hcju/isabelleps huggingface dataset:
IH: assumes "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"   and "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"   and "2 \<le> aff_dim S"   and "openin (top_of_set (affine hull S)) S"   and "S \<subseteq> T"   and "T \<subseteq> affine hull S"   and "connected S" shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i\<in>K. f (x i) = y i) \<and> {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
exivl: shows "K \<subseteq> existence_ivl t0 x0"
VVar: shows "\<forall>xs. |V xs| <o |UNIV| \<or> finite (V xs)"
exhaust: fixes y :: "('f, 'g, 'h) new_thread_action" obtains x11 :: "'f" and x12 :: "'g" and x13 :: "'h" where "y = NewThread x11 x12 x13"   | x21 :: "'f" and x22 :: "bool" where "y = ThreadExists x21 x22"
vrestr_image_1: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a"   and A :: "'a set" assumes "x \<in> f --` A" shows "x \<in> space M"
simps: fixes a1 :: "lam"   and a2 :: "lam" shows " a1 \<longrightarrow>\<^sub>\<beta>\<^sup>* a2 = ((\<exists>M. a1 = M \<and> a2 = M) \<or> (\<exists>M1 M2 M3. a1 = M1 \<and> a2 = M3 \<and>  M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2 \<and>  M2 \<longrightarrow>\<^sub>\<beta> M3))"
ln_smallo_imp_flat': fixes f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and q :: "real"   and p :: "real" assumes "filterlim f at_top at_top"   and "filterlim g at_top at_top"   and "(\<lambda>x. ln (f x)) \<in> o(\<lambda>x. ln (g x))"   and "q < 0" shows "(\<lambda>x. g x powr q) \<in> o(\<lambda>x. f x powr p)"
intros_2: fixes s :: "dB"   and t :: "dB" assumes "s => t" shows "Abs s => Abs t"
finite_borel_measure_axioms: shows "finite_borel_measure M1"
forget_subst_fmla: fixes a :: "name"   and A :: "fmla"   and x :: "trm" assumes "atom a \<sharp> A" shows "A(a::=x) = A"
I_2: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"
compE\<^sub>1_pres_wt: fixes P :: "(char list list \<times> char list exp) prog"   and Vs :: "char list list"   and Ts :: "ty list"   and e :: "char list exp"   and U :: "ty"   and f :: "staticb \<Rightarrow> char list list \<times> char list exp \<Rightarrow> nat exp" assumes "P,[Vs [\<mapsto>] Ts] \<turnstile> e :: U"   and "length Ts = length Vs" shows "compP f P,Ts \<turnstile>\<^sub>1 compE\<^sub>1 Vs e :: U"
onll_step_to_invariantI: fixes A :: "('a \<times> ('b, 'c, 'd, 'e) seqp, 'f) automaton"   and I :: "'f \<Rightarrow> bool"   and \<Gamma> :: "'d \<Rightarrow> ('b, 'c, 'd, 'e) seqp"   and Q :: "('a \<times> 'e) \<times> 'f \<times> 'a \<times> 'e \<Rightarrow> bool"   and P :: "'a \<times> 'e \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (I \<rightarrow>) onll \<Gamma> Q"   and "wellformed \<Gamma>"   and "\<And>\<xi> l p. \<lbrakk>(\<xi>, p) \<in> init A; l \<in> labels \<Gamma> p\<rbrakk> \<Longrightarrow> P (\<xi>, l)"   and "\<And>\<xi> p l \<xi>' l' a. \<lbrakk>(\<xi>, p) \<in> reachable A I; l \<in> labels \<Gamma> p; P (\<xi>, l); Q ((\<xi>, l), a, \<xi>', l'); I a\<rbrakk> \<Longrightarrow> P (\<xi>', l')" shows "A \<TTurnstile> (I \<rightarrow>) onl \<Gamma> P"
convex_powr': fixes p :: "real" assumes "0 < p"   and "p \<le> 1" shows "convex_on {0..} (\<lambda>x. - (x powr p))"
map_nth_eq_take: fixes n :: "nat"   and xs :: "'c list" assumes "n < length xs" shows "map ((!) xs) [0..<n] = take n xs"
real_Gromov_completion_rel_PInf: fixes u :: "nat \<Rightarrow> real"   and v :: "nat \<Rightarrow> real" assumes "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity>"   and "(\<lambda>x. ereal (v x)) \<longlonglongrightarrow> \<infinity>" shows "Gromov_completion_rel u v"
remove_def: fixes x :: "'a"   and A :: "'a set" shows "Set.remove x A = A - {x}"
polyfun_matrix_deep_model: fixes rs :: "nat list"   and y :: "nat"   and i :: "nat"   and j :: "nat"   and shared_weights :: "bool" assumes "deep_model_correct_params_y rs y"   and "i < last rs ^ deep_model_correct_params.N_half rs"   and "j < last rs ^ deep_model_correct_params.N_half rs" shows "polyfun {..<deep_model_correct_params.weight_space_dim shared_weights rs} (\<lambda>f. deep_model_correct_params_y.A' shared_weights rs y f $$ (i, j))"
Carmichael_number_imp_squarefree_alt_weak: fixes n :: "nat" assumes "Carmichael_number n" shows "\<exists>p l. n = p * l \<and> prime p \<and> \<not> p dvd l"
card_of_mono1: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "|A| \<le>o |B|"
lappend_llist_of_llist_of: fixes xs :: "'c list"   and ys :: "'c list" shows "lappend (llist_of xs) (llist_of ys) = llist_of (xs @ ys)"
Re_i_times: fixes z :: "complex" shows "Re (\<i> * z) = - Im z"
le_utransI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> t P \<tturnstile> u P" shows "le_utrans t u"
exec_lub_def: fixes r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" shows "exec_lub r f x y = while (\<lambda>z. (x, z) \<notin> r\<^sup>*) f y"
same_on_vars_clause: fixes S :: "('b, 'a) Term.term literal multiset"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "\<forall>v\<in>vars_clause S. \<sigma> v = \<tau> v" shows "subst_cls S \<sigma> = subst_cls S \<tau>"
assms_1: shows "x \<Turnstile>\<^sub>c\<^sub>s set (from_index_geq0_vector i v)"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form multiset" shows "antec ( ant \<Rightarrow>* suc) = ant"
inline1'_def: shows "local.inline1' \<equiv> curry (spmf.fixp_fun (\<lambda>inline1' (gpv, s). the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1' (rpv x) y) (\<lambda>out' rpv'. return_spmf (Inr (out, out', rpv', rpv))))))"
dfs_invar_add_def: shows "dfs_invar_add \<equiv> {(\<Sigma>, W). distinct W}"
pairwise_insert: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and s :: "'a set" shows "pairwise r (insert x s) = ((\<forall>y. y \<in> s \<and> y \<noteq> x \<longrightarrow> r x y \<and> r y x) \<and> pairwise r s)"
simps_2: fixes \<delta> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'd"   and i :: "nat" shows "run \<delta> q\<^sub>0 w (Suc i) = \<delta> (run \<delta> q\<^sub>0 w i) (w i)"
simps_4: fixes v :: "'d strand_label"   and vb :: "('e, 'f) Term.term"   and vc :: "('e, 'f) Term.term" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p (v, insert\<langle>vb,vc\<rangle>) = (v, insert\<langle>vb,vc\<rangle>)"
one_red: fixes a :: "name"   and t1 :: "lam"   and t2 :: "lam"   and M :: "lam" assumes " App (Lam [a].t1) t2 \<longrightarrow>\<^sub>1 M"   and "a \<sharp> (t2, M)" shows "(\<exists>s1 s2. M = App (Lam [a].s1) s2 \<and>  t1 \<longrightarrow>\<^sub>1 s1 \<and>  t2 \<longrightarrow>\<^sub>1 s2) \<or> (\<exists>s1 s2. M = s1[a::=s2] \<and>  t1 \<longrightarrow>\<^sub>1 s1 \<and>  t2 \<longrightarrow>\<^sub>1 s2)"
reflects_iso: fixes f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : a' \<rightarrow> a\<guillemotright>"   and "local.iso (cod \<iota> \<otimes> f)" shows "local.iso f"
cong_iff_dvd_diff: fixes a :: "'a"   and b :: "'a"   and m :: "'a" shows "[a = b] (mod m) = (m dvd a - b)"
simps_1: fixes next_round :: "nat"   and votes :: "nat \<Rightarrow> process \<Rightarrow> val option"   and decisions :: "process \<Rightarrow> val option"   and more :: "'a" shows "next_round \<lparr>next_round = next_round, votes = votes, decisions = decisions, \<dots> = more\<rparr> = next_round"
squarefree_part_def: fixes n :: "'b" shows "squarefree_part n = (if n = (0::'b) then 1::'b else n div (square_part n)\<^sup>2)"
d_OUT_Sup: fixes Y :: "('a \<times> 'a \<Rightarrow> ennreal) set"   and x :: "'a" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow (\<Squnion> Y))" shows "d_OUT (\<Squnion> Y) x = (\<Squnion>f\<in>Y. d_OUT f x)"
projs_6: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>7 (dsn, dsk, flag, hops, nhip, pre) = pre"
q2: shows "\<forall>x. (\<forall>i<n. x i \<le> p) \<longrightarrow> (\<forall>i<n. x i = 0 \<longrightarrow> (label (\<Sum>i\<in>Basis. (real (x (b' i)) / real p) *\<^sub>R i) \<circ> b) i = 0)"
fdia_add2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<langle>x + y| z = \<langle>x| z + \<langle>y| z"
size_int_def: fixes n :: "int" shows "size n = nat \<bar>n\<bar>"
runit_eqI: fixes f :: "'a"   and \<mu> :: "'a" assumes "B.ide f"   and "\<guillemotleft>\<mu> : f \<star>\<^sub>B src\<^sub>B f \<rightarrow>\<^sub>B f\<guillemotright>"   and "\<mu> \<star>\<^sub>B src\<^sub>B f = (f \<star>\<^sub>B \<i>[src\<^sub>B f]) \<cdot>\<^sub>B \<a>\<^sub>B[f, src\<^sub>B f, src\<^sub>B f]" shows "\<mu> = B.runit f"
fragments_vcg_fragments': fixes c :: "('g, 'h, 'i, 'j) com"   and aft :: "'j \<Rightarrow> 'h set"   and c' :: "('g, 'h, 'i, 'j) com"   and aft' :: "'h set" assumes "(c, aft) \<in> fragments c' aft'"   and "\<not> isResponse c" shows "(c, aft) \<in> vcg_fragments' c' aft'"
ext_inject: fixes states :: "hf set"   and init :: "hf"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more :: "'c"   and states' :: "hf set"   and init' :: "hf"   and final' :: "hf set"   and nxt' :: "hf \<Rightarrow> 'b \<Rightarrow> hf"   and more' :: "'c" shows "(\<lparr>dfa.states = states, init = init, final = final, nxt = nxt, \<dots> = more\<rparr> = \<lparr>dfa.states = states', init = init', final = final', nxt = nxt', \<dots> = more'\<rparr>) = (states = states' \<and> init = init' \<and> final = final' \<and> nxt = nxt' \<and> more = more')"
mono2mono: fixes ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd llist"   and orda :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and t :: "'e \<Rightarrow> 'c" assumes "monotone ordb (\<sqsubseteq>) f"   and "monotone orda ordb t" shows "monotone orda (\<sqsubseteq>) (\<lambda>x. f (t x))"
sign_simps_25: fixes an :: "real"   and bn :: "real"   and x :: "'a" shows "(an + bn) *\<^sub>R x = an *\<^sub>R x + bn *\<^sub>R x"
singleton_conv2: fixes a :: "'g" shows "{x. a = x} = {a}"
if\<^sub>S\<^sub>E_cond_cong: fixes f :: "'a \<Rightarrow> bool"   and \<sigma> :: "'a"   and g :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> ('b \<times> 'a) option"   and d :: "'a \<Rightarrow> ('b \<times> 'a) option" assumes "f \<sigma> = g \<sigma>" shows "(_if_SE f c d) \<sigma> = (_if_SE g c d) \<sigma>"
set_of_embed_is: fixes xs :: "'c list" shows "set (List_Supplement.embed xs) = {[x] |x. x \<in> set xs}"
Int_emptyI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
ltake_zero: fixes lst :: "'b llist" shows "ltake (enat 0) lst = LNil"
a_repr_independence': fixes y :: "'a"   and x :: "'a" assumes "y \<in> I +> x"   and "x \<in> carrier R" shows "I +> x = I +> y"
row: fixes nr :: "nat"   and nc :: "nat"   and m :: "'c list list"   and i :: "nat" assumes "mat nr nc m"   and "i < nr" shows "vec nc (row m i)"
run_shift: fixes A :: "('a, 'b) nba"   and w :: "'a stream"   and V :: "(nat \<times> 'b) set"   and r :: "(nat \<times> 'b) list"   and p :: "nat \<times> 'b"   and s :: "(nat \<times> 'b) stream" assumes "gpath A w V r p"   and "grun A w V s (gtarget r p)" shows "grun A w V (r @- s) p"
IV1: shows "(\<not> check_eq l1 l2 T) = (\<exists>bs. bddh (length bs) l1 \<and> bddh (length bs) l2 \<and> (\<exists>n<m. dist_nodes M n v (bdd_lookup l1 bs) (bdd_lookup l2 bs)))"
Poly_map: fixes a :: "'b"   and p :: "'b list" shows "Poly (map ((*) a) p) = smult a (Poly p)"
in_set_butlastI: fixes xs :: "'a list"   and x :: "'a" assumes "xs \<noteq> []"   and "x \<in> set xs"   and "x \<noteq> last xs" shows "x \<in> set (butlast xs)"
conj_simps_1: fixes Q :: "fm" shows "Ferrack.conj F Q = F"
in_set_member: fixes x :: "'x"   and xs :: "'x list" shows "(x \<in> set xs) = List.member xs x"
thetaSeqZO_ZObis: shows "thetaSeqZO \<subseteq> ZObis"
union_inter: fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "finite B" shows "sum g (A \<union> B) + sum g (A \<inter> B) = sum g A + sum g B"
small_by_big: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"   and "final e'" shows "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"
atLeastatMost_subset_iff: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b"   and c :: "'a \<times> 'b"   and d :: "'a \<times> 'b" shows "(prod.atLeastAtMost a b \<subseteq> prod.atLeastAtMost c d) = (\<not> less_eq_prod a b \<or> less_eq_prod c a \<and> less_eq_prod b d)"
vs_conf_conv_vs_conf: fixes typeof_addr :: "'a \<Rightarrow> htype option"   and P :: "'c prog"   and h :: "'b" shows "heap_base.vs_conf (\<lambda>_. typeof_addr) P h = heap_base'.vs_conf TYPE('b) typeof_addr P"
set_rotate1: fixes xs :: "'e list" shows "set (rotate1 xs) = set xs"
less_eq_top_def: fixes x :: "'a top"   and y :: "'a top" shows "(x \<le> y) = (case y of top.Value y \<Rightarrow> case x of top.Value x \<Rightarrow> x \<le> y | Top \<Rightarrow> False | Top \<Rightarrow> True)"
Min_Un: fixes A :: "'c Word.word set"   and B :: "'c Word.word set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "signed.Min (A \<union> B) = signed.min (signed.Min A) (signed.Min B)"
config'_snoc: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list \<times> 'b"   and rs :: "'a list"   and r :: "'a" shows "Partial_Cost_Model.config' A s (rs @ [r]) = Partial_Cost_Model.Step A (Partial_Cost_Model.config' A s rs) r"
cSup_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set"   and a :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "S \<noteq> {}"   and "preorder.bdd_above less_eq S" shows "less_eq (Sup S) a = (\<forall>x\<in>S. less_eq x a)"
ex_poincare_line_two_points: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "\<exists>l. is_poincare_line l \<and> z \<in> circline_set l \<and> w \<in> circline_set l"
assms_1: shows "inv_approx S X (\<Delta> + 1)"
abs_summable_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "(f abs_summable_on A) = integrable (count_space A) f"
hyps: fixes p :: "'b list" assumes "length p < length pa__"   and "awalk u p v" shows "P p"
connected_convex_1_gen: fixes S :: "'b set" assumes "DIM('b) = 1" shows "connected S = convex S"
mult_im_sheaf_def: shows "mult_im_sheaf \<equiv> \<lambda>V. \<cdot>\<^bsub>f \<^sup>\<inverse> S V\<^esub>"
Gcd_group_closure_eq_Gcd: fixes S :: "int set" shows "Gcd (group_closure S) = Gcd S"
wp1_aux1a: fixes xs :: "('b, 'c) Combinators list"   and x :: "('b, 'c) Combinators" assumes "xs \<noteq> []"   and "wellformed_policy1_strong (xs @ [x])" shows "wellformed_policy1_strong xs"
RSQPS_def: shows "RSQPS = {p \<in> ps\<^sub>R. (case p of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<in> RSQ}"
const_mono: fixes ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "'d spmf" shows "monotone ord (ord_spmf (=)) (\<lambda>f. c)"
\<tau>rtrancl3p_Nil_eq_\<tau>moves: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "mthr.\<tau>rtrancl3p s [] s' = mthr.silent_moves s s'"
gy_def: fixes x :: "'a"   and y :: "'a" shows "gy x y = (6::'a) / (10::'a) - (8::'a) / (10::'a)\<^sup>2 * y - x\<^sup>2 * y"
Zp_x_is_UP_cring: shows "UP_cring Zp"
coplanar_perm_5: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "Tarski_neutral_dimensionless.Coplanar Bet A D C B"
gt_ge_tpoly_trans: fixes r :: "('v pvar, hmultiset) tpoly"   and q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" assumes "r >\<^sub>p q"   and "q \<ge>\<^sub>p p" shows "r >\<^sub>p p"
coeff_eq: fixes p :: "'b mpoly"   and q :: "'b mpoly" shows "(coeff p = coeff q) = (p = q)"
Expr_lemma2: fixes s :: "Var \<Rightarrow> Val"   and h :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and e :: "Expr"   and X :: "Var set"   and \<beta> :: "(Location \<times> Location) set"   and t :: "Var \<Rightarrow> Val"   and k :: "(Location \<times> Class \<times> (Field \<times> Val) list) list"   and x :: "Var" assumes "noLowDPs (s, h)"   and "EVars e \<subseteq> X"   and "Expr_noLoc e"   and "\<And>x. x \<in> X \<Longrightarrow> CONTEXT x = low"   and "s  \<approx>\<^sub>\<beta> t"   and "twiddleHeap \<beta> h k" shows "noLowDPs (\<lambda>y. if x = y then evalE e s else s y, h)"
valid_ownership_and_sharing_def: fixes \<S> :: "nat \<Rightarrow> bool option"   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" shows "valid_ownership_and_sharing \<S> ts \<equiv> (valid_ownership \<S> ts \<and> outstanding_non_volatile_writes_unshared \<S> ts) \<and> sharing_consis \<S> ts \<and> no_outstanding_write_to_read_only_memory \<S> ts"
preprocess_distinct: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list" assumes "preprocess cs = (t, as, trans_v, U)"   and "distinct_indices_ns (set cs)" shows "distinct_indices_atoms (set as)"
standard_numbering: fixes A :: "'d set" assumes "finite A" obtains v :: "'d \<Rightarrow> nat" and n :: "nat" where "bij_betw v A {1..n}" and "\<forall>c\<in>A. 0 < v c" and "\<forall>c. c \<notin> A \<longrightarrow> n < v c"
stable_on_range': fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and X :: "'d set"   and x :: "'d" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_on Pd Ch ds X"   and "x \<in> X" shows "x \<in> Field (Pd (Xd x))"
powerset_lift_def: fixes f :: "'d \<rightarrow> 'e set" shows "\<^ps> f = (\<Lambda> S. \<Union> (Rep_cfun f ` S))"
H: shows "to_Gromov_completion x = to_Gromov_completion y"
DBM_and_sound1: fixes v :: "'c \<Rightarrow> nat"   and u :: "'c \<Rightarrow> 'd"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and n :: "nat" assumes "u \<turnstile>\<^bsub>v,n\<^esub> And M1 M2" shows "u \<turnstile>\<^bsub>v,n\<^esub> M1"
min_maxsimpchainD_min_betw: fixes x :: "'a set"   and xs :: "'a set list"   and y :: "'a set"   and ys :: "'a set list" assumes "min_maxsimpchain (x # xs @ [y])"   and "maxsimpchain (x # ys @ [y])" shows "order.greater_eq (length ys) (length xs)"
pp_total: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_bounded_idempotent_semiring.total (\<ominus> \<ominus> x) = matrix_p_algebra.dense (x \<odot> mtop)"
StaticCastRedsThrow: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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 E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> TargetTerm T2"
case_1: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a" shows "(case true\<^sub>n of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f1"
cos_Arg2pi: fixes z :: "complex" shows "cmod z * cos (Arg2pi z) = Re z"
meet_shunt_var: fixes x :: "'a"   and y :: "'a" shows "(x - y = \<bottom>) = (x \<le> y)"
c: fixes i :: "nat" assumes "i < max_Var_floatariths fas" shows "vs ! i = ws ! i"
assign_global_def: fixes upd :: "('b \<Rightarrow> 'b) \<Rightarrow> 'a control_state_scheme \<Rightarrow> 'a control_state_scheme"   and rhs :: "'a control_state_scheme \<Rightarrow> 'b" shows "assign_global upd rhs = assign (\<lambda>\<sigma>. upd (\<lambda>_. rhs \<sigma>) \<sigma>)"
piecewise_differentiable_on_imp_continuous_on: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "f piecewise_differentiable_on S" shows "continuous_on S f"
lfinite_inf_llist: fixes f :: "nat \<Rightarrow> 'b" shows "\<not> lfinite (inf_llist f)"
span_INT_subset: fixes f :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "span (\<Inter> (f ` A)) \<subseteq> (\<Inter>a\<in>A. span (f a))"
simps_1: fixes \<xi> :: "nat \<Rightarrow> 'b set" shows "\<xi> \<Turnstile>\<^sub>n true\<^sub>n = True"
add_less_imp_less_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus a c) (plus b c)" shows "less a b"
countable_funpow: fixes f :: "'d set \<Rightarrow> 'd set"   and A :: "'d set"   and n :: "nat" assumes "\<And>A. countable A \<Longrightarrow> countable (f A)"   and "countable A" shows "countable ((f ^^ n) A)"