hcju/isabelleps huggingface dataset:
sbh_step_preserves_valid: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')"   and "valid ts" shows "valid (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')])"
eql4: shows "st.pmat_extension (H_k k) = st2.ptensor_mat (1\<^sub>m 2) (st3.ptensor_mat (H_k k) (1\<^sub>m st4.d2))"
ceiling_mono: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "\<lceil>y\<rceil> \<le> \<lceil>x\<rceil>"
vbrelation_vconverse: shows "vbrelation (vint_uminus\<inverse>\<^sub>\<circ>)"
nat_of_string_def: shows "nat_of_string \<equiv> SOME f. bij f"
comp_def: fixes g :: "'a"   and f :: "'a" shows "Cop.comp g f \<equiv> f \<cdot>\<^sub>A g"
bij_lens_iff_pbij_vwb: fixes X :: "'d \<Longrightarrow> 'e" shows "bij_lens X = (pbij_lens X \<and> vwb_lens X)"
Atm_transC_simp: fixes atm :: "'atom"   and s :: "'state"   and cf :: "('test, 'atom) com \<times> 'state" shows "\<not> (Atm atm, s) \<rightarrow>c cf"
func_to_img: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set" assumes "f \<in> A \<rightarrow> B" shows "f \<in> A \<rightarrow> f ` A"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and sample :: "'c pmf \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_prob return bind sample" shows "monad_prob_axioms bind sample"
tab_simps_4: shows "local.dom r.tab = tab\<^sub>1 r"
INF_le_SUP: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "less_eq (Inf (f ` A)) (Sup (f ` A))"
rep_list_minus: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and s :: "'t \<Rightarrow>\<^sub>0 'b" shows "rep_list (r - s) = rep_list r - rep_list s"
simps_2: fixes m :: "nat"   and t :: "'a FOL_Fitting.term"   and ts :: "'a FOL_Fitting.term list" shows "closedts m (t # ts) = (closedt m t \<and> closedts m ts)"
countable_all: fixes S :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "countable S" shows "(\<forall>s\<in>S. P s) = (\<forall>n. from_nat_into S n \<in> S \<longrightarrow> P (from_nat_into S n))"
hyps_2: shows "\<langle>c',s'\<rangle> \<rightarrow>* \<langle>Skip,s\<^sub>1\<rangle>"
LLL_invariant_weak_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and fs :: "int vec list" shows "LLL.LLL_invariant_weak n m fs_init fs = (cof_vec_space.lin_indpt_list n (map of_int_hom.vec_hom fs) \<and> vec_module.lattice_of n fs = LLL.L n fs_init \<and> length fs = m)"
simps: fixes A :: "'d atMost \<Rightarrow> 'f atMost \<Rightarrow> 'c"   and B :: "'f atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c" shows "Matrix A *\<^sub>M Matrix B = Matrix (\<lambda>i j. \<Sum>k\<in>UNIV. A i k \<cdot> B k j)"
view_determination: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and u :: "'c"   and \<rho> :: "'d"   and v :: "'c" assumes "weak_lens x"   and "put\<^bsub>x\<^esub> \<sigma> u = put\<^bsub>x\<^esub> \<rho> v" shows "u = v"
LyndonD_nemp: fixes w :: "'a list" assumes "Lyndon w" shows "w \<noteq> \<epsilon>"
simps_280: fixes x7 :: "'a ltln" shows "atoms_ltln (X\<^sub>n x7) = atoms_ltln x7"
llist_all2_conv_all_lnth: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a llist"   and ys :: "'b llist" shows "llist_all2 P xs ys = (llength xs = llength ys \<and> (\<forall>n. enat n < llength ys \<longrightarrow> P (lnth xs n) (lnth ys n)))"
iT_Div_1: fixes I :: "nat set" shows "I \<oslash> Suc 0 = I"
SBHGhost: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(Ghost A L R W # is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>, sb @ [Ghost\<^sub>s\<^sub>b A L R W], m, \<D>, \<O>, \<R>, \<S>)"
transp_less: fixes ko :: "'a nat_term_order" shows "transp (ko_ntm.lt ko)"
oalist_inv_def: fixes xs :: "('a \<times> 'e) list \<times> 'o" shows "oalist_inv xs = oalist_inv_raw (snd xs) (fst xs)"
div_comp_temp_V_def: fixes i :: "sparc_operation \<times> inst_operand list"   and w32 :: "word32"   and w33 :: "word33" shows "div_comp_temp_V i w32 w33 \<equiv> if fst i = arith_type UDIV \<or> fst i = arith_type UDIVcc then if w32 = 0 then 0 else 1 else if w33 = 0 \<or> w33 = 8589934591 then 0 else 1"
reachable_without_loops: fixes x :: "'a" shows "x\<^sup>\<star> = (x \<sqinter> - (1::'a))\<^sup>\<star>"
isCont_char: fixes t :: "real" shows "isCont (char M') t"
square_lesseq_square: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) \<le> y" shows "(x * x \<le> y * y) = (x \<le> y)"
rep_eq: fixes x :: "'a['b, 'c]"   and xa :: "'a['d, 'c]" shows "(x \<cong>\<^sub>Q xa) = (fromQ x = fromQ xa)"
IH_2: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, F, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n"
inv_Seed: fixes p :: "nat" shows "Invariants.inv Seed\<^bsub>p\<^esub>"
lnth_lmap_ldropn: fixes n :: "nat"   and xs :: "'a llist"   and f :: "'a \<Rightarrow> 'b" assumes "enat n < llength xs" shows "lnth (lmap f (ldropn n xs)) 0 = lnth (lmap f xs) n"
ord_isom_minimum: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e"   and S :: "'a set"   and a :: "'a" assumes "Order E"   and "ord_isom D E f"   and "S \<subseteq> carrier D"   and "a \<in> carrier D"   and "minimum_elem D S a" shows "minimum_elem E (f ` S) (f a)"
cINF_greatest: fixes A :: "'c set"   and m :: "'d multiset"   and f :: "'c \<Rightarrow> 'd multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># Inf (f ` A)"
HIm_Octo: fixes w :: "real"   and x :: "real"   and y :: "real"   and z :: "real"   and u :: "real"   and v :: "real"   and q :: "real"   and g :: "real" shows "HIm (Octo w x y z u v q g) = vector [x, y, z, u, v, q, g]"
coeff_zero: fixes p :: "linear_poly"   and x :: "nat" shows "(Abstract_Linear_Poly.coeff p x \<noteq> 0) = (x \<in> vars p)"
indaut_halfmorph_fopp_fim: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "ChamberComplexMorphism (ChamberComplexFolding.opp_half_apartment X f) (f \<turnstile> X) (OpposedThinChamberComplexFoldings.induced_automorphism X f g)"
mult_less_le_imp_less: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less_eq c d"   and "less_eq zero a"   and "less zero c" shows "less (times a c) (times b d)"
substitution_lemma_automatic: fixes x :: "name"   and y :: "name"   and L :: "lam"   and N :: "lam"   and M :: "lam" assumes "x \<noteq> y"   and "x \<sharp> L" shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
degree_finite_fourier_poly': fixes ws :: "nat \<Rightarrow> complex"   and k :: "nat" shows "degree (finite_fourier_poly' ws k) \<le> k - 1"
conversion_def: fixes A :: "('b \<times> 'b) set" shows "A\<^sup>\<leftrightarrow>\<^sup>* = (A\<^sup>\<leftrightarrow>)\<^sup>*"
SeqSubstFormP_eq: fixes H :: "fm set"   and v :: "tm"   and tm :: "tm"   and x :: "tm"   and z :: "tm"   and s :: "tm"   and k :: "tm"   and y :: "tm"   and A :: "fm" assumes "H \<turnstile> SeqSubstFormP v tm x z s k"   and "insert (SeqSubstFormP v tm y z s k) H \<turnstile> A" shows "insert (x EQ y) H \<turnstile> A"
sd_exists_witness: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and Is :: "'b set"   and V1 :: "'b set"   and V2 :: "'b set"   and V3 :: "'b set" assumes "hasw [x, y, z] A"   and "Is = V1 \<union> V2 \<union> V3 \<and> V1 \<inter> V2 = {} \<and> V1 \<inter> V3 = {} \<and> V2 \<inter> V3 = {}"   and "Is \<noteq> {}" obtains P :: "'b \<Rightarrow> ('a \<times> 'a) set" where "profile A Is P" and "\<forall>i\<in>V1. x \<^bsub>(P i)\<^esub>\<prec> y \<and> y \<^bsub>(P i)\<^esub>\<prec> z" and "\<forall>i\<in>V2. z \<^bsub>(P i)\<^esub>\<prec> x \<and> x \<^bsub>(P i)\<^esub>\<prec> y" and "\<forall>i\<in>V3. y \<^bsub>(P i)\<^esub>\<prec> z \<and> z \<^bsub>(P i)\<^esub>\<prec> x"
simps_8: fixes x21 :: "nat"   and x22 :: "nat"   and x11 :: "nat set"   and x12 :: "'d" shows "x21:deliver(x22) \<noteq> x11:*cast(x12)"
is_ccpo: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "class.ccpo Sup less_eq less"
choose_smaller_in_sort: fixes c :: "String.literal"   and s :: "String.literal set" assumes "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" obtains c' :: "String.literal" where "c' \<in> s" and "class_les cs c' c"
evaln_elim_cases_24: fixes G :: "prog"   and s :: "st"   and C :: "qtname"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (NewC C)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains s1 :: "state" and a :: "loc" where "v = In1 (Addr a)" and "G\<turnstile>Norm s \<midarrow>Init C\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>halloc CInst C\<succ>a\<rightarrow> s'"
fps_XD_nth: fixes a :: "'a fps"   and n :: "nat" shows "fps_nth (fps_XD a) n = of_nat n * fps_nth a n"
minus_unique: fixes y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "y + x = (0::'a)"   and "x + y' = (0::'a)" shows "y = y'"
isomorphic_unit_right: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.isomorphic (f \<circ> Span.src f) f"
7: shows "\<forall>ab b. ab \<noteq> b \<and> ab \<in> set (net_list_aux [a, aa]) \<and> b \<in> set (net_list_aux [a, aa]) \<longrightarrow> netsDistinct ab b"
convolution_distrib_left: fixes g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "g \<in> aezfun_set"   and "h \<in> aezfun_set" shows "convolution f (g + h) = convolution f g + convolution f h"
contraction_is_pmc: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div> \<phi> = K \<div>\<^bsub>\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi>"
\<Sigma>_prot: fixes \<eta> :: "nat" shows "Schnorr_\<Sigma>.\<Sigma>_protocol \<eta>"
compseriesTr2: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "compseries G n f" shows "f 0 = carrier G"
fps_of_ratfps_eq_zero_iff: fixes p :: "'b ratfps" shows "(fps_of_ratfps p = 0) = (p = 0)"
assms_3: shows "integrable M (\<lambda>x. f (T x) * g x)"
ucast_up_inj: fixes x :: "'c word"   and y :: "'c word" assumes "ucast x = ucast y"   and "LENGTH('c) \<le> LENGTH('b)" shows "x = y"
drop_bit_nat_def: fixes n :: "nat"   and m :: "nat" shows "drop_bit n m = m div 2 ^ n"
assms_1: shows "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')"
kronecker_one: fixes x :: "nat"   and y :: "nat" shows "kronecker_product (1\<^sub>m x) (1\<^sub>m y) = 1\<^sub>m (x * y)"
lookup_update_neq: fixes k :: "'a"   and k' :: "'a"   and v :: "'b"   and m :: "('a, 'b) mapping" assumes "k \<noteq> k'" shows "Mapping.lookup (Mapping.update k v m) k' = Mapping.lookup m k'"
expand: fixes tree :: "'a tree"   and tree' :: "'a tree" assumes "root tree = root tree' \<and> left tree = left tree' \<and> right tree = right tree'" shows "tree = tree'"
le_diff_conv2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "(c \<le> b - a) = (c + a \<le> b)"
rq_suf: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v" shows "v\<^sup><\<inverse>u  \<cdot> u = v"
inject: fixes x21 :: "Unary_PCF.type"   and x22 :: "Unary_PCF.type"   and y21 :: "Unary_PCF.type"   and y22 :: "Unary_PCF.type" shows "(x21 \<rightarrow> x22 = y21 \<rightarrow> y22) = (x21 = y21 \<and> x22 = y22)"
abs_summable_finiteI0_2: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "real" assumes "\<And>F. \<lbrakk>finite F; F \<subseteq> S\<rbrakk> \<Longrightarrow> (\<Sum>x\<in>F. norm (f x)) \<le> B"   and "0 \<le> B" shows "(\<Sum>\<^sub>ax\<in>S. norm (f x)) \<le> B"
isInjAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c" shows "isInjAbs hA \<equiv> \<forall>us s A B. wlsAbs (us, s) A \<and> wlsAbs (us, s) B \<longrightarrow> hA A = hA B \<longrightarrow> A = B"
mult_isol: fixes x :: "'a list set"   and y :: "'a list set"   and z :: "'a list set" assumes "x \<subseteq> y" shows "p_prod z x \<subseteq> p_prod z y"
components_are_iso: fixes a :: "'b" assumes "ide a" shows "local.iso (\<ll> a)"
jordan_matrix_def: fixes n_as :: "(nat \<times> 'b) list" shows "jordan_matrix n_as = diag_block_mat (map (\<lambda>(n, a). jordan_block n a) n_as)"
open_exceptI_2: fixes x :: "'c"   and y :: "'c" assumes "x \<noteq> y" shows "x \<in> open_except x y"
UN_N: shows "\<Union> (range N) \<in> null_sets M"
bigo_bounded_alt: fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>x. (0::'b) \<le> f x"   and "\<forall>x. f x \<le> c * g x" shows "f \<in> O(g)"
wt_st: shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (App s t))) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
refl: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "blinding_of_on A h bo"   and "x \<in> A" shows "bo x x"
lspasl_iu_der: fixes h1 :: "'a"   and h2 :: "'a" assumes "h1,h2\<triangleright>0::'a" shows "(0::'a,0::'a\<triangleright>0::'a) \<and> h1 = (0::'a) \<and> h2 = (0::'a)"
simps_1: fixes v0 :: "(char list \<times> v) list"   and c0 :: "(char list, char list, nat \<times> tid_or_exn) namespace" shows "is_cupcake_all_env (make_sem_env (Bind v0 []) c0) = (c0 = static_cenv \<and> list_all (is_cupcake_value \<circ> snd) v0)"
has_integral_const_real: fixes c :: "'a"   and a :: "real"   and b :: "real" shows "((\<lambda>x. c) has_integral Henstock_Kurzweil_Integration.content {a..b} *\<^sub>R c) {a..b}"
simps_2: fixes \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<sigma> :: "(name list \<times> 'a list) list" shows "caseListSeqSubst ((\<phi>, P) # Cs) \<sigma> = (substCond.seqSubst \<phi> \<sigma>, P[<\<sigma>>]) # caseListSeqSubst Cs \<sigma>"
gfp_fusion: fixes h :: "'b \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'a" assumes "\<exists>f. f \<stileturn> h"   and "mono f"   and "mono g"   and "h \<circ> f = g \<circ> h" shows "h (gfp f) = gfp g"
lookup_Mapping: fixes t :: "('c, 'd) RBT.rbt" shows "Mapping.lookup (RBT_Mapping.Mapping t) = RBT.lookup t"
sorted_start_nth: fixes i :: "nat" assumes "i < n" shows "Sorted_Less.sorted (rev (start ! i))"
ef: shows "c = gcd c d * e \<and> d = gcd c d * f \<and> coprime e f"
isnpoly_fst_split0: fixes p :: "tm" assumes "allpolys isnpoly p" shows "isnpoly (fst (split0 p))"
HEndPhase2_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p" shows "allBlocks s' \<subseteq> allBlocks s"
intros: fixes s :: "'a + 'b"   and x :: "'a" assumes "s = Inl x" shows "setlp s x"
S_transform_Not: fixes x :: "('a, 'c, 'b) formula" shows "S_transform (Formula.Not x) = Formula.Not (S_transform x)"
list_all_eq_const_imp_replicate: fixes y :: "'l"   and xs :: "'l list" assumes "list_all (\<lambda>x. x = y) xs" shows "xs = replicate (length xs) y"
dbproj_Cons_1: fixes i :: "'d"   and d :: "'e"   and D :: "('d \<times> 'e) list" shows "dbproj i ((i, d) # D) = (i, d) # dbproj i D"
distinct_p: fixes p :: "nat"   and ss :: "nat list list" shows "p < n \<longrightarrow> distinct (map fst (step p (ss ! p)))"
a_absorb2: fixes x :: "'a"   and y :: "'a" shows "ad x \<oplus> ad x \<cdot> ad y = ad x"
IH: fixes x3aa :: "'a tree" assumes "x3aa \<in> set ts"   and "invar_btree x3aa" shows "size (mset_tree x3aa) = 2 ^ rank x3aa"
uminus_rep_def: fixes p :: "int extended \<times> int extended" shows "uminus_rep p = (let (l, h) = p in (- h, - l))"
Inf_dual: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "dual \<circ> Inf = Sup \<circ> (`) dual"
r_trans: fixes s :: "'a"   and t :: "'a"   and T :: "('a, 'b) TS_scheme" assumes "(s, t) \<in> TS.trans T"   and "s \<in> reach T" shows "t \<in> reach T"
continuous_on_Ln: fixes S :: "complex set" assumes "\<And>z. z \<in> S \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "continuous_on S Ln"
rank_eq_suc_to_nat_greatest: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "A \<noteq> 0" shows "rank A = mod_type_class.to_nat (GREATEST a. \<not> is_zero_row a (Gauss_Jordan A)) + 1"
signsE: fixes signs :: "signs" obtains "signs = Neg"   | "signs = Pos"
1: shows "\<guillemotleft>mkArr' a b F : a \<rightarrow> b\<guillemotright>"
cases: fixes ds :: "'d set"   and a :: "'x set"   and aa :: "'x set" assumes "COP.iterates_abovep ds a aa" obtains "aa = a"   | x :: "'x set" where "aa = cop_F ds x" and "COP.iterates_abovep ds a x"   | M :: "'x set set" where "aa = \<Union> M" and "M \<in> Chains {(x, y). x \<subseteq> y}" and "M \<noteq> {x. bot x}" and "\<And>x. x \<in> M \<Longrightarrow> COP.iterates_abovep ds a x"
bigo_compose1: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and k :: "'e \<Rightarrow> 'c" assumes "f \<in> O(g)" shows "(\<lambda>x. f (k x)) \<in> O(\<lambda>x. g (k x))"
nonabort_e: shows "chaos \<sqsubseteq> (e::'a::refinement_lattice)"
hom_zero: fixes a :: "'b" shows "poly 0 a = (0::'b)"
simps_2: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and n :: "'c \<Rightarrow> bool"   and P :: "'b" shows "normalized_n_primitive (disc, sel) n (Match P) = (if disc P then n (sel P) else True)"
retractionE: fixes S :: "'a set"   and T :: "'a set"   and r :: "'a \<Rightarrow> 'a" assumes "retraction S T r" obtains "T = r ` S" and "r ` S \<subseteq> S" and "continuous_on S r" and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x"
fps_cutoff_nth: fixes n :: "nat"   and f :: "'a fps"   and i :: "nat" shows "fps_nth (fps_cutoff n f) i = (if i < n then fps_nth f i else (0::'a))"
assms_5: shows "c = fps_nth F (subdegree G) / fps_nth G (subdegree G)"
sorted_wrt_append: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "sorted_wrt P (xs @ ys) = (sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
one_relative_homology_group: fixes p :: "nat"   and X :: "'a topology"   and S :: "'a set" shows "\<one>\<^bsub>relative_homology_group (int p) X S\<^esub> = singular_relboundary_set p X S"
discrete_diagram_def: fixes J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'b \<Rightarrow> 'c" shows "discrete_diagram J C D \<equiv> category J \<and> diagram J C D \<and> discrete_diagram_axioms J"
update_def: shows "update \<equiv> map_fun rep_st (map_fun id (map_fun id St)) (\<lambda>ps x a. (x, a) # ps)"
ys: fixes ys :: "'a llist" assumes "zs \<sqsubseteq> ys"   and "ys \<sqsubseteq> xs"   and "lfinite ys" shows "P ys"
All_E: fixes A :: "fm"   and i :: "name"   and x :: "tm"   and H :: "fm set"   and B :: "fm" assumes "insert (A(i::=x)) H \<turnstile> B" shows "insert (SyntaxN.All i A) H \<turnstile> B"
K: shows "(xs @ [y], {}) \<in> failures P \<and> None \<noteq> y"
cSup_eq_Max: fixes X :: "'b set" assumes "finite X"   and "X \<noteq> {}" shows "Sup X = Max X"
random_member_def: fixes f :: "'a fset" shows "random_member f = (if f = {||} then None else Some (SOME x. x |\<in>| f))"
lossless_def: fixes \<A> :: "'grp \<Rightarrow> 'grp \<Rightarrow> 'grp set spmf" shows "lossless \<A> = (\<forall>\<alpha> \<beta>. lossless_spmf (\<A> \<alpha> \<beta>))"
le_sup_iff: fixes x :: "'b upred"   and y :: "'b upred"   and z :: "'b upred" shows "(z \<sqsubseteq> (x \<or> y)) = (z \<sqsubseteq> x \<and> z \<sqsubseteq> y)"
epred_llength: fixes xs :: "'c llist" shows "epred (llength xs) = llength (ltl xs)"
intros_5: fixes P :: "complex mat"   and Q :: "complex mat"   and S :: "com"   and M :: "nat \<Rightarrow> complex mat" assumes "is_quantum_predicate P"   and "is_quantum_predicate Q"   and "\<turnstile>\<^sub>p {Q} S {adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1}" shows "\<turnstile>\<^sub>p {adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1} While M S {P}"
list_cong_rev_iff: fixes xs :: "'a list"   and ys :: "'a list" shows "rev xs \<cong> rev ys = xs \<cong> ys"
fashoda_unit_path: fixes f :: "real \<Rightarrow> (real, 2) vec"   and g :: "real \<Rightarrow> (real, 2) vec" assumes "path f"   and "path g"   and "path_image f \<subseteq> cbox (- 1) 1"   and "path_image g \<subseteq> cbox (- 1) 1"   and "pathstart f $ 1 = - 1"   and "pathfinish f $ 1 = 1"   and "pathstart g $ 2 = - 1"   and "pathfinish g $ 2 = 1" obtains z :: "(real, 2) vec" where "z \<in> path_image f" and "z \<in> path_image g"
polyfun_rootbound: fixes c :: "nat \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "c k \<noteq> (0::'b)"   and "k \<le> n" shows "finite {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)} \<and> card {z. (\<Sum>i\<le>n. c i * z ^ i) = (0::'b)} \<le> n"
parse_ipv6_address_compressed_identity: fixes ipv6_syntax :: "ipv6addr_syntax_compressed" shows "parse_ipv6_address_compressed (ipv6addr_syntax_compressed_to_list ipv6_syntax) = Some ipv6_syntax"
rel_set_OO: fixes R :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and S :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_set R OO rel_set S = rel_set (R OO S)"
nonemptyCn: fixes A :: "'a set" shows "Cn A \<noteq> {}"
le_neq_trans: fixes a :: "'b word"   and b :: "'b word" assumes "a \<le>s b"   and "a \<noteq> b" shows "a <s b"
pure_fresh: fixes a :: "atom"   and x :: "'c" shows "a \<sharp> x"
simps_3: fixes x :: "'b"   and xs :: "'b list"   and oid :: "'b"   and ref :: "'b" shows "insert_spec (x # xs) (oid, Some ref) = (if x = ref then x # oid # xs else x # insert_spec xs (oid, Some ref))"