hcju/isabelleps huggingface dataset:
normalize_manual_order_def: fixes p :: "((int \<times> int) set set, int) Combinators"   and l :: "(int \<times> int) set set set list" shows "normalize_manual_order p l = removeAllDuplicates (insertDenies (separate (FWNormalisationCore.sort (removeShadowRules2 (remdups (rm_MT_rules C (insertDeny (removeShadowRules1 (policy2list p)))))) l)))"
lcm_proj2_if_dvd_nat: fixes x :: "nat"   and y :: "nat" assumes "x dvd y" shows "lcm x y = y"
Rep_star_inverse: fixes x :: "'b star" shows "Abs_star (Rep_star x) = x"
Suc_pairwise: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b stream" assumes "\<forall>i. P (xs !! i) (xs !! Suc i)" shows "PTA_Reachability.pairwise P xs"
assms: shows "\<forall>i<length xs. a \<le> xs ! i"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "item set" assumes "\<J> x xa = y" obtains "x = 0" and "xa = 0" and "y = \<pi> 0 {} Init"   | k :: "nat" and u :: "nat" where "x = ka__" and "xa = Suc u" and "y = \<pi> ka__ (\<T> ka__ (Suc u)) (\<J> ka__ u)"   | k :: "nat" where "x = Suc k" and "xa = 0" and "y = \<pi> (Suc k) {} (\<I> k)"
condition_false: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c"   and A :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool"   and B :: "'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" assumes "\<not> P s" shows "condition P   A   B s = B s"
comp_ide_arr: fixes b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide b"   and "seq b f" shows "b \<cdot> f = f"
Resolution_weaken: fixes S :: "'a literal set set"   and D :: "'a literal set"   and T :: "'a literal set set" assumes "S \<turnstile> D" shows "T \<union> S \<turnstile> D"
FunctorId3Cod: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## id\<^bsub>CatDom F\<^esub> (cod\<^bsub>CatDom F\<^esub> f) = id\<^bsub>CatCod F\<^esub> (cod\<^bsub>CatCod F\<^esub> (F ## f))"
pwd_spy_sep_map_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_sep_map n)"
rbt_insertwk_is_rbt: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) RBT_Impl.rbt"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and x :: "'c" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_insert_with_key less f k x t)"
state_ext_def: fixes heap :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and assocs :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>"   and more :: "'e" shows "\<lparr>heap = heap, assocs = assocs, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons state_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple heap assocs) more"
sv01_2: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1"   and "n \<noteq> 0"   and "inconsist (amalg01 f) (f \<triangleright> n) \<down>= Suc n" shows "s\<^bsub>01\<^esub> (f \<triangleright> n) \<down>= amalg01 f"
distinct_1: fixes x1 :: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a \<Rightarrow> 'b option"   and x2 :: "('a \<Rightarrow> 'b option) \<Rightarrow> bool" shows "\<Up>x1 \<noteq> (x2)\<^sub>\<surd>"
Rep_Abs_mod: fixes m :: "int" shows "Rep_bit0 (Abs_bit0 (m mod int CARD('b bit0))) = m mod int CARD('b bit0)"
eq_symmetric_ax_def: shows "eq_symmetric_ax \<equiv> mk_eq Core.x Core.y \<longmapsto> mk_eq Core.y Core.x"
orbit_altdef: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "orbit f x = {(f ^^ n) x |n. 0 < n}"
pred_succ: fixes j :: "'a" assumes "local.aboveS j \<noteq> {}" shows "pred (succ j) = j"
start_of_eq: fixes ap :: "abc_inst list"   and as :: "nat" assumes "length ap < as" shows "start_of (layout_of ap) as = start_of (layout_of ap) (length ap)"
Infinitesimal_interval: fixes e :: "real star"   and e' :: "real star"   and x :: "real star" assumes "e \<in> Infinitesimal"   and "e' \<in> Infinitesimal"   and "e' < x"   and "x < e" shows "x \<in> Infinitesimal"
dense_lattice_char_1: shows "(\<forall>x y. x \<sqinter> y = bot \<longrightarrow> x = bot \<or> y = bot) = (\<forall>x. x \<noteq> bot \<longrightarrow> dense x)"
min_set_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" shows "ord.min_set less_eq X = {y \<in> X. \<forall>x\<in>X. less_eq x y \<longrightarrow> x = y}"
pop_count_def: fixes w :: "'b word" shows "pop_count w \<equiv> length (filter id (to_bl w))"
\<omega>_limit_set_invariant: fixes x :: "'a" shows "invariant (\<omega>_limit_set x)"
igSwapCls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) Iteration.model_scheme" shows "igSwapCls MOD \<equiv> igSwapIGVar MOD \<and> igSwapIGAbs MOD \<and> igSwapIGOp MOD"
assms_2: shows "\<forall>c\<in>set cs. v c \<le> n"
Inf_eq_Sup: fixes A :: "'a set" shows "\<Sqinter> A = \<Squnion> {b. \<forall>a\<in>A. b \<le> a}"
group_assoc: fixes inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.group inverse one times" shows "times (times x y) z = times x (times y z)"
induct: fixes P :: "'c rexp \<times> ('b val \<Rightarrow> 'b val) \<Rightarrow> 'c rexp \<times> ('b val \<Rightarrow> 'b val) \<Rightarrow> bool"   and a0 :: "'c rexp \<times> ('b val \<Rightarrow> 'b val)"   and a1 :: "'c rexp \<times> ('b val \<Rightarrow> 'b val)" assumes "\<And>f\<^sub>1 r\<^sub>2 f\<^sub>2. P (Zero, f\<^sub>1) (r\<^sub>2, f\<^sub>2)"   and "\<And>f\<^sub>1 f\<^sub>2. P (One, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Atom v, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Plus v va, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Times v va, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Star v, f\<^sub>1) (Zero, f\<^sub>2)"   and "\<And>f\<^sub>1 f\<^sub>2. P (One, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>f\<^sub>1 v f\<^sub>2. P (One, f\<^sub>1) (Atom v, f\<^sub>2)"   and "\<And>f\<^sub>1 v va f\<^sub>2. P (One, f\<^sub>1) (Plus v va, f\<^sub>2)"   and "\<And>f\<^sub>1 v va f\<^sub>2. P (One, f\<^sub>1) (Times v va, f\<^sub>2)"   and "\<And>f\<^sub>1 v f\<^sub>2. P (One, f\<^sub>1) (Star v, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Atom v, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Atom v, f\<^sub>1) (Atom va, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Atom v, f\<^sub>1) (Plus va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Atom v, f\<^sub>1) (Times va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Atom v, f\<^sub>1) (Star va, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Plus v va, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Plus v va, f\<^sub>1) (Atom vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Plus v va, f\<^sub>1) (Plus vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Plus v va, f\<^sub>1) (Times vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Plus v va, f\<^sub>1) (Star vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 f\<^sub>2. P (Times v va, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Times v va, f\<^sub>1) (Atom vb, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Times v va, f\<^sub>1) (Plus vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb vc f\<^sub>2. P (Times v va, f\<^sub>1) (Times vb vc, f\<^sub>2)"   and "\<And>v va f\<^sub>1 vb f\<^sub>2. P (Times v va, f\<^sub>1) (Star vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 f\<^sub>2. P (Star v, f\<^sub>1) (One, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Star v, f\<^sub>1) (Atom va, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Star v, f\<^sub>1) (Plus va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va vb f\<^sub>2. P (Star v, f\<^sub>1) (Times va vb, f\<^sub>2)"   and "\<And>v f\<^sub>1 va f\<^sub>2. P (Star v, f\<^sub>1) (Star va, f\<^sub>2)" shows "P a0 a1"
FOREACHoci_rule: fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>c \<sigma>; x \<in> it; it \<subseteq> S; I it \<sigma>; \<forall>y\<in>it - {x}. R x y; \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<noteq> {}; it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>; \<forall>x\<in>it. \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,I\<^esup> S c f \<sigma>0 \<le> SPEC P"
is_Prog\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Prog\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
assms_2: shows "is_map_to_list Rk Rv Rm tsl"
try_spmf_cong: fixes p :: "'a spmf"   and p' :: "'a spmf"   and q :: "'a spmf"   and q' :: "'a spmf" assumes "p = p'"   and "\<not> lossless_spmf p' \<Longrightarrow> q = q'" shows "TRY p ELSE q = TRY p' ELSE q'"
scope_split: fixes v :: "'d"   and ps :: "'d \<times> 'b"   and v' :: "'d"   and pth :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" assumes "v \<in> scope ps"   and "path v v' pth"   and "terminal_vertex v'" obtains pth1 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" and e :: "('d \<times> 'c) \<times> 'd \<times> 'b" and pth2 :: "(('d \<times> 'c) \<times> 'd \<times> 'b) list" where "pth = (pth1 @ [e]) @ pth2" and "path v (fst ps) (pth1 @ [e])" and "path (fst ps) v' pth2" and "snd e = ps" and "ps \<notin> snd ` set pth1"
assms: shows "\<forall>c. c el F \<longrightarrow> c el F'"
PR_CONST_def: fixes x :: "'j" shows "PR_CONST x \<equiv> x"
partition_random_permutations: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "finite A" shows "map_pmf (partition P) (pmf_of_set (permutations_of_set A)) = pair_pmf (pmf_of_set (permutations_of_set {x \<in> A. P x})) (pmf_of_set (permutations_of_set {x \<in> A. \<not> P x}))"
prefr': shows "u \<le>p r' \<^sup>@ k"
funcset_id: fixes A :: "'f set" shows "(\<lambda>x. x) \<in> A \<rightarrow> A"
CondT\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>1 :: "nat exp"   and e' :: "nat exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
rwof_leof_init: fixes m :: "'a nres"   and c :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres" shows "m \<le>\<^sub>n SPEC (rwof m c f)"
restrict_top_IF_id: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex"   and v' :: "'a"   and val :: "bool" assumes "ifex_ordered (IF v t e)"   and "v' < v" shows "restrict_top (IF v t e) v' val = IF v t e"
Int_insert_left_if0: fixes a :: "'e"   and C :: "'e set"   and B :: "'e set" assumes "a \<notin> C" shows "insert a B \<inter> C = B \<inter> C"
assms_1: shows "\<chi> \<noteq> principal_dchar n"
abs_Im_le_cmod: fixes x :: "complex" shows "\<bar>Im x\<bar> \<le> cmod x"
poly_eq_0_iff_dvd: fixes p :: "'a poly"   and c :: "'a" shows "(poly p c = (0::'a)) = ([:- c, 1::'a:] dvd p)"
LNil_eq_lappend_iff: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(LNil = xs @@ ys) = (xs = LNil \<and> ys = LNil)"
dtor2_def: shows "dtor2 = fold2 (F1map id ctor1 ctor2) (F2map id ctor1 ctor2)"
surj_pair: fixes p :: "'b \<times> 'd" shows "\<exists>x y. p = (x, y)"
comp_FSbImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'l, 'm, 'n) model_scheme"   and h :: "'l \<Rightarrow> 'o"   and hA :: "'m \<Rightarrow> 'p"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'o, 'p, 'q) model_scheme"   and h' :: "'o \<Rightarrow> 'r"   and hA' :: "'p \<Rightarrow> 's"   and MOD'' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'r, 's, 't) model_scheme" assumes "igWlsAbsIsInBar MOD"   and "FSbImorph h hA MOD MOD'"   and "FSbImorph h' hA' MOD' MOD''" shows "FSbImorph (h' \<circ> h) (hA' \<circ> hA) MOD MOD''"
sim_while_wp_aux: fixes b :: "bexp"   and b' :: "bexp"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and c' :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "bval b = bval b'"   and "wp \<pi> c = wp \<pi> c'"   and "\<pi>: (WHILE b DO c, s) \<Rightarrow> t" shows "\<pi>: (WHILE b' DO c', s) \<Rightarrow> t"
dtor_unfold_o_map: fixes g :: "'g \<Rightarrow> 'f"   and h :: "'e \<Rightarrow> ('g, 'e) coF0" shows "map_coF g \<circ> dtor_unfold_coF h = dtor_unfold_coF (map_pre_coF g id \<circ> h)"
ext1_ext2_is_not_act2: fixes A1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and A2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set"   and a :: "'e" assumes "compat_ioas A1 A2"   and "a \<in> externals (asig_of A1)"   and "a \<notin> externals (asig_of A2)" shows "a \<notin> actions (asig_of A2)"
atlas_is_atlas: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and a1 :: "('c, 'd) chart"   and a2 :: "('c, 'd) chart" assumes "c_manifold charts k"   and "a1 \<in> c_manifold.atlas charts k"   and "a2 \<in> c_manifold.atlas charts k" shows "k-smooth_compat a1 a2"
red_single_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd"   and f :: "'a \<Rightarrow>\<^sub>0 'd"   and t :: "'a" shows "punit.red_single p q f t = (f \<noteq> 0 \<and> lookup p (local.punit.splus t (punit.lt f)) \<noteq> (0::'d) \<and> q = p - local.punit.monom_mult (lookup p (local.punit.splus t (punit.lt f)) / punit.lc f) t f)"
param_map: fixes R1 :: "('g \<times> 'i) set"   and R2 :: "('h \<times> 'j) set" shows "(map, map) \<in> (R1 \<rightarrow> R2) \<rightarrow> \<langle>R1\<rangle>list_rel \<rightarrow> \<langle>R2\<rangle>list_rel"
banach_steinhaus: fixes f :: "'c \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>x. bounded (range (\<lambda>n. f n *\<^sub>v x))" shows "bounded (range f)"
f: shows "\<forall>\<^sub>F n in F. (f n \<longlongrightarrow> g n) (at x within S)"
ca_bound: fixes n :: "nat" assumes "N \<le> n" shows "real_of_int \<bar>c (n + 1)\<bar> < real_of_int (a n) / 2"
linearI: fixes f :: "'d \<Rightarrow> 'e" assumes "\<And>b1 b2. f (b1 + b2) = f b1 + f b2"   and "\<And>r b. f (r *\<^sub>R b) = r *\<^sub>R f b" shows "linear f"
collapse: fixes prod :: "'b \<times> 'c" shows "(fst prod, snd prod) = prod"
simps_5: fixes p :: "pf"   and q :: "pf"   and xs :: "nat list" shows "eval_pf (Imp p q) xs = (eval_pf p xs \<longrightarrow> eval_pf q xs)"
replicate_mset_0: fixes x :: "'k" shows "replicate_mset 0 x = {#}"
prod_z_x1: shows "poly (prod_list qs) x1 = 0"
sw: fixes a :: "atom"   and b :: "atom" assumes "\<And>x. x \<in># M \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" shows "(a \<rightleftharpoons> b) \<bullet> M = M"
E4: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init}. \<Sum>i | i < length qs \<and> qs ! i = y. ALG' A qs init i x) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x \<noteq> y}. \<Sum>i | i < length qs \<and> qs ! i = y. ALG' A qs init i x)"
pm_of_poly_in_Polys: fixes x :: "'c"   and p :: "'d poly" shows "pm_of_poly x p \<in> P[{x}]"
Cons: fixes xs :: "ct list" shows "ct_list_eq xs ys = (\<exists>zs. ct_list_eq xs (to_ct_list zs) \<and> ct_list_eq (to_ct_list zs) ys)"
simps_4: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula"   and j :: "nat" shows "Monitor.progress \<sigma> (formula.Or \<phi> \<psi>) j = min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)"
complex_mod_triangle_sub: fixes w :: "complex"   and z :: "complex" shows "cmod w \<le> cmod (w + z) + cmod z"
step_3_def: fixes A :: "'b mat" shows "??.Jordan_Normal_Form_Existence.step_3 A = step_3_main (dim_row A) 1 A"
t'_2: shows "t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs))"
Trg_HcompNml: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Nml t"   and "E.Nml u"   and "E.Src t = E.Trg u" shows "E.Trg (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = E.Trg t"
iMOD_Min: fixes r :: "nat"   and m :: "nat" shows "iMin [ r, mod m ] = r"
empty: shows "Gcd\<^sub>f\<^sub>i\<^sub>n {} = (0::'b)"
finite_measure_axioms: shows "finite_measure (lebesgue_on (cbox x y))"
Rep_star_inject: fixes x :: "'b star"   and y :: "'b star" shows "(Rep_star x = Rep_star y) = (x = y)"
has_ell2_norm_ket: fixes a :: "'c" shows "has_ell2_norm (\<lambda>i. if a = i then 1 else 0)"
cfg_onI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and cfg :: "'a cfg"   and s :: "'a" assumes "Markov_Decision_Process K"   and "state cfg = s"   and "action cfg \<in> K s"   and "\<And>t. t \<in> set_pmf (action cfg) \<Longrightarrow> cont cfg t \<in> Markov_Decision_Process.cfg_on K t" shows "cfg \<in> Markov_Decision_Process.cfg_on K s"
I2: shows "I2 = (\<lambda>l s. 0 \<le> s ''a'' \<and> 0 \<le> s ''b'' \<and> s ''z'' < s ''a'' \<and> 0 \<le> s ''z'' \<and> s ''g'' \<le> s ''b'' \<and> 0 \<le> s ''g'' \<and> s ''y'' = s ''z'' * s ''b'' + s ''g'')"
pinf_2: fixes P :: "'a \<Rightarrow> bool"   and P' :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and Q' :: "'a \<Rightarrow> bool" assumes "\<exists>z. \<forall>x>z. P x = P' x"   and "\<exists>z. \<forall>x>z. Q x = Q' x" shows "\<exists>z. \<forall>x>z. (P x \<or> Q x) = (P' x \<or> Q' x)"
adjuster_carrier: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us \<in> carrier_vec n"
get_child_nodes_is_strongly_scdom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child \<in> set sc) = (ptr' \<in> set sc)"
cod_comp: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and g :: "'ag"   and f :: "'ag" assumes "category C"   and "partial_magma.arr C (C g f)" shows "partial_magma.cod C (C g f) = partial_magma.cod C g"
assms_10: shows "\<langle>c,[cf\<^sub>1]\<rangle> \<Rightarrow> \<langle>c',s\<^sub>1\<rangle>"
singles: fixes s :: "'a" shows "k [pairtrue s] = f s"
errMOD_igSubstAllIPresIGWlsAllSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igSubstAllIPresIGWlsAll MOD"   and "igWlsAllDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igSubstAllIPresIGWlsAllSTR (errMOD MOD)"
parts_crypt_num_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s \<turnstile> s'"   and "Crypt (SesK SK) (Num 0) \<in> parts (used s')"   and "Crypt (SesK SK) (Num 0) \<notin> parts (used s)" shows "(\<exists>n. (Asset n, Crypt (SesK SK) (Num 0)) \<in> s') \<or> SesKey SK \<in> spied s'"
quasi_inverse_transpose_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "f \<star> g \<cong> h"   and "equivalence_map f" shows "g \<cong> f\<^sup>~ \<star> h"
assms_1: shows "wf_mstate \<phi> \<pi> R st"
ordertype_VWF_\<omega>: fixes A :: "V set" assumes "finite A" shows "tp A \<in> elts \<omega>"
nth_via_drop: fixes n :: "nat"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" assumes "drop n xs = y # ys" shows "xs ! n = y"
compact_Times: fixes s :: "'c set"   and t :: "'d set" assumes "compact s"   and "compact t" shows "compact (s \<times> t)"
x_in_span_yB: shows "x \<in> local.span (insert y B)"
allAllocationsIntersection: fixes G :: "'d set"   and N :: "'c set" shows "allAllocations N G = allocationsUniverse \<inter> {a. Domain a \<subseteq> N \<and> \<Union> (Range a) = G}"
basis_in_indep_in: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "basis_in \<E> X" shows "indep_in \<E> X"
measurable_marg_dens2: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and x :: "nat"   and y :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "density_context V V' \<Gamma> \<delta>"   and "x \<in> V"   and "y \<in> V"   and "x \<noteq> y"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "marg_dens2 (V, V', \<Gamma>, \<delta>) x y \<rho> \<in> borel_measurable (stock_measure (PRODUCT (\<Gamma> x) (\<Gamma> y)))"
neq_iff: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(x \<noteq> y) = (x \<prec> y \<or> y \<prec> x)"
mult_mod_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "times c (modulo a b) = modulo (times c a) (times c b)"
assms_2: fixes p :: "'s1 spmf"   and q :: "'s2 spmf"   and a :: "'a"   and res :: "'s2"   and b :: "'b"   and s' :: "'s2" assumes "S p q"   and "a \<in> A"   and "res \<in> set_spmf q"   and "(b, s') \<in> set_spmf (callee2 res a)" shows "trace_eq_simcl S (cond_spmf_fst (p \<bind> (\<lambda>s. callee1 s a)) b) (cond_spmf_fst (q \<bind> (\<lambda>s. callee2 s a)) b)"
visible_method_exists: 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 C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' fs ms. class P D = \<lfloor>(D', fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>(Ts, T, m)\<rfloor>"
transp_def: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
indicator_pos_le: fixes S :: "'b set"   and x :: "'b" shows "(0::'a) \<le> indicator S x"
inv_unique_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b" assumes "f \<circ> g = id"   and "g \<circ> f = id" shows "inv f = g"
intro: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexMorphism X Y f"   and "ChamberComplexIsomorphism_axioms X Y f" shows "ChamberComplexIsomorphism X Y f"