hcju/isabelleps huggingface dataset:
HINT_iff: fixes b :: "hf"   and A :: "hf"   and B :: "hf \<Rightarrow> hf" shows "(b \<^bold>\<in> \<Sqinter>RepFun A B) = ((\<forall>x\<^bold>\<in>A. b \<^bold>\<in> B x) \<and> A \<noteq> 0)"
usc_at_def: fixes x0 :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "usc_at x0 f = (\<forall>X l. X \<longlonglongrightarrow> x0 \<and> (f \<circ> X) \<longlonglongrightarrow> l \<longrightarrow> l \<le> f x0)"
consume_c_types_agree: fixes ts :: "t list"   and cts :: "ct list"   and ts' :: "t list"   and ctn :: "checker_type" assumes "consume (Type ts) cts = Type ts'"   and "c_types_agree ctn ts" shows "\<exists>c_t'. consume ctn cts = c_t' \<and> c_types_agree c_t' ts'"
add_tvarsT'_def: shows "add_tvarsT' = fold_atyps (\<lambda>x l. case x of Ty literal list \<Rightarrow> l | Tv idn s \<Rightarrow> List.insert (idn, s) l)"
cod_inv: fixes f :: "'c" assumes "C.iso f" shows "C.cod (C.inv f) = C.dom f"
join_def: fixes x :: "nat"   and y :: "nat" shows "nat.join x y = (THE sup. nat.is_sup x y sup)"
mod_integer_le: fixes a :: "integer"   and b :: "integer"   and x :: "integer" assumes "a \<le> b"   and "0 < a" shows "x mod (a + 1) \<le> b"
ex_locs_def: shows "ex_locs \<equiv> \<lambda>x. if x = cloud then (''free'', {((Actor ''Patient'', {Actor ''Doctor''}), 42)}) else ([], {})"
compP2_def: shows "compP2 \<equiv> compP (\<lambda>C M Ts T. compMb2)"
IdNatTrans_defs_2: fixes F :: "('g, 'h, 'i, 'j, 'k, 'l) Functor" shows "IdNatTrans' F \<equiv> \<lparr>NTDom = F, NTCod = F, NatTransMap = \<lambda>X. id\<^bsub>CatCod F\<^esub> (F @@ X)\<rparr>"
zero_pdevs_def: shows "zero_pdevs \<equiv> Abs_pdev (\<lambda>_. 0::'b)"
min_eq_2: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" shows "min a b = b"
exhaust: fixes y :: "('l, 'm) result" obtains x1 :: "'l" where "y = Error x1"   | x2 :: "'m" where "y = Ok x2"
distinct_13: fixes x21 :: "variable"   and x22 :: "typ"   and x51 :: "Core.term"   and x52 :: "Core.term" shows "Fv x21 x22 \<noteq> x51 $ x52"
F: shows "\<not> 0 < offs_num (length ns) xs index key mi ma 0"
tinst_def: fixes t1 :: "Core.term"   and t2 :: "Core.term" shows "tinst t1 t2 \<equiv> \<exists>\<rho>. tsubst t2 \<rho> = t1"
infinite: fixes A :: "'k set" assumes "infinite A" shows "sorted_list_of_set A = []"
admI: fixes P :: "'a \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i)\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)" shows "adm P"
vec_to_iarray_least_non_zero_position_of_vector_from_index': fixes i :: "'e"   and j :: "'f"   and A :: "(('d, 'f) vec, 'e) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "least_non_zero_position_of_vector_from_index (vec_to_iarray (column j A)) (mod_type_class.to_nat i) = mod_type_class.to_nat (LEAST n. A $ n $ j \<noteq> (0::'d) \<and> i \<le> n)"
red_single_diff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "red_single (p - q) r f t" shows "red_single p (r + q) f t \<or> red_single q (p - r) f t \<or> (\<exists>p' q'. red_single p p' f t \<and> red_single q q' f t \<and> r = p' - q')"
has_subprob_densityI: fixes f :: "'c \<Rightarrow> ennreal"   and N :: "'c measure"   and M :: "'c measure" assumes "f \<in> borel_measurable N"   and "M = density N f"   and "subprob_space M" shows "has_subprob_density M N f"
randomEl_def: fixes list :: "'a list"   and random :: "integer" shows "randomEl list random = list ! (nat_of_integer random mod length list)"
induct: fixes P :: "(char list list \<times> int) list \<Rightarrow> bool"   and a0 :: "(char list list \<times> int) list" assumes "P []"   and "\<And>xs n. P [(xs, n)]"   and "\<And>xs n ys m p. \<lbrakk>\<lbrakk>xs = ys; n + m \<noteq> 0\<rbrakk> \<Longrightarrow> P ((xs, n + m) # p); \<lbrakk>xs = ys; \<not> n + m \<noteq> 0\<rbrakk> \<Longrightarrow> P p; \<lbrakk>xs \<noteq> ys; n = 0\<rbrakk> \<Longrightarrow> P ((ys, m) # p); \<lbrakk>xs \<noteq> ys; n \<noteq> 0\<rbrakk> \<Longrightarrow> P ((ys, m) # p)\<rbrakk> \<Longrightarrow> P ((xs, n) # (ys, m) # p)" shows "P a0"
strict_prefixI: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>p ys"   and "xs \<noteq> ys" shows "xs <p ys"
assms_1: shows "\<R> = {region X I r |I r. valid_region X k I r}"
orderE: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" obtains "a = sup a b"
simps: fixes a :: "'a list"   and b :: "'a list" shows "get_text (BUF a b) = a @ b"
sel_2: fixes x1 :: "rat \<Rightarrow> rat \<Rightarrow> nat"   and x2 :: "rat \<Rightarrow> nat" shows "root_info.number_root (Root_Info x1 x2) = x2"
fg_e2: shows "LINT x|lebesgue_on {- pi..pi}. \<bar>f x - g x\<bar> < e / 2"
open_scaling_nonzero: fixes c :: "real"   and s :: "'b nonzero set" assumes "c \<noteq> 0"   and "open s" shows "open ((*\<^sub>R) c ` s)"
not_is_Tensor_TensorDiagE: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "\<not> is_Tensor (t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u)"   and "Diag t"   and "Diag u"   and "t \<noteq> \<^bold>\<I>"   and "u \<noteq> \<^bold>\<I>" shows "False"
exp_2dist_positive: fixes a :: "proj2"   and b :: "proj2" assumes "a \<in> hyp2"   and "b \<in> hyp2" shows "0 < exp_2dist a b"
less_dag_Node': fixes x :: "dag"   and l :: "dag"   and a :: "ref"   and r :: "dag" shows "(x < Node l a r) = (x = l \<or> x = r \<or> x < l \<or> x < r)"
Acts_eq: fixes init :: "'d set"   and acts :: "('d \<times> 'd) set set"   and allowed :: "('d \<times> 'd) set set" shows "Acts (mk_program (init, acts, allowed)) = insert Id acts"
assms_2: shows "i < \<^bold>|w\<^bold>|"
antilinear_axioms_def: fixes f :: "'d \<Rightarrow> 'e" shows "antilinear_axioms f \<equiv> \<forall>r x. f (r *\<^sub>C x) = cnj r *\<^sub>C f x"
gcd_dvd_1: fixes a :: "'a"   and b :: "'a" shows "(gcd a b dvd (1::'a)) = comm_monoid_mult_class.coprime a b"
Assign_le: fixes x :: "char list"   and e :: "aexp"   and S :: "'a"   and c :: "'a acom" shows "(x ::= e {S} \<le> c) = (\<exists>S'. c = x ::= e {S'} \<and> S \<le> S')"
inv_o_cancel: fixes f :: "'h \<Rightarrow> 'i" assumes "inj f" shows "inv f \<circ> f = id"
correctCompositionKS_PQ: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ks :: "KS" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionKS PQ"   and "ks \<in> specKeysSecrets PQ" shows "ks \<in> specKeysSecrets P \<or> ks \<in> specKeysSecrets Q"
has_products_preserved_by_bijection: fixes I :: "'c set"   and \<phi> :: "'c \<Rightarrow> 'd"   and I' :: "'d set" assumes "has_products I"   and "bij_betw \<phi> I I'"   and "I' \<noteq> UNIV" shows "has_products I'"
rt_fresher_update: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and ip :: "nat" assumes "dip \<in> kD rt"   and "1 \<le> the (dhops rt dip)"   and "update_arg_wf r" shows "rt \<sqsubseteq>\<^bsub>dip\<^esub> update rt ip r"
Says_to_Gets: fixes A :: "agent"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says A B X # evs \<in> set_pur" shows "Gets B X # Says A B X # evs \<in> set_pur"
le_fin_trivial: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
Un_Int_eq_3: fixes S :: "'c set"   and T :: "'c set" shows "S \<inter> (S \<union> T) = S"
\<phi>_prop: shows "\<phi> = (\<lambda>\<alpha>. make\<o> (\<lambda>w s. \<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> \<alpha> \<and> o\<^sub>1 = o\<^sub>2))"
emeasure_lborel_ball_finite: fixes c :: "'c"   and r :: "real" shows "emeasure lborel (ball c r) < \<infinity>"
ord_embed_inj: fixes f :: "'c \<Rightarrow> 'd" assumes "ord_embed f" shows "inj f"
llength_llist_of: fixes xs :: "'b list" shows "llength (llist_of xs) = enat (length xs)"
map_le_antisym: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" assumes "f \<subseteq>\<^sub>m g"   and "g \<subseteq>\<^sub>m f" shows "f = g"
eOp_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> 'c option"   and binp :: "'bindex \<Rightarrow> 'd option" assumes "igWlsInp MOD delta inp"   and "igWlsBinp MOD delta binp" shows "eOp MOD delta (OKI inp) (OKI binp) = OK (igOp MOD delta inp binp)"
dgk: fixes k :: "nat" assumes "k < n" shows "g k \<in> carrier_mat d d"
all_flows_def: fixes \<T> :: "'a stateful_policy" shows "all_flows \<T> \<equiv> flows_fix \<T> \<union> flows_state \<T> \<union> backflows (flows_state \<T>)"
conc_empty_2: fixes A :: "'b list set" shows "A \<cdot> {} = {}"
snd_image_times': shows "B = snd ` (A \<times> B)"
distinct_59: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi' :: "pi" shows "name1<name2>.pi \<noteq> !pi'"
lift_opts_exists: fixes ty_ty'_list :: "(ty \<times> ty) list"   and P :: "cld list" assumes "\<forall>x\<in>set ty_ty'_list. case x of (ty, ty') \<Rightarrow> is_sty_one P ty ty' = Some True" shows "\<exists>bools. lift_opts (map (\<lambda>(ty, ty'). is_sty_one P ty ty') ty_ty'_list) = Some bools"
zero_less_abs_iff: fixes a :: "'c" shows "((0::'c) < \<bar>a\<bar>) = (a \<noteq> (0::'c))"
simps_5: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and as :: "Arity list"   and v :: "Vars.var"   and S :: "stack_elem list" assumes "CardinalityArityTransformation Aexp Aheap" shows "CardinalityArityTransformation.transform_alts Aheap as (Upd v # S) = Upd v # CardinalityArityTransformation.transform_alts Aheap as S"
add_less_cancel_right: fixes a :: "'c multiset"   and c :: "'c multiset"   and b :: "'c multiset" shows "(a + c \<subset># b + c) = (a \<subset># b)"
boolean_algebra_axioms: shows "boolean_algebra (\<^bold>\<squnion>) (\<^bold>\<sqinter>) compl \<one> \<zero>"
cycle_free_alt: fixes g :: "'b ugraph" shows "cycle_free g = (\<forall>e\<in>edges g. e \<notin> (edges (restrict_edges g (- {e, prod.swap e})))\<^sup>*)"
assms: fixes x :: "'a" assumes "x \<in> A" shows "f x = g x"
equality: fixes r :: "'a state_scheme"   and r' :: "'a state_scheme" assumes "heap r = heap r'"   and "locals r = locals r'"   and "state.more r = state.more r'" shows "r = r'"
char_to_digit_simps_5: shows "char_to_digit CHR ''4'' = 4"
hyps: fixes "is" :: "'a set" assumes "\<And>n. n < length ias \<Longrightarrow> interrupt_action_ok' (redT_updIs is (take n ias)) (ias ! n)" shows "interrupt_actions_ok' is ias"
cospan: shows "trg u\<^sub>1 = trg t\<^sub>0"
Extend_SNOCs_zero_satisfies: fixes k :: "'k"   and idx :: "'n"   and \<phi> :: "('a, 'k) aformula"   and \<AA> :: "'i"   and i :: "nat"   and P :: "'v" assumes "local.wf (SUC k idx) \<phi>"   and "#\<^sub>V \<AA> = idx"   and "lformula \<phi>" shows "(Extend k 0 ((SNOC (zero (#\<^sub>V \<AA>)) ^^ i) \<AA>) P \<Turnstile> \<phi>) = (Extend k 0 \<AA> P \<Turnstile> \<phi>)"
2: shows "conjunct (pred c) \<sigma>\<^sub>s\<^sub>y\<^sub>m"
simps_1: fixes carrier :: "'c set"   and eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and more :: "'d" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
less_le_trans: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" assumes "x \<subset># y"   and "y \<subseteq># z" shows "x \<subset># z"
normalize_poly_eq_map_poly: fixes p :: "'a poly" shows "normalize p = map_poly (\<lambda>x. x div unit_factor (lead_coeff p)) p"
SubstTermP_imp_TermP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm" shows "{SubstTermP v i t u} \<turnstile> TermP i"
matchs_neq_length_none: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "term list"   and ys :: "'c list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "length xs \<noteq> length ys" shows "term.matchs unapp unconst xs ys = None"
eq_reflection: fixes x :: "'j"   and y :: "'j" assumes "x = y" shows "x \<equiv> y"
read_only_unowned_instructions_read_value_independent: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and i :: "nat"   and p :: "'b"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'b"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "read_only_reads_unowned ts"   and "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "read_only_reads_unowned (ts[i := (p', is', \<theta>', sb, \<D>', \<O>, \<R>')])"
mult_less_cancel_left1: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times c b) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
distinct_card: fixes xs :: "'d list" assumes "distinct xs" shows "card (set xs) = length xs"
double_compl: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus (uminus x) = x"
II: fixes it :: "'d set"   and \<sigma> :: "'c" assumes "it \<subseteq> dom m"   and "it \<noteq> {}"   and "\<not> c \<sigma>"   and "I it \<sigma>" shows "P \<sigma>"
assn_aci_1: fixes x :: "assn"   and y :: "assn" shows "x \<and>\<^sub>A y = y \<and>\<^sub>A x"
add_edge_delete_edges_list: fixes a :: "'c \<times> 'c"   and G :: "'c graph"   and ff :: "('c \<times> 'c) list" shows "add_edge (fst a) (snd a) (delete_edges_list G (a # ff)) = add_edge (fst a) (snd a) (delete_edges G (set ff))"
rep_eq: fixes bound :: "int" shows "generator (upto_generator bound) = (\<lambda>n. n \<le> bound, \<lambda>n. (n, n + 1))"
dagger_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<dagger> \<le> y\<^sup>\<dagger>"
ss1: shows "s - real_of_int \<lfloor>s\<rfloor> < 1"
single_zero: fixes k :: "'a" shows "Poly_Mapping.single k (0::'b) = 0"
fvar_upd_no_jump: fixes upd :: "val \<Rightarrow> state \<Rightarrow> state"   and statDeclC :: "qtname"   and stat :: "bool"   and fn :: "vname"   and a :: "val"   and s' :: "state"   and s :: "state"   and j :: "jump"   and val :: "val" assumes "upd = snd (fst (fvar statDeclC stat fn a s'))"   and "fst s \<noteq> Some (Jump j)" shows "fst (upd val s) \<noteq> Some (Jump j)"
assms_3: shows "i < dim_vec (b @\<^sub>v c)"
at_append: fixes pi1 :: "('a \<times> 'a) list"   and pi2 :: "('a \<times> 'a) list"   and c :: "'a" assumes "at TYPE('a)" shows "(pi1 @ pi2) \<bullet> c = pi1 \<bullet> pi2 \<bullet> c"
binomial_ge_n_over_k_pow_k: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "(of_nat n / of_nat k) ^ k \<le> of_nat (n choose k)"
arcsin_def: fixes y :: "real" shows "arcsin y = (THE x. - (pi / 2) \<le> x \<and> x \<le> pi / 2 \<and> sin x = y)"
fmerge_bind: fixes A :: "'a fset"   and m' :: "'m"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'm" shows "local.fmerge A (\<lambda>x. bind m' (\<lambda>A'. local.fmerge A' (f x))) = bind m' (\<lambda>A'. local.fmerge A (\<lambda>x. local.fmerge A' (f x)))"
simps_1: fixes q :: "('c monom \<times> 'd) list" shows "poly_add [] q = q"
isGlb_def: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" shows "isGlb R S x = greatestP (isLb R S) x"
intros: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and ts\<^sub>h :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "length ts = length ts\<^sub>h"   and "\<forall>i<length ts. \<exists>\<O>' \<D>' \<R>'. let (p, is, \<theta>, sb, \<D>, \<O>, \<R>) = ts\<^sub>h ! i in ts ! i = (p, is, \<theta>, filter is_Write\<^sub>s\<^sub>b sb, \<D>', \<O>', \<R>') \<and> (filter is_Write\<^sub>s\<^sub>b sb = [] \<longrightarrow> sb = [])" shows "ts \<sim>\<^sub>h ts\<^sub>h "
simps_1: fixes cl :: "nat \<Rightarrow> 'l"   and cs :: "nat \<Rightarrow> 'c" shows "arun cl cs 0 = as\<^sub>0"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<sim> <\<nu>x>P \<parallel> Q"
from_list_correct_2: fixes l :: "(('v \<times> 'v) \<times> nat) list" assumes "valid_wgraph_repr l" shows "nodes (\<alpha>g (from_list l)) = fst ` fst ` set l \<union> snd ` fst ` set l"
index_set_lm_1: fixes A :: "nat set"   and n :: "nat"   and m :: "nat" assumes "index_set A"   and "n \<in> A"   and "nat_to_ce_set n = nat_to_ce_set m" shows "m \<in> A"
mat_of_row_def: fixes y :: "'b Matrix.vec" shows "mat_of_row y = Matrix.mat 1 (dim_vec y) (\<lambda>ij. y $v snd ij)"
redT_updT_Some1: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h"   and xw :: "'f \<times> 'g \<Rightarrow>f nat"   and ta :: "('h, 'f, 'i) new_thread_action" assumes "ts t = \<lfloor>xw\<rfloor>" shows "\<exists>xw. redT_updT ts ta t = \<lfloor>xw\<rfloor>"
bequiv_up_to_def: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and b' :: "bexp" shows "(P \<Turnstile> b <\<sim>> b') = (\<forall>s. P s \<longrightarrow> bval b s = bval b' s)"
row_transpose: fixes j :: "nat"   and A :: "'a mat" assumes "j < dim_col A" shows "row A\<^sup>T j = col A j"
hoare\<^sub>3'_def: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "\<lbrace>P\<rbrace> M\<dagger> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> True | Some a \<Rightarrow> False)"
Module_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme" shows "R module M \<equiv> aGroup M \<and> Module_axioms M R"
S_steps: shows "\<forall>i. i \<le> f i \<and> (t (f i), t (Suc (f i))) \<in> S"
partition_filter_conv: fixes f :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "List.partition f xs = (filter f xs, filter (Not \<circ> f) xs)"
update_arg_wf_tuples_2: fixes n :: "nat"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "update_arg_wf (Suc n, kno, val, hops, nhip, pre)"
disjoint_family_subset: fixes A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" assumes "disjoint_family A"   and "\<And>x. B x \<subseteq> A x" shows "disjoint_family B"
subprob_space_distr: fixes f :: "'s stream \<Rightarrow> 'a"   and s :: "'s"   and M' :: "'a measure" assumes "K.T.random_variable s M' f"   and "space M' \<noteq> {}" shows "subprob_space (distr (K.T s) M' f)"
a_get_shadow_root_def: fixes element_ptr :: "(_) element_ptr" shows "get_shadow_root element_ptr = get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr shadow_root_opt"
quasinorm_subsetD: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" assumes "N1 \<subseteq>\<^sub>N N2" shows "\<exists>C\<ge>0. \<forall>f. eNorm N2 f \<le> ennreal C * eNorm N1 f"
Ctxt_valid_validn: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and n :: "nat" assumes " \<Turnstile> G " shows " \<Turnstile>\<^sub>n G  "
effect_shiftr_index: fixes a :: "'b array"   and key :: "'b"   and i :: "nat ref"   and h :: "heap"   and h' :: "heap"   and u :: "unit" assumes "effect (shiftr a key i) h h' u" shows "Ref.get h' i \<le> Ref.get h i"
fmdrop_fmupd_same: fixes x :: "'c"   and y :: "'d"   and m :: "('c, 'd) fmap" shows "fmdrop x (fmupd x y m) = fmdrop x m"
distinct_9: fixes x2 :: "'a" shows "Abrupt x2 \<noteq> Stuck"
sub1: fixes C :: "'a set" assumes "C \<in> \<G>" shows "\<exists>D. D \<in> \<F> \<and> C \<subseteq> D"
keys_dom_lookup: fixes m :: "('b, 'c) mapping" shows "keys m = dom (lookup m)"
intros: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and conv :: "('a, 'b, 'c, 'd) converter" assumes "\<And>a. a \<in> outs_\<I> \<I> \<Longrightarrow> plossless_gpv \<I>' (run_converter conv a)"   and "\<And>a b conv'. \<lbrakk>a \<in> outs_\<I> \<I>; (b, conv') \<in> results_gpv \<I>' (run_converter conv a)\<rbrakk> \<Longrightarrow> plossless_converter \<I> \<I>' conv'" shows "plossless_converter \<I> \<I>' conv"
evaluate_list_mono_strong: fixes R :: "'b state \<Rightarrow> exp0 \<Rightarrow> 'b state \<times> (v, v) result \<Rightarrow> bool"   and s :: "'b state"   and es :: "exp0 list"   and r :: "'b state \<times> (v list, v) result"   and Q :: "'b state \<Rightarrow> exp0 \<Rightarrow> 'b state \<times> (v, v) result \<Rightarrow> bool" assumes "Big_Step_Unclocked_Single.evaluate_list R s es r"   and "\<And>s e r. \<lbrakk>e \<in> set es; R s e r\<rbrakk> \<Longrightarrow> Q s e r" shows "Big_Step_Unclocked_Single.evaluate_list Q s es r"
AsDynComExpr: fixes x :: "'d"   and r :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and Q :: "'d set"   and A :: "'d set"   and ad :: "('d, 'e, 'f) ann"   and f :: "'d \<Rightarrow> ('d, 'e, 'f) com"   and a :: "'d set" assumes "x \<in> r"   and "assertionsR \<Gamma> \<Theta> Q A ad (f x) a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnRec r ad) (DynCom f) a"
add_node_def: fixes v :: "'d"   and g :: "('d, 'e, 'f) graph_scheme" shows "add_node v g \<equiv> \<lparr>nodes = insert v (nodes g), edges = edges g\<rparr>"
pick_NE: fixes X :: "'c set" assumes "X \<noteq> {}" shows "pick X \<in> X"
rfr_step: fixes r :: "real"   and n :: "nat"   and w :: "'c" shows "disc_rfr_proc r (Suc n) w = (1 + r) * disc_rfr_proc r n w"
section_def: fixes f :: "'a \<times> 'a" shows "CC.section f \<equiv> \<exists>g. CC.ide (CC.comp g f)"
lsc_hull_le: fixes f :: "'b \<Rightarrow> ereal"   and x :: "'b" shows "lsc_hull f x \<le> f x"
d_gchainTr2: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and l :: "nat"   and j :: "nat" assumes "Group G"   and "0 < n"   and "d_gchain G n f"   and "l \<le> n"   and "j \<le> n"   and "l \<le> j" shows "f j \<subseteq> f l"
perm_3: fixes pi :: "(name \<times> name) list"   and x1 :: "name"   and x2 :: "trm" shows "pi \<bullet> \<Lambda> x1 . x2 = \<Lambda> pi \<bullet> x1 . (pi \<bullet> x2)"
h2: shows "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h2"
lang_subset_lists: fixes \<Sigma> :: "nat \<Rightarrow> 'a set"   and project :: "'a \<Rightarrow> 'a"   and wf_atom :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "nat"   and r :: "'b rexp"   and lookup :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "project \<Sigma> project"   and "alphabet.wf wf_atom n r" shows "project.lang \<Sigma> project lookup n r \<subseteq> lists (\<Sigma> n)"