hcju/isabelleps huggingface dataset:
phis'_codem_correct: fixes g :: "'g"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping"   and "next" :: "'node \<times> 'val" assumes "g \<turnstile> nodes_of_phis \<approx>\<^sub>\<phi> uninst_code.ssa.phiNodes_of phis g"   and "next \<in> Mapping.keys (phis g)" shows "phis'_codem g next (uninst_code.substitution_code phis g next) nodes_of_phis = phis'_code g next"
D_eq_2: fixes r :: "('b \<times> 'b) set"   and \<tau> :: "'b list"   and \<sigma> :: "'b list"   and \<sigma>' :: "'b list"   and \<tau>' :: "'b list" assumes "trans r"   and "irrefl r"   and "D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<sigma>'| -s r \<down>l \<tau>, r|\<sigma>|) \<in> mul_eq r"
extremum_uniqueI: fixes a :: "'c multiset" assumes "a \<subseteq># {#}" shows "a = {#}"
assms_2: shows "netgmap sr s2 = netmask (net_tree_ips n2) (\<sigma>, snd (netgmap sr s2))"
finsum_closed: fixes f :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "f \<in> A \<rightarrow> carrier M" shows "finsum M f A \<in> carrier M"
ProblemIkTpartG_axioms: shows "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl"
cond_pmf_impl_code_alt: fixes A :: "'b set"   and p :: "'b pmf" assumes "finite A" shows "cond_pmf_impl p A = (let C = A \<inter> set_pmf p; prob = sum (pmf p) C in if prob = 0 then None else Some (Mapping.map_values (\<lambda>_ y. y / prob) (Mapping.filter (\<lambda>k _. k \<in> C) (mapping_of_pmf p))))"
distr_bind: fixes N :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and K :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and R :: "'e measure" assumes "N \<in> M \<rightarrow>\<^sub>M subprob_algebra K"   and "space M \<noteq> {}"   and "f \<in> K \<rightarrow>\<^sub>M R" shows "distr (M \<bind> N) R f = M \<bind> (\<lambda>x. distr (N x) R f)"
exists_gen: fixes I :: "'a list set" assumes "ideal I (K [X])" shows "\<exists>a\<in>carrier (K [X]). I = PIdl\<^bsub>K [X]\<^esub> a"
prems_5: shows "pre_strong_term_class.wellformed (name $$ ts)"
inversions_altdef: fixes xs :: "'a list" shows "inversions xs = {(i, j). i < j \<and> j < length xs \<and> xs ! j < xs ! i}"
confs_def: fixes P :: "'b prog" shows "confs P = h.confs P undefined"
h: shows "set (get_signs_R (reduce_system_R p (qs, M_mat_R [[1], [0], [- 1]] [([], []), ([0], []), ([], [0])], [([], []), ([0], []), ([], [0])], [[1], [0], [- 1]]))) = set (characterize_consistent_signs_at_roots p qs)"
empty_offending_contra: fixes F :: "('a \<times> 'a) set"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" assumes "F \<in> set_offending_flows G nP"   and "F = {}" shows "False"
g: shows "g \<in> N \<rightarrow>\<^sub>M P"
gj_3: shows "gauss_jordan (mat_of_rows n G)\<^sup>T (1\<^sub>m n) = (B, C)"
Not: fixes F :: "'effect fs_set"   and x :: "('a, 'pred, 'act, 'effect) formula" assumes "x \<in> \<A>[F]" shows "FL_Formula.Not x \<in> \<A>[F]"
block_complement_def: fixes b :: "'a set" shows "b\<^sup>c \<equiv> \<V> - b"
module_vec_simps_3: fixes ty :: "'a itself"   and n :: "nat" shows "carrier (module_vec ty n) = carrier_vec n"
WT_sim: fixes \<eta> :: "nat" shows "\<I>_real \<eta>, \<I>_middle \<eta> \<turnstile>\<^sub>C sim_outer \<eta> \<surd>"
union_iff: fixes a :: "'d"   and A :: "'d multiset"   and B :: "'d multiset" shows "(a \<in># A + B) = (a \<in># A \<or> a \<in># B)"
hyps_2: shows "\<Gamma> \<turnstile> a2 : t"
ideal_nsum_closed: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and n :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "Ring R"   and "ideal R I"   and "\<forall>j\<le>n. f j \<in> I" shows "\<Sigma>\<^sub>e R f n \<in> I"
dg_prod_is_arrD_5: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "vsv b"
IH: fixes ts :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option" assumes "redT_updTs ts TAS t = \<lfloor>(x, w)\<rfloor>"   and "thread_oks ts TAS"   and "ts t = None" shows "\<exists>m. NewThread t x m \<in> set TAS \<and> w = no_wait_locks"
lenext_snoc: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b" shows "lenext gts (xs @ [x]) xs"
v_trans_def: shows "v_trans = (\<Union>r v_f. \<Union> (range (v_round r v_f)))\<^sup>="
LEAST_FP_def: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" shows "LFP\<^bsub>L\<^esub> f = Lattice.inf L {u. u \<in> carrier L \<and> le L (f u) u}"
abs_eq: fixes xa :: "int"   and x :: "int" shows "Word.Word xa smod Word.Word x = Word.Word (signed_take_bit (LENGTH('b) - Suc 0) xa smod signed_take_bit (LENGTH('b) - Suc 0) x)"
conv_involutive: fixes x :: "'a" shows "id (id x) = x"
pcr_moebius_mat_def: shows "pcr_moebius_mat \<equiv> rel_prod (=) (rel_prod (=) (rel_prod (=) (=))) OO cr_moebius_mat"
isomorphicE: fixes a :: "'d"   and a' :: "'d" assumes "D.isomorphic a a'" obtains f :: "'d" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f"
cdeclaredmethd_def: fixes G :: "prog"   and C :: "qtname" shows "cdeclaredmethd G C = (case class G C of None \<Rightarrow> Map.empty | Some c \<Rightarrow> table_of (methods c))"
canceled_def: fixes l :: "(bool \<times> 'e) list" shows "canceled l = (\<not> Domainp cancels_to_1 l)"
b2: shows "real_of_int (sint NEG_INF) < real_of_int (sint ra)"
SLC_top_relaxed: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "(s\<^sub>1, s\<^sub>2) \<in> [\<leadsto>]\<^sup>="   and "s\<^sub>1 \<leadsto> s\<^sub>2'"   and "reachable s\<^sub>1"   and "infinite UNIV"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (s\<^sub>2, s\<^sub>3) \<in> [\<leadsto>]\<^sup>= \<and> (s\<^sub>2', s\<^sub>3') \<in> [\<leadsto>]\<^sup>="
check_CDMode_CTRL: shows "(CDMode_CTRL_States, CDMode_CTRL_Init, CDMode_CTRL_Labels, CDMode_CTRL_Delta) \<in> seqauto"
wf_RP_filtered_relation: shows "wf (natLess <*lex*> natLess <*lex*> natLess)"
map_def: fixes f :: "'a" shows "C.map f = (if C.arr f then f else C.null)"
subset_empt_def: shows "subset_empt = (Leaf None, [])"
lhseq: shows "(\<exists>B. A ** B = mat (1::'a)) = (\<forall>y. \<exists>x. (\<Sum>i\<in>UNIV. x $ i *s column i A) = y)"
ymax_1: shows "y \<in> {y. x \<noteq> y \<and> y pd\<rightarrow> x}"
star_inductr: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
eta: fixes A :: "'a trm"   and A' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A >> A'" shows "Fn x T A >> Fn x T A'"
insert_3: shows "card {s. fmdom' s = F} = 2 ^ card F"
sum_emeasure: fixes F :: "'c \<Rightarrow> 'b set"   and I :: "'c set"   and M :: "'b measure" assumes "F ` I \<subseteq> sets M"   and "disjoint_family_on F I"   and "finite I" shows "(\<Sum>i\<in>I. emeasure M (F i)) = emeasure M (\<Union> (F ` I))"
cring_simprules_4: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<ominus>\<^bsub>R\<^esub> y \<in> carrier R"
Diff_Int: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "A - B \<inter> C = A - B \<union> (A - C)"
msteps0_msteps0_stateless: fixes w :: "((char list \<times> 'a list) set \<times> nat) list"   and st :: "'a mstate" shows "fst (msteps0 w st) = msteps0_stateless w st"
mc_keys: shows "Mapping.keys mc \<subseteq> Mapping.keys m \<union> Mapping.keys m'"
count_image_mset_lt_imp_lt: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset"   and b :: "'b"   and N :: "'c multiset" assumes "count (image_mset f M) b < count (image_mset f N) b" shows "\<exists>x. f x = b \<and> count M x < count N x"
birkhoff_sum_cmult: fixes c :: "real"   and f :: "'a \<Rightarrow> real"   and n :: "nat"   and x :: "'a" shows "birkhoff_sum (\<lambda>x. c * f x) n x = c * birkhoff_sum f n x"
distinct_147: fixes x71 :: "char list"   and x72 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "x71:=x72 \<noteq> x121;; x122"
get_scdom_component_def: shows "Core_DOM_SC_DOM_Components.get_scdom_component \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component Core_DOM_Functions.get_owner_document get_disconnected_nodes Core_DOM_Functions.to_tree_order"
matrix_to_iarray_snd_Gauss_Jordan_det_P: fixes A :: "(('b, 'c) vec, 'd) vec" shows "matrix_to_iarray (snd (Gauss_Jordan_det_P A)) = snd (Gauss_Jordan_det_P_iarrays (matrix_to_iarray A))"
Inr_Inl_False: fixes x :: "'g"   and y :: "'f" shows "(Inr x = Inl y) = False"
BIT_step_def: fixes s :: "'a list \<times> bool list \<times> 'a list"   and q :: "'a" shows "BIT_step s q = (let a = (if fst (snd s) ! index (snd (snd s)) q then 0 else length (fst s), []) in return_pmf (a, flip (index (snd (snd s)) q) (fst (snd s)), snd (snd s)))"
fbox_add1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows " |x] (d y \<cdot> d z) =  |x] y \<cdot>  |x] z"
poly_prod_root_unity: fixes ns :: "nat list"   and x :: "'a" shows "(poly (prod_root_unity ns) x = (0::'a)) = (\<exists>k\<in>set ns. x ^ k = (1::'a))"
l_get_root_node_axioms: shows "l_get_root_node Shadow_DOM.get_root_node Shadow_DOM.get_parent"
env_restr_belowD: fixes S :: "'a set"   and m\<^sub>1 :: "'a \<Rightarrow> 'b"   and m\<^sub>2 :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "m\<^sub>1 f|` S \<sqsubseteq> m\<^sub>2 f|` S"   and "x \<in> S" shows "m\<^sub>1 x \<sqsubseteq> m\<^sub>2 x"
card_of_UNIV: fixes A :: "'b set" shows "|A| \<le>o |UNIV|"
discounted_cong': fixes X :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and n :: "nat"   and w :: "'c"   and Y :: "nat \<Rightarrow> 'c \<Rightarrow> real"   and r :: "real" assumes "X n w = Y n w" shows "discounted_value r X n w = discounted_value r Y n w"
simps_1: fixes a :: "nat"   and b :: "nat" shows "swap_fm a b TrueF = TrueF"
lnth_subset_Sup_upto_llist: fixes j :: "nat"   and Xs :: "'a set llist" assumes "enat j < llength Xs" shows "lnth Xs j \<subseteq> Sup_upto_llist Xs (enat j)"
image_mset_mset_set: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "image_mset f (mset_set A) = mset_set (f ` A)"
trans_is_node: fixes q :: "'a"   and a :: "'b" assumes "is_node q"   and "is_alpha a" shows "is_node (trans q a)"
\<tau>_ide: fixes b :: "'c" assumes "Cop.ide b" shows "\<T> e b = S.mkArr (Hom.set (b, a)) (Ya'.SET b) (\<lambda>x. Ya'.FUN (\<psi> (b, a) x) (\<tau>.FUN a (\<phi> (a, a) a)))"
mconn_trans: fixes E :: "('b \<times> 'b) set" shows "mconn E O mconn E = mconn E"
p_fix1_def: fixes x1 :: "real" shows "p_fix1 x1 = replace_coeff (insertion (\<lambda>_. x1)) (extract_var p n)"
zero_V_def: shows "[]\<^sub>\<circ> \<equiv> ZFC_in_HOL.set {}"
assms_3: shows "P \<turnstile> C sees clinit, b :  Ts\<rightarrow>T = m in D"
aux5a: fixes b :: "'a"   and a :: "'a"   and p :: "('a, 'b) Combinators list" assumes "DenyAllFromTo b a \<in> set p" shows "a \<in> set (net_list p)"
mbind'_concat: fixes S :: "'g list"   and T :: "'g list"   and f :: "'g \<Rightarrow> 'd \<Rightarrow> ('f \<times> 'd) option"   and P :: "'f list \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p (S @ T) f) P = _bind_SE os (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p S f) (_bind_SE os' (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p T f) (P (os @ os')))"
of_real_eq_0_iff: fixes x :: "real" shows "(of_real x = (0::'d)) = (x = 0)"
ta_prod_correct_aux2: fixes TA1 :: "('a, 'b) tree_automaton_rec"   and TA2 :: "('c, 'b) tree_automaton_rec" assumes "tree_automaton TA1"   and "tree_automaton TA2" shows "tree_automaton (ta_prod TA1 TA2)"
le_supE: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "class.semilattice_sup sup less_eq less"   and "less_eq (sup a b) x" obtains "less_eq a x" and "less_eq b x"
eq_\<I>_gpvD_WT1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv" assumes "eq_\<I>_gpv A \<I> gpv gpv'"   and "\<I> \<turnstile>g gpv \<surd>" shows "\<I> \<turnstile>g gpv' \<surd>"
assms: shows "prefix_of \<pi>' (map_\<Gamma> f \<sigma>)"
elims: fixes x :: "('b, 'c, 'a) interp"   and xa :: "('b, 'a) ODE"   and xb :: "(real, 'a) vec"   and y :: "(real, 'a) vec \<times> (real, 'a) vec" assumes "mk_xode x xa xb = y" obtains I :: "('b, 'c, 'a) interp" and ODE :: "('b, 'a) ODE" and sol :: "(real, 'a) vec" where "x = I" and "xa = ODE" and "xb = sol" and "y = (sol, ODE_sem I ODE sol)"
no_leading_def: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "no_leading P xs = (xs \<noteq> [] \<longrightarrow> \<not> P (hd xs))"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat" shows "fw m n (Suc k) 0 0 = fw_upd (fw m n k n n) (Suc k) 0 0"
lm099: fixes x :: "'a"   and X :: "'a set"   and Y :: "'a set" assumes "x \<in> X" shows "1 \<in> (X <|| Y) `` {x}"
sum_list_map_cong: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "class.monoid_add plus zero"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "monoid_add.sum_list plus zero (map f xs) = monoid_add.sum_list plus zero (map g xs)"
summable_erf: fixes z :: "'a" shows "summable (\<lambda>n. of_real (erf_coeffs n) * z ^ n)"
GF_advice_restricted_subformulas_inner: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "restricted_subformulas_inner (\<phi>[X]\<^sub>\<nu>) = {}"
sel: fixes x2 :: "'a" shows "the (\<lfloor>x2\<rfloor>) = x2"
stake_sdrop: fixes n :: "nat"   and s :: "'a stream" shows "stake n s @- sdrop n s = s"
id2: shows "subresultant (n (i - 1) - 1) (F (i - 2)) (F (i - 1)) = Polynomial.smult (inverse (\<alpha> i) * ((- 1) ^ (\<delta> (i - 2) + 1) * (f (i - 1) ^ (\<delta> (i - 2) + 1) * \<beta> i))) (F i)"
trancl_mono_set: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s" shows "r\<^sup>+ \<subseteq> s\<^sup>+"
p2: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All j' (SyntaxN.All k' (SeqSubstFormP v a x (Var j) s (Var k) IMP SeqSubstFormP v a x (Var j') s' (Var k') IMP Var j' EQ Var j)))(j::=y)"
t'_zcount: shows "0 < zcount (c_imp c loc1) t'"
HEndPhase0_dblock_allBlocksRead: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase0 s s' p"   and "Inv1 s" shows "dblock s' p \<in> mbal_update (\<lambda>_. mbal (dblock s' p)) ` allBlocksRead s p"
hyps: shows "\<forall>P Q R. P \<longmapsto>Cal\<^bsup>n1\<^esup> Q \<and> Q \<longmapsto>Cal\<^bsup>n\<^esup> R \<longrightarrow> P \<longmapsto>Cal\<^bsup>(n1 + n)\<^esup> R"
holomorphic_on_Polygamma: fixes A :: "complex set"   and n :: "nat" assumes "A \<inter> \<int>\<^sub>\<le>\<^sub>0 = {}" shows "Polygamma n holomorphic_on A"
add_0: fixes a :: "'g" shows "(0::'g) + a = a"
len_lexext_rev_map_strong: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt (f y) (f x)"   and "len_lexext_rev gt ys xs" shows "len_lexext_rev gt (map f ys) (map f xs)"
Ints_vec_rows_Ints_mat: fixes A :: "'b mat" shows "(set (rows A) \<subseteq> \<int>\<^sub>v) = (A \<in> \<int>\<^sub>m)"
injectivity_image_union: fixes F :: "'a set \<Rightarrow> 'a set"   and P :: "'a set set"   and G :: "'a set \<Rightarrow> 'a set"   and P' :: "'a set set"   and A :: "'a set" assumes "(\<lambda>X. X \<union> F X) ` P = (\<lambda>X. X \<union> G X) ` P'"   and "\<forall>X\<in>P. X \<subseteq> A"   and "\<forall>X\<in>P'. X \<subseteq> A"   and "\<forall>X\<in>P. \<forall>y\<in>F X. y \<notin> A"   and "\<forall>X\<in>P'. \<forall>y\<in>G X. y \<notin> A" shows "P = P'"
assms_2: assumes "\<And>u v. u \<in> verts G \<and> v \<in> verts G \<Longrightarrow> u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "P"
fresh_def: fixes a :: "'d"   and x :: "'e" shows "a \<sharp> x = (a \<notin> supp x)"
CSP_is_R: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "is_CSP_process P" shows "P is R healthy"
convex_convex_hull: fixes s :: "'c set" shows "convex (convex hull s)"
cos_eq_1: fixes z :: "complex" shows "(cos z = 1) = (\<exists>x. z = complex_of_real (2 * real_of_int x * pi))"
nres_order_simps_21: fixes x :: "'b"   and Y :: "'b set" shows "(RETURN x \<le> RES Y) = (x \<in> Y)"
PiF_def: fixes I :: "'c set set"   and M :: "'c \<Rightarrow> 'd measure" shows "Pi\<^sub>F I M \<equiv> sigma (\<Union>J\<in>I. \<Pi>' j\<in>J. space (M j)) {Pi' J X |X J. J \<in> I \<and> X \<in> (\<Pi> j\<in>J. sets (M j))}"
truncate_down_rat_precision: fixes prec :: "nat"   and x :: "nat"   and y :: "nat" shows "truncate_down prec (real x / real y) = round_down (rat_precision prec (int x) (int y)) (real x / real y)"
get_get: fixes f :: "'s \<Rightarrow> 's \<Rightarrow> 'm" shows "get (\<lambda>s. get (f s)) = get (\<lambda>s. f s s)"
flush_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 m' :: "nat \<Rightarrow> nat"   and sb' :: "'a memref list"   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 "(m, sb, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>f (m', sb', \<O>', \<R>', \<S>')"   and "valid ts" shows "valid (ts[i := (p, is, \<theta>, sb', \<D>, \<O>', \<R>')])"
assms: assumes "\<Phi>" shows "hn_refine \<Gamma> c \<Gamma>' R c'"
col_def: fixes m :: "'a list list"   and i :: "nat" shows "Matrix_Legacy.col m i \<equiv> m ! i"
rprI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "complete A r"   and "refl_on A r"   and "trans r" shows "rpr A r"
Buchberger_criterion_finite: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "finite F"   and "\<And>p q. \<lbrakk>p \<in> F; q \<in> F; p \<noteq> 0; q \<noteq> 0; p \<noteq> q; component_of_term (lt p) = component_of_term (lt q)\<rbrakk> \<Longrightarrow> (red F)\<^sup>*\<^sup>* (spoly p q) 0" shows "is_Groebner_basis F"
GF_advice_a2: fixes X :: "'b ltln set"   and \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" assumes "X \<subseteq> \<G>\<F> \<phi> w"   and "w \<Turnstile>\<^sub>n \<phi>[X]\<^sub>\<Pi>\<^sub>1" shows "w \<Turnstile>\<^sub>n \<phi>"
sub: shows "subresultant_prs div_exp G1 G2 = (Gk, hk)"
phi01: shows "((*) (1 / pi) \<circ> (\<lambda>x. if - 1 \<le> x \<and> x \<le> 1 then arccos x else if x < - 1 then - x + pi else 1 - x) \<circ> (\<lambda>t. 2 * x_coord (1 - t))) -` {0..1} \<subseteq> {0..1}"
integral_mono': fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "\<And>x. x \<in> space M \<Longrightarrow> g x \<le> f x"   and "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "integral\<^sup>L M g \<le> integral\<^sup>L M f"
inv_t1: shows "tree_invar (Node e1 a1 r ts1)"
independent_insertI: fixes a :: "'b"   and S :: "'b set" assumes "a \<notin> vs1.span S"   and "vs1.independent S" shows "vs1.independent (insert a S)"
assms_3: shows "partial_saturation S A (S \<union> R)"
le_INF_iff: fixes u :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(u \<le> \<Sqinter>(f ` A)) = (\<forall>i\<in>A. u \<le> f i)"
pseudo_proj_True_Suc_proj: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_True (Suc n) (pseudo_proj_True n w) = pseudo_proj_True n w"
d_IN_def: fixes g :: "'k \<times> 'k \<Rightarrow> ennreal"   and y :: "'k" shows "d_IN g y = (\<Sum>\<^sup>+ x. g (x, y))"
orthogonal_transformation_surj: fixes f :: "'a \<Rightarrow> 'a" assumes "orthogonal_transformation f" shows "surj f"
lookup_map_val: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and k :: "'c"   and xs :: "('c, 'd) oalist_ntm" assumes "f k (0::'d) = (0::'b)" shows "OAlist_lookup_ntm (OAlist_map_val_ntm f xs) k = f k (OAlist_lookup_ntm xs k)"
ProofStepInduct_NonTrivial_P_k_pre_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat"   and P_k_pre :: "'c list"   and y :: "'c"   and P_k_post :: "'c list" shows "ProofStepInduct_NonTrivial_P_k_pre G v0 v1 paths P_new sep_size P_k_pre y P_k_post \<equiv> ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size \<and> ProofStepInduct_NonTrivial_P_k_pre_axioms G v0 v1 paths P_new sep_size P_k_pre y P_k_post"
infty_inf_unbox_1: fixes a :: "'a" shows "Num a \<noteq> top"
inv_char: fixes f :: "'a" assumes "S.iso f" shows "S.inv f = local.inv f"
to_fun_monom': fixes c :: "'a"   and x :: "'a"   and n :: "nat" assumes "c \<in> carrier R"   and "c \<noteq> \<zero>"   and "x \<in> carrier R" shows "to_fun (c \<odot>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> n) x = c \<otimes> x [^] n"
subst_domI: fixes \<sigma> :: "'d \<Rightarrow> ('c, 'd) Term.term"   and v :: "'d" assumes "\<sigma> v \<noteq> Var v" shows "v \<in> subst_domain \<sigma>"
T_on_rand'_append: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and t :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> nat"   and A :: "('a \<Rightarrow> 'd pmf) \<times> ('a \<times> 'd \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) pmf)"   and s :: "('a \<times> 'd) pmf"   and xs :: "'b list"   and ys :: "'b list" shows "On_Off.T_on_rand' step t A s (xs @ ys) = On_Off.T_on_rand' step t A s xs + On_Off.T_on_rand' step t A (On_Off.config'_rand step A s xs) ys"
simps_3: fixes x :: "name"   and t :: "nterm"   and s' :: "name"   and t' :: "nterm" shows "subst_single (\<Lambda>\<^sub>n x. t) s' t' = (\<Lambda>\<^sub>n x. (if x = s' then t else subst_single t s' t'))"
CNV_eqD: fixes x :: "'e"   and y :: "'e" assumes "CNV x y" shows "x = y"
nths_dimsAw: shows "nths (dims Aw) (Collect even) = nths (dims Aw) {n. odd n}"
frameEqFresh: fixes x :: "name"   and F :: "'d frame"   and y :: "name"   and G :: "'d frame" assumes "\<lparr>\<nu>x\<rparr>F = \<lparr>\<nu>y\<rparr>G"   and "x \<sharp> F" shows "y \<sharp> G"
lockstep_backward_simulation_def: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and match :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "lockstep_backward_simulation step1 step2 match \<equiv> \<forall>s1 s2 s2'. match s1 s2 \<longrightarrow> step2 s2 s2' \<longrightarrow> (\<exists>s1'. step1 s1 s1' \<and> match s1' s2')"
comm_ring_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "comm_ring_ow U plus zero minus uminus times \<equiv> comm_semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
simps_1: fixes p :: "'b \<Rightarrow> bool" shows "\<guillemotleft>p\<guillemotright> [] = False"
type_less_eq_x_Set: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "\<tau> \<le> Set \<sigma>" obtains \<rho> :: "'a OCL_Types.type" where "\<tau> = Set \<rho>" and "\<rho> \<le> \<sigma>"
hyps: assumes "is_tangle_diagram xs" shows "is_tangle_diagram (xs \<otimes> basic (make_vert_block n))"