hcju/isabelleps huggingface dataset:
mon_hom_boundary_inclusion: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and p :: "int" assumes "homotopic_with (\<lambda>x. True) X X id f"   and "f ` topspace X \<subseteq> S" shows "hom_boundary p X S \<in> mon (relative_homology_group p X S) (homology_group (p - 1) (subtopology X S))"
at_neg_neg2: fixes c :: "cars"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. \<not> (\<forall>v'. v = c > v' \<longrightarrow> \<not> \<phi> ts v') \<longrightarrow> (\<forall>v'. v = c > v' \<longrightarrow> \<phi> ts v')"
simps_7: fixes lcl' :: "lenv \<Rightarrow> lenv"   and prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'b" shows "lcl_update lcl' \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = \<lparr>prg = prg, cls = cls, lcl = lcl' lcl, \<dots> = more\<rparr>"
ltln_expand_Until: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> or\<^sub>n (\<phi> and\<^sub>n X\<^sub>n (\<phi> U\<^sub>n \<psi>))"
invalidate_sqn: fixes dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "\<forall>rsn. dests dip = Some rsn \<longrightarrow> sqn rt dip \<le> rsn" shows "sqn rt dip \<le> sqn (invalidate rt dests) dip"
lran_empty_2: fixes a :: "int \<Rightarrow> 'b"   and l :: "int"   and h :: "int" shows "(lran a l h = []) = (h \<le> l)"
S_intersections_endpoints_in_S: fixes a :: "(real, 3) vec"   and b :: "(real, 3) vec" assumes "a \<noteq> 0"   and "b \<noteq> 0"   and "proj2_abs a \<noteq> proj2_abs b"   and "proj2_abs a \<in> hyp2"   and "proj2_abs b \<in> hyp2 \<union> S" shows "S_intersection1 a b = endpoint_in_S (proj2_abs a) (proj2_abs b) \<and> S_intersection2 a b = endpoint_in_S (proj2_abs b) (proj2_abs a) \<or> S_intersection2 a b = endpoint_in_S (proj2_abs a) (proj2_abs b) \<and> S_intersection1 a b = endpoint_in_S (proj2_abs b) (proj2_abs a)"
commute: fixes a :: "'d"   and b :: "'d" shows "a \<sqinter> b = b \<sqinter> a"
confluent_Un: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "confluent R"   and "confluent S"   and "commute R\<^sup>*\<^sup>* S\<^sup>*\<^sup>*" shows "confluent (sup R S)"
assms_2: shows "\<forall>n. trace (complex_of_real c \<cdot>\<^sub>m f n) \<le> 1"
meet_ord_def: fixes x :: "'a"   and y :: "'a" shows "(ad x \<le> ad y) = (ad x \<cdot> ad y = ad x)"
Limsup_eq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and net :: "'b filter" assumes "\<forall>\<^sub>F x in net. f x = g x" shows "Limsup net f = Limsup net g"
rep_eq: fixes x :: "'b word" shows "bit x = (\<lambda>n. n < LENGTH('b) \<and> bit (Word.rep x) n)"
analz_insert_Decrypt: fixes K :: "key"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz (insert X H))"
distinct_1: fixes x11 :: "name"   and x12 :: "name \<Rightarrow> pi noption"   and x21 :: "freeRes"   and x22 :: "pi" shows "BoundOutputR_Rep x11 x12 \<noteq> FreeR_Rep x21 x22"
invertibleI: fixes A :: "(('a, 'c) vec, 'b) vec"   and B :: "(('a, 'b) vec, 'c) vec" assumes "A ** B = mat (1::'a)"   and "B ** A = mat (1::'a)" shows "invertible A"
map_eq_appendE: fixes f :: "'d \<Rightarrow> 'c"   and ls :: "'d list"   and fl :: "'c list"   and fl' :: "'c list" assumes "map f ls = fl @ fl'" obtains l :: "'d list" and l' :: "'d list" where "ls = l @ l'" and "map f l = fl" and "map f l' = fl'"
PTrans: shows "P \<Longrightarrow>\<^sub>l\<^sup>^a[b] \<prec> P'"
dagger_iso: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>\<star> \<le> y\<^sup>\<star>"
elims_3: fixes x :: "ground_action"   and xa :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" assumes "\<not> sound_opr x xa" obtains pre :: "object atom formula" and add :: "object atom formula list" and del :: "object atom formula list" and f :: "(predicate \<times> object list \<Rightarrow> bool) \<Rightarrow> (predicate \<times> object list \<Rightarrow> bool) option" where "x = Ground_Action pre (Effect add del)" and "xa = f" and "\<not> ((\<forall>s. close_eq s \<Turnstile> pre \<longrightarrow> (\<exists>s'. f s = Some s' \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set del \<and> close_eq s \<Turnstile> atm \<longrightarrow> close_eq s' \<Turnstile> atm) \<and> (\<forall>atm. is_predAtom atm \<and> atm \<notin> set add \<and> close_eq s \<Turnstile> \<^bold>\<not> atm \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> atm) \<and> (\<forall>fmla. fmla \<in> set add \<longrightarrow> close_eq s' \<Turnstile> fmla) \<and> (\<forall>fmla. fmla \<in> set del \<and> fmla \<notin> set add \<longrightarrow> close_eq s' \<Turnstile> \<^bold>\<not> fmla))) \<and> Ball (set add) is_predAtom)"
atLeastAtMost_singleton': fixes a :: "'a"   and b :: "'a" assumes "a = b" shows "{a..b} = {a}"
SUP_mono': fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
rtranclp_into_tranclp2: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
pmf_integral_code_unfold: fixes p :: "'a pmf" shows "pmf_integral p = measure_pmf.expectation p"
p_def: shows "p \<equiv> \<lambda>i. (pcs i, pcss i, S (f i))"
in_hhomI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "arr \<mu>"   and "src \<mu> = a"   and "trg \<mu> = b" shows "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"
intros_5: shows "IK_nf (K \<circ> I)"
compl_inf_bot: fixes x :: "'a" shows "- x \<sqinter> x = \<bottom>"
nonneg: fixes x :: "'a"   and y :: "'a" shows "0 \<le> f x y"
qSwap_preserves_qGood: fixes zs :: "'e"   and x :: "'d"   and y :: "'d"   and X :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)" shows "FixVars.qGood (X #[[x \<and> y]]_zs) = FixVars.qGood X"
Sigma_msetI: fixes a :: "'a"   and A :: "'a multiset"   and b :: "'b"   and B :: "'a \<Rightarrow> 'b multiset" assumes "a \<in># A"   and "b \<in># B a" shows "(a, b) \<in># Sigma_mset A B"
env_add_dom: fixes e :: "'b environment"   and x :: "char list"   and X :: "'b" assumes "ok e"   and "x \<notin> env_dom e" shows "env_dom (e\<lparr>x:X\<rparr>) = env_dom e \<union> {x}"
vcomp_def: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "\<nu> \<bullet> \<mu> \<equiv> if arrow_of_spans (\<odot>) \<mu> \<and> arrow_of_spans (\<odot>) \<nu> \<and> Dom \<nu> = Cod \<mu> then \<lparr>Chn = Chn \<nu> \<odot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else Span.Null"
delete_abs_trieP: fixes ks :: "bool list"   and t :: "trieP" shows "delete ks (abs_trieP t) = abs_trieP (deleteP ks t)"
heap_changed_first_el: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and heap :: "int \<Rightarrow> 'a"   and l :: "int"   and r :: "int"   and b :: "'a" assumes "is_heap cmp heap l r"   and "l \<le> r" shows "is_heap_except_down cmp (heap(l := b)) l r l"
critical_set_fixed: fixes F :: "ordinal \<Rightarrow> ordinal"   and z :: "ordinal" assumes "normal F"   and "0 < z" shows "range (oVeblen F z) = {x. \<forall>y<z. oVeblen F y x = x}"
Cons_1: fixes S :: "('b, 'd) Term.term list" assumes "\<forall>i<length T. \<exists>s. P (T ! i) s \<and> S ! i = s \<cdot> \<delta>"   and "length T = length S" shows "\<exists>U. length T = length U \<and> (\<forall>i<length T. P (T ! i) (U ! i)) \<and> S = U \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>"
dims2_def: shows "st.dims2 = nths dims {Suc k}"
vpsubsetE: fixes A :: "V"   and B :: "V" assumes "A \<subset>\<^sub>\<circ> B" obtains x :: "V" where "A \<subseteq>\<^sub>\<circ> B" and "x \<notin>\<^sub>\<circ> A" and "x \<in>\<^sub>\<circ> B"
rep_number_g0_exists: fixes B :: "'b set multiset"   and x :: "'b" assumes "0 < B rep x" obtains b :: "'b set" where "b \<in># B" and "x \<in> b"
Gcd_greatest_int: fixes A :: "int set"   and a :: "int" assumes "\<And>b. b \<in> A \<Longrightarrow> a dvd b" shows "a dvd Gcd A"
set_intros_2: fixes x2 :: "'a" shows "x2 \<in> set_Literal (Neg x2)"
between_symmetry: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C" shows "Bet C B A"
hyps: shows "sum_list ls = Le_Constraint (sum_list (map lec_rel ls)) (sum_list (map lec_poly ls)) (sum_list (map lec_const ls))"
subst_subst: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and \<rho> :: "'d \<Rightarrow> 'e"   and e :: "('b, 'e) uexpr" shows "\<sigma> \<dagger> \<rho> \<dagger> e = (\<rho> \<circ> \<sigma>) \<dagger> e"
\<a>_nonempty_unique: fixes X :: "'x set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "finite X"   and "valid_decomp X ps"   and "standard_decomp k ps"   and "ps\<^sub>+ \<noteq> []" shows "\<a> ps = k"
intros_14: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and sx :: "sx"   and k :: "i32"   and off :: "nat"   and tp :: "tp"   and t :: "t"   and bs :: "bytes"   and vs :: "v list"   and a :: "nat" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "load_packed sx m (nat_of_int k) off (tp_length tp) (t_length t) = Some bs" shows "\<lparr>s;vs;[$C ConstInt32 k, $Load t (Some (tp, sx)) a off]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C wasm_deserialise bs t]\<rparr>"
assms_1: shows "extends g subs (v1, v2) subs'"
p_pow_car: fixes n :: "int" assumes "0 \<le> n" shows "[p] \<cdot> \<one> [^] n \<in> carrier Zp"
subst_type_term_empty_no_change: fixes t :: "Core.term" shows "subst_type_term [] [] t = t"
bounded_sphere: fixes a :: "'d"   and r :: "real" shows "bounded (sphere a r)"
hcomp_simps\<^sub>W\<^sub>C_3: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "cod (\<nu> \<star> \<mu>) = cod \<nu> \<star> cod \<mu>"
assms_3: fixes x :: "'n"   and k :: "nat" assumes "x \<in> S" shows "norm (f k x) \<le> h x"
Arg2pi_cnj_eq_inverse: fixes z :: "complex" assumes "z \<noteq> 0" shows "Arg2pi (cnj z) = Arg2pi (inverse z)"
prems_1: shows "Fun f T \<sqsubseteq> Fun f' T'"
0: shows "s \<odot>\<^bsub>P\<^esub> f = s \<odot>\<^bsub>P\<^esub> monom P (f 0) 0 \<oplus>\<^bsub>P\<^esub> X_poly R \<otimes>\<^bsub>P\<^esub> poly_shift (s \<odot>\<^bsub>P\<^esub> f)"
mult_monom: fixes a :: "'c"   and m :: "nat"   and b :: "'c"   and n :: "nat" shows "monom a m * monom b n = monom (a * b) (m + n)"
runit_hcomp_gen: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "\<r>[f \<star> g] \<star> h = ((f \<star> \<r>[g]) \<star> h) \<cdot> (\<a>[f, g, src g] \<star> h)"
simps_1: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "row_add_iterate A 0 i j = (if i = (0::'c) then A else row_add A (0::'c) i (- A $ (0::'c) $ j))"
simps_2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
foldl_spmf_Cons: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'a spmf"   and p :: "'a spmf"   and x :: "'b"   and xs :: "'b list" shows "foldl_spmf f p (x # xs) = foldl_spmf f (p \<bind> (\<lambda>a. f a x)) xs"
join_meet_absorb: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "dlat le" shows "dlat.join le x (dlat.meet le x y) = x"
Lindelof_spaceD: fixes X :: "'a topology"   and \<U> :: "'a set set" assumes "Lindelof_space X"   and "\<And>U. U \<in> \<U> \<Longrightarrow> openin X U"   and "\<Union> \<U> = topspace X" shows "\<exists>\<V>. countable \<V> \<and> \<V> \<subseteq> \<U> \<and> \<Union> \<V> = topspace X"
list_update_length: fixes xs :: "'f list"   and x :: "'f"   and ys :: "'f list"   and y :: "'f" shows "(xs @ x # ys)[length xs := y] = xs @ y # ys"
40: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> (x \<sqinter> y) = bot"
SatP: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and a :: "'b"   and i :: "'b"   and p :: "('c, 'b) fm" assumes "Hintikka A H"   and "(\<^bold>@ i p) at a in' H" shows "p at i in' H"
contour_integrable_part_circlepath_iff: fixes a :: "real"   and b :: "real"   and f :: "complex \<Rightarrow> complex"   and c :: "complex"   and r :: "real" assumes "a < b" shows "(f contour_integrable_on part_circlepath c r a b) = ((\<lambda>t. f (c + complex_of_real r * cis t) * complex_of_real r * \<i> * cis t) integrable_on {a..b})"
Int_Diff_disjoint: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B \<inter> (A - B) = {}"
assms_2: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<BB>"
monad_state_altc_stateT: fixes return :: "'a \<times> 'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('a \<times> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_altc return bind altc" shows "monad_state_altc (return_state return) (bind_state bind) get_state put_state (altc_state altc)"
DynCastThrow: fixes 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 e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
lookup_monic: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "lookup (monic p) v = lookup p v / lc p"
f_def: shows "f \<equiv> \<lambda>i x. real_of_int \<lfloor>min (real i) (u x) * 2 ^ i\<rfloor> / 2 ^ i"
strict_dominate_trans1: fixes n1 :: "nat"   and n2 :: "nat"   and n3 :: "nat" assumes "strict_dominate n1 n2"   and "dominate n2 n3" shows "strict_dominate n1 n3"
path_to_then_exec_serial: fixes \<pi>s :: "char list list"   and s :: "nat \<Rightarrow> nat option"   and s' :: "nat \<Rightarrow> nat option" assumes "\<forall>\<pi>\<in>set \<pi>s. lookup_operator \<pi> \<noteq> None"   and "path_to s \<pi>s s'" shows "s' \<subseteq>\<^sub>m execute_serial_plan_sas_plus s (abs_ast_plan \<pi>s)"
RedInit: fixes e :: "char list exp"   and C :: "char list"   and b :: "bool"   and s :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "\<not> sub_RI e" shows "P \<turnstile> \<langle>INIT C ([],b) \<leftarrow> e,s,b'\<rangle> \<rightarrow> \<langle>e,s,icheck P C e\<rangle>"
agree: shows "Vagree (mk_v I (OSing vid1 (f1 fid1 vid1)) (ab, bb) (sol t)) (mk_xode I (OSing vid1 (f1 fid1 vid1)) (sol t)) {Inl vid1, Inr vid1}"
t_stop_pos: assumes "v\<^sub>e \<noteq> 0" shows "0 < ego.t_stop"
state_succ_pair: fixes s :: "('d, 'e) fmap"   and p :: "('d, 'e) fmap"   and e :: "('d, 'e) fmap" shows "state_succ s (p, e) = (if p \<subseteq>\<^sub>f s then e ++ s else s)"
get_object_ptr_simp1: fixes object_ptr :: "(_) object_ptr"   and object :: "(_) Object"   and h :: "(_) heap" shows "get object_ptr (put object_ptr object h) = Some object"
simps: fixes G :: "'b list set" shows "\<BB> G = {x.  x \<in>B G }"
abs_eq: fixes x :: "int" assumes "eq_onp (\<lambda>x. x \<in> {0..<int CARD('b)}) x x" shows "- Abs_mod_ring x = Abs_mod_ring (if x = 0 then 0 else int CARD('b) - x)"
simps_1: fixes f :: "'c \<Rightarrow> 'b err" shows "strict f Err = Err"
Un_insert_right: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A \<union> ({a} \<union> B) = {a} \<union> (A \<union> B)"
assms: shows "f1 \<subseteq>\<^sub>f f2"
CompFun_HAInitStates_HAStates: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto"   and SA :: "('d, 'e, 'f) seqauto" assumes "S \<in> HAStates A"   and "SA \<in> the (CompFun A S)" shows "InitState SA \<in> HAInitStates A"
1: shows "\<forall>(k, \<gamma>)\<in>one_chain1 - ps1. line_integral_exists F basis \<gamma>"
summable_mult: fixes f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "summable f" shows "summable (\<lambda>n. c * f n)"
mod_ring_divide_integer: shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (divide_p_integer pp) (/)"
ud: shows "u = wordinterval_setminus r (prefix_to_wordinterval s)"
bits_stable_imp_add_self: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "divide a (numeral.numeral one plus (num.Bit0 num.One)) = a" shows "plus a (modulo a (numeral.numeral one plus (num.Bit0 num.One))) = zero"
nth_Basis_list_in_Basis: fixes n :: "nat" assumes "n < length Basis_list" shows "Basis_list ! n \<in> Basis"
c_nth_is_pr: shows "c_nth \<in> PrimRec2"
\<psi>_\<phi>: fixes f :: "'c"   and b :: "'c"   and a :: "'c" assumes "\<guillemotleft>f : b \<rightarrow>\<^sub>C a\<guillemotright>" shows "\<psi>C (b, a) (\<phi>C (b, a) f) = f"
Ball_Collect: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "Ball A P = (A \<subseteq> Collect P)"
del_unEdge_com: fixes v :: "'a"   and w :: "'b"   and v' :: "'a"   and n :: "'a"   and e :: "'b"   and n' :: "'a"   and g :: "('a, 'b, 'c) graph_scheme" shows "del_unEdge v w v' (del_unEdge n e n' g) = del_unEdge n e n' (del_unEdge v w v' g)"
a: shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
pseudo_proj_False_neq_True: fixes n :: "nat"   and w :: "bool stream" shows "pseudo_proj_False n w \<noteq> pseudo_proj_True n w"
assms: shows "\<forall>C. S \<notin> DSources i C"
integrable_meissel_mertens: fixes a :: "real" assumes "1 < a" shows "set_integrable lborel {a<..} (\<lambda>t. (\<MM> t - ln t) / (t * (ln t)\<^sup>2))"
rel_spmf_reflI: fixes p :: "'h spmf"   and P :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
map_simp_ide: fixes a :: "'c" assumes "Cop.ide a" shows "\<Phi>.map a = \<Phi>o a"
dgrad_p_set_exhaust: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and d :: "'a \<Rightarrow> nat" assumes "finite F" obtains m :: "nat" where "F \<subseteq> dgrad_p_set d m"
cet_inverse: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e : REAL" shows "\<Gamma> \<turnstile>\<^sub>c inverse\<^sub>c e : REAL"
multrow_mat_carrier: fixes n :: "nat"   and k :: "nat"   and a :: "'a" shows "multrow_mat n k a \<in> carrier_mat n n"
Basis_imp: fixes k :: "real"   and a :: "real"   and b :: "real"   and c :: "real" assumes "k \<in> Basis" shows "g (cbox a b) = f (g (cbox a b \<inter> {x. x \<bullet> k \<le> c})) (g (cbox a b \<inter> {x. c \<le> x \<bullet> k}))"
min_distI_eq: fixes v :: "nat"   and d :: "nat"   and v' :: "nat" assumes "dist v d v'"   and "\<And>d'. dist v d' v' \<Longrightarrow> d \<le> d'" shows "min_dist v v' = d"
UN_insert: fixes B :: "'c \<Rightarrow> 'b set"   and a :: "'c"   and A :: "'c set" shows "\<Union> (B ` (a \<triangleright> A)) = B a \<union> \<Union> (B ` A)"
const0_in_RPred1: shows "0\<^sup>\<infinity> \<in> \<R>\<^sub>0\<^sub>1"
R: shows "R \<in> {[x, y], [y, x]}"
abs_eq: fixes x :: "32 word" shows "uint64_of_uint32 (Abs_uint32 x) = Abs_uint64 (ucast x)"
integrable_on_def: fixes f :: "'a \<Rightarrow> 'c"   and i :: "'a set" shows "(f integrable_on i) = (\<exists>y. (f has_integral y) i)"
LSCategory_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme" shows "LSCategory C \<equiv> Category C \<and> LSCategory_axioms C"
finite_subst_img_if_finite_dom': fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "finite (subst_domain \<sigma>)" shows "finite (subst_range \<sigma>)"
eventually_imp_conv_disj: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<diamond>(\<lambda>s. P s \<longrightarrow> Q s) = (\<lambda>s. (\<diamond>(\<lambda>s. \<not> P s)) s \<or> (\<diamond>Q) s)"
empty_replicate: fixes n :: "nat"   and x :: "'a" shows "([] = x \<up> n) = (n = 0)"
bset_eq_iff: fixes A :: "'c set['d]"   and B :: "'c set['d]" shows "(A = B) = (\<forall>a. bmember a A = bmember a B)"
cases: fixes a1 :: "('a, 'b, 'c, 'd) converter"   and a2 :: "enat" assumes "interaction_any_bounded_converter a1 a2" obtains conv :: "('a, 'b, 'c, 'd) converter" and n :: "enat" where "a1 = conv" and "a2 = n" and "\<And>a. interaction_any_bounded_by (run_converter conv a) n" and "\<And>a b conv'. (b, conv') \<in> results'_gpv (run_converter conv a) \<Longrightarrow> interaction_any_bounded_converter conv' n"
sgn_one: shows "sgn (1::'b) = (1::'b)"
existence_ivl_cong: fixes Y :: "'a set"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and x0 :: "'a" assumes "Y = Y"   and "UNIV = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> g x = g t x" shows "Y.existence_ivl0 x0 = ll_on_open.existence_ivl S g Y 0 x0"
env_restr_HSem: fixes \<Gamma> :: "('var \<times> 'exp) list"   and S :: "'var set"   and \<rho> :: "'var \<Rightarrow> 'value" assumes "domA \<Gamma> \<inter> S = {}" shows "(\<lbrace> \<Gamma> \<rbrace>\<rho>) f|` S = \<rho> f|` S"
vars1'_def: shows "d.vars1' = ind_in_set d.vars0 ` vars1"
path_eq: fixes p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "path p"   and "\<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t" shows "path q"
Number_Sign: fixes X :: "nat"   and X' :: "msg"   and Y' :: "msg" shows "Number X \<noteq> Sign X' Y'"
insertI1: fixes a :: "'a"   and B :: "'a set" shows "a \<in> insert a B"
HFail_allBlocks: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HFail s s' p" shows "allBlocks s' \<subseteq> allBlocks s \<union> {dblock s' p}"
subspace_set_in_sets: fixes A :: "('d \<Rightarrow>\<^sub>F 'e) set"   and J :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure"   and I :: "'d set set" assumes "A \<in> sets (Pi\<^sub>F J M)"   and "J \<subseteq> I"   and "\<And>j. j \<in> J \<Longrightarrow> finite j" shows "A \<in> sets (Pi\<^sub>F I M)"
points_index_block_image_add_eq: fixes x :: "'b"   and ps :: "'b set"   and B :: "'b set multiset"   and l :: "nat" assumes "x \<notin> ps"   and "B index ps = l" shows "image_mset (insert x) B index ps = l"
LeastI_ex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"