hcju/isabelleps huggingface dataset:
normalize_ports_generic_normalize_positive_ports_step_erule: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and P :: "'c \<times> 'd \<Rightarrow> 'b common_primitive match_expr \<Rightarrow> bool"   and disc2 :: "'c"   and sel2 :: "'d"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc (Prot a)"   and "P (disc2, sel2) m"   and "\<And>a. \<lbrakk>normalized_nnf_match a; a \<in> set (normalize_match (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one m))\<rbrakk> \<Longrightarrow> P (disc2, sel2) a"   and "\<And>a dpts rst. \<lbrakk>normalized_nnf_match a; primitive_extractor (disc, sel) a = (dpts, rst); getNeg dpts = []; P (disc2, sel2) a\<rbrakk> \<Longrightarrow> P (disc2, sel2) rst"   and "\<And>a spt rst. P (disc2, sel2) rst \<Longrightarrow> P (disc2, sel2) (MatchAnd (Match (C spt)) rst)"   and "m' \<in> set (normalize_ports_generic (normalize_positive_ports_step (disc, sel) C) (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one) m)" shows "P (disc2, sel2) m'"
inconsist_converg_2: fixes i :: "nat"   and e :: "nat" assumes "inconsist i e \<down>" shows "\<forall>x<e_length e. \<phi> i x \<down>"
contractible_imp_path_connected_space: fixes X :: "'a topology" assumes "contractible_space X" shows "path_connected_space X"
QuadRes_neg: fixes p :: "int"   and a :: "int" shows "QuadRes (- p) a = QuadRes p a"
orthogonal_def: fixes x :: "('a, 'cols) vec"   and y :: "('a, 'cols) vec" shows "COLS.v.orthogonal x y = (inner_cols x y = (0::'a))"
maps_def: fixes f :: "'b \<Rightarrow> 'a list"   and xs :: "'b list" shows "List.maps f xs = concat (map f xs)"
prems_4: shows "\<forall>a\<in>set (vs @ [v]). case a of (idn, a) \<Rightarrow> typ_ok \<Theta> a"
not_less_eq: fixes m :: "nat"   and n :: "nat" shows "(\<not> order.greater n m) = order.greater (Suc m) n"
elims_1: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set"   and y :: "bool" assumes "validate_clause x xa = y" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and C :: "'b literal set" where "x = I" and "xa = C" and "y = (\<forall>s. ground_clause (subst_cl C s) \<longrightarrow> validate_ground_clause I (subst_cl C s))"
compat_append_right: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and zs :: "'a list" assumes "ext gt ys xs" shows "ext gt (ys @ zs) (xs @ zs)"
group: fixes S :: "'b set"   and T :: "'c set"   and g :: "'b \<Rightarrow> 'c"   and h :: "'b \<Rightarrow> 'd" assumes "finite S"   and "finite T"   and "g ` S \<subseteq> T" shows "(\<Sum>y\<in>T. sum h {x \<in> S. g x = y}) = sum h S"
2: shows "backward_terminating_path (end_points x ; q\<^sup>T)"
rem_effectless_works_5_i: fixes as :: "('d \<times> ('e, 'f) fmap) list" shows "subseq (rem_effectless_act as) as"
iMin_subset: fixes A :: "'a set"   and B :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "iMin B \<le> iMin A"
eval_wp_SetPC: fixes f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" shows "wp (SetPC f p) P = (\<lambda>s. \<Sum>a\<in>supp (p s). p s a * wp (f a) P s)"
irrefl_def: fixes r :: "('a \<times> 'a) set" shows "irrefl r = (\<forall>a. (a, a) \<notin> r)"
bounded_w': fixes i :: "nat" shows "w i \<in> \<Sigma>"
nths_take: fixes iset :: "nat set"   and n :: "nat"   and xs :: "'a list" assumes "\<forall>i\<in>iset. i < n" shows "nths (take n xs) iset = nths xs iset"
succeed_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" shows "mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i = {x. \<exists>n. semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some i \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n \<notin> F - {q\<^sub>0} \<and> semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc n) \<in> F}"
hid_is_CSP2: fixes P :: "('c, 'd) action"   and cs :: "'c set" shows "hid P cs is CSP2 healthy"
inject_5: fixes x51 :: "linear_poly"   and x52 :: "rat"   and y51 :: "linear_poly"   and y52 :: "rat" shows "(EQ x51 x52 = EQ y51 y52) = (x51 = y51 \<and> x52 = y52)"
vs_facts: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> F" shows "set (vs \<phi>) \<subseteq> var \<and> set (vs \<phi>) \<inter> Fvars \<phi> = {} \<and> set (vs \<phi>) \<inter> \<Union> (FvarsT ` fst ` set txs) = {} \<and> set (vs \<phi>) \<inter> snd ` set txs = {} \<and> length (vs \<phi>) = length txs \<and> distinct (vs \<phi>)"
simps_4: fixes tst :: "'test"   and c :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and i :: "nat" shows "wt_cont_eff (While tst c) s i = (if tval tst s then (1, c ;; While tst c, s) else (1, Done, s))"
r_mul_prim: shows "prim_recfn 2 r_mul"
One_le_of_nat_ennreal: fixes x :: "nat" shows "(1 \<le> of_nat x) = (1 \<le> x)"
power_one_right: fixes a :: "'b" shows "a ^ 1 = a"
law_restrict_def: fixes R :: "('c, 'd) ring_scheme" shows "law_restrict R \<equiv> ring.truncate R\<lparr>mult := \<lambda>a\<in>carrier R. restrict ((\<otimes>\<^bsub>R\<^esub>) a) (carrier R), add := \<lambda>a\<in>carrier R. restrict ((\<oplus>\<^bsub>R\<^esub>) a) (carrier R)\<rparr>"
range_coeff: fixes p :: "'a poly" shows "range (coeff p) = insert (0::'a) (set (coeffs p))"
inj_on_Abs_list: shows "inj_on Abs_List (list (range Leaf))"
ninv_r_inv: fixes f :: "nat \<Rightarrow> nat"   and n :: "nat"   and b :: "nat" assumes "f \<in> {i. i \<le> n} \<rightarrow> {i. i \<le> n}"   and "inj_on f {i. i \<le> n}"   and "b \<le> n" shows "f (ninv n f b) = b"
polyfun_def: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" shows "polyfun N f = (\<exists>p. vars p \<subseteq> N \<and> (\<forall>x. insertion x p = f x))"
cases: fixes a1 :: "trm"   and a2 :: "trm" assumes "a1 \<Down> a2" obtains s :: "trm" and t :: "trm" and u :: "trm" where "a1 = s" and "a2 = u" and "s \<leadsto> ta__" and "ta__ \<Down> u"   | t :: "trm" where "a1 = t" and "a2 = t" and "t \<leadsto>|"
lm12: fixes x :: "'c"   and y :: "'d"   and P :: "('c \<times> 'd) set" shows "((x, y) \<in> P) = (y \<in> P `` {x})"
f9: fixes ss :: "symbol list" shows "take j ss = take j (take i ss)"
continuous_closed_vimage: fixes S :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "closed S"   and "\<And>x. isCont f x" shows "closed (f -` S)"
T_on_on': fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and s0 :: "'a"   and qs :: "'b list"   and A :: "('a \<Rightarrow> 'd) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'd)"   and x :: "'d" shows "On_Off.T step t s0 qs (On_Off.off2 step A (s0, x) qs) = On_Off.T_on' step t A (s0, x) qs"
class_leq_refl: fixes cs :: "(String.literal \<times> String.literal) set"   and c :: "String.literal" assumes "wf_subclass_loc cs"   and "class_ex cs c" shows "class_leq cs c c"
wp1_prec_Seq_correct: fixes c1 :: "com"   and c2 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and l :: "char list \<Rightarrow> nat"   and s :: "char list \<Rightarrow> int" assumes "wp\<^sub>1 (c1;; c2) Q l s" shows "\<down>\<^sub>t (c1, s) + prec c1 (\<lambda>s. \<down>\<^sub>t (c2, s)) s \<le> \<down>\<^sub>t (c1;; c2, s)"
assms: shows "sort_of (atom a) \<noteq> sort_of (atom b)"
loc_fin: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.locally_finite_partial_semigroup times D" shows "finite (partial_semigroup.gR_downset times D x)"
subs_Bot_G_entails: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Inf_G :: "'b inference set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and sB :: "'b set"   and N :: "'b set" assumes "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"   and "sB \<noteq> {}"   and "sB \<subseteq> Bot_G" shows "entails_G sB N"
induct: fixes x1 :: "(Vars.var \<times> exp) list"   and x2 :: "exp"   and x3 :: "Vars.var list"   and x4 :: "(Vars.var \<times> exp) list"   and x5 :: "exp"   and P :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Vars.var list \<Rightarrow> (Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> bool" assumes "x1 : x2 \<Down>\<^bsub>x3\<^esub> x4 : x5"   and "\<And>\<Gamma> x e L. P \<Gamma> (Lam [x]. e) L \<Gamma> (Lam [x]. e)"   and "\<And>y \<Gamma> e x L \<Delta> \<Theta> z e'. \<lbrakk>atom y \<sharp> (\<Gamma>, e, x, L, \<Delta>, \<Theta>, z); \<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : Lam [y]. e'; P \<Gamma> e L \<Delta> (Lam [y]. e'); \<Delta> : e'[y::=x] \<Down>\<^bsub>L\<^esub> \<Theta> : z; P \<Delta> e'[y::=x] L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (App e x) L \<Theta> z"   and "\<And>\<Gamma> x e L \<Delta> z. \<lbrakk>map_of \<Gamma> x = Some e; delete x \<Gamma> : e \<Down>\<^bsub>x # L\<^esub> \<Delta> : z; P (delete x \<Gamma>) e (x # L) \<Delta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (Var x) L ((x, z) # \<Delta>) z"   and "\<And>\<Delta> \<Gamma> L body \<Theta> z. \<lbrakk>atom ` domA \<Delta> \<sharp>* (\<Gamma>, L); \<Delta> @ \<Gamma> : body \<Down>\<^bsub>L\<^esub> \<Theta> : z; P (\<Delta> @ \<Gamma>) body L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (Terms.Let \<Delta> body) L \<Theta> z"   and "\<And>\<Gamma> b L. P \<Gamma> (Bool b) L \<Gamma> (Bool b)"   and "\<And>\<Gamma> scrut L \<Delta> b e\<^sub>1 e\<^sub>2 \<Theta> z. \<lbrakk>\<Gamma> : scrut \<Down>\<^bsub>L\<^esub> \<Delta> : Bool b; P \<Gamma> scrut L \<Delta> (Bool b); \<Delta> : (if b then e\<^sub>1 else e\<^sub>2) \<Down>\<^bsub>L\<^esub> \<Theta> : z; P \<Delta> (if b then e\<^sub>1 else e\<^sub>2) L \<Theta> z\<rbrakk> \<Longrightarrow> P \<Gamma> (scrut ? e\<^sub>1 : e\<^sub>2) L \<Theta> z" shows "P x1 x2 x3 x4 x5"
distinct_card: fixes xs :: "'a list" assumes "distinct xs" shows "card (list.set xs) = length xs"
simps: fixes a1 :: "'tp"   and a2 :: "('c, 'd) lit list" shows "(a1 \<turnstile> a2) = ((\<exists>\<sigma> c. a1 = \<sigma> \<and> a2 = c \<and> infTp \<sigma>) \<or> (\<exists>c \<sigma>. a1 = \<sigma> \<and> a2 = c \<and> (\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>)))"
chamber_preimage: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and C :: "'b set" assumes "ChamberComplexIsomorphism X Y f"   and "C \<in> X"   and "SimplicialComplex.maxsimp Y (f ` C)" shows "SimplicialComplex.maxsimp X C"
Catch: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c\<^sub>1 :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and c\<^sub>1' :: "('e, 'd, 'f) com"   and s' :: "('e, 'f) xstate"   and c\<^sub>2 :: "('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> (c\<^sub>1, s) \<rightarrow> (c\<^sub>1', s')" shows "\<Gamma>\<turnstile> (Catch c\<^sub>1 c\<^sub>2, s) \<rightarrow> (Catch c\<^sub>1' c\<^sub>2, s')"
assms_1: shows "\<guillemotleft>g : a' \<rightarrow> a\<guillemotright>"
butlast_take: fixes n :: "nat"   and xs :: "'e list" assumes "n \<le> length xs" shows "butlast (take n xs) = take (n - 1) xs"
bal_split_last_1: fixes ls :: "('b btree \<times> 'b) list"   and sub :: "'b btree"   and sep :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "bal (Node (ls @ (sub, sep) # rs) t)" shows "bal (Node (ls @ rs) t)"
fst_mono: fixes x :: "'d \<times> 'e"   and y :: "'d \<times> 'e" assumes "x \<le> y" shows "fst x \<le> fst y"
Bernstein_Poly_span1: fixes c :: "real"   and d :: "real"   and P :: "real poly"   and p :: "nat" assumes "c \<noteq> d"   and "degree P \<le> p" shows "P \<in> poly_vs.span {Bernstein_Poly x p c d |x. x \<le> p}"
convert_obs_initial_simps_1: fixes ta :: "('d, 'e, 'f, 'g, 'h, 'c) thread_action" shows "\<lbrace>convert_obs_initial ta\<rbrace>\<^bsub>o\<^esub> = map NormalAction \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>"
Can_implies_Arr: 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.Can V src trg t" shows "bicategorical_language.Arr V src trg t"
elims_1: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a tree"   and y :: "bool" assumes "bst_wrt x xa = y" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = P__" and "xa = \<langle>\<rangle>" and "y"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and l :: "'a tree" and a :: "'a" and r :: "'a tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "y = ((\<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)"
Ys: shows "{vec_first y m |y. y \<in> X} = set (map (\<lambda>y. vec_first y m) Xs)"
swap_swap: fixes p :: "'b \<times> 'd" shows "prod.swap (prod.swap p) = p"
Abs_Int_fun_axioms: shows "Abs_Int_fun num' plus' \<gamma>"
br_if_true: fixes n :: "i32"   and i :: "nat" assumes "\<not> int_eq n 0" shows "\<lparr>[$C ConstInt32 n, $Br_if i]\<rparr> \<leadsto> \<lparr>[$Br i]\<rparr>"
ads_d_def: fixes x :: "('a \<times> 'a) set" shows "rdom x = rel_ad (rel_ad x)"
1: shows "invertible_mat P'' \<and> invertible_mat Q'' \<and> isDiagonal_mat D \<and> D = P'' * A * Q'' \<and> P'' \<in> carrier_mat m m \<and> Q'' \<in> carrier_mat n n \<and> D \<in> carrier_mat m n"
assms_3: shows "J.qmpt_factor proj2 M3 T3"
eq_numeral_iff_iszero_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "num"   and y :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(uminus (numeral.numeral one plus x) = uminus (numeral.numeral one plus y)) = ring_1.iszero zero (neg_numeral.sub minus plus one y x)"
used_Cons: fixes ev :: "event"   and evs :: "event list" shows "used (ev # evs) = (case ev of Says A B X \<Rightarrow> parts {X} \<union> used evs | Gets A X \<Rightarrow> used evs | Notes A X \<Rightarrow> parts {X} \<union> used evs)"
inverse_2_1: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "real_of_2 (inverse_2 x) = inverse (real_of_2 x)"
r_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<otimes> \<one> = x"
lookup_pair_map_pair: fixes xs :: "('a \<times> 'd) list"   and f :: "'a \<times> 'd \<Rightarrow> 'a \<times> 'e"   and k :: "'a"   and v :: "'d" assumes "oalist_inv_raw xs"   and "snd (f (k, 0::'d)) = (0::'e)"   and "\<And>a b. comp (fst (f a)) (fst (f b)) = comp (fst a) (fst b)" shows "lookup_pair (map_pair f xs) (fst (f (k, v))) = snd (f (k, lookup_pair xs k))"
DeltaNegForall: fixes A :: "('a, 'b) form"   and n :: "'a"   and G :: "('a, 'b) form list" assumes "\<stileturn> Neg (A[App n []/0]) # G"   and "news n (A # G)" shows "\<stileturn> Neg (Forall A) # G"
ODE_bound_effect: fixes s :: "real"   and t :: "real"   and sol :: "real \<Rightarrow> (real, 'b) vec"   and I :: "('c, 'd, 'b) interp"   and ODE :: "('c, 'b) ODE"   and X :: "(real, 'b) vec set"   and b :: "(real, 'b) vec" assumes "s \<in> {0..t}"   and "(sol solves_ode (\<lambda>_. ODE_sem I ODE)) {0..t} X" shows "Vagree (sol 0, b) (sol s, b) (- BVO ODE)"
length_xs: shows "length xs = length (nths xs A) + length (nths xs (- A))"
is_iso_def: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" shows "G \<cong> H = (Group.iso G H \<noteq> {})"
chine_in_hom: shows "\<guillemotleft>assoc\<mu>\<nu>\<pi>.chine : assoc\<mu>\<nu>\<pi>.dom.apex \<rightarrow>\<^sub>C assoc\<mu>\<nu>\<pi>.cod.apex\<guillemotright>"
elims: fixes x :: "nat"   and y :: "nat" assumes "nlen x = y" obtains "x = 0" and "y = 0"   | v :: "nat" where "x = Suc v" and "y = 1 + nlen (Suc v div 10)"
diff_cancel: fixes A :: "'b multiset" shows "A - A = {#}"
red_subst: fixes p :: "trm"   and p' :: "trm"   and x :: "name"   and m :: "trm" assumes "p \<mapsto> p'" shows " m[x::=p] \<mapsto>\<^sup>* m[x::=p'] "
w_simps_4: shows "trg TTfgh_THfgh.chine = src (tab\<^sub>0 h \<star> THfgh.p\<^sub>0)"
push_bit_of_1: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n one = power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n"
borel_measurable_iff_halfspace_ge: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>i\<in>Basis. \<forall>a. {w \<in> space M. a \<le> f w \<bullet> i} \<in> sets M)"
select_convs_10: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "undef \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = undef"
intro: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "mpt M T"   and "ergodic_qmpt M T" shows "ergodic_mpt M T"
s2: fixes f :: "face" assumes "f \<in> set (facesAt g a)" shows "f \<bullet> a \<notin> V"
eq: fixes x :: "(nat, nat) pp"   and y :: "(nat, nat) pp" shows "(comp_of_ord lex_pp x y = Eq) = (x = y)"
hyps: fixes y :: "'a poly" assumes "degree y < degree p" shows "P y"
simpnum_numbound0: fixes t :: "Cooper.num" assumes "numbound0 t" shows "numbound0 (simpnum t)"
distinct_35: fixes x41 :: "nat"   and x42 :: "nat" shows "connect(x41, x42) \<noteq> \<tau>\<^sub>n"
cobounded2: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b \<le> b"
mrw_before: fixes \<E> :: "('a \<times> ('b, 'a) obs_event action) llist set"   and P :: "'c prog"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and r :: "nat"   and w :: "nat" assumes "executions_aux \<E> P"   and "E \<in> \<E>"   and "P \<turnstile> (E, ws) \<surd>"   and "P,E \<turnstile> r \<leadsto>mrw w"   and "\<And>a. \<lbrakk>a < r; a \<in> read_actions E\<rbrakk> \<Longrightarrow> P,E \<turnstile> a \<leadsto>mrw ws a" shows "w < r"
total_weight_lowerbound: fixes g :: "graph"   and w :: "face \<Rightarrow> nat" assumes "Invariants.inv g"   and "final g"   and "tame g"   and "admissible w g"   and "ListSum (faces g) w < squanderTarget" shows "squanderLowerBound g \<le> ListSum (faces g) w"
unitary: fixes n :: "nat"   and A :: "complex Matrix.mat" assumes "gate n A" shows "unitary A"
infinite_rev_existence_ivl0_rewrites_1: fixes x :: "'a" shows "({0..} \<subseteq> rev.existence_ivl0 x) = ({..0} \<subseteq> existence_ivl0 x)"
hyps: fixes Ts :: "ty list"   and vs :: "val list"   and l :: "char list \<Rightarrow> val option"   and l' :: "char list \<Rightarrow> val option" assumes "length ps' = length Ts"   and "length ps' = length vs"   and "P \<turnstile> \<langle>blocks (ps', Ts, vs, e),(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>" shows "\<exists>l''. P \<turnstile> \<langle>e,(h, l(ps' [\<mapsto>] vs))\<rangle> \<Rightarrow> \<langle>e',(h', l'')\<rangle>"
da_elim_cases_55: fixes Env :: "env"   and B :: "lname set"   and e1 :: "expr"   and e2 :: "expr"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>e1.[e2]\<rangle>\<guillemotright> A" obtains E1 :: "assigned" where "Env\<turnstile> B \<guillemotright>\<langle>e1\<rangle>\<guillemotright> E1" and "Env\<turnstile> nrm E1 \<guillemotright>\<langle>e2\<rangle>\<guillemotright> A"
setmult_rcos_assoc: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "'c set"   and x :: "'c" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H <#>\<^bsub>G\<^esub> (K #>\<^bsub>G\<^esub> x) = H <#>\<^bsub>G\<^esub> K #>\<^bsub>G\<^esub> x"
wf_problem_def: shows "wf_problem \<equiv> wf_domain \<and> distinct (map fst (objects P) @ map fst (consts D)) \<and> (\<forall>(n, y)\<in>set (objects P). wf_type y) \<and> distinct (init P) \<and> wf_world_model (set (init P)) \<and> wf_fmla objT (goal P)"
perm_weak_contr_mono: fixes P :: "'a list \<Rightarrow> bool"   and x :: "'a list"   and y :: "'a list" assumes "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"   and "\<forall>x xs. P (x # x # xs) \<longrightarrow> P (x # xs)"   and "\<forall>x xs. P xs \<longrightarrow> P (x # xs)"   and "set x \<subseteq> set y"   and "P x" shows "P y"
isMinimalBackjumpLevelGetBackjumpLevel: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantUniqC (getC state)"   and "getConflictFlag state"   and "isUIP (opposite (getCl state)) (getC state) (getM state)"   and "0 < currentLevel (getM state)" shows "isMinimalBackjumpLevel (getBackjumpLevel state) (opposite (getCl state)) (getC state) (getM state)"
unit_circle_fix_iff_unit_circle_set: fixes M :: "moebius" shows "unit_circle_fix M = (moebius_pt M ` unit_circle_set = unit_circle_set)"
gcd_zero_iff': fixes a :: "'a"   and b :: "'a" shows "((0::'a) = gcd a b) = (a = (0::'a) \<and> b = (0::'a))"
ivl_cong: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "a = c"   and "b = d"   and "\<And>x. \<lbrakk>c \<le> x; x < d\<rbrakk> \<Longrightarrow> g x = h x" shows "sum g {a..<b} = sum h {c..<d}"
free_ag_single: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "free_gen_condition f i a z"   and "n \<noteq> m" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<noteq> m\<Odot>a\<^bsub>f,i,z\<^esub>"
mset_factors_imp_nonzero: fixes F :: "'a multiset"   and p :: "'a" assumes "mset_factors F p" shows "p \<noteq> (0::'a)"
f4: shows "fst (memory_read 10 (address + 4) (snd (fst (write_reg w curr_win (rd AND 30) s1)))) = fst (memory_read 10 (address + 4) (snd (fst (write_reg w curr_win (rd AND 30) s2))))"
rep_eq: fixes x :: "'b list" shows "fset (fset_from_list x) = set x"
ou_witness_closure_simps_2: shows "closure {ou_witness.b} = {ou_witness.a, ou_witness.b}"
simps_1: fixes t :: "nat"   and v :: "nat" shows "eval_expr t (Const v) = ({}, \<lambda>\<theta>. v)"
add: shows "count (image_mset f A) x = sum (count A) (f -` {x} \<inter> set_mset A)"