And>u. u \<in> keys p \<Longrightarrow> v \<preceq> u" shows "punit.tt p = v"
atom_ordering_total: shows "\<forall>x y. x \<noteq> y \<longrightarrow> (x, y) \<in> atom_ordering \<or> (y, x) \<in> atom_ordering"
theI: fixes P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
Zero_not_eq_j0: shows "quiver.Zero \<noteq> parallel_pair.j0"
sub_num_simps_5: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (num.Bit1 k) num.One = numeral.numeral one plus (num.Bit0 k)"
product_topology_def: fixes T :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "product_topology T I = topology_generated_by {Pi\<^sub>E I X |X. (\<forall>i. openin (T i) (X i)) \<and> finite {i. X i \<noteq> topspace (T i)}}"
span_is_subset: fixes S :: "complex vec set"   and N :: "complex vec set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule class_ring N V" shows "local.span S \<subseteq> N"
map_comp: fixes g :: "'e \<Rightarrow> 'c"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f set['d]" shows "map_bset g (map_bset f v) = map_bset (g \<circ> f) v"
if_case_mvup: fixes b :: "bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and p :: "'d \<times> 'e"   and e :: "'c" shows "(if b then case p of (x1, x2) \<Rightarrow> f x1 x2 else e) = (case p of (x1, x2) \<Rightarrow> if b then f x1 x2 else e)"
N_1: shows "N 1 = 1 / (q * (q - 1))"
mod: fixes a :: "nat" shows "\<chi> (a mod n) = \<chi> a"
filter_True: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "filter P xs = xs"
star_induct: fixes w :: "'b list"   and A :: "'b list set"   and P :: "'b list \<Rightarrow> bool" assumes "w \<in> star A"   and "P []"   and "\<And>u v. \<lbrakk>u \<in> A; v \<in> star A; P v\<rbrakk> \<Longrightarrow> P (u @ v)" shows "P w"
bounded: shows "\<exists>c. \<forall>x\<in>F. \<bar>f x\<bar> \<le> c * \<parallel>x\<parallel>"
simps_5: fixes r :: "'c rexp"   and s :: "'c rexp" shows "\<guillemotleft>Plus r s\<guillemotright> = PLUS (sorted_list_of_set (toplevel_summands (Plus \<guillemotleft>r\<guillemotright> \<guillemotleft>s\<guillemotright>)))"
degb: shows "b = 0 \<or> degree b < degree (#v)"
table_wf_table: fixes n :: "nat"   and A :: "nat set"   and X :: "'a option list set" shows "table n A X = wf_table n A (\<lambda>v. v \<in> X) X"
cEnum_sum_def: shows "CENUM('a + 'b) = (case ID CENUM('a) of None \<Rightarrow> None | Some (enum_a, enum_all_a, enum_ex_a) \<Rightarrow> case ID CENUM('b) of None \<Rightarrow> None | Some (enum_b, enum_all_b, enum_ex_b) \<Rightarrow> Some (map Inl enum_a @ map Inr enum_b, \<lambda>P. enum_all_a (\<lambda>x. P (Inl x)) \<and> enum_all_b (\<lambda>x. P (Inr x)), \<lambda>P. enum_ex_a (\<lambda>x. P (Inl x)) \<or> enum_ex_b (\<lambda>x. P (Inr x))))"
coplanar_perm_18: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Coplanar A B C D" shows "Coplanar D A B C"
assms_2: fixes x :: "'b"   and xs :: "'b list" assumes "l = x # xs" shows "fc x xs \<le> fc' x xs"
finite_measure_finite_Union: fixes S :: "'c set"   and A :: "'c \<Rightarrow> 'a set" assumes "finite S"   and "A ` S \<subseteq> events"   and "disjoint_family_on A S" shows "prob (\<Union> (A ` S)) = (\<Sum>i\<in>S. prob (A i))"
simps: fixes a1 :: "recf"   and a2 :: "nat" shows "primerec a1 a2 = (a1 = z \<and> a2 = Suc 0 \<or> a1 = s \<and> a2 = Suc 0 \<or> (\<exists>n m. a1 = recf.id m n \<and> a2 = m \<and> n < m) \<or> (\<exists>f k gs m n. a1 = Cn n f gs \<and> a2 = m \<and> primerec f k \<and> length gs = k \<and> (\<forall>i<length gs. primerec (gs ! i) m) \<and> m = n) \<or> (\<exists>f n g m. a1 = Pr n f g \<and> a2 = m \<and> primerec f n \<and> primerec g (Suc (Suc n)) \<and> m = Suc n))"
and_pre_def: fixes P1 :: "'e \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool" shows "and_pre P1 P2 \<equiv> \<lambda>x. P1 x \<and> P2 x"
complex_mult_cnj: fixes z :: "complex" shows "z * Complex.cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
lemDotCommute: fixes u :: "'a Vector"   and v :: "'a Vector" shows "(u dot v) = (v dot u)"
cases: fixes a1 :: "('b \<times> nat) option"   and a2 :: "'b" assumes "may_lock a1 a2" obtains t :: "'b" where "a1 = None" and "a2 = tb__"   | t :: "'b" and n :: "nat" where "a1 = \<lfloor>(t, n)\<rfloor>" and "a2 = t"
pw_ESPEC_1: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool" shows "nofail (ESPEC \<Phi> \<Psi>)"
AE_distrD: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and M' :: "'c measure"   and P :: "'c \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "almost_everywhere (distr M M' f) P" shows "AE x in M. P (f x)"
wf_clinit_def: fixes ms :: "'a mdecl list" shows "wf_clinit ms = (\<exists>m. (clinit, Static, [], Void, m) \<in> set ms)"
map_nth_eq_conv: fixes xs :: "'a list"   and ys :: "'b list"   and f :: "'a \<Rightarrow> 'b" assumes "length xs = length ys" shows "(map f xs = ys) = (\<forall>i<length ys. f (xs ! i) = ys ! i)"
simps_2: fixes v :: "'a list"   and va :: "'a list list" shows "UP.normalize (v # va) = (if lead_coeff (v # va) \<noteq> \<zero>\<^bsub>K [X]\<^esub> then v # va else UP.normalize (tl (v # va)))"
int_of_uint32_inv: fixes x :: "int" assumes "0 \<le> x"   and "x < 4294967296" shows "int_of_uint32 (uint32_of_int x) = x"
l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<equiv> (get_ancestors = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors get_parent \<and> get_ancestors_locs = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_ancestors_locs get_parent_locs) \<and> get_root_node = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node get_parent \<and> get_root_node_locs = l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs get_parent_locs"
measurable_inequality_set_2: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. f x < g x} \<in> sets M"
Dg1_is_bot_edge_pathimg: shows "path_image (\<lambda>x. twoC (1, x)) = {p. \<exists>x. x \<in> cbox a b \<and> p = (g1 x, x)}"
sp_from_tgt_in_extends_is_Nil: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and e :: "('d \<times> nat) edge"   and g' :: "('d \<times> nat, 'e) rgraph_scheme"   and es :: "('d \<times> nat) edge list"   and v :: "'d \<times> nat" assumes "sub_rel_of g subs"   and "ArcExt.extends g e g'"   and "SubRel.subpath g' (tgt e) es v subs" shows "es = []"
summable_mult_D: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "summable (\<lambda>n. c * f n)"   and "c \<noteq> (0::'b)" shows "summable f"
sum_squares_eq_zero_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "(plus (times x x) (times y y) = zero) = (x = zero \<and> y = zero)"
Red_I_\<G>_def: fixes N :: "'f set" shows "Red_I_\<G> N = {\<iota> \<in> Inf_F. \<G>_I \<iota> \<noteq> None \<and> the (\<G>_I \<iota>) \<subseteq> Red_I_G (\<G>_Fset N) \<or> \<G>_I \<iota> = None \<and> \<G>_F (concl_of \<iota>) \<subseteq> \<G>_Fset N \<union> Red_F_G (\<G>_Fset N)}"
extended_hyperb_ineq': fixes e :: "'a"   and x :: "'a Gromov_completion"   and y :: "'a Gromov_completion"   and z :: "'a Gromov_completion" shows "min (extended_Gromov_product_at e x y) (extended_Gromov_product_at e y z) \<le> extended_Gromov_product_at e x z + ereal (deltaG TYPE('a))"
floor_unique: fixes z :: "int"   and x :: "'b" assumes "of_int z \<le> x"   and "x < of_int z + (1::'b)" shows "\<lfloor>x\<rfloor> = z"
simps: fixes xs :: "'e"   and x :: "'d"   and X :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "FixVars.qGoodAbs (qAbs xs x X) = FixVars.qGood X"
dns1: fixes x :: "('a \<times> 'a) set" shows "rdom x ; x = x"
reads_bind_pure: fixes f :: "('x, 'y, 'z) prog"   and h :: "'x"   and h' :: "'x"   and S :: "('x \<Rightarrow> 'x \<Rightarrow> bool) set"   and g :: "'z \<Rightarrow> ('x, 'y, 'aa) prog" assumes "pure f h"   and "pure f h'"   and "reads S f h h'"   and "\<And>x. h \<turnstile> f \<rightarrow>\<^sub>r x \<Longrightarrow> reads S (g x) h h'" shows "reads S (Heap_Error_Monad.bind f g) h h'"
sel_2: fixes x1 :: "'d"   and x2 :: "'d stream" shows "stl (x1 ## x2) = x2"
A: fixes a :: "('a, 'b) OrdSum"   and b :: "('a, 'b) OrdSum" shows "(a l\<rightarrow> b) * a = a * (a r\<rightarrow> b)"
env_restr_env_restr: fixes d1 :: "'a set"   and d2 :: "'a set"   and x :: "'a \<Rightarrow> 'b" shows "x f|` d2 f|` d1 = x f|` (d1 \<inter> d2)"
semiring_norm_70: fixes m :: "num" shows "(num.Bit1 m \<le> num.One) = False"
B1: fixes x :: "'a" assumes "x \<in> B" shows "norm x = 1"
limit: shows "(\<lambda>b. (\<Sum>k = a..b. f (real k)) - F (real b) - (\<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i (real b))) \<longlonglongrightarrow> C"
valid_path_rev_aux_split_1: fixes cs :: "'edge list"   and as :: "'edge list"   and as' :: "'edge list" assumes "valid_path_rev_aux cs (as @ as')" shows "valid_path_rev_aux cs as'"
interleavings_filter: fixes X :: "'a set"   and Y :: "'a set"   and zs :: "'a list" assumes "X \<inter> Y = {}"   and "set zs \<subseteq> X \<union> Y" shows "interleavings (filter (\<lambda>z. z \<in> X) zs) (filter (\<lambda>z. z \<in> Y) zs) zs"
comp_simp: fixes g :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2"   and f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr \<times> 'a2" assumes "A2_B.seq (fst g) (fst f)"   and "A2.seq (snd g) (snd f)" shows "g \<cdot>\<^sub>[\<^sub>A\<^sub>2\<^sub>,\<^sub>B\<^sub>]\<^sub>x\<^sub>A\<^sub>2 f = (fst g \<cdot>\<^sub>[\<^sub>A\<^sub>2,\<^sub>B\<^sub>] fst f, snd g \<cdot>\<^sub>A\<^sub>2 snd f)"
Int: fixes a :: "'a set"   and M :: "'a measure"   and b :: "'a set" assumes "a \<in> sets M"   and "b \<in> sets M" shows "a \<inter> b \<in> sets M"
sup_right_top: fixes x :: "'b" shows "top_homomorphism ((\<squnion>) x)"
vreal_of_real_impl'_bij_betw: shows "bij_betw vreal_of_real_impl' (range vreal_of_real_impl) (elts vreal_impl)"
atms_empty_atms: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex" assumes "safe_regex m g r" shows "(atms r = {}) = (regex.atms r = {})"
28422: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) = - (- y \<sqinter> - x)"
cil_subset_il: fixes w :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and w' :: "'a list" shows "w \<otimes>\<^bsub>\<alpha>\<^esub> w' \<subseteq> w \<otimes> w'"
simps_2: fixes t :: "Core.term"   and u :: "Core.term"   and k :: "nat" shows "loose_bvar (t $ u) k = (loose_bvar t k \<or> loose_bvar u k)"
serial_strips_equivalent_to_serial_sas_plus: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<pi> :: "('a \<times> 'b) strips_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "STRIPS_Semantics.is_serial_solution_for_problem (\<phi> \<Psi> ) \<pi>" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (map (strips_op_to_sasp \<Psi>) \<pi>)"
id_iso: fixes r :: "('c \<times> 'c) set" shows "iso r r id"
le_ordertype_obtains_subset: fixes \<beta> :: "V"   and \<alpha> :: "V"   and H :: "V set" assumes "\<beta> \<le> \<alpha>"   and "tp H = \<alpha>"   and "small H"   and "Ord \<beta>" obtains G :: "V set" where "G \<subseteq> H" and "tp G = \<beta>"
ext: fixes g :: "'s"   and f :: "'s" assumes "g \<cdot> f \<noteq> null" shows "seq g f"
oguard_NF_wp: fixes f :: "'c \<Rightarrow> bool"   and P :: "unit \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalidNF (\<lambda>s. f s \<and> P () s) (oguard f) P"