hcju/isabelleps huggingface dataset:
lift_lemma: fixes x :: "V"   and y :: "V" assumes "x \<noteq> 0"   and "y \<noteq> 0" shows "\<not> lift (x * y) x \<le> x"
fun_prod_commute: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "f (1::'b) = (1::'a)"   and "\<And>x y. f (x * y) = f x * f y" shows "f (prod g A) = (\<Prod>a\<in>A. f (g a))"
simps_1: shows "interpret_slp [] = (\<lambda>xs. xs)"
representant_exists: fixes A :: "'a set" assumes "A \<in> Partition" shows "\<exists>a\<in>M. a \<in> A \<and> A = Class a"
A: fixes x :: "'a" assumes "x \<in># A" shows "prime x"
long_list_tl: fixes us :: "'b list" assumes "1 < \<^bold>|us\<^bold>|" shows "tl us \<noteq> \<epsilon>"
assms_2: shows "\<exists>x\<in>set ns. P x"
pr_id2_2': shows "pr_conv_2_to_1 (\<lambda>x y. y) \<in> PrimRec1'"
tllist_all2_rsp: fixes R1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and T :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and R2 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S' :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and T' :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "('e, 'g) tllist"   and ys :: "('f, 'h) tllist"   and xs' :: "('e, 'g) tllist"   and ys' :: "('f, 'h) tllist" assumes "\<forall>x y. R1 x y \<longrightarrow> (\<forall>a b. R1 a b \<longrightarrow> S x a = T y b)"   and "\<forall>x y. R2 x y \<longrightarrow> (\<forall>a b. R2 a b \<longrightarrow> S' x a = T' y b)"   and "tllist_all2 R1 R2 xs ys"   and "tllist_all2 R1 R2 xs' ys'" shows "tllist_all2 S S' xs xs' = tllist_all2 T T' ys ys'"
simps_5: fixes x1 :: "nat" shows "Exit \<noteq> Label x1"
merge_nth: shows "merges f qs ss ! q = map snd (filter (\<lambda>(p', t'). p' = q) qs) \<Squnion>\<^bsub>f\<^esub> ss ! q"
id_univalent: fixes x :: "'e set" shows "univalent (Id_on x)"
rv_minus_rv: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "(\<lambda>t. f t - g t) \<in> rv M"
intros_5: fixes u :: "'e \<Rightarrow> 'd"   and r :: "'e"   and c :: "'e"   and d :: "'d" assumes "u r - u c \<le> d" shows "dbm_entry_val u (Some r) (Some c) (Le d)"
eval_def: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "eval b c = mkArr (local.set (HF'.prod (exp b c) b)) (local.set c) (\<lambda>x. UP (happ (hfst (DOWN x)) (hsnd (DOWN x))))"
0: fixes x :: "'b" assumes "x \<notin> S" shows "f x = (0::'a)"
sb: shows "carrier (relative_homology_group (int n) (nsphere n) (equator n)) \<subseteq> (\<lambda>(x, y). hom_induced (int n) (lsphere n) (equator n) (nsphere n) (equator n) id x \<otimes>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> hom_induced (int n) (usphere n) (equator n) (nsphere n) (equator n) id y) ` carrier (relative_homology_group (int n) (lsphere n) (equator n) \<times>\<times> relative_homology_group (int n) (usphere n) (equator n))"
vec_norm_ge_0: fixes v :: "complex vec"   and n :: "nat" assumes "v \<in> carrier_vec n"   and "v \<noteq> 0\<^sub>v n" shows "0 < vec_norm v"
ccSup_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set"   and b :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
NewFail: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and h :: "heap"   and P :: "(char list list \<times> char list exp) prog"   and l :: "sfields" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "new_Addr h = None"   and "is_class P C" shows "P \<turnstile> \<langle>new C,(h, l, sh)\<rangle> \<Rightarrow> \<langle>THROW OutOfMemory,(h, l, sh)\<rangle>"
simps_4: fixes n :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound' n (F \<^bold>\<and> G) = cut_bound' (cut_bound' n F) G"
zero_closed: shows "\<zero>\<^bsub>M\<^esub> \<in> carrier M"
One_not_eq_fromArr: shows "One \<notin> fromArr ` {False, True}"
list_subset_iff_projection_neutral: fixes l :: "'b list"   and X :: "'b set" shows "(set l \<subseteq> X) = (l \<upharpoonleft> X = l)"
h_def: fixes x :: "'c"   and y :: "'c" shows "h x y = (if Regions.isIntv (I x) \<and> Regions.isIntv (I y) then if (y, x) \<in> r \<and> (x, y) \<notin> r then Lt (real_of_int (int (intv_const (I x)) - int (intv_const (I y)) + 1)) else if (x, y) \<in> r \<and> (y, x) \<notin> r then Lt (real_of_int (int (intv_const (I x)) - int (intv_const (I y)))) else Le (real_of_int (int (intv_const (I x)) - int (intv_const (I y)))) else if Regions.isConst (I x) \<and> Regions.isConst (I y) then Le (real_of_int (int (intv_const (I x)) - int (intv_const (I y)))) else if Regions.isIntv (I x) \<and> Regions.isConst (I y) then Lt (real_of_int (int (intv_const (I x)) + 1 - int (intv_const (I y)))) else if Regions.isConst (I x) \<and> Regions.isIntv (I y) then Lt (real_of_int (int (intv_const (I x)) - int (intv_const (I y)))) else \<infinity>)"
Sup_comp: fixes X :: "'a set"   and z :: "'a" shows "\<Squnion>X * z = (\<Squnion>x\<in>X. x * z)"
span_eq_real2: shows "module_on.span (*\<^sub>R) = span"
assms: shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"
scale_cancel_left: fixes a :: "'c"   and x :: "'c"   and y :: "'c" shows "(a * x = a * y) = (x = y \<or> a = (0::'c))"
desargues_config_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" shows "desargues_config A B C A' B' C' M N P R \<equiv> distinct7 A B C A' B' C' R \<and> \<not> col A B C \<and> \<not> col A' B' C' \<and> distinct3l (line A A') (line B B') (line C C') \<and> meet_3_in (line A A') (line B B') (line C C') R \<and> line A B \<noteq> line A' B' \<and> line B C \<noteq> line B' C' \<and> line A C \<noteq> line A' C' \<and> meet_in (line B C) (line B' C') M \<and> meet_in (line A C) (line A' C') N \<and> meet_in (line A B) (line A' B') P"
limitin_atin: fixes Y :: "'c topology"   and f :: "'d \<Rightarrow> 'c"   and y :: "'c"   and X :: "'d topology"   and x :: "'d" shows "limitin Y f y (atin X x) = (y \<in> topspace Y \<and> (x \<in> topspace X \<longrightarrow> (\<forall>V. openin Y V \<and> y \<in> V \<longrightarrow> (\<exists>U. openin X U \<and> x \<in> U \<and> f ` (U - {x}) \<subseteq> V))))"
1: shows "- Abs_freelistfst (x # xs) + Abs_freeword (x # xs) = - Abs_freelistfst xs + - Abs_freeletter (fst x) + Abs_freeword [x] + Abs_freeword xs"
induct: fixes P :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> nat) \<Rightarrow> ('a, 'b) Term.term set \<Rightarrow> ('a, 'b) Term.term \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> bool"   and a1 :: "'a \<Rightarrow> nat"   and a2 :: "('a, 'b) Term.term set"   and a3 :: "('a, 'b) Term.term" assumes "\<And>pu ar M x. P pu ar M (Var x)"   and "\<And>pu ar M f T. (\<And>z. z \<in> set T \<Longrightarrow> P pu ar M z) \<Longrightarrow> P pu ar M (Fun f T)" shows "P a0 a1 a2 a3"
hyps: fixes s :: "'n list"   and c1 :: "'n list multiset"   and c2 :: "'n list multiset"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c1 + c2), w, s', c') \<in> trcl (ntrp fg)"   and "valid fg ({#s#} + c1 + c2)" shows "\<exists>w1 w2 c1' c2'. w \<in> w1 \<otimes>\<^bsub>\<alpha>nl fg\<^esub> map ENV w2 \<and> c' = c1' + c2' \<and> ((s, c1), w1, s', c1') \<in> trcl (ntrp fg) \<and> (c2, w2, c2') \<in> trcl (ntr fg) \<and> mon_ww fg (map le_rem_s w1) \<inter> mon_c fg c2 = {} \<and> mon_ww fg w2 \<inter> mon_c fg ({#s#} + c1) = {}"
Fourier_sum_limit_Dirichlet_kernel: fixes f :: "real \<Rightarrow> real"   and t :: "real"   and l :: "real" assumes "f absolutely_integrable_on {- pi..pi}"   and "\<And>x. f (x + 2 * pi) = f x" shows "(\<lambda>n. \<Sum>k\<le>n. Fourier_coefficient f k * trigonometric_set k t) \<longlonglongrightarrow> l = (\<lambda>n. LINT x|lebesgue_on {- pi..pi}. Dirichlet_kernel n x * f (x + t)) \<longlonglongrightarrow> pi * l"
dim_mult_mat_vec: fixes A :: "'a mat"   and v :: "'a Matrix.vec" shows "dim_vec (A *\<^sub>v v) = dim_row A"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_self_inverse: fixes A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A) = A"
False: shows "\<nexists>f. t = Fun f [] \<or> t' = Fun f []"
simps_3: fixes u :: "'b"   and v :: "'b"   and w :: "'b"   and e1 :: "'b \<times> 'b"   and e2 :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" shows "co_path (u, v) w (e1 # e2 # es) = (if e1 = (u, w) \<and> e2 = (w, v) then (u, v) # co_path (u, v) w es else if e1 = (v, w) \<and> e2 = (w, u) then (v, u) # co_path (u, v) w es else e1 # co_path (u, v) w (e2 # es))"
delete_edge_valid: fixes g :: "('a, 'b) graph"   and v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "valid_graph g" shows "valid_graph (delete_edge v e v' g)"
face_of_convex_hull_aux: fixes x :: "real"   and p :: "'b"   and u :: "real"   and a :: "'b"   and v :: "real"   and b :: "'b"   and w :: "real"   and c :: "'b"   and S :: "'b set" assumes "x *\<^sub>R p = u *\<^sub>R a + v *\<^sub>R b + w *\<^sub>R c"   and "u + v + w = x"   and "x \<noteq> 0"   and "affine S"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S" shows "p \<in> S"
1: shows "ipd (\<pi> j) pd\<rightarrow> \<pi> k"
f_2_lemma_2: fixes x :: "int" shows "[f_2 x = int p - x] (mod int p)"
lincomb_0coeffs: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and ms :: "'b list"   and rs :: "'a list" assumes "RModule R smult M"   and "set ms \<subseteq> M"   and "\<forall>s\<in>set rs. s = (0::'a)" shows "scalar_mult.lincomb smult rs ms = (0::'b)"
iteci_lu_rule: fixes i :: "nat"   and t :: "nat"   and e :: "nat"   and bdd :: "bdd"   and p :: "nat"   and bdd' :: "bdd"   and bddi :: "bddi" shows "mi.ite_impl_lu i t e bdd = Some (p, bdd') \<longrightarrow> <is_bdd_impl bdd bddi> iteci_lu i t e bddi <\<lambda>r. case r of (pi, bddi') \<Rightarrow> is_bdd_impl bdd' bddi' * \<up> (pi = p)>\<^sub>t"
span_finite: fixes S :: "'b poly set" assumes "finite S" shows "poly_vs.span S = range (\<lambda>u. \<Sum>v\<in>S. smult (u v) v)"
3: shows "dl \<in> set Di \<or> \<lbrakk>M; [\<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd dl))]\<rangle>\<^sub>s\<^sub>t]\<rbrakk>\<^sub>d \<I>"
conjuncts_NFAND: fixes X :: "('a, 'b) aformula set" assumes "finite X"   and "X \<noteq> {}"   and "\<forall>x\<in>X. \<not> is_FAnd x" shows "conjuncts (NFAND X) = X"
gen_analz_insert_eq: fixes X :: "msg"   and H :: "msg set"   and G :: "msg set" assumes "X \<in> analz H"   and "H \<subseteq> G" shows "analz (insert X G) = analz G"
id: shows "(gcd_poly_i ff_ops (of_int_poly_i ff_ops (Mp f)) (of_int_poly_i ff_ops (Mp g)) = one_poly_i ff_ops) = comm_monoid_mult_class.coprime f'' g''"
name_set_avoiding: fixes X :: "name set"   and c :: "'a" assumes "finite X" obtains pi :: "(name \<times> name) list" where "(pi \<bullet> X) \<sharp>* c" and "distinctPerm pi" and "set pi \<subseteq> X \<times> pi \<bullet> X"
degree_eq_degreen0: fixes p :: "'b poly" shows "Polynomial_Expression.degree p = degreen p 0"
sum_squares_bound: fixes x :: "'a"   and y :: "'a" shows "(2::'a) * x * y \<le> x\<^sup>2 + y\<^sup>2"
set_cases: fixes e :: "'b"   and a :: "'b option" assumes "e \<in> set_option a" obtains "a = Some e"
sq_f_bnd: shows "\<parallel>fs ! i\<parallel>\<^sup>2 \<le> int (4 ^ (m - 1) * N ^ m * m * m)"
symE: fixes r :: "('e \<times> 'e) set"   and b :: "'e"   and a :: "'e" assumes "sym r"   and "(b, a) \<in> r" obtains "(a, b) \<in> r"
soadisj: fixes x :: "'b set set"   and a :: "'b"   and y :: "'b set set" assumes "x \<in> subnetsOfAdr a"   and "y \<in> subnetsOfAdr a" shows "\<not> netsDistinct x y"
assms_2: shows "v \<notin> subst_domain \<delta>"
False: shows "mem_context_val_w32 9 (ucast (ptp OR ucast (ucast (va >> 18)))) (mem s1) \<noteq> None"
distinct_19: fixes x4 :: "'a"   and x51 :: "nat"   and x52 :: "'a"   and x53 :: "'a" shows "Snapshot x4 \<noteq> RecvMarker x51 x52 x53"
Ok_def: shows "Ok \<equiv> \<Lambda> a. error_abs\<cdot>(sinr\<cdot>(up\<cdot>a))"
exhaust: fixes y :: "tcolor" obtains "y = Red"   | "y = Black"
frac_less_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "y \<noteq> zero"   and "z \<noteq> zero" shows "less (divide x y) (divide w z) = less (divide (minus (times x z) (times w y)) (times y z)) zero"
deg: shows "degree_m f = degree_m (Polynomial.smult c (\<Prod>\<^sub># fs))"
subset_gen2: fixes I :: "'a set"   and J :: "'a set"   and P :: "'a set set" assumes "{I, J} \<subseteq> P"   and "gen P I \<inter> gen P J \<noteq> {}" shows "I \<subseteq> gen P J"
cc_restr_twist: fixes S :: "Vars.var set"   and S' :: "Vars.var set"   and G :: "CoCalls" shows "G G|` S' G|` S = G G|` S G|` S'"
beukers_aux_integral_transform2: fixes x :: "real"   and y :: "real" assumes "x \<in> {0<..<1}"   and "y \<in> {0<..<1}" shows "LBINT z=0..1. (x * y * z) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * z) ^ (n + 1) = LBINT w=0..1. (1 - w) ^ n * (1 - y) ^ n / (1 - (1 - x * y) * w)"
mix_in_lot: fixes x :: "'a pmf"   and outcomes :: "'a set"   and y :: "'a pmf"   and \<alpha> :: "real" assumes "x \<in> \<P> outcomes"   and "y \<in> \<P> outcomes"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> x y \<in> \<P> outcomes"
assms_2: shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n \<le> n'"
inv_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "L'.C.iso f"   and "L'.C.in_hom f a b" shows "L'.C.in_hom (L'.C.inv f) b a"
Inl_Inr_False: fixes x :: "'e"   and y :: "'f" shows "(Inl x = Inr y) = False"
static_operation_det: fixes classes :: "'a fset"   and associations :: "String.literal \<rightharpoonup>\<^sub>f String.literal \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and operations :: "(String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option) list"   and \<tau> :: "'b"   and name :: "String.literal"   and \<pi> :: "'b list"   and oper\<^sub>1 :: "String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option"   and oper\<^sub>2 :: "String.literal \<times> 'b \<times> (String.literal \<times> 'b \<times> param_dir) list \<times> 'b \<times> bool \<times> 'c option" assumes "object_model classes associations"   and "unique_static_operation operations \<tau> name \<pi> oper\<^sub>1"   and "unique_static_operation operations \<tau> name \<pi> oper\<^sub>2" shows "oper\<^sub>1 = oper\<^sub>2"
cases: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'d list" assumes "for_all2 r a1 a2" obtains "a1 = []" and "a2 = []"   | x :: "'c" and y :: "'d" and xs :: "'c list" and ys :: "'d list" where "a1 = x # xs" and "a2 = y # ys" and "r x y" and "for_all2 r xs ys"
fract_poly_1: shows "map_poly to_fract 1 = 1"
trivial_FGModule: fixes smult :: "('a, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>a\<in>ActingGroup.group_ring. smult a (0::'b) = (0::'b)" shows "FGModule G smult 0"
cblinfun_image_ccspan: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and G :: "'b set" shows "A *\<^sub>S ccspan G = ccspan ((*\<^sub>V) A ` G)"
cyclic_group_def: fixes G :: "('a, 'b) cyclic_group_scheme" shows "cyclic_group G \<equiv> Group.group G \<and> cyclic_group_axioms G"
identify_blocks_cong: fixes k :: "nat"   and n :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "k < n"   and "??.Jordan_Normal_Form_Existence.same_diag n A B"   and "same_upto n k A B" shows "??.Jordan_Normal_Form_Existence.identify_blocks A k = ??.Jordan_Normal_Form_Existence.identify_blocks B k"
V\<^sub>0_def: fixes n :: "'b" shows "V\<^sub>0 n \<equiv> monomial (1::'c) (single_pp n 1)"
helper: shows "D = E @ F \<and> \<gamma> = \<gamma>1 @ \<gamma>2 \<and> LeftDerivationLadder \<alpha> E L' \<gamma>1 \<and> derivation_ge F (length \<gamma>1) \<and> LeftDerivation \<delta> (derivation_shift F (length \<gamma>1) 0) \<gamma>2 \<and> L' = ladder_cut L (length E)"
Cons_2: assumes "rtrancl_path r x ((a # as) @ [y]) y"   and "rtrancl_path r y ys z" shows "thesis"
IH_1: fixes ass :: "'a \<Rightarrow> bool" assumes "ifex_no_twice t"   and "ifex_sat t = Some ass" shows "val_ifex t ass = True"
DSource_level: fixes S :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" assumes "S \<in> DSources i C" shows "C \<in> AbstrLevel i"
acq_mode_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and SomeMode :: "Mode"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (Skip@[x +=\<^sub>m SomeMode]) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Skip@[Eg2_var\<^sub>C_of_Eg1 x +=\<^sub>m SomeMode]) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
diff_less_mono2: fixes m :: "nat"   and n :: "nat"   and l :: "nat" assumes "m < n"   and "m < l" shows "l - n < l - m"
seqI: fixes f :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr"   and g :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr" assumes "S.arr f"   and "S.arr g"   and "S.dom g = S.cod f" shows "S.seq g f"
prems: shows "\<forall>r\<in>set (x # sb). is_Read\<^sub>s\<^sub>b r \<or> is_Prog\<^sub>s\<^sub>b r \<or> is_Ghost\<^sub>s\<^sub>b r"
finite_product_sigma_finite_def: fixes M :: "'d \<Rightarrow> 'e measure"   and I :: "'d set" shows "finite_product_sigma_finite M I \<equiv> product_sigma_finite M \<and> finite_product_sigma_finite_axioms I"
signed_real_of_word_sum_less_numeral_iff: fixes v :: "'a Word.word"   and w :: "'b Word.word"   and n :: "num" shows "(signed_real_of_word v + signed_real_of_word w < numeral n) = (sint v + sint w < numeral n)"
cont_fun_lub_Sup: fixes M :: "('b \<Rightarrow> 'a) set" assumes "Complete_Partial_Order.chain (fun_ord (\<le>)) M"   and "\<forall>f\<in>M. mcont \<Or> (\<sqsubseteq>) Sup (\<le>) f" shows "cont \<Or> (\<sqsubseteq>) Sup (\<le>) (fun_lub Sup M)"
igFreshIGAbs2STR_def: fixes MOD :: "('c, 'd, 'e, 'f, 'g, 'h, 'i, 'j, 'k) model_scheme" shows "igFreshIGAbs2STR MOD \<equiv> \<forall>ys y xs x X. igFresh MOD ys y X \<longrightarrow> igFreshAbs MOD ys y (igAbs MOD xs x X)"
add_eq_conv_diff: fixes a :: "'b"   and M :: "'b multiset"   and b :: "'b"   and N :: "'b multiset" shows "(a, M = b, N) = (M = N \<and> a = b \<or> M = b, N - (a, {#}) \<and> N = a, M - (b, {#}))"
Rep_enat_inverse: fixes x :: "enat" shows "Abs_enat (Rep_enat x) = x"
closed_invs_1: fixes f :: "db"   and x :: "db" shows "closed (DBApp f x) = (closed f \<and> closed x)"
assms_1: shows "(mulex_on P A)\<^sup>=\<^sup>= M N"
set_sort_oalist: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw ko' xs" shows "set (sort_oalist ko xs) = set xs"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_new_ptr: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (new_shadow_root_ptr, h')" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr|}"
Healthy_def': fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and H :: "('c \<times> 'c \<Rightarrow> bool) \<Rightarrow> 'c \<times> 'c \<Rightarrow> bool" shows "P is H healthy = (H P = P)"
nth_via_drop: fixes n :: "nat"   and xs :: "'d list"   and y :: "'d"   and ys :: "'d list" assumes "drop n xs = y # ys" shows "xs ! n = y"
analz_Un_implAuthSet: fixes G :: "msg set"   and H :: "msg set" assumes "Enc_keys_clean (G \<union> H)" shows "analz ({sym.implAuth A B M |A B M. M \<in> G} \<union> H) \<subseteq> synth (analz (G \<union> H)) \<union> - payload"
intros_3: fixes T :: "('c, 'd) Term.term list"   and S :: "('c, 'd) Term.term list"   and P :: "'c \<Rightarrow> 'c list"   and f :: "'c" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)" shows "term_variants_pred P (Fun f T) (Fun f S)"
ordered_map_def: fixes S :: "('a \<times> 'b) set" shows "ordered_map S = map (get_map S) (sorted_list_of_set (fst ` S))"
FieldI1: fixes i :: "'d"   and j :: "'d"   and R :: "('d \<times> 'd) set" assumes "(i, j) \<in> R" shows "i \<in> Field R"
elims: fixes x :: "opubx"   and y :: "Unboxed.type option list \<times> Unboxed.type option" assumes "typeof_opubx x = y" obtains "x = OpAddNumUbx" and "y = ([Some Ubx1, Some Ubx1], Some Ubx1)"
expand: fixes quat :: "quat"   and quat' :: "quat" assumes "quat.Re quat = quat.Re quat' \<and> Im1 quat = Im1 quat' \<and> Im2 quat = Im2 quat' \<and> Im3 quat = Im3 quat'" shows "quat = quat'"
dim_def: fixes V :: "'c set" shows "dim V = (if \<exists>b. independent b \<and> span b = span V then card (SOME b. independent b \<and> span b = span V) else 0)"
distinct_35: fixes x31 :: "nat"   and x32 :: "'e" shows "unicast x31 x32 \<noteq> \<tau>\<^sub>s"
subset: fixes B :: "'a set"   and A :: "'a set" assumes "set_option (local.MaxR.MaxR_opt B) \<subseteq> A"   and "A \<subseteq> B" shows "local.MaxR.MaxR_opt B = local.MaxR.MaxR_opt A"
m_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.submonoid H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
trans_reflclI: fixes r :: "('c \<times> 'c) set" assumes "trans r" shows "trans (r\<^sup>=)"
convert_def: fixes fml :: "'a fml" shows "convert fml = (let ps = remdups (poly_list fml) in (map_fml (index_of ps) fml, ps))"