hcju/isabelleps huggingface dataset:
assms_1: shows "(Ast, Bst) \<in> code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"
path_append: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and l :: "'b list"   and v :: "'b"   and w :: "'b" assumes "path E u l v"   and "(v, w) \<in> E" shows "path E u (l @ [v]) w"
prems: shows "Suc k \<le> mod_type_class.to_nat a"
ordinal_1_exp: fixes x :: "ordinal" shows "oSuc 0 ** x = oSuc 0"
len_eq: shows "length (fst gc\<^sub>C) = length cms"
liouville_lambda_power_odd: fixes m :: "nat"   and n :: "nat" assumes "odd m" shows "liouville_lambda (n ^ m) = liouville_lambda n"
has_integral_is_0_cbox: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = (0::'b)" shows "(f has_integral (0::'b)) (cbox a b)"
prems_2: shows "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) (D @ [di])\<rbrakk>\<^sub>d \<I>"
poly_altdef: fixes p :: "'b poly"   and x :: "'b" shows "poly p x = (\<Sum>i\<le>Polynomial.degree p. poly.coeff p i * x ^ i)"
c_def: shows "c = (\<lambda>n. (- 1) ^ Suc j * bernoulli' n * pochhammer (real (Suc n)) j')"
coeff_keys: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and p :: "'b mpoly" shows "(m \<in> keys (mapping_of p)) = (coeff p m \<noteq> (0::'b))"
semiring_norm_87: fixes m :: "num"   and n :: "num" shows "(num.Bit0 m = num.Bit0 n) = (m = n)"
eq1: shows "concat ((Dec G u) \<cdot> (Dec G v)) = u \<cdot> v"
correctness: fixes x0 :: "'grp"   and n :: "nat"   and x1 :: "'grp"   and \<sigma> :: "bool" assumes "x0 \<in> carrier (\<G> n)"   and "x1 \<in> carrier (\<G> n)" shows "mal_def.correct n (x0, x1) \<sigma>"
l11_17: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"
intro: fixes \<alpha> :: "V"   and \<NN> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V" assumes "\<Z> \<alpha>"   and "vfsequence \<NN>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "is_tdghm_axioms \<AA> \<BB> \<FF> \<GG> \<NN>" shows "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
vors_1: shows "pre_split_face f ram1 ram2 vs"
assms: shows "(f \<longlongrightarrow> fx) (at x within S)"
halfspace_simps_10: shows "gt_halfspace \<equiv> \<lambda>sctn x. pstn sctn < x \<bullet> normal sctn"
nn_integral_mono_AE: fixes u :: "'b \<Rightarrow> ennreal"   and v :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "AE x in M. u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
composite_simps_4: shows "trg ((f \<star> g) \<star> h) = trg f"
select_convs_3: fixes heap :: "heap"   and locals :: "locals"   and more :: "'a" shows "state.more \<lparr>heap = heap, locals = locals, \<dots> = more\<rparr> = more"
that: assumes "nop A []\<^sub>\<circ> f" shows "thesis"
simps_3: fixes m :: "modality"   and g :: "safety"   and r :: "'a regex"   and s :: "'a regex" shows "local.safe_regex m g (Plus r s) = ((g = Lax \<or> fv_regex fv r = fv_regex fv s) \<and> local.safe_regex m g r \<and> local.safe_regex m g s)"
f: fixes i :: "'a"   and j :: "'a"   and x :: "'b" assumes "i \<in> I"   and "j \<in> I"   and "x \<in> topspace X \<inter> T i \<inter> T j" shows "f i x = f j x"
set_der_stack_lt: fixes k :: "nat"   and i\<^sub>0 :: "nat" assumes "k \<in> set (local.der_stack i\<^sub>0)" shows "k < i\<^sub>0"
le_less_linear: fixes xa :: "'o"   and x :: "'a"   and y :: "'a" shows "le xa x y \<or> lt xa y x"
RedSFAccNone: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool" assumes "\<nexists>b t. P \<turnstile> C has F,b:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh),b\<rangle> \<rightarrow> \<langle>THROW NoSuchFieldError,(h, l, sh),False\<rangle>"
consec_un_max: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> maximum j = maximum (i \<squnion> j)"
inv_i_iso_\<iota>_2: fixes f :: "'a" assumes "\<guillemotleft>f : \<I> \<rightarrow> \<I>\<^sub>1\<guillemotright>"   and "local.iso f"   and "f \<cdot> \<iota> = \<iota>\<^sub>1 \<cdot> (f \<otimes>\<^sub>1 f)" shows "local.iso (local.inv i \<cdot> f)"
lset_ltake: fixes n :: "enat"   and xs :: "'c llist" shows "lset (ltake n xs) \<subseteq> lset xs"
qs_def: shows "qs = (SOME qs'. valid_decomp X qs' \<and> standard_decomp 0 qs' \<and> monomial_decomp qs' \<and> cone_decomp N qs' \<and> exact_decomp X 0 qs' \<and> (\<forall>g\<in>punit.reduced_GB F. poly_deg g \<le> \<b> qs' 0))"
nz2: shows "1 + (tan z)\<^sup>2 \<noteq> 0"
elims_2: fixes x :: "real_max"   and xa :: "real_max" assumes "x < xa" obtains x :: "real" and y :: "real" where "xa__ = Rmax x" and "xa = Rmax y" and "x < y"
NONCE_imp_eq: fixes m :: "nat"   and n :: "nat" assumes "NONCE m \<sim> NONCE n" shows "m = n"
OUTfromChCorrect_data8: shows "OUTfromChCorrect data8"
mset_less_eq_Bag0: fixes xs :: "('b, nat) alist"   and A :: "'b multiset" shows "(Bag xs \<subseteq># A) = (\<forall>(x, n)\<in>set (alist.impl_of xs). count_of (alist.impl_of xs) x \<le> count A x)"
intro: fixes S :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf set" assumes "\<And>s D. \<lbrakk>s \<in> S; D \<in> K s\<rbrakk> \<Longrightarrow> \<forall>s'\<in>set_pmf D. s' \<in> S" shows "Markov_Decision_Process_Invariant_axioms K S"
rel_boundary_of_polyhedron: fixes S :: "'b set" assumes "polyhedron S" shows "S - rel_interior S = \<Union> {F. F facet_of S}"
K_def: fixes c_k :: "int \<Rightarrow> int \<Rightarrow> int \<Rightarrow> nat"   and c :: "int \<Rightarrow> int \<Rightarrow> nat"   and w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "QI c_k c w" shows "QI.K c_k i j \<equiv> if i = j then i else Max (Args_min_on (c_k i j) {i + 1..j})"
top_unique: fixes a :: "'a" shows "(1 \<le> a) = (a = 1)"
D: shows "t \<equiv> \<lambda>\<pp>\<in>U. s (cover_of_subset.select_index I V \<pp>) \<pp>"
symmetric_inf_closed: fixes x :: "'a"   and y :: "'a" assumes "symmetric x"   and "symmetric y" shows "symmetric (x \<sqinter> y)"
adj_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^bsub>G\<^esub> v" shows "u \<in> verts G"
assms_4: shows "u \<turnstile>\<^bsub>v,n\<^esub> M'"
Collect_disj_eq: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{x. P x \<or> Q x} = {x. P x} \<union> {x. Q x}"
DT_D: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T Q" shows "P \<sqsubseteq>\<^sub>D Q"
tabulate_Mapping: fixes ks :: "'c list"   and f :: "'c \<Rightarrow> 'd" shows "Mapping.tabulate ks f = AList_Mapping.Mapping (map (\<lambda>k. (k, f k)) ks)"
divide_right: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)"   and "f \<in> L F g" shows "(\<lambda>x. f x / h x) \<in> L F (\<lambda>x. g x / h x)"
least_squares_approximation3: fixes S :: "'a set"   and v :: "'a" assumes "subspace S" shows "\<exists>p\<in>S. \<forall>y\<in>S. norm (v - p) \<le> norm (v - y) \<and> v - p \<in> orthogonal_complement S"
lm043: fixes pair :: "'c \<times> 'd set" shows "omega pair = {(fst pair, {y}) |y. y \<in> snd pair}"
point_not_equal_2: fixes p :: "'a"   and q :: "'a" assumes "is_point p"   and "is_point q" shows "(p \<le> - q) = (p ; q\<^sup>T \<le> - 1')"
fw_invariant: fixes k' :: "nat"   and k :: "nat"   and i :: "nat"   and n :: "nat"   and j :: "nat"   and j'' :: "nat"   and i'' :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i' :: "nat"   and j' :: "nat" assumes "k' \<le> k"   and "i \<le> n"   and "j \<le> n"   and "k \<le> n"   and "j'' \<le> j"   and "i'' \<le> i" shows "fw m n k i j i' j' \<le> fw m n k' i'' j'' i' j'"
lookup_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> {f. finite {x. f x \<noteq> (0::'b)}}" obtains x :: "'a \<Rightarrow>\<^sub>0 'b" where "y = lookup x"
spec_ops_empty: fixes deps :: "'c \<Rightarrow> 'b set" shows "spec_ops [] deps"
eq_\<I>_attach_on: fixes \<I>' :: "('y, 'z) \<I>"   and res :: "('y, 'z) resource"   and A :: "'aa set"   and conv :: "('aa, 'ab, 'y, 'z) converter"   and conv' :: "('aa, 'ab, 'y, 'z) converter" assumes "\<I>' \<turnstile>res res \<surd>"   and "\<I>_uniform A UNIV,\<I>' \<turnstile>\<^sub>C conv \<sim> conv'" shows "A \<turnstile>\<^sub>R conv \<rhd> res \<sim> conv' \<rhd> res"
trss_find_call': fixes sp :: "'n"   and c :: "'n list multiset"   and w :: "('p, 'ba) label list"   and p :: "'p"   and u' :: "'n"   and c' :: "'n list multiset" assumes "(([sp], c), w, [return fg p, u'], c') \<in> trcl (trss fg)" obtains uh :: "'n" and ch :: "'n list multiset" and wa :: "('p, 'ba) label list" and wba :: "('p, 'ba) label list" where "w = wa @ LCall p # wba" and "(([sp], c), wa, [uh], ch) \<in> trcl (trss fg)" and "(([uh], ch), LCall p, [entry fg p, u'], ch) \<in> trss fg" and "(uh, Call p, u') \<in> edges fg" and "(([entry fg p], ch), wba, [return fg p], c') \<in> trcl (trss fg)"
nemp_pow: fixes t :: "'b list"   and m :: "nat" assumes "t \<^sup>@ m \<noteq> \<epsilon>" shows "m \<noteq> 0"
finite_fold1_fold1_keys: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "('b, unit) RBT.rbt" assumes "semilattice f"   and "\<not> RBT.is_empty t" shows "semilattice_set.F f (Set t) = fold1_keys f t"
timpl_closure_set_def: fixes M :: "(('e, 'f, 'g) prot_fun, 'h) Term.term set"   and TI :: "('g set \<times> 'g set) set" shows "timpl_closure_set M TI \<equiv> \<Union>t\<in>M. timpl_closure t TI"
subterms_subset: fixes t' :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "t' \<sqsubseteq> t" shows "subterms t' \<subseteq> subterms t"
mr_cos_sprod_mem: fixes H :: "'a set"   and a :: "'b"   and X :: "'a set" assumes "submodule R M H"   and "a \<in> carrier R"   and "X \<in> set_mr_cos M H" shows "mr_cos_sprod M H a X \<in> set_mr_cos M H"
nn_integral_indicator_singleton: fixes y :: "'d"   and M :: "'d measure"   and f :: "'d \<Rightarrow> ennreal" assumes "{y} \<in> sets M" shows "set_nn_integral M {y} f = f y * emeasure M {y}"
clearjunk_update: fixes k :: "'c"   and v :: "'d"   and al :: "('c \<times> 'd) list" shows "clearjunk (update k v al) = update k v (clearjunk al)"
weakCongPar1: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and Q :: "ccs" assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P' \<parallel> Q"
crit_pair_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "crit_pair p q = (if component_of_term (lt p) = component_of_term (lt q) then (monom_mult ((1::'c) / lc p) (lcs (lp p) (lp q) - lp p) (tail p), monom_mult ((1::'c) / lc q) (lcs (lp p) (lp q) - lp q) (tail q)) else (0, 0))"
absorb_iff1: fixes a :: "'b"   and b :: "'b" shows "(a \<le> b) = (min a b = a)"
return_partition_basics_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "qmpt M T"   and "A \<in> sets M" shows "qmpt.return_partition M T A n \<in> sets M"
ArrayStore_def: shows "ArrayStore = STR ''java/lang/ArrayStoreException''"
assms: shows "((X\<^sub>r \<mu> \<in> old (fst n_ns) \<longrightarrow> \<mu> \<in> next (fst n_ns)) \<and> (\<forall>nd\<in>snd n_ns. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming (fst n_ns) \<longrightarrow> \<mu> \<in> old (fst n_ns) \<union> new (fst n_ns))) \<and> (\<forall>nd\<in>snd n_ns. \<forall>nd'\<in>snd n_ns. X\<^sub>r \<mu> \<in> old nd \<and> name nd \<in> incoming nd' \<longrightarrow> \<mu> \<in> old nd') \<and> ((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"
cases: fixes x :: "'a list" obtains "x = []"   | y :: "'a" and ys :: "'a list" where "x = y # ys"
prj_simps_7: shows "C.cod \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 = \<mu>.apex"
eSubstAbs_simp2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and ys :: "'varSort"   and Y :: "'c"   and y :: "'var"   and eA :: "'d withERR" assumes "\<not> igWls MOD (asSort ys) Y" shows "eSubstAbs MOD ys (OK Y) y eA = ERR"
nextl_app: fixes Q :: "hf set"   and xs :: "'a list"   and ys :: "'a list" shows "ST.nextl Q (xs @ ys) = ST.nextl (ST.nextl Q xs) ys"
C1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "(\<Psi>, P, Q) \<in> Rel"
exec_gpv_plus_oracle_right: fixes oracle1 :: "'b \<Rightarrow> 'c \<Rightarrow> ('e \<times> 'b) spmf"   and oracle2 :: "'b \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'b) spmf"   and gpv :: "('a, 'd, 'f) gpv"   and s :: "'b" shows "exec_gpv (oracle1 \<oplus>\<^sub>O oracle2) (right_gpv gpv) s = exec_gpv oracle2 gpv s"
spickI: fixes n :: "nat"   and w :: "bool stream"   and v :: "bool" shows "stake n (spick w n v) = stake n w \<and> spick w n v !! n = v"
strip_map2_acom: fixes c1 :: "'a acom"   and c2 :: "'a acom"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "strip c1 = strip c2" shows "strip (map2_acom f c1 c2) = strip c1"
continuous_on_closed_UN: fixes S :: "'a set set"   and f :: "'a \<Rightarrow> 'b" assumes "finite S"   and "\<And>s. s \<in> S \<Longrightarrow> closed s"   and "\<And>s. s \<in> S \<Longrightarrow> continuous_on s f" shows "continuous_on (\<Union> S) f"
supp_minus_perm: fixes p :: "perm" shows "supp (- p) = supp p"
vcons_in_vcpowerE1: fixes xs :: "V"   and x :: "V"   and A :: "V"   and n :: "V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "n \<in>\<^sub>\<circ> \<omega>" obtains "xs \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs" and "x \<in>\<^sub>\<circ> A" and "n = vcard (xs #\<^sub>\<circ> x)"
judge_weak_subst: fixes a :: "Syntax.var"   and \<tau>' :: "ty"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty"   and e' :: "Syntax.term" assumes "\<Gamma>(a $$:= \<tau>') \<turnstile>\<^sub>W e : \<tau>"   and "{$$} \<turnstile>\<^sub>W e' : \<tau>'" shows "\<Gamma> \<turnstile>\<^sub>W e[e' / a] : \<tau>"
a_set_child_nodes_def: fixes ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list" shows "local.a_set_child_nodes ptr children = invoke a_set_child_nodes_tups ptr children"
pdevs_val_zero: fixes x :: "'b pdevs" shows "pdevs_val (\<lambda>_. 0) x = (0::'b)"
simps_14: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PPair x51 x52 \<noteq> PUnit"
dim_ty_sem_DimTimes_def: fixes x :: "('a \<cdot> 'b) itself" shows "dim_ty_sem x = QD('a) \<cdot> QD('b)"
\<Z>_sgrp_def: fixes \<alpha> :: "V"   and \<SS> :: "V" shows "\<Z>_sgrp \<alpha> \<SS> \<equiv> \<Z>_sgrp_basis \<alpha> \<SS> \<and> \<Z>_sgrp_axioms \<SS>"
IntD2: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>A b\<guillemotright>" shows "\<guillemotleft>F f : F a \<rightarrow>\<^sub>B F b\<guillemotright>"
bool_eq_splitI: fixes x :: "bool \<Longrightarrow> 'a"   and P :: "('b, 'a) uexpr"   and Q :: "('b, 'a) uexpr" assumes "vwb_lens x"   and "P\<lbrakk>true/x\<rbrakk> = Q\<lbrakk>true/x\<rbrakk>"   and "P\<lbrakk>false/x\<rbrakk> = Q\<lbrakk>false/x\<rbrakk>" shows "P = Q"
iso_finfun_sup: fixes A :: "'a \<Rightarrow>f 'b"   and B :: "'a \<Rightarrow>f 'b" shows "sup (($) A) (($) B) = ($) (sup A B)"
equivpartp_def: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" shows "equivpartp r x y \<equiv> x = y \<or> r x y \<and> r y x"
fps_shift_subdegree_zero_iff: fixes f :: "'a fps" shows "(fps_shift (subdegree f) f = 0) = (f = 0)"
IH: assumes "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> (V \<union> fv u)" shows "wf\<^sub>s\<^sub>t (V \<union> fv u) (S @ S')"
spec_refines_impl': assumes "finite SYN" shows "SYSTEM' \<sqsubseteq> COPY'"
borel_cantelli_AE1: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "\<And>n. A n \<in> sets M"   and "\<And>n. emeasure M (A n) < \<infinity>"   and "summable (\<lambda>n. Sigma_Algebra.measure M (A n))" shows "AE x in M. \<forall>\<^sub>F n in sequentially. x \<in> space M - A n"
dom_absorp_2: fixes x :: "'a"   and y :: "'a" shows "d x + d x \<cdot> d y = d x"
flush_data_cache_def: fixes state :: "'b sparc_state" shows "flush_data_cache state \<equiv> state\<lparr>cache := cache state\<lparr>dcache := empty_cache\<rparr>\<rparr>"
word_or_not: fixes x :: "'b word" shows "x || ~~ x = - 1"
of_real_arcsin: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "complex_of_real (arcsin x) = Arcsin (complex_of_real x)"
gt_imp_wt_ge: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "wt s \<le> wt t"
hg: fixes x :: "'a" assumes "x \<in> \<Union> (\<G> \<inter> Pow V)" shows "h x = g x"
fmdom'_restrict_fset_precise: fixes A :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdom (fmrestrict_fset A m) = fmdom m |\<inter>| A"
disj2': shows "carrier (Fstack as S) \<subseteq> - domA \<Delta>"
op_hom: shows "f \<in> hom (G\<lparr>carrier := H\<rparr>) (G\<lparr>carrier := N <#> H\<rparr> Mod N)"
fundchamber_S_adjacent: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "C0 \<sim> s `\<rightarrow> C0"
fals: shows "falsifies\<^sub>g G C\<^sub>1'"
that: assumes "SPLIT x1 x2"   and "SPLIT y1 y2" shows "thesis"
rel_funI: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
hom_1_iff: fixes x :: "int" shows "(of_int x = (1::'a)) = (x = 1)"
disjnt_subset2: fixes X :: "'a set"   and Y :: "'a set"   and Z :: "'a set" assumes "disjnt X Y"   and "Z \<subseteq> Y" shows "disjnt X Z"
inputPermFrameSubject: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* P"   and "Ys \<sharp>* P" shows "p \<bullet> \<Psi> \<rhd> P \<longmapsto> (p \<bullet> M)\<lparr>N\<rparr> \<prec> P'"
append_Cons_rev: fixes x :: "'a"   and xs :: "'a list"   and ys :: "'a list" shows "x # xs @ ys = (x # xs) @ ys"
fold_bind_random_permutation_foldl: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b pmf"   and x :: "'c" assumes "finite A" shows "fold_bind_random_permutation f g x A = pmf_of_set (permutations_of_set A) \<bind> (\<lambda>xs. g (foldl (\<lambda>x y. f y x) x xs))"
exy: shows "\<exists>y. ((\<lambda>x. inverse (\<gamma> x - z) * vector_derivative \<gamma> (at x)) has_integral y) {a..b}"
sees_methods_idemp: fixes P :: "'a prog"   and C :: "String.literal"   and Mm :: "String.literal \<Rightarrow> ((ty list \<times> ty \<times> 'a option) \<times> String.literal) option"   and M :: "String.literal"   and m :: "ty list \<times> ty \<times> 'a option"   and D :: "String.literal" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "\<exists>Mm'. P \<turnstile> D sees_methods Mm' \<and> Mm' M = \<lfloor>(m, D)\<rfloor>"
canonical_saturated_1: fixes X :: "'b set"   and v :: "'b \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'b"   and r :: "'c"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and c1 :: "'b" assumes "Beta_Regions' X v n not_in_X"   and "Le r \<le> M (v c1) 0"   and "Le (- r) \<le> M 0 (v c1)"   and "cycle_free M n"   and "canonical M n"   and "v c1 \<le> n"   and "0 < v c1"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" obtains u :: "'b \<Rightarrow> 'c" where "u \<in> [M]\<^bsub>v,n\<^esub>" and "u c1 = r"
curve_elt_opp: fixes R :: "('c, 'd) ring_scheme"   and p\<^sub>1 :: "'c point"   and x\<^sub>1 :: "'c"   and y\<^sub>1 :: "'c"   and p\<^sub>2 :: "'c point"   and x\<^sub>2 :: "'c"   and y\<^sub>2 :: "'c"   and a :: "'c"   and b :: "'c" assumes "ell_field R"   and "p\<^sub>1 = Point x\<^sub>1 y\<^sub>1"   and "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"   and "ell_field.on_curve R a b p\<^sub>1"   and "ell_field.on_curve R a b p\<^sub>2"   and "x\<^sub>1 = x\<^sub>2" shows "p\<^sub>1 = p\<^sub>2 \<or> p\<^sub>1 = ell_field.opp R p\<^sub>2"
ct_2: fixes sa :: "'s" assumes "sa \<in> S"   and "sa \<in> S1" shows "lfp F_inf sa = integral\<^sup>N (measure_pmf (ct sa)) (lfp F_inf)"
nth_root_nat_aux_correct: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k ^ m \<le> n"   and "0 < m" shows "nth_root_nat_aux m k (k ^ m) n = nth_root_nat m n"
simps_12: fixes va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b" shows "balance (Branch B va vb vc vd) w x (Branch R (Branch R b s t c) y z Empty) = Branch R (Branch B (Branch B va vb vc vd) w x b) s t (Branch B c y z Empty)"
canonical_inj_is_hom: fixes H :: "'a set" assumes "subgroup H G" shows "group_hom (G\<lparr>carrier := H\<rparr>) G id"
mat_kernel_berlekamp_resulting_mat_degree_0: fixes u :: "'b mod_ring poly" assumes "degree u = 0" shows "mat_kernel (berlekamp_resulting_mat u) = {0\<^sub>v 0}"
ExprI: fixes G :: "prog"   and s :: "state"   and e :: "expr"   and v :: "val"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>e-\<succ>v\<rightarrow> s'" shows "G\<turnstile>s \<midarrow>Expr e\<rightarrow> s'"
mat_row_last_def: fixes A :: "'b mat"   and n :: "nat" shows "mat_row_last A n \<equiv> mat n (dim_col A) (\<lambda>(i, j). A $$ (dim_row A - n + i, j))"
iso_inv_iso: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.iso (CC.inv f)"
rtrancl_mono_mp: fixes U :: "('b \<times> 'b) set"   and V :: "('b \<times> 'b) set"   and x :: "'b \<times> 'b" assumes "U \<subseteq> V"   and "x \<in> U\<^sup>*" shows "x \<in> V\<^sup>*"
bij_betw_UNION_disjoint: fixes A' :: "'c \<Rightarrow> 'd set"   and I :: "'c set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'c \<Rightarrow> 'e set" assumes "disjoint_family_on A' I"   and "\<And>i. i \<in> I \<Longrightarrow> bij_betw f (A i) (A' i)" shows "bij_betw f (\<Union> (A ` I)) (\<Union> (A' ` I))"
cinner_scaleC_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and r :: "complex"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (scaleC r y) = r * cinner x y"
preserves_ide: fixes a :: "'c \<times> 'c" assumes "C.CC.ide a" shows "C.CC.ide (C.CC.map a)"