hcju/isabelleps huggingface dataset:
abs_int_poly_inv: fixes p :: "int poly" shows "smult (sgn (lead_coeff p)) (abs_int_poly p) = p"
the_s: shows "(THE f. \<forall>i. v $ i = (\<Sum>x\<in>UNIV. f x * X $ x $ i)) = s"
inv_iso: shows "OrderingSetIso (\<^bold>\<le>*) (\<^bold><*) (\<^bold>\<le>) (\<^bold><) (f ` P) (the_inv_into P f)"
subsumption_and_max_literal: fixes C1 :: "'at Literal set"   and C2 :: "'at Literal set"   and L1 :: "'at Literal"   and L2 :: "'at Literal"   and A1 :: "'at"   and A2 :: "'at" assumes "subsumes C1 C2"   and "strictly_maximal_literal C1 L1"   and "strictly_maximal_literal C2 L2"   and "A1 = atom L1"   and "A2 = atom L2" shows "A1 = A2 \<or> (A1, A2) \<in> atom_ordering"
AE_iff_nn_integral: fixes M :: "'b measure"   and P :: "'b \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M" shows "almost_everywhere M P = (integral\<^sup>N M (indicator {x. \<not> P x}) = 0)"
strategy_def: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" shows "strategy p \<sigma> \<equiv> \<forall>v\<in>VV p. \<not> deadend v \<longrightarrow> v \<rightarrow> \<sigma> v"
of_rat_eq_0_iff: fixes a :: "rat" shows "(of_rat a = (0::'a)) = (a = 0)"
simps_2: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "ProblemIkTpartG.GE_wtFsym wtFsym resOf protCl (Wit \<sigma>) = (\<not> ProblemIkTpart.isRes wtFsym resOf \<sigma> \<or> protCl \<sigma>)"
sum_rel_simp_1: fixes a :: "'a"   and a' :: "'c"   and Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "((Inl a, Inl a') \<in> \<langle>Rl, Rr\<rangle>sum_rel) = ((a, a') \<in> Rl)"
deriv_A: fixes i :: "nat" shows "(deriv ^^ i) f z = (if zn \<le> int i then A i else 0)"
Amin_Suc: fixes n :: "nat"   and f :: "nat \<Rightarrow> ant" shows "Amin (Suc n) f = amin (Amin n f) (f (Suc n))"
pwF: shows "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) \<F>"
step_deterministic: fixes s1 :: "('fenv, 'henv, ('fun, 'dyn) frame) state"   and s2 :: "('fenv, 'henv, ('fun, 'dyn) frame) state"   and s3 :: "('fenv, 'henv, ('fun, 'dyn) frame) state" assumes "s1 \<rightarrow> s2"   and "s1 \<rightarrow> s3" shows "s2 = s3"
ffield_iff: fixes a :: "V"   and r :: "V" shows "(a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r) = (\<exists>b. [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<or> [b, a]\<^sub>\<circ> \<in>\<^sub>\<circ> r)"
prob_space_pair: fixes M :: "'c measure"   and N :: "'d measure" assumes "prob_space M"   and "prob_space N" shows "prob_space (M \<Otimes>\<^sub>M N)"
fresherI: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and r' :: "nat \<times> k \<times> f \<times> nat \<times> nat" assumes "nsqn\<^sub>r r < nsqn\<^sub>r r' \<or> nsqn\<^sub>r r = nsqn\<^sub>r r' \<and> \<pi>\<^sub>5 r' \<le> \<pi>\<^sub>5 r" shows "r \<sqsubseteq> r'"
sum_dcharacter_block': shows "sum (principal_dchar n) {Suc 0..n} = (if principal_dchar n = principal_dchar n then of_nat (totient n) else 0)"
ruler_equal: fixes u :: "'b list"   and w :: "'b list"   and v :: "'b list" assumes "u \<le>p w"   and "v \<le>p w"   and "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|" shows "u = v"
map_pair_subset: fixes f :: "'d \<times> 'f \<Rightarrow> 'd \<times> 'e"   and xs :: "('d \<times> 'f) list" shows "set (map_pair f xs) \<subseteq> f ` set xs"
takeWhile_filter: fixes xs :: "(nat \<times> 'a) list"   and nt :: "nat"   and c :: "enat" assumes "sorted (map fst xs)"   and "\<forall>t\<in>fst ` set xs. t \<le> nt" shows "takeWhile (\<lambda>(t, X). c < enat (nt - t)) xs = filter (\<lambda>(t, X). c < enat (nt - t)) xs"
M_def: shows "M = Max {m. \<exists>lam. (m, lam) \<in> set n_as \<and> cmod lam = 1}"
lmeasurable_iff_integrable_on: fixes S :: "'c set" shows "(S \<in> lmeasurable) = ((\<lambda>x. 1) integrable_on S)"
f: shows "f = (\<lambda>s. 3 + 4 * nat (s ''a'') + 3 * (nat (s ''a'') * nat (s ''b'')))"
in_idem: fixes A :: "'b sat set"   and x :: "'b sat" assumes "finite A"   and "x \<in> A" shows "max x (Sup A) = Sup A"
derived_of_subgroup_minimal: fixes K :: "'a set"   and H :: "'a set" assumes "K \<lhd> G\<lparr>carrier := H\<rparr>"   and "subgroup H G"   and "comm_group (G\<lparr>carrier := H\<rparr> Mod K)" shows "derived G H \<subseteq> K"
\<pi>1_geq: fixes k :: "nat" assumes "i < k" shows "\<pi>1 !! k = (toSink (fst (\<pi> !! i)), L (toSink (fst (\<pi> !! i))))"
carrier_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "carrier (Ring_Divisibility.mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
SeqConst_imp_SeqTerm: fixes s :: "hf"   and k :: "hf"   and x :: "hf" assumes "SeqConst s k x" shows "SeqTerm s k x"
max_spec2mheads: fixes G :: "prog"   and S :: "qtname"   and rt :: "ref_ty"   and mn :: "mname"   and pTs :: "ty list"   and mh :: "ref_ty \<times> mhead"   and pTs' :: "ty list"   and A :: "((ref_ty \<times> mhead) \<times> ty list) set" assumes "max_spec G S rt \<lparr>name = mn, parTs = pTs\<rparr> = insert (mh, pTs') A" shows "mh \<in> mheads G S rt \<lparr>name = mn, parTs = pTs'\<rparr> \<and> G\<turnstile>pTs[\<preceq>]pTs'"
set_to_map_set_is_map_of: fixes l :: "('a \<times> 'b) list" assumes "distinct (map fst l)" shows "set_to_map (set l) = map_of l"
LeftDerivation_delta_early: fixes index :: "nat" assumes "0 < index"   and "index + 1 < length L" shows "LeftDerivation (ladder_\<alpha> \<alpha> E L' index) (drop (ladder_n L' (index - Suc 0)) (take (ladder_n L' index) E)) (ladder_\<gamma> \<alpha> E L' index)"
assoc_def: shows "assoc \<equiv> \<lambda>\<mu> \<nu> \<pi>. \<alpha>.map (\<mu>, \<nu>, \<pi>)"
finite_insert: fixes a :: "'g"   and A :: "'g set" shows "finite (insert a A) = finite A"
functions_of_eq: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "finite A"   and "f \<in> {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A}" shows "functions_of A (f ` A) = domain_permutation A B `` {f}"
assigns_SNOC_zero: fixes k :: "'k"   and m :: "nat"   and \<AA> :: "'i" assumes "LESS k m (#\<^sub>V \<AA>)" shows "m\<^bsup>SNOC (zero (#\<^sub>V \<AA>)) \<AA>\<^esup>k = m\<^bsup>\<AA>\<^esup>k"
simps_6: fixes m :: "nat"   and p :: "('a, 'b) form"   and q :: "('a, 'b) form" shows "closed m (Impl p q) = (closed m p \<and> closed m q)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>B b\<guillemotright>" obtains "B.arr f" and "B.dom f = a" and "B.cod f = b"
lim_const_over_n: fixes a :: "'b" shows "(\<lambda>n. a / of_nat n) \<longlonglongrightarrow> (0::'b)"
AE: fixes e :: "real" assumes "0 < e"   and "e < k" shows "\<exists>d>0. d < e \<and> (\<forall>z\<in>sphere 0 d. cmod (f z) \<le> B)"
conjugate_in_unit_disc: fixes z :: "complex_homo" assumes "z \<in> unit_disc" shows "conjugate z \<in> unit_disc"
assms: shows "llength a = llength b \<and> (\<forall>n. enat n < llength a \<longrightarrow> a $ n = b $ n)"
assms_3: shows "g maps\<^bsub>C\<^esub> X to Y"
comp_null_2: fixes f :: "'c" shows "f \<cdot>\<^sub>C C.null = C.null"
add_mset_commute: fixes x :: "'k"   and y :: "'k"   and M :: "'k multiset" shows "add_mset x (add_mset y M) = add_mset y (add_mset x M)"
weakTransitionFrameImp: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and R :: "('a, 'b, 'c) psi" assumes "\<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "insertAssertion (extractFrame R) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q) \<Psi>" shows "\<Psi> : R \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"
sort_key_by_quicksort_code: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.linorder less_eq less" shows "linorder.sort_key less_eq f xs = (case xs of [] \<Rightarrow> [] | [x] \<Rightarrow> xs | [x, y] \<Rightarrow> if less_eq (f x) (f y) then xs else [y, x] | x # y # ab # list \<Rightarrow> let (lts, eqs, gts) = linorder.part less f (f (xs ! (length xs div 2))) xs in linorder.sort_key less_eq f lts @ eqs @ linorder.sort_key less_eq f gts)"
rel_eq_onp: shows "(=) = eq_onp pred_DEADID"
function_ring_car_eqI: fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and g :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int" assumes "f \<in> carrier (function_ring (carrier Zp) Zp)"   and "g \<in> carrier (function_ring (carrier Zp) Zp)"   and "\<And>a. a \<in> carrier Zp \<Longrightarrow> f a = g a" shows "f = g"
assms_5: shows "A' B' C' D' E' F' SumA G' H' I'"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'j" assumes "l_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
2: fixes t :: "'a \<times> 'a" assumes "t \<in> cl (cl ts)" shows "t \<in> cl ts"
M_index: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "gram_schmidt_fs_Rn n fs"   and "i < length fs"   and "j < length fs" shows "gram_schmidt_fs_Rn.M n fs (length fs) $$ (i, j) = gram_schmidt_fs.\<mu> n fs i j"
inj_setminus: fixes A :: "'a set set" shows "inj_on uminus A"
all_self_eq_2: fixes v :: "i" shows "[\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<alpha> \<^bold>= \<alpha> in v]"
lformula_disjuncts: fixes \<phi> :: "('a, 'k) aformula" shows "Ball (disjuncts \<phi>) lformula = lformula \<phi>"
space_K: fixes M :: "'b measure"   and K :: "'b \<Rightarrow> 'b measure"   and x :: "'b" assumes "discrete_Markov_process M K"   and "x \<in> space M" shows "space (K x) = space M"
simps_1: shows "size1 \<langle>\<rangle> = 1"
InitBlockRed: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and v' :: "val"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and 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 v'' :: "val"   and v :: "val" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e,(h, l(V \<mapsto> v'))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = \<lfloor>v''\<rfloor>"   and "P \<turnstile> T casts v to v' " shows "P,E \<turnstile> \<langle>{V:T; V:=Val v;; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; V:=Val v'';; e'},(h', l'(V := l V))\<rangle>"
sig_trd_term_wf: fixes d :: "'a \<Rightarrow> nat" assumes "dickson_grading d" shows "wf (sig_trd_term d)"
OUrename_preserves_ODE_vars: fixes ODE :: "('sf, 'sz) ODE"   and x :: "'sz"   and y :: "'sz"   and I :: "('sf, 'b, 'sz) interp" assumes "ORadmit ODE" shows "{z. swap x y z \<in> ODE_vars I ODE} = ODE_vars I (OUrename x y ODE)"
Fault_end: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate"   and f :: "'f" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "s = Fault f" shows "t = Fault f"
splitFace_edges_disj2: fixes g :: "graph"   and u :: "nat"   and v :: "nat"   and f :: "face"   and vs :: "nat list" assumes "minGraphProps g"   and "pre_splitFace g u v f vs" shows "edges_disj (snd (snd (splitFace g u v f vs)))"
constraints_of_mat_vec_def: fixes A :: "rat mat"   and b :: "rat vec" shows "constraints_of_mat_vec A b = (\<lambda>i. Le_Constraint Leq_Rel (poly_of_vec (row A i)) (b $ i)) ` {0..<dim_row A}"
finite_span: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier V" shows "local.module.span S = {lincomb a S |a. a \<in> S \<rightarrow> carrier K}"
subset_plus_right: fixes A :: "'g set" shows "A \<subseteq> G + A"
transp_sequence: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and seq :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "transp r"   and "\<And>i. r (seq (Suc i)) (seq i)"   and "i < j" shows "r (seq j) (seq i)"
satisfies_norm: fixes \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i" assumes "lformula \<phi>" shows "(\<AA> \<Turnstile> norm \<phi>) = (\<AA> \<Turnstile> \<phi>)"
shadow_root_eq_h2: fixes ptr' :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option" shows "h2 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt = h3 \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt"
nsv_hom_par: fixes x :: "'a"   and y :: "'a" shows "x \<parallel> y \<cdot> nc = x \<cdot> nc \<parallel> (y \<cdot> nc)"
ff_f: shows "f \<circ>\<^sub>A\<^bsub>\<BB>\<^esub> f = f"
simps_4: fixes v :: "'b formula" shows "is_disj (\<^bold>\<not> v) = is_lit_plus (\<^bold>\<not> v)"
mk_rtrancl_list_main: fixes fin :: "'a set"   and finl :: "'a list"   and todo :: "'a list" assumes "fin = set finl" shows "set (mk_rtrancl_list_main subsumes r todo finl) = mk_rtrancl_main todo fin"
unifiers_insert_swap: fixes s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "unifiers (insert (s, t) E) = unifiers (insert (t, s) E)"
trapped_sol_right: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and K :: "'b set"   and x :: "'b" assumes "auto_ll_on_open f X"   and "compact K"   and "K \<subseteq> X"   and "x \<in> X"   and "auto_ll_on_open.trapped_forward f X x K" shows "{0..} \<subseteq> auto_ll_on_open.existence_ivl0 f X x"
eq_iff_diff_eq_0: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(a = b) = (minus a b = zero)"
iso_relative_homology_of_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "topspace X \<inter> S \<noteq> {}" shows "hom_boundary p X S \<in> Group.iso (relative_homology_group p X S) (reduced_homology_group (p - 1) (subtopology X S))"
noStucknI': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> Stuck \<Longrightarrow> False" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Stuck}"
rk2_increment_rk2_fas_err: fixes cx :: "floatarith list"   and vs :: "real list"   and x0 :: "floatarith list"   and h :: "floatarith"   and s1' :: "real"   and s2 :: "floatarith"   and rkp :: "floatarith" assumes "einterpret cx vs = flow0 (einterpret x0 vs) (interpret_floatarith h vs * s1')"   and "length x0 = DIM('d)"   and "length cx = DIM('d)"   and "DIM('d) \<le> length vs"   and "einterpret x0 vs \<in> Csafe"   and "einterpret cx vs \<in> Csafe"   and "einterpret x0 vs + (interpret_floatarith s2 vs * interpret_floatarith h vs * interpret_floatarith rkp vs) *\<^sub>R ode (einterpret x0 vs) \<in> Csafe" shows "einterpret (rk2_fas_err rkp x0 h cx s2) vs = heun_remainder1 (flow0 (einterpret x0 vs)) ode_na ode_d_na ode_d2_na 0 (interpret_floatarith h vs) s1' - heun_remainder2 (interpret_floatarith rkp vs) (flow0 (einterpret x0 vs)) ode_na ode_d2_na 0 (interpret_floatarith h vs) (interpret_floatarith s2 vs)"
add2_comm: fixes x :: "'b"   and y :: "'b" shows "Sep_Eq.add2 x y = Sep_Eq.add2 y x"
subset_strictly_subsumes: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<subset># D" shows "strictly_subsumes C D"
power_eq_top_ennreal: fixes x :: "ennreal"   and n :: "nat" shows "(x ^ n = top) = (n \<noteq> 0 \<and> x = top)"
\<alpha>ah_hd: fixes m :: "'b"   and e :: "'b set \<times> 'b set"   and x :: "'b"   and w :: "('b set \<times> 'b set) list" assumes "m \<in> fst e"   and "x \<in> fst e \<union> snd e \<union> mon_pl w" shows "x \<in> \<alpha>ah (e # w) m"
mHom_inv: fixes N :: "('g, 'b, 'h) Module_scheme"   and m :: "'a"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "m \<in> carrier M"   and "f \<in> mHom R M N" shows "f (-\<^sub>a m) = -\<^sub>a\<^bsub>N\<^esub> f m"
GuaranteeAsGuard: fixes P :: "'d set"   and g :: "'d set"   and R :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and f :: "'f" assumes "P \<subseteq> g \<inter> R"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> R c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P guaranteeStrip f g c Q,A"
markedElementsToAppend: fixes e :: "'b"   and M1 :: "('b \<times> bool) list"   and M2 :: "('b \<times> bool) list" shows "markedElementsTo e (M1 @ M2) = (if e \<in> set (elements M1) then markedElementsTo e M1 else markedElements M1 @ markedElementsTo e M2)"
nat_numeral: fixes k :: "num" shows "nat (numeral k) = numeral k"
one_is_polynomial: fixes K :: "'a list set" assumes "subring K (poly_ring R)" shows "polynomial\<^bsub>poly_ring R\<^esub> K [\<one>\<^bsub>poly_ring R\<^esub>]"
Allowed_def: fixes F :: "'c program" shows "Allowed F \<equiv> {G. Acts G \<subseteq> AllowedActs F}"
incomp_ext: fixes w1 :: "'b list"   and w2 :: "'b list"   and z :: "'b list"   and z' :: "'b list" assumes "\<not> w1 \<bowtie> w2" shows "\<not> w1 \<cdot> z \<bowtie> w2 \<cdot> z'"
kbo_trans: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" shows "(S s t \<longrightarrow> NS t u \<longrightarrow> S s u) \<and> (NS s t \<longrightarrow> S t u \<longrightarrow> S s u) \<and> (NS s t \<longrightarrow> NS t u \<longrightarrow> NS s u)"
Lyndon_suf_char: fixes w :: "'a list" assumes "w \<noteq> \<epsilon>" shows "Lyndon w = (\<forall>s. s \<le>ns w \<longrightarrow> s \<noteq> w \<longrightarrow> w <lex s)"
mult_powr_eq: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "x * b powr y = b powr (y + log b x)"
dnf_diff: fixes n :: "nat" shows "(deriv ^^ n) f field_differentiable at w"
ts''_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [$Grow_memory] : ts'' _> ts'"
extends_subst_def: fixes \<sigma> :: "'c \<Rightarrow> 'd option"   and \<tau> :: "'c \<Rightarrow> 'd option" shows "extends_subst \<sigma> \<tau> = (\<forall>x\<in>dom \<sigma>. \<sigma> x = \<tau> x)"
sel: fixes r :: "('d, 'b, 'c) gpv"   and f :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "the_gpv (r \<bind> f) = map_spmf (map_generat id id ((\<circ>) (case_sum id (\<lambda>r. r \<bind> f)))) (the_gpv r \<bind> case_generat (\<lambda>x. map_spmf (map_generat id id ((\<circ>) Inl)) (the_gpv (f x))) (\<lambda>out c. return_spmf (IO out (\<lambda>input. Inr (c input)))))"
phi_equiv_mapping_def: fixes p :: "'g \<Rightarrow> ('node \<times> 'val, 'val list) mapping"   and g :: "'g"   and nou\<^sub>1 :: "('val, 'b set) mapping"   and nou\<^sub>2 :: "('val, 'b set) mapping" shows "uninst_code.phi_equiv_mapping p g nou\<^sub>1 nou\<^sub>2 \<equiv> \<forall>v\<in>Mapping.keys (uninst_code.ssa.phidefNodes p g). (case Mapping.lookup nou\<^sub>1 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x) = (case Mapping.lookup nou\<^sub>2 v of None \<Rightarrow> {} | Some x \<Rightarrow> id x)"
simps_8: fixes document_element' :: "(_) element_ptr option \<Rightarrow> (_) element_ptr option"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "document_element_update document_element' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element' document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr>"
ereal_between_2: fixes x :: "ereal"   and e :: "ereal" assumes "\<bar>x\<bar> \<noteq> \<infinity>"   and "0 < e" shows "x < x + e"
word_ao_dist2: fixes x :: "'b word"   and y :: "'b word"   and z :: "'b word" shows "x AND (y OR z) = x AND y OR x AND z"
Gauss_Jordan_in_ij_det_P_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and i :: "'f"   and j :: "'e" shows "Gauss_Jordan_in_ij_det_P A i j = (let n = LEAST n. A $ n $ j \<noteq> (0::'d) \<and> i \<le> n in (if i = n then (1::'d) / A $ i $ j else - (1::'d) / A $ n $ j, Gauss_Jordan_in_ij A i j))"
1: shows "[\<lbrace>a\<^sup>P,\<^bold>\<lambda>z. \<lparr>R,a\<^sup>P,z\<^sup>P\<rparr>\<rbrace> in v]"
pr0_dup: fixes a :: "'a" assumes "ide a" shows "\<r>[a] \<cdot> (a \<otimes> \<t>[a]) \<cdot> \<d>[a] = a"
\<pi>E_def: shows "cipher.\<pi>E \<equiv> 1\<^sub>C"
simps_960: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f6 :: "tf \<Rightarrow> b_e list \<Rightarrow> 'a"   and f7 :: "tf \<Rightarrow> b_e list \<Rightarrow> b_e list \<Rightarrow> 'a"   and f8 :: "nat \<Rightarrow> 'a"   and f9 :: "nat \<Rightarrow> 'a"   and f10 :: "nat list \<Rightarrow> nat \<Rightarrow> 'a"   and f11 :: "'a"   and f12 :: "nat \<Rightarrow> 'a"   and f13 :: "nat \<Rightarrow> 'a"   and f14 :: "nat \<Rightarrow> 'a"   and f15 :: "nat \<Rightarrow> 'a"   and f16 :: "nat \<Rightarrow> 'a"   and f17 :: "nat \<Rightarrow> 'a"   and f18 :: "nat \<Rightarrow> 'a"   and f19 :: "t \<Rightarrow> (tp \<times> sx) option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f20 :: "t \<Rightarrow> tp option \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f21 :: "'a"   and f22 :: "'a"   and f23 :: "v \<Rightarrow> 'a"   and f24 :: "t \<Rightarrow> unop_i \<Rightarrow> 'a"   and f25 :: "t \<Rightarrow> unop_f \<Rightarrow> 'a"   and f26 :: "t \<Rightarrow> binop_i \<Rightarrow> 'a"   and f27 :: "t \<Rightarrow> binop_f \<Rightarrow> 'a"   and f28 :: "t \<Rightarrow> testop \<Rightarrow> 'a"   and f29 :: "t \<Rightarrow> relop_i \<Rightarrow> 'a"   and f30 :: "t \<Rightarrow> relop_f \<Rightarrow> 'a"   and f31 :: "t \<Rightarrow> cvtop \<Rightarrow> t \<Rightarrow> sx option \<Rightarrow> 'a"   and x61 :: "tf"   and x62 :: "b_e list" shows "(case Loop x61 x62 of Unreachable \<Rightarrow> f1 | Nop \<Rightarrow> f2 | Drop \<Rightarrow> f3 | Select \<Rightarrow> f4 | Block x xa \<Rightarrow> f5 x xa | Loop x xa \<Rightarrow> f6 x xa | b_e.If x xa xb \<Rightarrow> f7 x xa xb | Br x \<Rightarrow> f8 x | Br_if x \<Rightarrow> f9 x | Br_table x xa \<Rightarrow> f10 x xa | Return \<Rightarrow> f11 | Call x \<Rightarrow> f12 x | Call_indirect x \<Rightarrow> f13 x | Get_local x \<Rightarrow> f14 x | Set_local x \<Rightarrow> f15 x | Tee_local x \<Rightarrow> f16 x | Get_global x \<Rightarrow> f17 x | Set_global x \<Rightarrow> f18 x | Load x xa xb xc \<Rightarrow> f19 x xa xb xc | Store x xa xb xc \<Rightarrow> f20 x xa xb xc | Current_memory \<Rightarrow> f21 | Grow_memory \<Rightarrow> f22 | C x \<Rightarrow> f23 x | Unop_i x xa \<Rightarrow> f24 x xa | Unop_f x xa \<Rightarrow> f25 x xa | Binop_i x xa \<Rightarrow> f26 x xa | Binop_f x xa \<Rightarrow> f27 x xa | Testop x xa \<Rightarrow> f28 x xa | Relop_i x xa \<Rightarrow> f29 x xa | Relop_f x xa \<Rightarrow> f30 x xa | Cvtop x xa xb xc \<Rightarrow> f31 x xa xb xc) = f6 x61 x62"
c: fixes x :: "real" shows "norm (pbernpoly (Suc n) x) \<le> c"
space_measure_pmf: fixes p :: "'c pmf" shows "space (measure_pmf p) = UNIV"