hcju/isabelleps huggingface dataset:
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)" shows "\<exists>a\<in>elts (\<AA>\<lparr>Obj\<rparr>). \<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = b"
stake_run: fixes r :: "'transition stream"   and p :: "'state" assumes "\<And>k. path (stake k r) p" shows "run r p"
surj_pair: fixes p :: "'b \<times> 'c" shows "\<exists>x y. p = (x, y)"
indep_in_subset: fixes \<E> :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X_insert.indep_in \<E> X"   and "Y \<subseteq> X" shows "X_insert.indep_in \<E> Y"
DERIV_isconst_all: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and y :: "real" assumes "\<forall>x. (f has_real_derivative 0) (at x)" shows "f x = f y"
le_funE: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "f \<le> g" obtains "f x \<le> g x"
fvP: fixes P :: "'s \<Rightarrow> real" assumes "sound P" shows "t P (v P) \<tturnstile> v P"
simps_10: fixes \<phi> :: "'a ltl"   and \<psi> :: "'a ltl"   and \<nu> :: "'a set" shows "af_G_letter (\<phi> U \<psi>) \<nu> = \<phi> U \<psi> and af_G_letter \<phi> \<nu> or af_G_letter \<psi> \<nu>"
matrix_to_iarray_nrows: fixes A :: "(('a, 'b) vec, 'c) vec" shows "nrows A = nrows_iarray (matrix_to_iarray A)"
assms: shows "run_one_step d i (s, vs, ves, $Nop) = (s', vs', res)"
distr_const: fixes c :: "'c"   and N :: "'c measure" assumes "c \<in> space N" shows "distr M N (\<lambda>x. c) = return N c"
V_def: shows "cf.V \<equiv> {u. \<exists>v. (u, v) \<in> cf.E \<or> (v, u) \<in> cf.E}"
fconverse_fconverse: shows "(r\<inverse>\<^sub>\<bullet>)\<inverse>\<^sub>\<bullet> = r"
subgroupI: fixes H :: "'a set" assumes "H \<subseteq> carrier R"   and "H \<noteq> {}"   and "\<And>a. a \<in> H \<Longrightarrow> \<ominus> a \<in> H"   and "\<And>a b. \<lbrakk>a \<in> H; b \<in> H\<rbrakk> \<Longrightarrow> a \<oplus> b \<in> H" shows "subgroup H (add_monoid R)"
f_Exec_Stream_nth: fixes n :: "nat"   and xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e" assumes "n < length xs" shows "f_Exec_Comp_Stream trans_fun xs c ! n = f_Exec_Comp trans_fun (xs \<down> Suc n) c"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus> y \<in> carrier R"
regular_closed_inf: fixes x :: "'a"   and y :: "'a" assumes "regular x"   and "regular y" shows "regular (x \<sqinter> y)"
orbit_perm_swap: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "orbit (perm_swap a b f) x = (a \<rightleftharpoons>\<^sub>F b) ` orbit f ((a \<rightleftharpoons>\<^sub>F b) x)"
submonoid_axioms: shows "submonoid Ker G (\<cdot>) \<one>"
stable_sort: fixes cmp :: "'b comparator"   and x :: "'b"   and xs :: "'b list" shows "filter (\<lambda>y. compare cmp x y = Equiv) (Sorting_Algorithms.sort cmp xs) = filter (\<lambda>y. compare cmp x y = Equiv) xs"
eq: shows "h \<circ> hom_induced p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0} (nsphere n) {x. x k \<le> 0} id \<circ> f = hom_induced p (subtopology (nsphere n) {x. x k \<le> 0}) {x. x k = 0} (nsphere n) {x. 0 \<le> x k} id"
image_chamber_system_image: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f" shows "ChamberComplex.chamber_system (f \<turnstile> X) = ChamberComplex.chamber_system Y \<inter> (f \<turnstile> X)"
eval\<kappa>_inject: fixes x :: "\<kappa>"   and y :: "\<kappa>" shows "(eval\<kappa> x = eval\<kappa> y) = (x = y)"
write_actions_change_prefix: fixes w :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and n :: "enat"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "w \<in> write_actions E"   and "ltake n E [\<approx>] ltake n E'"   and "enat w < n" shows "w \<in> write_actions E'"
polytope_convex_hull: fixes S :: "'b set" assumes "finite S" shows "polytope (convex hull S)"
simps_1: fixes e :: "'b equation" shows "subterms_of_lit (Pos e) = subterms_of_eq e"
impr_N_def: fixes a :: "'b N"   and b :: "'b N" shows "a r\<rightarrow> b \<equiv> Abs_N ((- Rep_N a + Rep_N b) \<sqinter> (0::'b))"
refl: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "dpo le" shows "le x x"
tab_simps_2: shows "src Tgh.\<rho>\<sigma>.tab = src (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0)"
s_top_l_unit: fixes G :: "('e, 'f) Group_scheme"   and K :: "'e set" assumes "Group G"   and "G \<guillemotright> K " shows "{\<one>\<^bsub>G\<^esub>} \<diamondop>\<^bsub>G\<^esub> K = K"
zip_option_elims_2: fixes xs :: "'e list"   and ys :: "'f list" assumes "zip_option xs ys = None" obtains "length xs \<noteq> length ys"
degree_offset_poly: fixes p :: "'a poly"   and h :: "'a" shows "degree (offset_poly p h) = degree p"
assms: shows "K\<^bsub>m,n\<^esub> (with_proj G)"
span_insert_zero: fixes B :: "'a set" shows "ideal (insert (0::'a) B) = ideal B"
inject_2: fixes x21 :: "tyvrs"   and x22 :: "ty"   and y21 :: "tyvrs"   and y22 :: "ty" shows "(TVarB_Rep x21 x22 = TVarB_Rep y21 y22) = (x21 = y21 \<and> x22 = y22)"
vec_last_carrier: fixes v :: "'b vec"   and n :: "nat" shows "vec_last v n \<in> carrier_vec n"
invariant_1_2_of_rat: fixes rai :: "int poly \<times> rat \<times> rat"   and x :: "rat" assumes "invariant_1_2 rai" shows "real_of_1 rai \<noteq> real_of_rat x"
us: shows "us = getFrN (map snd txs) (map fst txs) [imp \<phi>1 \<phi>2] (length txs)"
prod_mono2: fixes B :: "'e set"   and A :: "'e set"   and f :: "'e \<Rightarrow> 'f" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> (1::'f) \<le> f b"   and "\<And>a. a \<in> A \<Longrightarrow> (0::'f) \<le> f a" shows "prod f A \<le> prod f B"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "weak_lattice L"   and "weak_partial_order_bottom L"   and "weak_partial_order_top L" shows "weak_bounded_lattice L"
normalize_simps_11: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "Language.normalize (Catch c\<^sub>1 c\<^sub>2) = Catch (Language.normalize c\<^sub>1) (Language.normalize c\<^sub>2)"
elts_of_set: fixes X :: "V set" shows "elts (set X) = (if small X \<Rightarrow> X | otherwise \<Rightarrow> {})"
partition_between_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and lo :: "nat"   and hi :: "nat"   and xs0 :: "'c list" shows "partition_between R h lo hi xs0 = ASSERT (hi < length xs0 \<and> lo \<le> hi) \<bind> (\<lambda>_. choose_pivot R h xs0 lo hi \<bind> (\<lambda>k. ASSERT (k < length xs0) \<bind> (\<lambda>_. RETURN (swap xs0 k hi) \<bind> (\<lambda>xs. ASSERT (length xs = length xs0) \<bind> (\<lambda>_. partition_main R h lo hi xs)))))"
assms_2: shows "\<FF>\<lparr>ObjMap\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
INF_identity_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and A :: "'b set" shows "Inf {x |. x \<in> A} = Inf A"
exhaust: fixes y :: "primfun" obtains x :: "nat" where "y = LnChain x"
module_hom_scale_self: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'c" assumes "module scale" shows "module_hom scale scale (scale c)"
hv_add: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec" shows "Hv (x + y) = Hv x + Hv y"
w: shows "w \<equiv> the_inv_into (P .\<^bold>\<le> x) f (\<Union> (f ` a))"
hyps_2: assumes "#p = 0" shows "\<exists>p'. (p = 0) = (p' = 0) \<and> degree p' \<le> degree p \<and> p = Polynomial.smult (of_nat CARD('b)) (#p')"
paq_Int_cs: shows "path_image q \<inter> closed_segment (a - complex_of_real d) (a + complex_of_real e) = {a - complex_of_real d, a + complex_of_real e}"
map_comp: fixes g :: "'g \<Rightarrow> 'e"   and f :: "'h \<Rightarrow> 'g"   and v :: "'h set['f]" shows "map_bset g (map_bset f v) = map_bset (g \<circ> f) v"
new_def: fixes A :: "'c set" shows "new A = (SOME a. a \<notin> A \<and> a \<noteq> Null)"
cyclic: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "ccw p q r" shows "ccw q r p"
in_keys: fixes I :: "(nat \<Rightarrow>\<^sub>0 nat) set"   and h :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> int" assumes "finite I" shows "keys (mapping_of (\<Sum>x\<in>I. MPoly_Type.monom x (h x))) = (\<Union>x\<in>I. if h x = 0 then {} else {x})"
simps_2: fixes U :: "('b, 'g) aezfun set"   and Us :: "('b, 'g) aezfun set list" shows "remisodups (U # Us) = (if \<exists>W\<in>set Us. \<exists>T. FGModuleIso G (*) U (*) T W then remisodups Us else U # remisodups Us)"
le_Suc': fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "less_eq (plus a one) b"   and "mod_type.to_nat Rep a + 1 < CARD('d)" shows "less a b"
assms_6: shows "\<langle>w|w\<rangle> = 1"
distinct_17: shows "cospan_shape.AA \<noteq> cospan_shape.BT"
fprod_member: fixes x :: "'a"   and xs :: "'a fset"   and y :: "'b"   and ys :: "'b fset" assumes "x |\<in>| xs"   and "y |\<in>| ys" shows "(x, y) |\<in>| xs |\<times>| ys"
profile_non_empty: fixes A :: "'a set"   and Is :: "'b set"   and P :: "'b \<Rightarrow> ('a \<times> 'a) set" assumes "profile A Is P" shows "Is \<noteq> {}"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs"   and "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs known_ptr get_child_nodes get_child_nodes_locs"
AC_comm: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Squnion> b = b \<Squnion> a"
min_height_le_height: fixes t :: "'c tree" shows "min_height t \<le> height t"
act_needed_asses_submap_succ_submap: fixes a :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s2 :: "('a, 'b) fmap"   and s1 :: "('a, 'b) fmap" assumes "action_needed_asses a s2 \<subseteq>\<^sub>f action_needed_asses a s1"   and "s1 \<subseteq>\<^sub>f s2" shows "state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a"
nth_append_simps_1: fixes i :: "nat"   and xs :: "'a list"   and ys :: "'a list" assumes "i < length xs" shows "(xs @ ys) ! i = xs ! i"
bangPres: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Rel' :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> Rel"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>\<^sup>^<Rel> S"   and "\<And>R S T U. \<lbrakk>(R, S) \<in> Rel; (T, U) \<in> Rel'\<rbrakk> \<Longrightarrow> (R \<parallel> T, S \<parallel> U) \<in> Rel'"   and "bangRel Rel \<subseteq> Rel'"   and "\<And>R S. (R \<parallel> !R, S) \<in> Rel' \<Longrightarrow> (!R, S) \<in> Rel'" shows "!P \<leadsto>\<^sup>^<Rel'> !Q"
monad_fail_envT': assumes "monad_fail Monad_Overloading.return Monad_Overloading.bind fail" shows "monad_fail Monad_Overloading.return Monad_Overloading.bind fail"
monom_mult_eq_zero_iff: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "(monom_mult c t p = 0) = (c = (0::'c) \<or> p = 0)"
StatImp: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "insertAssertion (extractFrame T) \<Psi>' \<hookrightarrow>\<^sub>F insertAssertion (extractFrame S) \<Psi>'"
SemSkip: fixes s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "s = t" shows " s , Skip \<rightarrow>\<^sub>1  t "
vfst_def: fixes p :: "V" shows "vfst p \<equiv> THE x. \<exists>y. p = \<langle>x, y\<rangle>"
R': shows "R' \<subseteq> {..n} \<times> {..n}"
r_tl_prim: shows "prim_recfn 1 r_tl"
simps_1: fixes at :: "('fsym, 'psym) atm" shows "glitOfL (Pos at) = {grdLit x |x. x \<in> varsA at \<and> (prot (GE.tpOfV x) \<or> protFw (GE.tpOfV x) \<and> x \<in> nvA at)}"
f: shows "(f \<longlongrightarrow> - \<infinity>) F"
open_subopen: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "open S = (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
assms: shows "f \<in> space\<^sub>N bcontfun\<^sub>N"
NCol_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col A B C \<and> \<not> Col A C B \<and> \<not> Col B A C \<and> \<not> Col B C A \<and> \<not> Col C A B \<and> \<not> Col C B A"
set_def: fixes X :: "V set" shows "set X \<equiv> if small X then inv elts X else inv elts {}"
wt_elim_cases_10: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (BinOp binop e1 e2)\<Colon>T" obtains T1 :: "ty" and T2 :: "ty" where "T = Inl (PrimT (binop_type binop))" and "E,dt\<Turnstile>e1\<Colon>-T1" and "E,dt\<Turnstile>e2\<Colon>-T2" and "wt_binop (prg E) binop T1 T2"
lex_fun_trans: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c"   and u :: "'a \<Rightarrow> 'c" assumes "rlex_fun t s"   and "rlex_fun u t" shows "rlex_fun u s"
as_needed_asses_submap_exec_vii: fixes f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap" assumes "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v" shows "f \<subseteq>\<^sub>f g"
simp_el_el: fixes b :: "'b list"   and G :: "'b list set" assumes " b \<in>B G " shows "b \<in> G"
simps: fixes i :: "int"   and j :: "int" shows "opt_bst_wpl2 i j = (if j < i then (\<langle>\<rangle>, 0) else if i = j then (\<langle>\<langle>\<rangle>, i, \<langle>\<rangle>\<rangle>, w i i) else let l = root (fst (opt_bst_wpl2 i (j - 1))); r = root (fst (opt_bst_wpl2 (i + 1) j)) in argmin snd (map (\<lambda>k. let (tl, wl) = opt_bst_wpl2 i (k - 1); (tr, wr) = opt_bst_wpl2 (k + 1) j in (\<langle>tl, k, tr\<rangle>, wl + wr + w i j)) [l..r]))"
l_adopt_node_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and adopt_node :: "(_) document_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and adopt_node_locs :: "(_) object_ptr option \<Rightarrow> (_) document_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "l_adopt_node type_wf known_ptr known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_adopt_node_axioms type_wf known_ptrs get_parent adopt_node adopt_node_locs get_child_nodes get_owner_document"
sl'_def: shows "sl' = map fst (stake i0 \<pi>')"
iso_verts_iso_inv: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism"   and u :: "'e" assumes "digraph_isomorphism hom"   and "u \<in> verts (app_iso hom G)" shows "iso_verts hom (iso_verts (inv_iso hom) u) = u"
sorted_sort_id: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sort_key less_eq (\<lambda>x. x) xs = xs"
ring_hom_uminus: fixes S :: "('c, 'd) ring_scheme"   and f :: "'c \<Rightarrow> 'a"   and a :: "'c" assumes "ring S"   and "f \<in> ring_hom S R"   and "a \<in> carrier S" shows "f (\<ominus>\<^bsub>S\<^esub> a) = \<ominus> f a"
WellOrderedStrategies_axioms: shows "WellOrderedStrategies G (S - W) p good r"
ihpex': shows "hp1' ## hp2' \<and> hp' = hp1' + hp2' \<and> I hp1' \<and> I' hp2'"
fps_const_add_left: fixes c :: "'a"   and f :: "'a fps" shows "fps_const c + f = Abs_fps (\<lambda>n. if n = 0 then c + fps_nth f 0 else fps_nth f n)"
assms_3: fixes m :: "'a"   and n :: "'a" assumes "m \<le> n" shows "F n \<subseteq> F m"
unfold_congs_5: fixes r :: "('b, 'c) state_scheme"   and r' :: "('b, 'c) state_scheme"   and v' :: "'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "r = r'"   and "tortoise r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "tortoise_update f r = tortoise_update f' r'"
inverse_eq_iff_eq: fixes a :: "'a"   and b :: "'a" shows "(inverse a = inverse b) = (a = b)"
vpeq_symmetric: shows "\<forall>a b u. vpeq u a b \<longrightarrow> vpeq u b a"
fps_lr_inverse_eq0_imp_starting0_1: fixes f :: "'c fps"   and x :: "'c" assumes "fps_left_inverse f x = 0" shows "x = (0::'c)"
simps_5: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "summands ([a\<frown>b]P) = {}"
frag_extend_cmul: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and c :: "int"   and x :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend f (frag_cmul c x) = frag_cmul c (frag_extend f x)"
is_dyn_box_operand: fixes x :: "('dyn, 'ubx1, 'ubx2) unboxed" shows "is_dyn_operand (box_operand x)"
measure_empty: fixes M :: "'a measure" shows "Sigma_Algebra.measure M {} = 0"
disc_4: shows "\<not> is_NONSAT_PUSH' RELABEL'"
correct_subseqs_foldr_impl_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and impl :: "('c, 'd, 'e) subseqs_foldr_impl"   and invariant :: "'d \<Rightarrow> 'c list \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> bool" shows "correct_subseqs_foldr_impl f impl invariant \<equiv> (\<forall>base elements n out state. subseqs_foldr impl base elements n = (out, state) \<longrightarrow> invariant base elements n state \<and> set out = subseqs_foldr_impl.S f base elements n) \<and> (\<forall>state out state' base elements n. next_subseqs_foldr impl state = (out, state') \<longrightarrow> invariant base elements n state \<longrightarrow> invariant base elements (Suc n) state' \<and> set out = subseqs_foldr_impl.S f base elements (Suc n))"
nodes_edges_consistent_def: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" shows "nodes_edges_consistent V E \<equiv> finite V \<and> irrefl E \<and> sym E \<and> E \<subseteq> V \<times> V"
unfold_ind: fixes P :: "('b \<rightarrow> 'c LList) \<Rightarrow> bool"   and h :: "'b \<rightarrow> ('c, 'b) Step" assumes "adm P"   and "P \<bottom>"   and "\<And>u. P u \<Longrightarrow> P (unfoldF\<cdot>h\<cdot>u)" shows "P (unfold\<cdot>h)"
fall_intro: fixes \<pi>l :: "('state \<times> 'aprop set) stream list"   and F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<And>\<pi>. \<lbrakk>wfp AP' \<pi>; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0\<rbrakk> \<Longrightarrow> F \<pi> \<pi>l" shows "fall F \<pi>l"
paint_rbt_sorted: fixes t :: "('a, 'c) rbt"   and c :: "color" assumes "rbt_sorted t" shows "rbt_sorted (paint c t)"
Nadd: fixes x :: "int \<times> int"   and y :: "int \<times> int" shows "INum (x +\<^sub>N y) = INum x + INum y"
f12_def: shows "f12 = fst (split_face oldF ram1 ram2 [])"
rel_sum_simps_2: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c"   and b2 :: "'f" shows "rel_sum R1 R2 (Inl a1) (Inr b2) = False"
Pred\<^sub>\<alpha>: fixes \<phi> :: "'f" shows "hereditarily_fs (Pred\<^sub>\<alpha> \<phi>)"
all_imp_alw: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xs :: "'c stream" assumes "\<And>xs. \<phi> xs" shows "alw \<phi> xs"
Suc_less_eq: fixes m :: "nat"   and n :: "nat" shows "(Suc m < Suc n) = (m < n)"
\<delta>_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
assms: fixes M :: "nat bdd list \<times> bool list"   and T :: "bool list list" assumes "fst (iter M T) \<Longrightarrow> P M (snd (iter M T))" shows "P M T"
scale_left_diff_distrib: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "(a - b) * x = a * x - b * x"
convergent_subseq_def: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int" shows "convergent_subseq s = take_subseq s (convergent_subseq_fun s (acc_point s))"
orthogonal_transformation_inv: fixes f :: "'a \<Rightarrow> 'a" assumes "orthogonal_transformation f" shows "orthogonal_transformation (inv f)"
a_consistent_alts_on_stack: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and b :: "bool"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "ArityAnalysisLetSafe Aexp Aheap"   and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (\<Gamma>, Bool b, Alts e1 e2 # S)" obtains a' :: "Arity" and as' :: "Arity list" where "as = a' # as'" and "a = 0"
concI_if_Nil2: fixes B :: "'b list set"   and xs :: "'b list"   and A :: "'b list set" assumes "[] \<in> B"   and "xs \<in> A" shows "xs \<in> A @@ B"
mk_base_unit: fixes a :: "'b itself"   and s :: "'c itself" shows "is_base_unit (mk_base_unit a s)"
in_uminus_image_iff: fixes x :: "'b"   and S :: "'b set" shows "(x \<in> uminus ` S) = (- x \<in> S)"
sets_F_mono: fixes \<Omega> :: "'c set"   and F :: "'d \<Rightarrow> 'c measure"   and i :: "'d"   and j :: "'d" assumes "filtration \<Omega> F"   and "i \<le> j" shows "sets (F i) \<subseteq> sets (F j)"
int_of_rat_2: fixes z :: "rat" assumes "z \<in> \<int>" shows "rat_of_int (int_of_rat z) = z"
simps_3: fixes x :: "name"   and t :: "nterm"   and env :: "(name, nterm) fmap" shows "subst (\<Lambda>\<^sub>n x. t) env = (\<Lambda>\<^sub>n x. subst t (fmdrop x env))"
index_map_mat_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and f :: "'c \<Rightarrow> 'd" assumes "i < dim_row A"   and "j < dim_col A" shows "map_mat f A $$ (i, j) = f (A $$ (i, j))"
map_IInf_def: fixes F :: "'d upset \<Rightarrow> 'e upset" shows "map_IInf F = IInf \<circ> F \<circ> us"
showpre7: fixes q :: "int"   and p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and n :: "nat" assumes "0 < q"   and "0 < p"   and "\<forall>n. 0 < a n"   and "\<forall>n. 0 < b n"   and "(\<lambda>n. real_of_int (b (n + 1)) / real_of_int (a (n + 1))) sums (real_of_int p / real_of_int q)" shows "real_of_int (q * (\<Prod>j = 1..n. of_int (a j))) * (\<Sum>j. real_of_int (b (j + n + 1)) / real_of_int (a (j + n + 1))) = (\<Prod>j = 1..n. real_of_int (a j)) * (real_of_int p - real_of_int q * (\<Sum>j = 1..n. real_of_int (b j) / real_of_int (a j)))"
Trg_Dom: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Trg trg (bicategorical_language.Dom V src trg t) = bicategorical_language.Trg trg t"
assms_1: shows "LeftDerivation [\<SS>] D (N # v)"
intros_1: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" shows "r\<^sup>*\<^sup>* a a"