hcju/isabelleps huggingface dataset:
Heap_lub_empty: shows "Heap_lub {} = Heap Map.empty"
steps_Suc: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" shows "((x, y) \<in> steps rel (Suc n)) = ((x, y) \<in> steps rel n \<or> (x, n) \<in> steps rel n \<and> (n, y) \<in> steps rel n)"
lnth_llist_of: fixes xs :: "'c list" shows "lnth (llist_of xs) = (!) xs"
times_divide_eq_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times a (divide b c) = divide (times a b) c"
that: fixes basis :: "'b set" assumes "finite_dimensional_vector_space (*s) basis" shows "thesis"
simps_4: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalLinear var b c (Neq p) = neg (allZero p var)"
ring_hom_cring_def: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "ring_hom_cring R S h \<equiv> cring R \<and> cring S \<and> ring_hom_cring_axioms R S h"
dgrad_p_setD_lp: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<in> gd_term.dgrad_p_set pair_of_term d m"   and "p \<noteq> 0" shows "d (term_powerprod.pp_of_term pair_of_term (ordered_term.lt term_of_pair ord_term p)) \<le> m"
rpo_optim_axioms: shows "rpo_optim ground_heads_var (>\<^sub>s) extf arity_sym arity_var"
induct: fixes P :: "'b list \<Rightarrow> bool"   and a0 :: "'b list" assumes "P []"   and "\<And>a. P [a]"   and "\<And>a b as. P (a # as) \<Longrightarrow> P (a # b # as)" shows "P a0"
lub_is_join: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "{x, y} <<| z" shows "x \<squnion> y = z"
simps_2: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "Unf (\<phi> W\<^sub>n \<psi>) = (\<phi> W\<^sub>n \<psi> and\<^sub>n Unf \<phi>) or\<^sub>n Unf \<psi>"
Diff_UNIV: fixes A :: "'d set" shows "A - UNIV = {}"
rep_of_bound: fixes l :: "nat list"   and i :: "nat" assumes "ufa_invar l"   and "i < length l" shows "rep_of l i < length l"
ap_spec_inE: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and p :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and q :: "('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap"   and "(p, q) \<in> set (ap gs bs ps hs data)" obtains "(p, q) \<in> set ps"   | "p \<in> set hs" and "q \<in> set gs \<union> set bs \<union> set hs"
compare_double_swap: fixes x :: "double"   and y :: "double" shows "(0 < compare_double x y) = (compare_double y x < 0)"
bind_returns_result_E: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and y :: "'n" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y" obtains x :: "'o" and h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> g x \<rightarrow>\<^sub>r y"
assms_2: fixes dip :: "nat" assumes "dip \<in> kD (rt \<xi>)"   and "dip \<in> kD (rt \<xi>')" shows "rt \<xi> \<sqsubseteq>\<^bsub>dip\<^esub> rt \<xi>'"
foldr_Nil: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "foldr f [] = id"
simps_2: fixes t :: "('c, 'd) Term.term"   and \<theta> :: "'d \<Rightarrow> ('c, 'd) Term.term" shows "receive\<langle>t\<rangle>\<^sub>s\<^sub>t \<cdot>\<^sub>s\<^sub>t\<^sub>p \<theta> = receive\<langle>t \<cdot> \<theta>\<rangle>\<^sub>s\<^sub>t"
pull_negation: fixes x :: "trilean"   and y :: "trilean" shows "(x = \<not>? y) = (\<not>? x = y)"
initial_in_F_token_run: fixes x :: "nat"   and y :: "nat" assumes "q\<^sub>0 \<in> F" shows "token_run x y \<in> F"
tag_eq_def: fixes tag :: "'b list \<Rightarrow> 'c" shows "=tag= \<equiv> {(x, y). tag x = tag y}"
RCOSETS_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" shows "rcosets\<^bsub>G\<^esub> H = (\<Union>a\<in>carrier G. {H #>\<^bsub>G\<^esub> a})"
field_poly_irreducible_dvd_mult: fixes p :: "'b poly"   and q :: "'b poly"   and r :: "'b poly" assumes "irreducible p" shows "(p dvd q * r) = (p dvd q \<or> p dvd r)"
inject: fixes x21 :: "'a"   and x22 :: "'a list"   and y21 :: "'a"   and y22 :: "'a list" shows "(x21 \<Colon> x22 = y21 \<Colon> y22) = (x21 = y21 \<and> x22 = y22)"
norm_triangle_le_diff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (minus x y) \<le> e"
euler_incr_fas'_def: fixes D :: "nat" shows "euler_incr_fas' D = map fold_const_fa (euler_incr_fas (map floatarith.Var [0..<D]) (floatarith.Var D) (map floatarith.Var [Suc D..<Suc (2 * D)]))"
\<psi>_in_terms_of_\<epsilon>: fixes x :: "'b"   and g :: "'c"   and y :: "'c" assumes "B.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>C G' x\<guillemotright>" shows "\<psi>' x g = F'G'.\<epsilon> x \<cdot>\<^sub>B F' g"
fB_def: fixes x :: "'a" shows "fB x = (if x \<notin> A then f x else (0::'b))"
in_set_zip: fixes p :: "'a \<times> 'b"   and xs :: "'a list"   and ys :: "'b list" shows "(p \<in> set (zip xs ys)) = (\<exists>n. xs ! n = fst p \<and> ys ! n = snd p \<and> n \<sqsubset> length xs \<and> n \<sqsubset> length ys)"
semiring_norm_171: fixes v :: "num"   and w :: "num"   and y :: "'b" shows "numeral v * (- numeral w * y) = - numeral (v * w) * y"
Rep_bit_inverse: fixes x :: "bit" shows "Abs_bit (Rep_bit x) = x"
fls_compose_power_shift: fixes d :: "nat"   and n :: "int"   and f :: "'b fls" assumes "0 < d" shows "fls_compose_power (fls_shift n f) d = fls_shift (int d * n) (fls_compose_power f d)"
infinite_iff_alw_ev: fixes P :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream" shows "infinite {m. P (sdrop m \<omega>)} = alw (ev P) \<omega>"
cop_in_scop: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat"   and \<sigma>' :: "'var \<Rightarrow> 'val"   and k' :: "nat" assumes "((\<sigma>, k), \<sigma>', k') \<in> cop" shows "(path \<sigma>, k) \<in> scop \<and> (path \<sigma>', k') \<in> scp"
V_4: fixes i :: "nat" shows "closure (V i) \<subseteq> carrier"
write_store_barrier_pending_def: fixes b :: "bool"   and v :: "sparc_state_var" shows "write_store_barrier_pending b v \<equiv> v\<lparr>st_bar := b\<rparr>"
isint_c: fixes j :: "int"   and bs :: "real list" shows "isint (C j) bs"
add_divide_distrib: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide (plus a b) c = plus (divide a c) (divide b c)"
Plus_toplevel_summands: fixes r :: "'c rexp"   and s :: "'c rexp"   and t :: "'c rexp" assumes "Plus r s \<in> toplevel_summands t" shows "False"
fls_ex_antiderivative_residue_nonzero: fixes f :: "'b fls" assumes "\<exists>F. fls_deriv F = f" shows "fls_residue f = (0::'b)"
always_weakly_fair: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<box>weakly_fair enabled taken = weakly_fair enabled taken"
even_not_iff_int: fixes k :: "int" shows "even (NOT k) = odd k"
lookup_AList_clearjunk: fixes k :: "'b"   and al :: "('b \<times> 'a) list" shows "lookup (\<lambda>x. x = k) al = lookup (\<lambda>x. x = k) (AList.clearjunk al)"
sum_singleton: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.sum plus zero f {x} = f x"
l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs \<equiv> type_wf = DocumentClass.type_wf \<and> set_tag_name = l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name \<and> set_tag_name_locs = l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name_locs"
RBTid2_def: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real" shows "RBTid2 TYPE('b) bal_i e \<equiv> RBTi2 TYPE('b) bal_i e \<and> RBTid1 TYPE('b) bal_i"
errMOD_igAbsIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igAbsIPresIGWls MOD"   and "igWlsAbsDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igAbsIPresIGWlsSTR (errMOD MOD)"
nth_append: fixes xs :: "'e list"   and ys :: "'e list"   and n :: "nat" shows "(xs @ ys) ! n = (if n < length xs then xs ! n else ys ! (n - length xs))"
bigo_const: fixes c :: "'b"   and F :: "'a filter" shows "(\<lambda>_. c) \<in> O[F](\<lambda>_. 1::'b)"
times_ms_aux_altdef: fixes xs :: "('a \<times> real) msllist"   and ys :: "('a \<times> real) msllist" shows "times_ms_aux xs ys = (case (xs, ys) of (MSLNil, ys) \<Rightarrow> MSLNil | (MSLCons a msllist, MSLNil) \<Rightarrow> MSLNil | (MSLCons a msllist, MSLCons y ysa) \<Rightarrow> MSLCons (fst a * fst y, snd a + snd y) (plus_ms_aux (scale_shift_ms_aux a ysa) (times_ms_aux msllist (MSLCons y ysa))))"
flowD_source_IN: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "flow \<Delta> f" shows "d_IN f (source \<Delta>) = 0"
Rstep_CohideRR: fixes j :: "nat"   and A :: "('sf, 'sc, 'sz) formula list"   and S :: "('sf, 'sc, 'sz) formula list" shows "Rrule_result CohideRR j (A, S) = [([], [S ! j])]"
Nats_subset_Ints: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c" assumes "class.ring_1 one times plus zero minus uminus" shows "semiring_1.Nats one plus zero \<subseteq> ring_1.Ints one plus zero minus"
desargues_config_not_col_7: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "desargues_config A B C A' B' C' M N P R" shows "\<not> col A B B'"
\<Delta>_naturality: shows "(r \<star> tab\<^sub>0 r) \<cdot> r.tab = (r \<star> r.the_\<theta>) \<cdot> \<a>[r, tab\<^sub>0 r, r.chine] \<cdot> (r.tab \<star> r.chine) \<cdot> r.the_\<nu>"
vcard_def: fixes a :: "V" shows "vcard a \<equiv> LEAST i. Ord i \<and> i \<approx>\<^sub>\<circ> a"
iter_tensor_of_H_fst_pos: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < 2 ^ n \<or> j < 2 ^ n"   and "i < 2 ^ (n + 1) \<and> j < 2 ^ (n + 1)" shows "Matrix.mat (2 ^ Suc n) (2 ^ Suc n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>Suc n\<^esub>  j / sqrt 2 ^ Suc n)) $$ (i, j) = complex_of_real (1 / sqrt 2) * Matrix.mat (2 ^ n) (2 ^ n) (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> (- 1) ^ i \<cdot>\<^bsub>n\<^esub>  j / sqrt 2 ^ n)) $$ (i mod 2 ^ n, j mod 2 ^ n)"
ereal_INF_cmult: assumes "0 < c" shows "(\<Sqinter>i\<in>I. ereal c * f i) = ereal c * \<Sqinter> (f ` I)"
spec_ops_sorted: fixes ops :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set" assumes "spec_ops ops deps" shows "sorted (map fst ops)"
leq_path_asym: fixes Cs :: "char list list"   and Cs' :: "char list list"   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 C :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "(Cs, Cs') \<in> (leq_path1 P C)\<^sup>+"   and "wf_prog wf_md P" shows "(Cs', Cs) \<notin> (leq_path1 P C)\<^sup>+"
fib_rec_even: fixes n :: "nat" shows "fib (2 * n) = (fib (n - 1) + fib (n + 1)) * fib n"
tso_lock_invL: shows "\<lbrace>tso_lock_invL\<rbrace> mutator m"
odd_nodes_set_def: fixes g :: "('c, 'd) graph" shows "odd_nodes_set g \<equiv> {v \<in> nodes g. odd (degree v g)}"
intro: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>P a. (\<And>x. (\<And>y. less y x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a" shows "class.wellorder_axioms less"
P_gs_def: fixes x :: "'a" shows "P_gs x \<equiv> if x \<in> P then gs else []"
fmdrop_set_empty: fixes A :: "'c set" shows "fmdrop_set A {$$} = {$$}"
case_4: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case REAL of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f4"
mcont2mcont': fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lub' :: "'d set \<Rightarrow> 'd"   and ord' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd" assumes "class.ccpo Sup less_eq less"   and "\<And>x. mcont lub' ord' Sup less_eq (f x)"   and "\<And>y. mcont lub ord Sup less_eq (\<lambda>x. f x y)"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup less_eq (\<lambda>x. f x (t x))"
getParts_complete: fixes rs :: "'a simple_rule list" shows "wordinterval_list_to_set (getParts rs) = UNIV"
one_le_\<omega>\<^sub>z: shows "1 \<le> \<omega>\<^sub>z"
moebius_pt_neq_I: fixes z1 :: "complex_homo"   and z2 :: "complex_homo"   and M :: "moebius" assumes "z1 \<noteq> z2" shows "moebius_pt M z1 \<noteq> moebius_pt M z2"
empty_partial_heap: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int" shows "is_partial_heap cmp heap l r r"
fset_simps_2: fixes x :: "'c"   and xa :: "'c fset" shows "fset (finsert x xa) = insert x (fset xa)"
mono: fixes t :: "'a"   and s1 :: "'b \<times> 'c"   and s2 :: "'d \<times> 'e" assumes "bisim t s1 s2" shows "bisim' t s1 s2"
divide_le_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (divide b c) a = (if less zero c then less_eq b (times a c) else if less c zero then less_eq (times a c) b else less_eq zero a)"
Lim_trivial_limit: fixes net :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and l :: "'b" assumes "net = bot" shows "(f \<longlongrightarrow> l) net"
base: fixes A :: "'a set"   and c :: "'b" assumes "A \<in> sets M"   and "emeasure M A < \<infinity>" shows "P (\<lambda>x. indicat_real A x *\<^sub>R c)"
rref_upt_condition2: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'b)) = (1::'b)"
ww'_3: shows "w `\<rightarrow> C0 = w' `\<rightarrow> C0"
lr_trans_l: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset"   and B :: "'b multiset" assumes "locally_refl R (A + B)" shows "locally_refl R A"
ring_simprules_13: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
CSP_join: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and y :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process x"   and "is_CSP_process y" shows "is_CSP_process (x \<or> y)"
eq_fold: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "aggregation.sum_0 g A = Finite_Set.fold ((+) \<circ> g) zero A"
S'_def: fixes n :: "nat"   and x :: "real" shows "S' n x = 1 / (2 * x) + (\<Sum>r = 1..<n. 1 / (real r + x)) + 1 / (2 * (real n + x))"
Ostrowski_baseTr2: fixes n :: "nat"   and vv :: "nat \<Rightarrow> 'a \<Rightarrow> ant"   and l :: "nat"   and m :: "nat" assumes "vals_nonequiv K (Suc n) vv"   and "l \<le> Suc n"   and "m \<le> Suc n"   and "l \<noteq> m" shows "0 < vv m ((\<Omega>\<^bsub>K vv Suc n\<^esub>) l)"
simps_2: fixes P :: "'c"   and uu :: "'d"   and uv :: "('c, 'd) chain" shows "pre (\<lbrace> P \<rbrace> \<cdot> uu \<cdot> uv) = P"
twiddleObj_def: fixes \<beta> :: "(Location \<times> Location) set"   and o1 :: "Class \<times> (Field \<times> Val) list"   and o2 :: "Class \<times> (Field \<times> Val) list" shows "twiddleObj \<beta> o1 o2 = (fst o1 = fst o2 \<and> LowDom (snd o1) = LowDom (snd o2) \<and> (\<forall>f v w. lookup (snd o1) f = Some v \<longrightarrow> lookup (snd o2) f = Some w \<longrightarrow> GAMMA f = low \<longrightarrow> (\<beta>, v, w) \<in> twiddleVal))"
pdg_Hom_vifunion_in_Vset: fixes J :: "V"   and A :: "V"   and B :: "V" assumes "J \<subseteq>\<^sub>\<circ> I"   and "A \<subseteq>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>"   and "B \<subseteq>\<^sub>\<circ> dg_prod J \<AA>\<lparr>Obj\<rparr>"   and "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>" shows "\<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>A. \<Union>\<^sub>\<circ> (VLambda B (Hom (dg_prod J \<AA>) a) `\<^sub>\<circ> B)) `\<^sub>\<circ> A) \<in>\<^sub>\<circ> Vset \<alpha>"
fps_XD'_funpow_affine: fixes G :: "real fps"   and n :: "nat"   and H :: "real fps" assumes "fps_deriv G = 1" shows "(fps_XD' G ^^ n) H = (\<Sum>m\<le>n. fps_const (real (Stirling n m)) * G ^ m * (fps_deriv ^^ m) H)"
B1_def: shows "B1 = {x. jumpF f (at_left x) \<noteq> 0 \<and> a < x \<and> x \<le> b}"
plssk: shows "prod_list (nths dims ({0..<Suc k} \<union> {Suc k})) = 2 ^ Suc (Suc k)"
eucl_ivals_5: fixes a :: "'a"   and b :: "'a" shows "{a..b} \<in> sets borel"
nb\<^sub>2: shows "\<forall>op\<in>set ops. is_valid_operator_sas_plus \<Psi> op"
induced_function_integral_aux_2: fixes A :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "A \<in> sets M"   and "integrable M f"   and "\<And>x. 0 \<le> f x" shows "integral\<^sup>L M (induced_function A f) = set_lebesgue_integral M (\<Union>n. (T ^^ n) --` A) f"
derives_is_sentence: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list" assumes "derives \<alpha> \<beta>" shows "\<alpha> = \<beta> \<or> is_sentence \<alpha> \<and> is_sentence \<beta>"
fold_pair_upd_triv: fixes A :: "'a"   and U :: "('a \<times> 'b) list"   and F :: "'a \<Rightarrow> 'b" assumes "A \<notin> fst ` set U" shows "foldl pair_fun_upd F U A = F A"
convert_obs_initial_def: fixes ta :: "('c, 'd, 'e, 'f, 'g, 'h) thread_action" shows "convert_obs_initial ta = (\<lbrace>ta\<rbrace>\<^bsub>l\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>c\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>w\<^esub>, \<lbrace>ta\<rbrace>\<^bsub>i\<^esub>, map NormalAction \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)"
lattice_def: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.lattice inf less_eq less sup \<equiv> class.semilattice_inf inf less_eq less \<and> class.semilattice_sup sup less_eq less"
xor_zero_right: fixes x :: "'c" shows "x XOR (0::'c) = x"
assms_2: shows "A \<subseteq>\<^sub>\<circ> \<real>\<^sub>\<circ>"
axioms_1: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c" assumes "vmc_path G P v0 p \<sigma>" shows "vm_path G P v0"
qp_ineq: fixes k :: "int" shows "(int p * k \<le> (int p * int q - 1) div 2) = (k \<le> (int q - 1) div 2)"
iso_hom_induced_relativization_contractible: fixes X :: "'a topology"   and S :: "'a set"   and T :: "'a set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "contractible_space (subtopology X T)"   and "T \<subseteq> S"   and "topspace X \<inter> T \<noteq> {}" shows "hom_induced p X T X S id \<in> Group.iso (relative_homology_group p X T) (relative_homology_group p X S)"
sdrop_add: fixes n :: "nat"   and m :: "nat"   and s :: "'b stream" shows "sdrop n (sdrop m s) = sdrop (m + n) s"
lock_thread_ok_def: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option" shows "lock_thread_ok ls ts \<equiv> \<forall>l t. has_lock (ls $ l) t \<longrightarrow> (\<exists>xw. ts t = \<lfloor>xw\<rfloor>)"
Newman_Ingham_1_1: fixes F :: "complex fds"   and f :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "fds_nth F \<in> O(\<lambda>_. 1)"   and "f analytic_on {s. 1 \<le> Re s}"   and "\<And>s. 1 < Re s \<Longrightarrow> eval_fds F s = f s"   and "1 \<le> Re w" shows "fds_converges F w"
IH: shows "prod_list (nths xs A) = prod_list (nths xs B) * prod_list (nths xs (A - B))"
sim: shows "cis \<phi> *k cA = inv_D ** cA ** D"