hcju/isabelleps huggingface dataset:
cong_sum: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "\<And>x. x \<in> A \<Longrightarrow> unique_euclidean_semiring.cong modulo (f x) (g x) a" shows "unique_euclidean_semiring.cong modulo (comm_monoid_add.sum plus zero f A) (comm_monoid_add.sum plus zero g A) a"
distinct_7: fixes x2 :: "'a"   and x3 :: "'b" shows "Conclusion x2 \<noteq> Rule x3"
prm_apply_composition: fixes f :: "'b prm"   and g :: "'b prm"   and x :: "'b" shows "f \<diamondop> g $ x = f $ (g $ x)"
EF_lem1: fixes x :: "'a"   and f :: "'a set" assumes "x \<in> f \<or> x \<in> EX' (EF f)" shows "x \<in> EF f"
nz1: shows "1 - \<i> * z \<noteq> 0"
assms_1: shows "hermitean H1 \<and> H1 \<noteq> mat_zero"
hyps: assumes "\<not> c \<le> d"   and "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))" shows "card {q \<in> set (shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))). m < card (snd q)} \<le> card {q \<in> set (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)}))). m < card (snd q)}"
l10_4: fixes P :: "'p"   and P' :: "'p"   and A :: "'p"   and B :: "'p" assumes "P P' Reflect A B" shows "P' P Reflect A B"
assms: shows "a b \<congruent>\<^sub>K c d"
reversepath_simps_2: fixes g :: "real \<Rightarrow> 'a" shows "path_image (reversepath g) = path_image g"
cblinfun_apply: fixes x :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "(*\<^sub>V) x \<in> {f. bounded_clinear f}"
sg_unit_closed: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" assumes "Group G"   and "G \<guillemotright> H " shows "\<one>\<^bsub>G\<^esub> \<in> H"
prod_ineqs2: fixes k :: "nat"   and m :: "nat" assumes "0 < k" shows "m \<le> m * k"
arith_modZero1: fixes t :: "nat"   and n :: "nat" shows "(t + n * t) mod Suc n = 0"
mset_subsetD: fixes A :: "'b multiset"   and B :: "'b multiset"   and x :: "'b" assumes "A \<subset># B"   and "x \<in># A" shows "x \<in># B"
term_variants_pred_inv'_5: fixes P :: "'a \<Rightarrow> 'a list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and g :: "'a"   and h :: "'a" assumes "term_variants_pred P (Fun f T) t"   and "P \<equiv> (\<lambda>_. [])(g := [h])"   and "f \<noteq> the_Fun t" shows "f = g \<and> the_Fun t = h"
sum_hom: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "\<And>a b. f (a + b) = f a + f b"   and "f (0::'a) = (0::'b)" shows "sum (f \<circ> h) A = f (sum h A)"
runit_simps_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "src (bicategory.runit V H \<a> \<i> src f) = src f"
b: shows "!q \<cdot> (q \<cdot> r1 \<cdot> y1 + !q \<cdot> r2 \<cdot> y2)\<^sup>\<dagger> \<cdot> (q \<cdot> !r1 + !q \<cdot> !r2) = !q \<cdot> (r2 \<cdot> y2)\<^sup>\<dagger> \<cdot> !r2"
set_bit_eq: fixes a :: "'b"   and n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit a n b = (if b then ring_bit_operations_class.set_bit else unset_bit) n a"
snth_szip: fixes s1 :: "'d stream"   and s2 :: "'e stream"   and n :: "nat" shows "szip s1 s2 !! n = (s1 !! n, s2 !! n)"
inj_on_image_set_diff: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "order.greater_eq C (A - B)"   and "order.greater_eq C B" shows "f ` (A - B) = f ` A - f ` B"
raw_interp: fixes x :: "('a, 'b, 'c) good_interp" shows "raw_interp x \<in> {I. is_interp I}"
pairself_apply: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b \<times> 'b" shows "pairself f x = (f (fst x), f (snd x))"
Rep_token_run_af: fixes x :: "nat"   and n :: "nat" shows "Rep (token_run x n) \<equiv>\<^sub>P af\<^sub>G \<psi> (w [x \<rightarrow> n])"
simps_1: fixes iostep :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e" shows "mbind [] iostep \<sigma> = Some ([], \<sigma>)"
rcond_lift_true: shows "\<lceil>true\<rceil>\<^sub>\<leftarrow> = true"
elims_3: fixes x :: "term list" assumes "\<not> linears x" obtains t :: "term" and ts :: "term list" where "x = t # ts" and "\<not> (linear t \<and> fdisjnt (frees t) (freess ts) \<and> linears ts)"
Max_wrt_among_subset_2: fixes A :: "'a set" shows "Max_wrt_among le A \<subseteq> A"
sublens_obs_get: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d"   and \<rho> :: "'e" assumes "mwb_lens X"   and "Y \<subseteq>\<^sub>L X" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> v) = get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<rho> v)"
gt_or_eq_compat_App: fixes s' :: "('s, 'v) tm"   and s :: "('s, 'v) tm"   and t' :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "s' \<ge>\<^sub>t s"   and "t' \<ge>\<^sub>t t" shows "App s' t' \<ge>\<^sub>t App s t"
permutation_lemma: fixes S :: "'c set"   and p :: "'c \<Rightarrow> 'c" assumes "finite S"   and "bij p"   and "\<forall>x. x \<notin> S \<longrightarrow> p x = x" shows "permutation p"
rep_eq: shows "space_as_set \<bottom> = {0::'d}"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (local.dom \<tau>) \<tau>.r.tab (tab\<^sub>0 (local.dom \<tau>)) (tab\<^sub>1 (local.dom \<tau>)) (cod \<tau>) \<tau>.s.tab (tab\<^sub>0 (cod \<tau>)) (tab\<^sub>1 (cod \<tau>)) \<tau>"
add_right: fixes a :: "'a"   and b :: "'d"   and b' :: "'d" shows "a ** g (b + b') = a ** g b + a ** g b'"
lchild_def: fixes tree :: "'b tree" shows "lchild tree = (case tree of Leaf \<Rightarrow> Leaf | Node x21 x22 x23 x24 x25 \<Rightarrow> x24)"
smultfunsetD: fixes f :: "('r, 'g) aezfun \<Rightarrow> 'v"   and G :: "'g set"   and FH :: "('r, 'g) aezfun set"   and g :: "'g"   and x :: "('r, 'g) aezfun"   and a :: "'r" assumes "f \<in> smultfunset G FH"   and "g \<in> G"   and "x \<in> FH" shows "f (a \<delta>\<delta> g * x) = (a \<delta>\<delta> g) \<cdot> f x"
gx1_def: shows "gx1 \<equiv> max gx0 (SOME gx1. \<forall>x\<ge>gx1. \<forall>i<k. gx0 \<le> (ts ! i) x)"
distinct_by_mergesort: fixes xs :: "'a list" shows "(length (mergesort_remdups xs) = length xs) = distinct xs"
hyps_4: shows "p\<^sub>3 = ell_field_class.add a (ell_field_class.add a p p) (opp p)"
uminus_monom_mult: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "- p = local.punit.monom_mult (- (1::'c)) (0::'a) p"
env_restr_eq_subset: fixes S :: "'a set"   and S' :: "'a set"   and m1 :: "'a \<Rightarrow> 'b"   and m2 :: "'a \<Rightarrow> 'b" assumes "S \<subseteq> S'"   and "m1 f|` S' = m2 f|` S'" shows "m1 f|` S = m2 f|` S"
AbGroup_sum_list: fixes Gs :: "'b set list" assumes "\<forall>G\<in>set Gs. AbGroup G" shows "AbGroup (\<Sum>G\<leftarrow>Gs. G)"
nth_to_bl: assumes "LENGTH('a) - Suc 0 < LENGTH('a)" shows "to_bl w ! (LENGTH('a) - Suc 0) = bit w (LENGTH('a) - Suc (LENGTH('a) - Suc 0))"
continuous_map_openin_preimage_eq: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (f ` topspace X \<subseteq> topspace Y \<and> (\<forall>U. openin Y U \<longrightarrow> openin X (topspace X \<inter> f -` U)))"
RSpan_single: fixes m :: "'c" shows "GSpan [m] = {r \<cdot> m |r. r \<in> FG}"
axioms: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "semilattice_set f" shows "semilattice f"
in_set_butlast_concatI: fixes x :: "'b"   and xs :: "'b list"   and xss :: "'b list list" assumes "x \<in> set (butlast xs)"   and "xs \<in> set xss" shows "x \<in> set (butlast (concat xss))"
store_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {load_store_type STB, load_store_type STH, load_store_type ST, load_store_type STA, load_store_type STD}" shows "snd (store_instr i s) = False"
not_in_iff_zmset: fixes x :: "'a"   and M :: "'a zmultiset" shows "(x \<notin>#\<^sub>z M) = (zcount M x = 0)"
is_sig_GB_uptD2: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> nat"   and term_of_pair :: "'d \<times> nat \<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 fs :: "('d \<Rightarrow>\<^sub>0 'e) list"   and d :: "'d \<Rightarrow> nat"   and G :: "('c \<Rightarrow>\<^sub>0 'e) set"   and u :: "'c"   and v :: "'c" assumes "qpm_inf_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "qpm_inf_term.is_sig_GB_upt pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs d G u"   and "ord_term_strict v u" shows "qpm_inf_term.is_sig_GB_in pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict fs d G v"
lower_eq_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "(lower p v = lower q v) = (\<forall>u. u \<prec>\<^sub>t v \<longrightarrow> lookup p u = lookup q u)"
Box_def: fixes \<alpha> :: "game"   and P :: "fml" shows "([[\<alpha>]]P) = ! (\<langle> \<alpha> \<rangle> ! P)"
div_is_floor_divide_real: fixes n :: "int"   and y :: "int" shows "n div y = \<lfloor>real_of_int n / real_of_int y\<rfloor>"
is_mset_set_def: fixes X :: "'d multiset" shows "is_mset_set X \<equiv> \<forall>x\<in>#X. count X x = 1"
closed_Collect_neg: fixes "open" :: "'d set \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}" shows "topological_space.closed open {x. \<not> P x}"
preserves_reflects_arr: fixes f :: "'b" shows "B.arr (src\<^sub>B f) = B.arr f"
prj_char_2: shows "Maps.PRJ\<^sub>1 \<lbrakk>\<lbrakk>t\<^sub>0\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>u\<^sub>1\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>t\<^sub>0u\<^sub>1.p\<^sub>1\<rbrakk>\<rbrakk>"
cong_mult_lcancel: fixes k :: "'a"   and m :: "'a"   and a :: "'a"   and b :: "'a" assumes "coprime k m" shows "[k * a = k * b] (mod m) = [a = b] (mod m)"
bst_insert: fixes t :: "'a tree"   and x :: "'a" assumes "bst t" shows "bst (Tree_Set.insert x t)"
ok'_2: shows "term_ok \<Theta> (Abs \<tau> (g $ Bv 0))"
Dg1_is_bot_edge_pathimg: shows "path_image (\<lambda>x. twoC (x, 1)) = {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1 x)}"
is_constant_constructorsI_9: fixes x :: "'a ltln"   and y :: "'a ltln" assumes "is_constant x" shows "is_constant (mk_until y x)"
CCV_sat: fixes ls :: "lit set" assumes "consistent ls"   and "G `` ls \<subseteq> ls"   and "vars_of_cnf cnf \<subseteq> var ` ls" shows "sat_cnf (\<lambda>x. Pos x \<in> ls) cnf"
sets_point_measure: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" shows "sets (point_measure A f) = Pow A"
strict_boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "order.greater (inf b c) a" obtains "order.greater b a" and "order.greater c a"
fls_of_fpxs_of_fls: fixes f :: "'b fls" shows "fls_of_fpxs (fpxs_of_fls f) = f"
Int_atLeastAtMostL2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atLeast less_eq c = ord.atLeastAtMost less_eq (ord.max less_eq a c) b"
equiv_Eps_in: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and X :: "'b set" assumes "equiv A r"   and "X \<in> A // r" shows "(SOME x. x \<in> X) \<in> X"
not_Some_eq: fixes x :: "'d option" shows "(\<forall>y. x \<noteq> Some y) = (x = None)"
filterlim_tendsto_pos_mult_at_top: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "0 < c"   and "filterlim g at_top F" shows "LIM x F. f x * g x :> at_top"
set_insort_wrt: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "set (insort_wrt r x xs) = insert x (set xs)"
integral_Pair_const: fixes a :: "'b"   and c :: "'c"   and b :: "'b"   and d :: "'c"   and k :: "'a" shows "integral (cbox (a, c) (b, d)) (\<lambda>x. k) = integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. k))"
iso_tuple_UNIV_I: fixes x :: "'a" shows "x \<in> UNIV \<equiv> True"
scaleR_scaleR_nonzero: fixes b :: "real"   and a :: "real"   and x :: "'b nonzero" assumes "b \<noteq> 0" shows "a *\<^sub>R b *\<^sub>R x = (a * b) *\<^sub>R x"
HPT_Target_HAStates: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "Target TS \<subseteq> HAStates (HA ST)"
finsum_zero': fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = (0::'a)" shows "sum f A = (0::'a)"
signed_take_bit_int_less_eq: fixes n :: "nat"   and k :: "int" assumes "2 ^ n \<le> k" shows "signed_take_bit n k \<le> k - 2 ^ Suc n"
pp_of_lookup: fixes t :: "('b, 'c) pp" shows "pp_of_fun (lookup_pp t) = t"
sel_5: fixes x1 :: "('a, 'b) hd" shows "arg (Hd x1) = Hd x1"
payload_inductive_cases_7: fixes X :: "msg"   and Y :: "msg" assumes "Sign X Y \<in> payload" obtains "X \<in> payload" and "Y \<in> payload"
sel: fixes out :: "'d"   and c :: "('b, 'd, 'e) rpv" shows "the_gpv (Generative_Probabilistic_Value.Pause out c) = return_spmf (IO out c)"
resl_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<leftarrow> z \<le> y \<leftarrow> z"
Inf_empty: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<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" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Inf {} = top"
imageI: fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> A" shows "f x \<in> f ` A"
coset_mult_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "M #> g #> h = M #> g \<otimes> h"
fresh_psubst_updEnv: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho"   and "fresh xs x Y" shows "Y #[rho [x \<leftarrow> X]_xs] = Y #[rho]"
While: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c :: "com" assumes "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}" shows "\<turnstile>\<^sub>2\<^sub>' {\<lambda>s. I s + 1} WHILE b DO c {\<lambda>s. I s + \<up> (\<not> bval b s)}"
factorize_rat_poly_2: fixes alg :: "int_poly_factorization_algorithm"   and f :: "rat poly"   and c :: "rat"   and fs :: "(rat poly \<times> nat) list"   and fi :: "rat poly"   and i :: "nat" assumes "factorize_rat_poly_generic alg f = (c, fs)"   and "(fi, i) \<in> set fs" shows "irreducible fi"
tendsto_ident_at: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" assumes "class.topological_space open" shows "LIM x topological_space.at_within open a s. x :> topological_space.nhds open a"
m1_inv1r_cache_def: shows "m1_inv1r_cache \<equiv> {s. \<forall>Rb A B Kab Ts Ta nl. runs s Rb = Some (Resp, [A, B], aKey Kab # aNum Ts # aNum Ta # nl) \<longrightarrow> clk s < Ta + La \<longrightarrow> (B, Kab, Ta) \<in> cache s}"
mult_rel_well_defined_llbound: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b"   and z :: "'a set" assumes "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y" shows "\<forall>\<^sub>F w in llbound. \<lfloor>z, \<cdot>\<^bsub>z\<^esub> (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))\<rfloor> = \<lfloor>w, \<cdot>\<^bsub>w\<^esub> (\<rho> (fst x) w (snd x)) (\<rho> (fst y) w (snd y))\<rfloor>"
min_0_1_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq zero one = zero"
conf_widen: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and T :: "ty"   and T' :: "ty" assumes "P,h \<turnstile> v :\<le> T"   and "subtype P T T'" shows "P,h \<turnstile> v :\<le> T'"
select_convs_7: fixes cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "dwrite \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = dwrite"
F_in: fixes F :: "i \<Rightarrow> bool" assumes "\<exists>x. F x" shows "\<forall>y. F y \<longrightarrow> P y (\<sigma>x. F x)"
idem: fixes a :: "'g multiset" shows "a \<inter># a = a"
nonzero_mult_div_cancel_right: fixes b :: "'a"   and a :: "'a" assumes "b \<noteq> (0::'a)" shows "a * b / b = a"
mat_mult_row_code: fixes m :: "(('b, 'c) vec, 'd) vec"   and m' :: "(('b, 'a) vec, 'c) vec"   and f :: "'d" shows "($) (mat_mult_row m m' f) = (\<lambda>c. \<Sum>k\<in>UNIV. m $ f $ k * m' $ k $ c)"
sset_lists_fcs: fixes as :: "('b \<times> 'b) list" shows "sset (lists_fcs as) = {face_cycle_set a |a. a \<in> set as}"
some_gcd_ff_list_def: fixes xs :: "'b fract list" shows "some_gcd_ff_list xs = (SOME g. gcd_ff_list xs g)"
comp_mwb_lens: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'c \<Longrightarrow> 'd" assumes "mwb_lens x"   and "mwb_lens y" shows "mwb_lens (x ;\<^sub>L y)"
downward_completeI: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>C. \<lbrakk>chain_on P C A; C \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>x\<in>A. glb P C x" shows "dc_on P A"
lemScaleInverse: fixes k :: "'a"   and v :: "'a Vector"   and u :: "'a Vector" assumes "k \<noteq> (0::'a)"   and "v = k ** u" shows "u = inverse k ** v"
prime_power_cancel2: fixes p :: "'a"   and m :: "'a"   and k :: "nat"   and m' :: "'a"   and k' :: "nat" assumes "prime p"   and "m * p ^ k = m' * p ^ k'"   and "\<not> p dvd m"   and "\<not> p dvd m'" obtains "m = m'" and "k = k'"
order_eq_refl: fixes x :: "'c"   and y :: "'c" assumes "x = y" shows "x \<le> y"
g_def: shows "g \<equiv> \<lambda>x. f x - f' x0 x"
ipt_tcp_flags_Any_def: shows "ipt_tcp_flags_Any \<equiv> TCP_Flags {} {}"
assms_2: shows "q at a in (ps, a) # filter nonempty left @ right"
card_of_empty4: fixes A :: "'d set" shows "(|{}| <o |A|) = (A \<noteq> {})"
lprefix_llength_eq_imp_eq: fixes xs :: "'a llist"   and ys :: "'a llist" assumes "lprefix xs ys"   and "llength xs = llength ys" shows "xs = ys"
valid_reads_nth_update: fixes m :: "nat \<Rightarrow> nat"   and 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 \<O> :: "nat set"   and sb :: "'b memref list"   and p :: "'b"   and "is" :: "instr list"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<R> :: "nat \<Rightarrow> bool option" assumes "valid_reads m ts"   and "i < length ts"   and "reads_consistent False \<O> m sb" shows "valid_reads m (ts[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
norm_cmul_rule_thm: fixes x :: "'b"   and b :: "real"   and c :: "real" assumes "norm x \<le> b" shows "norm (c *\<^sub>R x) \<le> \<bar>c\<bar> * b"
bisimulation_def: fixes R :: "('a \<times> 'b) set"   and A :: "('a, 'c) graph_rec_scheme"   and B :: "('b, 'd) graph_rec_scheme" shows "bisimulation R A B \<equiv> (graph A \<and> graph B) \<and> simulation R A B \<and> simulation (R\<inverse>) B A"
SigTyId: fixes S :: "('a, 'b) Signature"   and A :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> Vx : A" shows "A \<in> Ty\<^bsub>S\<^esub>"
VPi_def: fixes A :: "V"   and B :: "V \<Rightarrow> V" shows "VPi A B \<equiv> set {f \<in> elts (VPow (VSigma A B)). elts A \<subseteq> Domain (pairs f) \<and> single_valued (pairs f)}"
surjective: fixes r :: "('d, 'e, 'f, 'g) FSM_scheme" shows "r = \<lparr>succ = succ r, inputs = inputs r, outputs = outputs r, initial = initial r, \<dots> = more r\<rparr>"
seqE: fixes g :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr"   and f :: "('a S.E.term, 'a) S.arr \<times> ('a S.E.term, 'a) S.arr" assumes "S.VV.seq g f" obtains "S.VV.arr f" and "S.VV.arr g" and "S.VV.dom g = S.VV.cod f"
wfp_stateOf_shift_stake_sconst: fixes \<pi> :: "('state \<times> 'aprop set) stream"   and i :: "nat" assumes "wfp AP' \<pi>" shows "wfp AP' (stake (Suc i) \<pi> @- sconst (toSink (fst (\<pi> !! i)), L (toSink (fst (\<pi> !! i))))) \<and> stateOf (stake (Suc i) \<pi> @- sconst (toSink (fst (\<pi> !! i)), L (toSink (fst (\<pi> !! i))))) = stateOf \<pi>"
bij_betw_trans: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and C :: "'f set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (g \<circ> f) A C"
f_def: fixes n :: "nat" shows "f n = a * complex_of_real (cos_coeff (n + 1) - sin_coeff (n + 2))"
reduction_signs_set_helper_lemma2: fixes B :: "rat list list"   and C :: "rat vec"   and A :: "rat list set" assumes "distinct B"   and "dim_vec C = length B"   and "A \<subseteq> set B"   and "\<forall>n<dim_vec C. C $ n \<noteq> 0 \<longrightarrow> B ! n \<in> A" shows "set (take_indices B (find_nonzeros_from_input_vec C)) \<subseteq> A"
assms_1: shows "M \<equiv> mult_row A i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)))"
r_and_def: shows "r_and \<equiv> Cn 2 r_not [r_nand]"
f_nn_integral_harm_1: shows "ennreal (- harm n / ln q) \<le> \<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV"
region_cover: fixes v :: "'c \<Rightarrow> real" assumes "\<forall>x\<in>X. 0 \<le> v x" shows "\<exists>R. R \<in> \<R> \<and> v \<in> R"
resPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and x :: "name" assumes "P \<leadsto><Rel> Q"   and "\<And>R S y. (R, S) \<in> Rel \<Longrightarrow> (\<lparr>\<nu>y\<rparr>R, \<lparr>\<nu>y\<rparr>S) \<in> Rel'" shows "\<lparr>\<nu>x\<rparr>P \<leadsto><Rel'> \<lparr>\<nu>x\<rparr>Q"
listsp_simps_1: fixes A :: "'a \<Rightarrow> bool" shows "listsp A [] = True"
satA_gA: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and at :: "('fsym, 'psym) atm" shows "GE.satA \<xi> (gA at) = Ik.satA \<xi> at"