hcju/isabelleps huggingface dataset:
simps_5: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and x1 :: "'e" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
ipassmt_disjoint_inj_k: fixes ipassmt :: "iface \<Rightarrow> ('b word \<times> nat) list option"   and ifce :: "iface"   and ips :: "('b word \<times> nat) list"   and k :: "iface"   and ips' :: "('b word \<times> nat) list"   and p :: "'b word" assumes "ipassmt_sanity_disjoint ipassmt"   and "ipassmt ifce = Some ips"   and "ipassmt k = Some ips'"   and "p \<in> ipcidr_union_set (set ips)"   and "p \<in> ipcidr_union_set (set ips')" shows "k = ifce"
sgnx_poly_nz_2: fixes p :: "real poly"   and x :: "real" assumes "poly p x \<noteq> 0" shows "sgnx (poly p) (at_right x) = sgn (poly p x)"
g1im: shows "g1 ` {0..1} \<subseteq> S"
distinct_conv_nth: fixes xs :: "'k list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
relevant_vars_subls: fixes L :: "fterm literal set"   and \<sigma>\<^sub>1 :: "char list \<Rightarrow> fterm"   and \<sigma>\<^sub>2 :: "char list \<Rightarrow> fterm" assumes "\<forall>x\<in>vars\<^sub>l\<^sub>s L. \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "L \<cdot>\<^sub>l\<^sub>s \<sigma>\<^sub>1 = L \<cdot>\<^sub>l\<^sub>s \<sigma>\<^sub>2"
vint_commutative_law_addition: fixes x :: "V"   and y :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x +\<^sub>\<int> y = y +\<^sub>\<int> x"
globEANow: fixes n :: "nat"   and i :: "nat"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "nat \<Rightarrow> cnf"   and tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'c"   and t' :: "nat \<Rightarrow> 'c"   and \<gamma> :: "(nat \<Rightarrow> 'c) \<Rightarrow> nat \<Rightarrow> bool" assumes "n \<le> i"   and "active c (t i)"   and "dynamic_component.eval tCMP active c t t' n (dynamic_component.glob \<gamma>)" shows "dynamic_component.eval tCMP active c t t' i \<gamma>"
emeasure_Collect_T: fixes P :: "'a cfg stream \<Rightarrow> bool"   and s :: "'a cfg" assumes "Measurable.pred (stream_space (count_space UNIV)) P" shows "emeasure (MC.T s) {x \<in> space (MC.T s). P x} = \<integral>\<^sup>+ t. emeasure (MC.T t) {x \<in> space (MC.T t). P (t ## x)} \<partial>measure_pmf (K_cfg s)"
binomial_deriv1: fixes n :: "nat"   and a :: "real"   and b :: "real" shows "(\<Sum>k\<le>n. real k * real (n choose k) * a ^ (k - 1) * b ^ (n - k)) = real n * (a + b) ^ (n - 1)"
X_def: shows "X = (SIGMA x:UNIV. set_pmf (J x))\<^sup>* `` {x}"
Ord_induct2: fixes k :: "hf"   and P :: "hf \<Rightarrow> bool" assumes "Ord k"   and "P 0"   and "\<And>k. \<lbrakk>Ord k; P k\<rbrakk> \<Longrightarrow> P (succ k)" shows "P k"
greedy_conforming_path_properties_5: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v0 \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>"   and "ParityGame.strategy G p** \<sigma>'" shows "ParityGame.path_conforms_with_strategy G p (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0) \<sigma>"
ans2: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> y) + ad (x \<cdot> ad (ad y)) = ad (x \<cdot> ad (ad y))"
rev_conv_fold: fixes xs :: "'b list" shows "rev xs = fold (#) xs []"
tab_in_hom: shows "\<guillemotleft>\<rho>\<sigma>.tab : tab\<^sub>1 r \<star> p\<^sub>1 \<Rightarrow> (r \<star> s) \<star> tab\<^sub>0 s \<star> p\<^sub>0\<guillemotright>"
id: shows "C *\<^sub>v w = 0\<^sub>v n"
M_def: shows "M = inv_into (carrier G) \<phi> ` N"
valid_modif: shows "\<forall>s\<in>P. \<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"
kronecker_factorization_rat_def: fixes p :: "rat poly" shows "kronecker_factorization_rat p = kronecker_factorization_rat_main divisors_int divisors_int_pos (degree p div 2) p"
top_add: fixes x :: "'a" assumes "(0::'a) \<le> x" shows "\<top> + x = \<top>"
component_constrains: fixes F :: "'c program"   and G :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "F \<le> G"   and "G \<in> A co B" shows "F \<in> A co B"
v: shows "P v y \<and> P v (\<midarrow> x)"
DSourcesS12: shows "DSources level2 sS12 = {sS10}"
pelims: fixes x :: "int \<times> (int \<times> int) \<times> int \<times> int"   and xa :: "int \<times> (int \<times> int) \<times> int \<times> int"   and xb :: "int"   and xc :: "(int \<times> int) list"   and y :: "int \<times> (int \<times> int) \<times> int \<times> int" assumes "combine_code x xa xb xc = y"   and "combine_code_dom (x, xa, xb, xc)" obtains \<delta>\<^sub>L :: "int" and p\<^sub>0\<^sub>L :: "int \<times> int" and p\<^sub>1\<^sub>L :: "int \<times> int" and \<delta>\<^sub>R :: "int" and p\<^sub>0\<^sub>R :: "int \<times> int" and p\<^sub>1\<^sub>R :: "int \<times> int" and l :: "int" and ps :: "(int \<times> int) list" where "x = (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L)" and "xa = (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R)" and "xb = l" and "xc = ps" and "y = (let (\<delta>, c\<^sub>0, c\<^sub>1) = if \<delta>\<^sub>L < \<delta>\<^sub>R then (\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) else (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) in Let (filter (\<lambda>p. (fst p - l)\<^sup>2 < \<delta>) ps) (find_closest_pair_code (\<delta>, c\<^sub>0, c\<^sub>1)))" and "combine_code_dom ((\<delta>\<^sub>L, p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L), (\<delta>\<^sub>R, p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R), l, ps)"
vsv_vrrestriction: fixes A :: "V" shows "vsv (r \<restriction>\<^sup>r\<^sub>\<circ> A)"
map_val2_conv: fixes xs :: "char list exp list"   and ys :: "val list"   and e :: "char list exp"   and zs :: "char list exp list" shows "(xs = map Val ys @ throw e # zs) = map_val2 xs ys (throw e # zs)"
red_inf_impl: fixes \<iota> :: "('f \<times> 'l) inference"   and NL :: "('f \<times> 'l) set" assumes "\<iota> \<in> Red_I_\<G> NL" shows "to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` NL)"
cases: fixes x :: "(bool list cstate \<times> (bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)) \<times> (bool list \<times> bool list) insec_query" obtains uu_ :: "(bool list \<times> bool list) option" and as :: "bool list \<Rightarrow> bool list option" and a' :: "bool list" and m' :: "bool list" where "x = ((Void, uu_, as__), Edit (a'__, m'__))"   | m :: "bool list" and uv_ :: "(bool list \<times> bool list) option" and as :: "bool list \<Rightarrow> bool list option" and a' :: "bool list" and m' :: "bool list" where "x = ((Store m__, uv_, asa__), Edit (a', m'))"   | m :: "bool list" and es :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((Store ma__, es__), Forward)"   | m :: "bool list" and es :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((Store mb__, es), Drop)"   | m :: "bool list" and as :: "bool list \<Rightarrow> bool list option" where "x = ((Store m, None, as), Look)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and uw_ :: "(bool list \<times> bool list) insec_query" where "x = ((cstate.Fail, va__), uw___)"   | vb :: "bool list" and vd :: "bool list \<times> bool list" and vc :: "bool list \<Rightarrow> bool list option" where "x = ((Store vb__, Some vd__, vc__), Look)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" and uw_ :: "(bool list \<times> bool list) insec_query" where "x = ((cstate.Collect vba__, vaa__), uw_)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((Void, vab__), Look)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Fail, vac__), Look)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Collect vbb__, vad__), Look)"   | v :: "bool list cstate" and vd :: "bool list \<times> bool list" and vc :: "bool list \<Rightarrow> bool list option" where "x = ((v, Some vd, vc), Look)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((Void, vae__), Forward)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Fail, vaf__), Forward)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Collect vbc__, vag__), Forward)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((Void, vah__), Drop)"   | va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Fail, vai__), Drop)"   | vb :: "bool list" and va :: "(bool list \<times> bool list) option \<times> (bool list \<Rightarrow> bool list option)" where "x = ((cstate.Collect vb, va), Drop)"
wpT_is_pre: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>a {wp\<^sub>3\<^sub>' c Q} c { Q}"
big_iff_small: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_prog P" shows "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle> = (P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle> \<and> final e')"
sum_distrib_left: fixes r :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
Quotient_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "Quotient R Abs Rep T = ((\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and> (\<forall>r s. R r s = (R r r \<and> R s s \<and> Abs r = Abs s)) \<and> T = (\<lambda>x y. R x x \<and> Abs x = y))"
homotopic_paths: fixes s :: "'c set"   and p :: "real \<Rightarrow> 'c"   and q :: "real \<Rightarrow> 'c" shows "homotopic_paths s p q = (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and> h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and> (\<forall>x\<in>{0..1}. h (0, x) = p x) \<and> (\<forall>x\<in>{0..1}. h (1, x) = q x) \<and> (\<forall>t\<in>{0..1}. pathstart (h \<circ> Pair t) = pathstart p \<and> pathfinish (h \<circ> Pair t) = pathfinish p))"
Osum_Well_order: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set" assumes "Field r \<inter> Field r' = {}"   and "Well_order r"   and "Well_order r'" shows "Well_order (r \<union>o r')"
it_simr: fixes y :: "'a"   and x :: "'a" assumes "\<lbrace>y\<rbrace>x\<lbrace>y\<rbrace>" shows "\<lbrace>y\<rbrace>it x\<lbrace>y\<rbrace>"
Suc: fixes f :: "nat \<Rightarrow> 'a" shows "poly_of_vec (vec n f) = Poly (rev (map f [0..<n]))"
natural_transformation_axioms: shows "natural_transformation A C (K \<circ> F) (L \<circ> F) (\<tau>' \<circ> F)"
simps: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and mgu :: "'b set set \<Rightarrow> 'c option"   and less_atm :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and N :: "'b literal multiset set"   and P :: "'b literal multiset set"   and Q :: "'b literal multiset set" assumes "FO_resolution_prover S subst_atm id_subst comp_subst renamings_apart atm_of_atms mgu less_atm" shows "FO_resolution_prover.Q_of_state (N, P, Q) = Q"
poly_mult_left_cancel: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b list"   and q :: "'b list"   and r :: "'b list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.poly plus zero times (semiring_0.pmult plus zero times p q) = semiring_0.poly plus zero times (semiring_0.pmult plus zero times p r)) = (semiring_0.poly plus zero times p = semiring_0.poly plus zero times [] \<or> semiring_0.poly plus zero times q = semiring_0.poly plus zero times r)"
domains_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "partial_magma C" shows "partial_magma.domains C f \<equiv> {a. partial_magma.ide C a \<and> C f a \<noteq> partial_magma.null C}"
gid: fixes x :: "'a" assumes "x \<in> sphere a r"   and "x \<in> T" shows "g x = x"
ennreal_minus_cancel_iff: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "(a - b = a - c) = (b = c \<or> a \<le> b \<and> a \<le> c \<or> a = top)"
ipresVar_def: fixes h :: "('d, 'e, 'varSort, 'var, 'f) qTerm set \<Rightarrow> 'g"   and MOD :: "('h, 'i, 'varSort, 'j, 'k, 'var, 'g, 'l, 'm) model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"
arc_to_block_def: fixes G :: "('a, 'b) pre_digraph"   and e :: "'b" assumes "graph G" shows "graph.arc_to_block G e \<equiv> {tail G e, head G e}"
strict_iff_order: fixes ko :: "'c key_order"   and b :: "'c"   and a :: "'c" shows "ko.lt ko b a = (ko.le ko b a \<and> a \<noteq> b)"
entropy_distr: fixes MX :: "'c measure"   and X :: "'a \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> real" assumes "distributed M MX X (\<lambda>x. ennreal (f x))"   and "\<And>x. x \<in> space MX \<Longrightarrow> 0 \<le> f x" shows "entropy b MX X = - (LINT x|MX. f x * log b (f x))"
distinct_indicesD: fixes as :: "('a \<times> 'b) list"   and i :: "'a"   and x :: "'b"   and y :: "'b" assumes "distinct_indices as"   and "(i, x) \<in> set as"   and "(i, y) \<in> set as" shows "x = y"
matrix_inv_matrix_mul: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'd) vec, 'b) vec" assumes "invertible A"   and "invertible B" shows "matrix_inv (A ** B) = matrix_inv B ** matrix_inv A"
indicator_simps_1: fixes x :: "'k"   and S :: "'k set" assumes "x \<in> S" shows "indicator S x = (1::'l)"
Int_extend_set_lemma: fixes A :: "'c set"   and B :: "'a set" shows "A \<inter> extend_set h (project_set h A \<inter> B) = A \<inter> extend_set h B"
arc_implies_awalk: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" assumes "wf_digraph G"   and "e \<in> arcs G" shows "pre_digraph.awalk G (tail G e) [e] (head G e)"
ltake_lappend: fixes n :: "enat"   and xs :: "'c llist"   and ys :: "'c llist" shows "ltake n (lappend xs ys) = lappend (ltake n xs) (ltake (n - llength xs) ys)"
join_equality: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and s :: "'a" assumes "dlat le"   and "dpo.is_sup le x y s" shows "dlat.join le x y = s"
take_cols_carrier_mat_strict: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and inds :: "nat list" assumes "A \<in> carrier_mat nr nc"   and "\<And>i. i \<in> set inds \<Longrightarrow> i < nc" shows "take_cols A inds \<in> carrier_mat nr (length inds)"
intros_2: fixes ws :: "letter list list" assumes "\<And>w. bar (w # ws)" shows "bar ws"
I_def: fixes f :: "complex poly"   and u :: "complex" shows "lindemann_weierstrass_aux.I f u = exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) - (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)"
tendsto_within_open: fixes a :: "'b"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "a \<in> S"   and "open S" shows "(f \<longlongrightarrow> l) (at a within S) = f \<midarrow>a\<rightarrow> l"
comm_monoid_cancelI: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.comm_monoid G"   and "\<And>a b c. \<lbrakk>a \<otimes>\<^bsub>G\<^esub> c = b \<otimes>\<^bsub>G\<^esub> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" shows "comm_monoid_cancel G"
peY: shows "chain (\<lambda>i. plotkin_exists (Y i))"
liftable_set_extension_axioms: shows "liftable_set_extension alts' alts set_ext"
symmetric_condition_2: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and index :: "int" assumes "symmetric_bibd point_set block_collection u_block_size index" shows "index * (int (card point_set) - 1) = u_block_size * (u_block_size - 1)"
wlsSEM_def: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a, 'b) semDom_scheme" shows "wlsSEM SEM \<equiv> sWlsNE SEM \<and> sWlsDisj SEM \<and> sOpPrSWls SEM"
cases: fixes x :: "Unboxed.type" obtains "x = Ubx1"   | "x = Ubx2"
SUCCEED_abort_def: fixes msg :: "String.literal"   and dm :: "'b nres"   and m :: "'b nres" shows "SUCCEED_abort msg dm m = (case m of FAILi \<Rightarrow> m | RES X \<Rightarrow> if X = {} then Code.abort msg (\<lambda>_. dm) else RES X)"
axioms_2: fixes nfv :: "nat"   and fv :: "nat set"   and sat :: "'c trace \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> bool" assumes "cosafety_fo_spec nfv fv sat" shows "cosafety_fo_spec_axioms sat"
nearly_healthy_wlp_Bind: fixes p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>s. nearly_healthy (wlp (p (f s)))" shows "nearly_healthy (wlp (Bind f p))"
assms_8: shows "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"
rmge2_in_R1: shows "rmge2 \<in> \<R>"
Rep_inject_sym: fixes x :: "'a"   and y :: "'a" shows "(x = y) = (Rep x = Rep y)"
lookup_except_eq_idI: fixes t :: "'a"   and S :: "'a set"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<notin> S" shows "lookup (except p S) t = lookup p t"
n_gt: fixes i :: "nat" assumes "2 \<le> i"   and "i < k" shows "n (Suc i) < n i"
either_empty2: fixes t :: "'b ttree" shows "t \<oplus>\<oplus> TTree.empty = t"
Cons: fixes ys :: "(real \<times> real) list"   and xa :: "real \<times> real"   and x :: "real \<times> real" assumes "\<forall>y\<in>set ys. ccw' xa x y"   and "ccw'.sortedP xa ys" shows "ccw'.sortedP xa (x # ys)"
orbit_altdef_permutation: fixes f :: "'c \<Rightarrow> 'c"   and s :: "'c" assumes "permutation f" shows "orbit f s = {(f ^^ n) s |n. True}"
assms: fixes x :: "val"   and \<sigma> :: "nat \<Rightarrow> val" assumes "x \<in> type_universe t"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "0 \<le> extract_real (cexpr_sem (case_nat x \<sigma>) e)"
overpower_lemma: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and g :: "real \<Rightarrow> real"   and d :: "real \<Rightarrow> real"   and x :: "real" assumes "f a \<le> g a"   and "\<And>x. a \<le> x \<Longrightarrow> ((\<lambda>x. g x - f x) has_real_derivative d x) (at x)"   and "\<And>x. a \<le> x \<Longrightarrow> 0 \<le> d x"   and "a \<le> x" shows "f x \<le> g x"
induct: fixes x1 :: "com list"   and x2 :: "state"   and x3 :: "com list"   and x4 :: "state"   and P :: "com list \<Rightarrow> state \<Rightarrow> com list \<Rightarrow> state \<Rightarrow> bool" assumes "(x1, x2) \<rightarrow> (x3, x4)"   and "\<And>t f s cs. t \<in> f s \<Longrightarrow> P (Do f # cs) s cs t"   and "\<And>c1 c2 cs s. P ((c1; c2) # cs) s (c1 # c2 # cs) s"   and "\<And>b s c1 c2 cs. b s \<Longrightarrow> P ((IF b THEN c1 ELSE c2) # cs) s (c1 # cs) s"   and "\<And>b s c1 c2 cs. \<not> b s \<Longrightarrow> P ((IF b THEN c1 ELSE c2) # cs) s (c2 # cs) s"   and "\<And>b s c cs. \<not> b s \<Longrightarrow> P ((WHILE b DO c) # cs) s cs s"   and "\<And>b s c cs. b s \<Longrightarrow> P ((WHILE b DO c) # cs) s (c # (WHILE b DO c) # cs) s"   and "\<And>p cs s. P (CALL p # cs) s (body p # cs) s"   and "\<And>f c g cs s. P ((LOCAL f; c; g) # cs) s (c # Do (\<lambda>t. {g s t}) # cs) (f s)" shows "P x1 x2 x3 x4"
intros_1: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelRPO Enc"
f_join_singleton_if: fixes xs :: "'b list"   and n :: "nat" shows "xs \<Join>\<^sub> {n} = (if n < length xs then [xs ! n] else [])"
dispatch_instruction_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "dispatch_instruction instr \<equiv> let instr_name = fst instr in do traps \<leftarrow> gets get_trap_set; if traps = {} then if instr_name \<in> {load_store_type LDSB, load_store_type LDUB, load_store_type LDUBA, load_store_type LDUH, load_store_type LD, load_store_type LDA, load_store_type LDD} then load_instr instr else if instr_name \<in> {load_store_type STB, load_store_type STH, load_store_type ST, load_store_type STA, load_store_type STD} then store_instr instr else if instr_name \<in> {sethi_type SETHI} then sethi_instr instr else if instr_name \<in> {nop_type NOP} then nop_instr instr else if instr_name \<in> {logic_type ANDs, logic_type ANDcc, logic_type ANDN, logic_type ANDNcc, logic_type ORs, logic_type ORcc, logic_type ORN, logic_type XORs, logic_type XNOR} then logical_instr instr else if instr_name \<in> {shift_type SLL, shift_type SRL, shift_type SRA} then shift_instr instr else if instr_name \<in> {arith_type ADD, arith_type ADDcc, arith_type ADDX} then add_instr instr else if instr_name \<in> {arith_type SUB, arith_type SUBcc, arith_type SUBX} then sub_instr instr else if instr_name \<in> {arith_type UMUL, arith_type SMUL, arith_type SMULcc} then mul_instr instr else if instr_name \<in> {arith_type UDIV, arith_type UDIVcc, arith_type SDIV} then div_instr instr else if instr_name \<in> {ctrl_type SAVE, ctrl_type RESTORE} then save_restore_instr instr else if instr_name \<in> {call_type CALL} then call_instr instr else if instr_name \<in> {ctrl_type JMPL} then jmpl_instr instr else if instr_name \<in> {ctrl_type RETT} then rett_instr instr else if instr_name \<in> {sreg_type RDY, sreg_type RDPSR, sreg_type RDWIM, sreg_type RDTBR} then read_state_reg_instr instr else if instr_name \<in> {sreg_type WRY, sreg_type WRPSR, sreg_type WRWIM, sreg_type WRTBR} then write_state_reg_instr instr else if instr_name \<in> {load_store_type FLUSH} then flush_instr instr else if instr_name \<in> {bicc_type BE, bicc_type BNE, bicc_type BGU, bicc_type BLE, bicc_type BL, bicc_type BGE, bicc_type BNEG, bicc_type BG, bicc_type BCS, bicc_type BLEU, bicc_type BCC, bicc_type BA, bicc_type BN} then branch_instr instr else fail () else return () od"
binary_chinese_remainder_unique_nat: fixes m1 :: "nat"   and m2 :: "nat"   and u1 :: "nat"   and u2 :: "nat" assumes "coprime m1 m2"   and "m1 \<noteq> 0"   and "m2 \<noteq> 0" shows "\<exists>!x. x < m1 * m2 \<and> [x = u1] (mod m1) \<and> [x = u2] (mod m2)"
l_inv_ex: fixes x :: "'c" assumes "x \<in> carrier H" shows "\<exists>y\<in>carrier H. y \<otimes>\<^bsub>H\<^esub> x = \<one>\<^bsub>H\<^esub>"
ex_map_conv: fixes ys :: "'c list"   and f :: "'b \<Rightarrow> 'c" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
scalar_prod_smult_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and a :: "'c" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (a \<cdot>\<^sub>v v\<^sub>2) = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
case_prod_beta': fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
bs_mset_of_empty: shows "BsSkewBinomialHeapStruc.bs_to_mset BsSkewBinomialHeapStruc.bs_empty = {#}"