hcju/isabelleps huggingface dataset:
minit0_And: fixes \<psi> :: "'a MFOTL.formula"   and \<phi> :: "'a MFOTL.formula"   and n :: "nat" assumes "\<not> (safe_formula (formula.Neg \<psi>) \<and> fv \<psi> \<subseteq> fv \<phi>)" shows "minit0 n (MFOTL.And \<phi> \<psi>) = MAnd (minit0 n \<phi>) True (minit0 n \<psi>) ([], [])"
games_gfp_unfold: shows "games_gfp = fixgames games_gfp"
map_sel_2: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c stream" shows "stl (smap f a) = smap f (stl a)"
siso0_transC_indis: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and s :: "'a"   and t :: "'a"   and c' :: "('b, 'c) com"   and s' :: "'a" assumes "PL_Indis indis"   and "PL_Indis.siso0 tval aval indis c"   and "PL.mustT tval aval c s"   and "PL.mustT tval aval c t"   and "PL.transC tval aval (c, s) (c', s')"   and "(s, t) \<in> indis" shows "\<exists>t'. (s', t') \<in> indis \<and> PL.transC tval aval (c, t) (c', t')"
map_iterator_rule_P: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "'b \<Rightarrow> 'c option"   and I :: "'b set \<Rightarrow> 'a \<Rightarrow> bool"   and \<sigma>0 :: "'a"   and c :: "'a \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" assumes "map_iterator it m"   and "I (dom m) \<sigma>0"   and "\<And>k v it \<sigma>. \<lbrakk>c \<sigma>; k \<in> it; m k = Some v; it \<subseteq> dom m; I it \<sigma>\<rbrakk> \<Longrightarrow> I (it - {k}) (f (k, v) \<sigma>)"   and "\<And>\<sigma>. I {} \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> it. \<lbrakk>it \<subseteq> dom m; it \<noteq> {}; \<not> c \<sigma>; I it \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
valuationContainsItsLiteralsVariable: fixes literal :: "Literal"   and valuation :: "Literal list" assumes "literal el valuation" shows "var literal \<in> vars valuation"
simps_1: fixes p :: "real \<times> real \<times> real" shows "getPoly (EqUni p) = p"
has_field_derivative_series: fixes S :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and f' :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a"   and x0 :: "'a" assumes "convex S"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"   and "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"   and "x0 \<in> S"   and "summable (\<lambda>n. f n x0)" shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
preserves_hom: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and G :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a \<Rightarrow> 'b"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.in_hom A f a b" shows "partial_magma.in_hom B (\<tau> f) (F a) (G b)"
ws_def: shows "ws = getFrN (xs @ us @ vs) (r # ts) [] (length xs)"
no_cond_effs: shows "\<forall>\<pi>\<in>set ast\<delta>. is_standard_operator' \<pi>"
vector_space_def: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e" shows "vector_space scale \<equiv> ((\<forall>a x y. scale a (x + y) = scale a x + scale a y) \<and> (\<forall>a b x. scale (a + b) x = scale a x + scale b x)) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'d) x = x)"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b" assumes "constant_functor A B b" shows "category A"
BuildSeq3_1: fixes B :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool"   and y :: "hf"   and y' :: "hf"   and y'' :: "hf"   and C :: "hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> hf \<Rightarrow> bool" assumes "B y y' y''" shows "BuildSeq3 B C \<lbrace>\<langle>0, y, y', y''\<rangle>\<rbrace> 0 y y' y''"
store_word_mem_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and addr :: "word32"   and data :: "word32"   and bm :: "word4"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> Some t1 = store_word_mem s1 addr data bm 10 \<and> Some t2 = store_word_mem s2 addr data bm 10" shows "low_equal t1 t2"
disc_2: fixes xs :: "'a llist"   and b :: "'b" assumes "\<not> lnull xs" shows "\<not> is_TNil (tllist_of_llist b xs)"
SeqAbstFormP_Eq: fixes s :: "name"   and k :: "name"   and x :: "tm"   and y :: "tm"   and x' :: "tm"   and y' :: "tm"   and v :: "tm"   and i :: "tm" assumes "atom s \<sharp> (k, x, y, x', y', v, i)"   and "atom k \<sharp> (x, y, x', y', v, i)" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i (Q_Eq x y) (Q_Eq x' y') (Var s) (Var k)))"
update_i_def: fixes ba :: "'a list \<times> 'a list"   and x :: "'a" shows "update_i ba x = (fst ba, x # tl (snd ba))"
in_Solutions_iff: fixes x :: "nat list"   and y :: "nat list" shows "((x, y) \<in> local.Solutions) = (length x = m \<and> length y = n \<and> a \<bullet> x = b \<bullet> y)"
timpls_transformable_to_pred_if_timpls_transformable_to': fixes TI :: "('sets set \<times> 'sets set) list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and s :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and A :: "'sets set set" assumes "timpls_transformable_to' TI t s"   and "wf\<^sub>t\<^sub>r\<^sub>m t"   and "\<forall>f\<in>funs_term t. is_Abs f \<longrightarrow> the_Abs f \<in> A" shows "timpls_transformable_to_pred (A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s"
distinct_7: fixes x21 :: "'a"   and x22 :: "'a form list" shows "Compound x21 x22 \<noteq> ff"
inducts: fixes x1 :: "bool"   and x2 :: "char list list"   and x3 :: "v sem_env"   and x4 :: "'b state"   and x5 :: "dec list"   and x6 :: "'b state \<times> (v sem_env, v) result"   and P :: "bool \<Rightarrow> char list list \<Rightarrow> v sem_env \<Rightarrow> 'b state \<Rightarrow> dec list \<Rightarrow> 'b state \<times> (v sem_env, v) result \<Rightarrow> bool" assumes "evaluate_decs x1 x2 x3 x4 x5 x6"   and "\<And>ck mn env s. P ck mn env s [] (s, Rval (make_sem_env nsEmpty nsEmpty))"   and "\<And>ck mn s1 s2 env d ds e. evaluate_dec ck mn env s1 d (s2, Rerr e) \<Longrightarrow> P ck mn env s1 (d # ds) (s2, Rerr e)"   and "\<And>ck mn s1 s2 s3 env d ds new_env r. evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r) \<and> P ck mn (extend_dec_env new_env env) s2 ds (s3, r) \<Longrightarrow> P ck mn env s1 (d # ds) (s3, combine_dec_result new_env r)" shows "P x1 x2 x3 x4 x5 x6"
Ds_def: shows "Ds = (ARG_MIN length Ds. codomain.gallery (f ` C # Ds @ [f ` D]))"
head_App: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" shows "head (App s t) = head s"
distance_unique: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "\<exists>!d. poincare_distance_pred u v d"
one_root_a_lt0: fixes a :: "real"   and b :: "real"   and r :: "real"   and c :: "real" assumes "a < 0"   and "b = 2 * a * r"   and "a * r\<^sup>2 - 2 * a * r * r + c = 0" shows "\<exists>y'>- r. \<forall>x\<in>{- r<..y'}. a * x\<^sup>2 + 2 * a * r * x + c < 0"
nonzero_one_closed: shows "\<one> \<in> nonzero R"
ordIso_iff_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" shows "(r =o r') = (r \<le>o r' \<and> r' \<le>o r)"
fmrestrict_agree: fixes X :: "'d set"   and f :: "('d, 'e) fmap"   and g :: "('d, 'e) fmap"   and x :: "'d" assumes "agree (fmrestrict_set X f) (fmrestrict_set X g)"   and "x \<in> X \<inter> fmdom' f \<inter> fmdom' g" shows "fmlookup (fmrestrict_set X f) x = fmlookup (fmrestrict_set X g) x"
F_3: fixes u :: "nat \<Rightarrow> 'a" assumes "\<forall>n. u n \<in> F n" shows "u \<longlonglongrightarrow> Inf A"
gallery_remdup_adj: fixes xs :: "'a set list"   and x :: "'a set"   and ys :: "'a set list" assumes "folding.gallery (xs @ [x, x] @ ys)" shows "folding.gallery (xs @ [x] @ ys)"
comm_inv_char: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "x \<oplus> y = \<zero>" shows "inv\<^bsub>add_monoid R\<^esub> x = y"
simps_1: fixes uu :: "'a" shows "typeof (OpDyn uu) = None"
inject_6: fixes x8 :: "('a, 'b) form"   and y8 :: "('a, 'b) form" shows "(Forall x8 = Forall y8) = (x8 = y8)"
swap_atom_simps_2: fixes a :: "atom"   and b :: "atom" assumes "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> b = a"
assms_3: shows "h \<turnstile> g x \<rightarrow>\<^sub>r y"
MON2_LEQ: shows "mon_ww fg w2 \<subseteq> mon_env fg ww"
plus_of_int: shows "fds_perzeta (frac q + real_of_int \<lfloor>q\<rfloor>) = fds_perzeta (frac q)"
shows_list_xml_def: fixes xs :: "xml list" shows "shows_list xs = showsp_list shows_prec 0 xs"
tendsto_ln_over_ln: fixes a :: "real"   and c :: "real" assumes "0 < a"   and "0 < c" shows "((\<lambda>x. ln (a * x) / ln (c * x)) \<longlongrightarrow> 1) at_top"
power_def: shows "(^) \<equiv> \<lambda>uu uua. rec_nat (\<lambda>a. 1::'a) (\<lambda>n na a. a * na a) uua uu"
finite_Diff: fixes A :: "'k set"   and B :: "'k set" assumes "finite A" shows "finite (A - B)"
all_chars_def: shows "all_chars \<equiv> enum_class.enum"
discI_2: fixes resumption :: "('f, 'g, 'h) Resumption.resumption"   and x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" assumes "resumption = Resumption.resumption.Pause x21 x22" shows "\<not> Resumption.resumption.is_Done resumption"
card_B: shows "|{p \<bullet> x |p. supp (F, P) \<sharp>* p}| <o natLeq +c |UNIV|"
ball_def: fixes x :: "'c"   and e :: "real" shows "ball x e = {y. dist x y < e}"
assms_1: shows "v \<in> set (add_effects_of op)"
enabled_iff: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c"   and \<omega> :: "'c stream" shows "MC_syntax.enabled K s \<omega> = (MC_syntax.enabled K (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
fw_shortest_path: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and i' :: "nat"   and i :: "nat"   and j' :: "nat"   and j :: "nat"   and k :: "nat" assumes "linordered_monoid.cycle_free m n"   and "i' \<le> i"   and "j' \<le> j"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n" shows "linordered_monoid.D m i' j' k = linordered_monoid.fw m n k i j i' j'"
intro\<^sub>3: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "lc \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'" shows "lc \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> lc'"
demonic_sup_inf: fixes Q :: "'b \<Rightarrow> 'a"   and Q' :: "'b \<Rightarrow> 'a" shows "[: Q \<squnion> Q' :] = [: Q :] \<sqinter> [: Q' :]"
simps_2: fixes v :: "'b" shows "\<^bold>\<langle>v\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<I> = \<^bold>\<langle>v\<^bold>\<rangle>"
wf3: shows "\<forall>j<nc. \<forall>i<nr. (A1 \<circ> A2 \<otimes> B1 \<circ> B2) ! j ! i = ((A1 \<otimes> B1) \<circ> (A2 \<otimes> B2)) ! j ! i"
zero_Nat_def: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and RepNat :: "'a \<Rightarrow> nat"   and AbsNat :: "nat \<Rightarrow> 'a" assumes "class.nat minus plus zero RepNat AbsNat" shows "zero = AbsNat 0"
rel_stacktraces_Cons: fixes get :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option"   and st1 :: "('a, 'dyn) frame list"   and st2 :: "('a, ('dyn, 'ubx1, 'ubx2) unboxed) frame list"   and f :: "'a"   and \<Sigma>1 :: "'dyn list"   and \<Sigma>2 :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and fd2 :: "('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef"   and n :: "nat"   and opt :: "'a option" assumes "rel_stacktraces get st1 st2 (Some f)"   and "\<Sigma>1 = norm_stack \<Sigma>2"   and "get f = Some fd2"   and "sp_fundef get fd2 (take n (body fd2)) = Ok (map typeof \<Sigma>2)"   and "pred_option (is_valid_fun_call get fd2 n \<Sigma>2) opt" shows "rel_stacktraces get (Frame f n \<Sigma>1 # st1) (Frame f n \<Sigma>2 # st2) opt"
node\<^sub>i_rule_app: fixes k :: "nat"   and c :: "nat"   and tsi' :: "('a btnode ref option \<times> 'a) list"   and li :: "'a btnode ref option"   and ai :: "'a"   and aa :: "('a btnode ref option \<times> 'a) array"   and al :: "nat"   and ls :: "('a btree \<times> 'a) list"   and l :: "'a btree"   and a :: "'a"   and r :: "'a btree"   and ri :: "'a btnode ref option" assumes "2 * k \<le> c"   and "c \<le> 4 * k + 1" shows "<is_pfa c (tsi' @ [(li, ai)]) (aa, al) * blist_assn k ls tsi' * btree_assn k l li * id_assn a ai * btree_assn k r ri> node\<^sub>i k (aa, al) ri <btupi_assn k (abs_split.node\<^sub>i k (ls @ [(l, a)]) r)>\<^sub>t"
IH: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and vtx2 :: "'a"   and vtx1 :: "'a" assumes "\<forall>x y. x \<le> f x y \<and> y \<le> f x y"   and "\<forall>x y. x \<le> g x y \<and> y \<le> g x y"   and "\<forall>vtx. ListMem vtx G \<longrightarrow> \<not> R vtx vtx"   and "R vtx2 vtx1"   and "ListMem vtx1 G"   and "top_sorted_abs R G" shows "f (w vtx2) (wlp R w g f vtx1 G) \<le> wlp R w g f vtx2 G"
is_Eulerian_trail_def: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" shows "is_Eulerian_trail v ps v' \<equiv> is_trail v ps v' \<and> edges (rem_unPath ps G) = {}"
Fourier_series_square_summable: fixes S :: "real set"   and w :: "nat \<Rightarrow> real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and I :: "nat set" assumes "orthonormal_system S w"   and "\<And>i. w i square_integrable S"   and "f square_integrable S" shows "summable (confine (\<lambda>i. (orthonormal_coeff S w f i)\<^sup>2) I)"
adal: shows "(ad, al) \<in> action_loc P E (ws a)"
LPDs_aux2: fixes i :: "nat"   and r :: "mregex" shows "LPDi i r \<subseteq> lfp (\<lambda>S. insert r S \<union> (insert r S \<bind> LPD))"
carr_prodag_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" shows "carr_prodag I A = {f \<in> extensional I. f \<in> I \<rightarrow> Un_carrier I A \<and> (\<forall>i\<in>I. f i \<in> carrier (A i))}"
zeroprimeideal: shows "primeideal {\<zero>\<^bsub>K [X]\<^esub>} (K [X])"
ordering_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "Orderings.ordering less_eq less \<equiv> ((\<forall>a b. less a b = (less_eq a b \<and> a \<noteq> b)) \<and> (\<forall>a. less_eq a a)) \<and> (\<forall>a b. less_eq a b \<longrightarrow> less_eq b a \<longrightarrow> a = b) \<and> (\<forall>a b c. less_eq a b \<longrightarrow> less_eq b c \<longrightarrow> less_eq a c)"
collinear_quadratic: fixes t :: "(real, 3) vec"   and i :: "real"   and a :: "(real, 3) vec"   and r :: "(real, 3) vec" assumes "t = i *\<^sub>R a + r" shows "t \<bullet> (M *v t) = a \<bullet> (M *v a) * i\<^sup>2 + 2 * (a \<bullet> (M *v r)) * i + r \<bullet> (M *v r)"
all_comm: fixes P :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
fmran_of_list: fixes m :: "('b \<times> 'a) list" shows "fmran (fmap_of_list m) = snd |`| fset_of_list (AList.clearjunk m)"
wf_subtypeE: fixes \<Gamma> :: "binding list"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> T <: U" obtains "\<Gamma> \<turnstile>\<^sub>w\<^sub>f" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f T" and "\<Gamma> \<turnstile>\<^sub>w\<^sub>f U"
f_def: shows "f \<equiv> \<lambda>ta xa. if xa \<in> Fvars \<phi> then FvarsT ta else {}"
g: fixes x :: "'a" assumes "x \<in> topspace X" shows "\<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
finI: fixes K :: "'c \<Rightarrow> 'c DBMEntry"   and g :: "'a \<times> 'c \<Rightarrow> 'd"   and f :: "(nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry) \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" assumes "theLe \<circ> K = id" shows "finite (g ` {(c, d) |c d. 0 < v c \<and> v c \<le> n \<and> f M (v c) = K d})"
emeasure_sigma: fixes \<Omega> :: "'b set"   and A :: "'b set set" shows "emeasure (sigma \<Omega> A) = (\<lambda>x. 0)"
topSeqSubst: fixes \<sigma> :: "(name list \<times> 'a list) list" shows "substCond.seqSubst (\<top>) \<sigma> = \<top>"
replace_def: fixes e1 :: "'a fset"   and old :: "'a"   and new :: "'a" shows "replace e1 old new = (\<lambda>x. if x = old then new else x) |`| e1"
InvariantNoDecisionsWhenConflictNorUnitAfterApplyLearn: fixes state :: "State" assumes "InvariantUniq (getM state)"   and "InvariantConsistent (getM state)"   and "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"   and "InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantClCharacterization (getCl 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 "let state' = applyLearn state in InvariantNoDecisionsWhenConflict (getF state) (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenUnit (getF state) (getM state') (currentLevel (getM state')) \<and> InvariantNoDecisionsWhenConflict [getC state] (getM state') (getBackjumpLevel state') \<and> InvariantNoDecisionsWhenUnit [getC state] (getM state') (getBackjumpLevel state')"
wt_elim_cases_16: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and C :: "qtname"   and sig :: "sig"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Methd C sig)\<Colon>T" obtains m :: "qtname \<times> methd" and T :: "ty" where "Tb__ = Inl T" and "is_class (prg E) C" and "methd (prg E) C sig = Some m" and "E,dt\<Turnstile>Body (declclass m) (stmt (mbody (mthd m)))\<Colon>-T"
s_def: shows "s = [:a / of_nat n, 1::'a:]"
assms_1: shows "(f \<longlongrightarrow> L) F"
mult_less_cancel_right1: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times b c) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
assms: shows "length (cs\<^bsup>\<pi>\<^esup> i) = 1"
countably_compactE: fixes s :: "'b set"   and C :: "'b set set" assumes "countably_compact s"   and "\<forall>t\<in>C. open t"   and "s \<subseteq> \<Union> C"   and "countable C" obtains C' :: "'b set set" where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union> C'"
real_divr_nonneg_neg_upper_bound: fixes x :: "real"   and y :: "real"   and prec :: "nat" assumes "0 \<le> x"   and "y \<le> 0" shows "real_divr prec x y \<le> 0"
product_intro: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "CM"   and "\<forall>w. P w z = (P w x \<and> P w y)" shows "x \<otimes> y = z"
Fubini_set_integrable: fixes A :: "'a set"   and B :: "'b set"   and f :: "'a \<times> 'b \<Rightarrow> 'c" assumes "set_borel_measurable (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f"   and "set_integrable lborel A (\<lambda>x. LBINT y:B. norm (f (x, y)))"   and "AE x\<in>A in lborel. set_integrable lborel B (\<lambda>y. f (x, y))" shows "set_integrable (lborel \<Otimes>\<^sub>M lborel) (A \<times> B) f"
add_2: fixes Y1 :: "'a multiset"   and Y2 :: "'a multiset" assumes "Y = Y1 + Y2"   and "(X1, Y1) \<in> multpw ns"   and "(add_mset x2 X2, Y2) \<in> multpw ns" shows "thesis"
vec_to_iarray_column': fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec" assumes "k < ncols A" shows "vec_to_iarray (column (mod_type_class.from_nat k) A) = column_iarray k (matrix_to_iarray A)"
elims_3: fixes x :: "'a Line"   and xa :: "'a Line" assumes "\<not> meets x xa" obtains line1 :: "'a Line" and line2 :: "'a Line" where "x = line1" and "xa = line2" and "\<nexists>p. inLine p line1 \<and> inLine p line2"
prefix_of_psnocE: fixes p :: "'a prefix"   and x :: "'a set \<times> nat"   and s :: "'a trace" assumes "prefix_of (psnoc p x) s"   and "last_ts p \<le> snd x" obtains "prefix_of p s" and "\<Gamma> s (plen p) = fst x" and "\<tau> s (plen p) = snd x"
mkM_simps_1: fixes T :: "('a, 'es, 'as) BEState Trace set" shows "worlds (SPR.mkM T) = T"
that: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "prime q"   and "dist (real p / real q) \<alpha> < \<epsilon>" shows "thesis"
simps_1: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b stream" shows "shd (sfilter P s) = shd (sdrop_while (Not \<circ> P) s)"
countable_as_injective_image: fixes A :: "'d set" assumes "countable A"   and "infinite A" obtains f :: "nat \<Rightarrow> 'd" where "A = range f" and "inj f"
inj_on_restrict_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "inj_on (restrict f A) A = inj_on f A"
HA2Kripke_def: fixes a :: "('d, 'e, 'f) hierauto" shows "HA2Kripke a = Abs_kripke ({ST. HA ST = a}, {InitStatus a}, StepRelSem a, LabelFunSem a)"
other: shows "\<forall>i p is \<theta> sb \<D> \<O> \<R>. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> a \<notin> \<O> \<union> all_acquired sb"
IH: shows "tick_count r c k = #\<^sub>\<le> r c k"
ChiPlus_HAStates: fixes A :: "('a, 'b, 'c) hierauto" shows "ChiPlus A \<subseteq> HAStates A \<times> HAStates A"
per_initN_refine: shows "(per_init' (Max (insert 0 V) + 1), per_init V) \<in> local.per_supset_rel"
IDE_def: fixes f :: "'c local.term set" shows "IDE f \<equiv> ARR f \<and> (\<forall>t. t \<in> f \<longrightarrow> Ide \<^bold>\<lfloor>t\<^bold>\<rfloor>) \<and> DOM f = COD f"
lst_def: fixes ss :: "'a \<times> ('b \<times> 'a) list" shows "lst ss = (if snd ss = [] then fst ss else snd (last (snd ss)))"
Union_Int_subset: fixes A :: "'b set set"   and B :: "'b set set" shows "\<Union> (A \<inter> B) \<subseteq> \<Union> A \<inter> \<Union> B"