hcju/isabelleps huggingface dataset:
proj2_pt_def: fixes v :: "(real, 2) vec" shows "proj2_pt v \<equiv> proj2_abs (vector2_append1 v)"
all_post: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and a3 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3" shows "\<forall>\<sigma> \<sigma>' \<sigma>''. ((\<sigma>, \<sigma>') \<Turnstile> PRE self a1 a2 a3) = ((\<sigma>, \<sigma>'') \<Turnstile> PRE self a1 a2 a3)"
dghm_ObjMap_vdomain: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
weakEmptyTrans: fixes P :: "ccs" shows "P \<Longrightarrow>\<^sup>^\<tau> \<prec> P"
nlists_not_empty: fixes A :: "'a set"   and n :: "nat" assumes "A \<noteq> {}" shows "nlists A n \<noteq> {}"
sgn_if: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "sgn x = (if x = zero then zero else if less zero x then one else uminus one)"
tranclD: fixes x :: "'c"   and y :: "'c"   and R :: "('c \<times> 'c) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R \<and> (z, y) \<in> R\<^sup>*"
find_lambda_def: shows "find_lambda \<equiv> (\<lambda>s. s\<lparr>carrots := 1, l := 1, tortoise := x0, hare := f x0\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) ((\<lambda>s. if carrots s = l s then s\<lparr>tortoise := hare s, carrots := 2 * carrots s, l := 0\<rparr> else id s) ;; (\<lambda>s. s\<lparr>hare := f (hare s), l := l s + 1\<rparr>))"
div_diff: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.idom_modulo times one plus zero minus uminus divide modulo"   and "dvd.dvd times c a"   and "dvd.dvd times c b" shows "divide (minus a b) c = minus (divide a c) (divide b c)"
set_mode_get_tag_name: fixes ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_mode_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
assms: shows "open_cover_of_open_subset S' is_open' U I C"
rel_distinct_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'d"   and y22 :: "'d list" shows "\<not> list_all2 R [] (y21 # y22)"
aboveS_def: fixes r :: "('c \<times> 'c) set"   and a :: "'c" shows "aboveS r a \<equiv> {b. b \<noteq> a \<and> (a, b) \<in> r}"
swaprows_mat_def: fixes n :: "nat"   and k :: "nat"   and l :: "nat" shows "swaprows_mat n k l = mat n n (\<lambda>(i, j). if k = i \<and> l = j \<or> k = j \<and> l = i \<or> i = j \<and> i \<noteq> k \<and> i \<noteq> l then 1::'a else (0::'a))"
simps: fixes n :: "char list"   and args :: "object list" shows "wf_plan_action (PAction n args) = (case resolve_action_schema n of None \<Rightarrow> False | Some a \<Rightarrow> action_params_match a args \<and> wf_effect_inst (ground_action.effect (instantiate_action_schema a args)))"
is_document_ptr_kind_not_element_ptr_kind: fixes ptr :: "(_) object_ptr" assumes "is_document_ptr_kind ptr" shows "\<not> is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"
star_prod_unfold: fixes x :: "'a"   and y :: "'a" shows "1' + x ; (y ; x)\<^sup>\<star> ; y = (x ; y)\<^sup>\<star>"
elims: fixes x :: "nat"   and y :: "item set" assumes "\<I> x = y" obtains k :: "nat" where "x = k" and "y = natUnion (\<J> k)"
ball_insert: fixes a :: "'b"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>insert a B. P x) = (P a \<and> (\<forall>x\<in>B. P x))"
RDirProd_list_hom2: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme" assumes "f \<in> ring_hom R S" shows "(\<lambda>a. [f a]) \<in> ring_hom R (RDirProd_list [S])"
restrict_star_unfold: fixes l :: "'c"   and f :: "'c \<times> 'c \<Rightarrow> 'd" shows "[l] \<langle>mone\<rangle> [l] \<oplus> [l] \<langle>f\<rangle> [l] \<odot> [l] \<langle>star \<circ> f\<rangle> [l] = [l] \<langle>star \<circ> f\<rangle> [l]"
alpha_Tree_eqvt_aux: fixes t :: "('a, 'b, 'c, 'd) Tree"   and p :: "perm" assumes "\<And>a b. ((a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> t) = (p \<bullet> (a \<rightleftharpoons> b) \<bullet> t =\<^sub>\<alpha> p \<bullet> t)" shows "p \<bullet> supp_rel (=\<^sub>\<alpha>) t = supp_rel (=\<^sub>\<alpha>) (p \<bullet> t)"
cyclic_on_orbit: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "f permutes S"   and "finite S" shows "cyclic_on f (orbit f x)"
fps_power_Suc_nth: fixes k :: "nat"   and f :: "'a fps"   and m :: "nat" assumes "0 < k" shows "fps_nth (f ^ Suc m) k = of_nat (Suc m) * (fps_nth f k * fps_nth f 0 ^ m) + (\<Sum>v | v \<in> natpermute k (m + 1) \<and> k \<notin> set v. \<Prod>j = 0..m. fps_nth f (v ! j))"
degree_X: assumes "\<one> \<noteq> \<zero>" shows "deg R (X_poly R) = 1"
preserves_comp_2: fixes A1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'b \<times> 'd \<Rightarrow> 'e"   and f2' :: "'d"   and f2 :: "'d" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
in_keys_iff: fixes x :: "'a"   and s :: "'a \<Rightarrow>\<^sub>0 'b" shows "(x \<in> keys s) = (lookup s x \<noteq> (0::'b))"
card_Pow: fixes A :: "'b set" assumes "finite A" shows "card (Pow A) = 2 ^ card A"
simps_8: fixes x1 :: "nat" shows "Em \<noteq> At x1"
neq: fixes u :: "real" assumes "u \<in> {0..1}" shows "(1 - x * y) * u \<noteq> 1"
MM2_def: shows "MM2 \<equiv> PM \<and> MM2_axioms"
well_ordered_iff_well_related: fixes X :: "'a set"   and a :: "'a" assumes "X \<subseteq> {a}" shows "well_ordered_set X (\<sqsubseteq>) = well_related_set X (\<sqsubseteq>)"
Grp_def: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
in_homE: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>1 b\<guillemotright>" obtains "C1.arr f" and "C1.dom f = a" and "C1.cod f = b"
dropWhile_eq_Nil_conv: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(dropWhile P xs = []) = (\<forall>x\<in>set xs. P x)"
stutter_equiv_def: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" shows "\<sigma> \<approx> \<tau> \<equiv> \<natural>\<sigma> = \<natural>\<tau>"
free_gen_mem: fixes P :: "('f \<times> 'g, 'a, 'h) Module_scheme"   and M :: "('f, 'a, 'h) Module_scheme"   and N :: "('g, 'a, 'h) Module_scheme"   and a :: "'f \<times> 'g" assumes "FM\<^bsub>R\<^esub> P M N"   and "a \<in> M \<times>\<^sub>c N" shows "a \<in> carrier P"
simulation2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tl2 :: "'t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "s1 \<approx>m s2"   and "r2.redT s2 tl2 s2'"   and "\<not> m\<tau>move2 s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. \<tau>mRed1 s1 s1' \<and> r1.redT s1' tl1 s1'' \<and> \<not> m\<tau>move1 s1' tl1 s1'' \<and> s1'' \<approx>m s2' \<and> tl1 \<sim>T tl2"
map_upt_add: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" shows "map f [0..<n + m] = map f [0..<n] @ map (\<lambda>i. f (i + n)) [0..<m]"
f_aggregate_empty_conv: fixes k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k" shows "(f_aggregate xs k ag = []) = (length xs < k)"
3: shows "carrier (add_monoid (residue_ring (p ^ k))) = carrier (residue_ring (p ^ k))"
enn2real_of_nat: fixes n :: "nat" shows "enn2real (of_nat n) = real n"
i_2: shows "segments_of_aform X ! i = (y, z)"
map_append: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'c list" shows "map f (xs \<cdot> ys) = map f xs \<cdot> map f ys"
eq_fin_alt_def: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2 = (traces.le_fin ind w\<^sub>1 w\<^sub>2 \<and> traces.le_fin ind w\<^sub>2 w\<^sub>1)"
lemScale1: fixes u :: "'a Vector" shows "(1::'a) ** u = u"
wall_crossings_snoc: fixes Cs :: "'a set list"   and D :: "'a set"   and E :: "'a set" shows "wall_crossings (Cs @ [D, E]) = wall_crossings (Cs @ [D]) @ [THE_default {} (\<lambda>H. H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}}) \<and> separated_by H D E)]"
3: shows "\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow> \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<rightarrow> \<langle>\<alpha>,Q'\<rangle> \<and> R P' Q'))"
comp\<^sub>2_correct: fixes P\<^sub>1 :: "nat exp prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and h :: "heap"   and ls :: "val list"   and sh :: "sheap"   and e' :: "nat exp"   and h' :: "heap"   and ls' :: "val list"   and sh' :: "sheap" assumes "wf_J\<^sub>1_prog P\<^sub>1"   and "P\<^sub>1 \<turnstile> C sees M, b :  Ts\<rightarrow>T = body in C"   and "P\<^sub>1 \<turnstile>\<^sub>1 \<langle>body,(h, ls, sh)\<rangle> \<Rightarrow> \<langle>e',(h', ls', sh')\<rangle>"   and "M \<noteq> clinit" shows "compP\<^sub>2 P\<^sub>1 \<turnstile> (None, h, [([], ls, C, M, 0, No_ics)], sh) -jvm\<rightarrow> (exception e', h', [], sh')"
IH: fixes x :: "nat"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and t' :: "pdf_type"   and e :: "cexpr" assumes "insert_var x \<Gamma> t \<turnstile>\<^sub>c e' : t'"   and "\<Gamma> \<turnstile>\<^sub>c e : t" shows "\<Gamma> \<turnstile>\<^sub>c cexpr_subst x e e' : t'"
ode_d_Suc_0_eq_ode_d2: fixes x :: "'d" assumes "x \<in> Csafe" shows "ode_d (Suc 0) x = blinfun_apply (ode_d2 x)"
ide_map\<^sub>0_obj: fixes a :: "'d" assumes "D.obj a" shows "D.ide (\<eta>\<^sub>0 a)"
Sups_least: fixes X :: "'a set"   and z :: "'a" assumes "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" shows "\<Squnion> X \<le> z"
matches_add_match_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>add_match m rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
sorted_add_spvec_helper2: fixes a :: "nat"   and b :: "'a"   and arr :: "(nat \<times> 'a) list"   and brr :: "(nat \<times> 'a) list"   and ab :: "nat"   and bb :: "'a"   and list :: "(nat \<times> 'a) list"   and aa :: "nat"   and ba :: "'a" assumes "add_spvec ((a, b) # arr) brr = (ab, bb) # list"   and "aa < a"   and "sorted_spvec ((aa, ba) # brr)" shows "aa < ab"
run_def: fixes e :: "exp"   and ob :: "obs" shows "e \<Down> ob \<equiv> (\<exists>v. e \<longrightarrow>* v \<and> observe v ob) \<or> (diverge e \<or> goes_wrong e) \<and> ob = OBad"
set_single_hd: fixes vs :: "'a list"   and v :: "'a" assumes "set vs = {v}" shows "hd vs = v"
we_n_df1: shows "we_n = nat_to_ce_set n"
poly_const_inv: fixes a :: "'d" shows "poly_inv (poly_const a)"
ins_ins2_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins2 n d x t = Bal2 t'" shows "ins n d x t = Bal t'"
axioms_1: fixes V\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>B :: "'c \<Rightarrow> 'c"   and src\<^sub>B :: "'c \<Rightarrow> 'c"   and trg\<^sub>B :: "'c \<Rightarrow> 'c"   and V\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a>\<^sub>C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i>\<^sub>C :: "'d \<Rightarrow> 'd"   and src\<^sub>C :: "'d \<Rightarrow> 'd"   and trg\<^sub>C :: "'d \<Rightarrow> 'd"   and V\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and H\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<a>\<^sub>D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<i>\<^sub>D :: "'e \<Rightarrow> 'e"   and src\<^sub>D :: "'e \<Rightarrow> 'e"   and trg\<^sub>D :: "'e \<Rightarrow> 'e"   and F :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>F :: "'c \<times> 'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<Phi>\<^sub>G :: "'c \<times> 'c \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'e"   and \<Phi>\<^sub>H :: "'d \<times> 'd \<Rightarrow> 'e"   and \<tau>\<^sub>0 :: "'c \<Rightarrow> 'd"   and \<tau>\<^sub>1 :: "'c \<Rightarrow> 'd" assumes "pseudonatural_transformation_whisker_left V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G H \<Phi>\<^sub>H \<tau>\<^sub>0 \<tau>\<^sub>1" shows "bicategory V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B"
Pythagoras: fixes A :: "'b"   and C :: "'b"   and B :: "'b" assumes "orthogonal (A - C) (B - C)" shows "(dist B C)\<^sup>2 + (dist C A)\<^sup>2 = (dist A B)\<^sup>2"
arcs_decomp: fixes xs :: "nat list"   and ys :: "nat list"   and y :: "nat"   and zs :: "nat list"   and x :: "nat"   and z :: "nat" assumes "xs = ys @ y # zs" shows "arcs x z xs = arcs x y ys @ arcs y z zs"
in_homI: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.dom f = a"   and "\<F>C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"
assm_3: fixes K :: "'a set" assumes "K \<in> d" shows "K \<noteq> {}"
r2: fixes x :: "'b" shows "(getRel S_Idt G `` {x} = {}) = (x \<notin> vertices G)"
simps_8: fixes v :: "('a, 'b) form" shows "compl (Forall v) = Neg (Forall v)"
sum_symmetric: fixes y :: "'a"   and x :: "'a" assumes "y = y\<^sup>T" shows "sum_class.sum (x\<^sup>T \<sqinter> y) = sum_class.sum (x \<sqinter> y)"
formula_signs_cases: fixes P :: "formula \<Rightarrow> bool"   and A :: "formula" assumes "\<And>p vs. P (FAtom Pos p vs)"   and "\<And>p vs. P (FAtom Neg p vs)"   and "\<And>f1 f2. P (FConj Pos f1 f2)"   and "\<And>f1 f2. P (FConj Neg f1 f2)"   and "\<And>f1. P (FAll Pos f1)"   and "\<And>f1. P (FAll Neg f1)" shows "P A"
comp_null_1: fixes f :: "('o, 'a) arr" shows "COMP null f = null"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t"
fdia_split: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x\<rangle> y = d z \<cdot>  |x\<rangle> y + ad z \<cdot>  |x\<rangle> y"
\<tau>\<phi>_1: shows "Some \<tau> = mgu (set_mset ` set (Map2.map2 add_mset As0' AAs0'))"
sub1_closed: fixes q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat"   and m :: "nat"   and n :: "nat" assumes "local.sub1 q k l (r, A) = (r', A')"   and "A \<in> carrier_mat m n" shows "A' \<in> carrier_mat m n"
return_prob_def: shows "Monad_Overloading.return = return_pmf"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (sup a b = b)"
widen_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty" shows "P \<turnstile> T \<le> T"
swapnorm_def: fixes n :: "nat"   and m :: "nat"   and t :: "poly" shows "swapnorm n m t = polynate (swap n m t)"
th: shows "(1 + fps_X) $ 0 \<noteq> (0::'a)"
emb_step_at_induct: fixes P :: "dir list \<Rightarrow> dir \<Rightarrow> ('c, 'd) tm \<Rightarrow> bool"   and a0 :: "dir list"   and a1 :: "dir"   and a2 :: "('c, 'd) tm" assumes "\<And>t s. P [] dir.Left (App t s)"   and "\<And>t s. P [] dir.Right (App t s)"   and "\<And>p dir t s. P p dir t \<Longrightarrow> P (dir.Left # p) dir (App t s)"   and "\<And>p dir t s. P p dir s \<Longrightarrow> P (dir.Right # p) dir (App t s)"   and "\<And>uu_ uv_ h. P uu_ uv_ (Hd h)" shows "P a0 a1 a2"
refl_slice1: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n \<in> sum_SDG_slice1 n"
in_set_upd_cases: fixes x :: "'a"   and l :: "'a list"   and i :: "nat"   and y :: "'a" assumes "x \<in> set (l[i := y])" obtains "i < length l" and "x = y"   | "x \<in> set l"
mono: fixes callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and x :: "('d, 'e, 'f) gpv \<times> 'g" shows "spmf.mono_body (\<lambda>exec_gpv. case x of (c, s) \<Rightarrow> the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). curry exec_gpv (c x) y)))"
simps: fixes x :: "'b"   and xs :: "'b list" shows "last (x # xs) = (if xs = [] then x else last xs)"
sum_Un2: fixes A :: "'a set"   and B :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite (A \<union> B)" shows "sum f (A \<union> B) = sum f (A - B) + sum f (B - A) + sum f (A \<inter> B)"
div_le_conv: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m" shows "(n div m \<le> k) = (n \<le> Suc k * m - Suc 0)"
simps_2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a tree"   and a :: "'a"   and r :: "'a tree" shows "bst_wrt P \<langle>l, a, r\<rangle> = ((\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r)"
The_on_Some_equality: fixes a :: "'c"   and U :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> U"   and "P a"   and "\<And>x. \<lbrakk>x \<in> U; P x\<rbrakk> \<Longrightarrow> x = a" shows "(THE x on U. P x) = Some a"
skipD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = []" shows "s = t"
word_leq_minus_one_le: fixes y :: "'b word"   and x :: "'b word" assumes "y \<noteq> 0"   and "x \<le> y - 1" shows "x < y"
insert_Diff: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
SUP_cong: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
sb_idem: fixes xs :: "'b"   and x :: "'a"   and y :: "'a"   and zs :: "'b"   and z :: "'a" shows "z @zs[x / y]_xs @zs[x / y]_xs = z @zs[x / y]_xs"
conv_radius_leI_ex': fixes R :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "0 \<le> R"   and "\<And>r. \<lbrakk>0 < r; R < ereal r\<rbrakk> \<Longrightarrow> \<not> summable (\<lambda>n. f n * of_real r ^ n)" shows "conv_radius f \<le> R"
minus_add_cancel: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.group_add minus plus zero uminus" shows "plus (uminus a) (plus a b) = b"
empty: shows "ord_term_lin.sorted_list_of_set {} = []"
NegInf: shows "NegInf = real_of_int (sint NEG_INF)"
cbnd_closed: fixes k :: "nat"   and n :: "nat" shows "cbnd k n = k * 2 ^ 2 ^ n"
fU'0: shows "(\<Sum>u'\<in>set us - {u}. (- (w \<bullet>c u') / (u' \<bullet>c u') \<cdot>\<^sub>v u') \<bullet>c u) = (0::'a)"
inv_into_def2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b" shows "inv_into A f x = (SOME y. y \<in> A \<and> f y = x)"
IFF: fixes b :: "BExpr"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<forall>s ss.  s \<approx> ss  \<longrightarrow> evalB b s = evalB b ss"   and "G \<rhd> c1 : Sec \<Phi>"   and "G \<rhd> c2 : Sec \<Psi>" shows "G \<rhd> Iff b c1 c2 : Sec (\<lambda>(s, t). (evalB b t \<longrightarrow> \<Phi> (s, t)) \<and> (\<not> evalB b t \<longrightarrow> \<Psi> (s, t)))"
proots_upper_def: fixes p :: "complex poly" shows "proots_upper p = proots_count p {z. 0 < Im z}"
pos_pos_linear: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a"   and "(0::'a) < b" shows "a < b \<or> a = b \<or> b < a"
append_self_conv2: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs @ ys = ys) = (xs = [])"
w0_3: shows "\<rho> (fst x) w0 (snd x) = \<rho> (fst y) w0 (snd y)"
assms_2: assumes "arr \<mu>"   and "obj a"   and "obj b"   and "src \<mu> = a"   and "trg \<mu> = b" shows "T"
typeofNew_def: fixes n :: "New" shows "typeofNew n = (case n of new_instance C \<Rightarrow> CClassT C | new_array T l \<Rightarrow> ArrT T)"
assms_1: shows "u \<le>p p \<cdot> w \<cdot> q"
boundedByMono: fixes m :: "nat"   and B :: "(nat \<times> 'b) set"   and M :: "nat" assumes "boundedBy m B"   and "m < M" shows "boundedBy M B"
low_mds_eq_dma: fixes mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2" shows "dma mem\<^sub>1 = dma mem\<^sub>2"
image_vimage_subset: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'c set" shows "f ` f -` A \<subseteq> A"
rho'_alpha_rho'': shows "rho' &= rho''"
shadow_root_ptr_kinds_eq_h: shows "shadow_root_ptr_kinds h2 = shadow_root_ptr_kinds h |\<union>| {|new_shadow_root_ptr|}"
optimize_sound: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and x :: "rat vec" assumes "maximize A b c = Some (Inr x)" shows "x \<in> max_lp A b c"
unfinished_nodes_subset: fixes S :: "'v set" shows "unfinished_dnodes S \<subseteq> all_dnodes"
M2: shows "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R"
PK_H: shows "(P, K) = local.reduce_column_aux div_op xs H (1\<^sub>m (dim_row H), H)"
Hermite_of_row_preserves_previous_cols: fixes b :: "'a"   and A :: "(('b, 'a) vec, 'c) vec"   and i :: "'c"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'c" assumes "b < (LEAST n. A $ i $ n \<noteq> (0::'b))"   and "\<not> is_zero_row i A"   and "echelon_form A" shows "Hermite_of_row_i ass res A i $ a $ b = A $ a $ b"
assms_2: shows "b - a = l *\<^sub>R (c - a)"
mult_1_right: fixes a :: "'c" shows "brouwer.comp_inf.vector a"