hcju/isabelleps huggingface dataset:
vrange_vsubset: fixes r :: "V"   and A :: "V" assumes "\<And>x y. \<langle>x, y\<rangle> \<in>\<^sub>\<circ> r \<Longrightarrow> y \<in>\<^sub>\<circ> A" shows "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A"
poly_deg_sum_le: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" shows "poly_deg (sum f A) \<le> Max (poly_deg ` f ` A)"
opp_power_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and n :: "nat" assumes "class.dioid_one_zero plus times one zero less_eq less" shows "power.power one (times.opp_mult times) x n = power.power one times x n"
bilateral_substitutes_def: fixes f :: "X5 set \<Rightarrow> X5 set" shows "BSI.bilateral_substitutes_on UNIV f = (\<forall>B a b. X5d a \<notin> X5d ` B \<and> X5d b \<notin> X5d ` B \<and> b \<notin> f (insert b B) \<longrightarrow> b \<notin> f (insert a (insert b B)))"
awlast_append: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and q :: "('a \<times> 'a) list" shows "last (pawalk_verts u (p @ q)) = last (pawalk_verts (last (pawalk_verts u p)) q)"
induct: fixes P :: "Reflection_Examples.num \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "Reflection_Examples.num"   and a1 :: "nat" assumes "\<And>j i. P (C j) i"   and "\<And>n c a i. (i \<noteq> 0 \<Longrightarrow> P a i) \<Longrightarrow> P (CN n c a) i"   and "\<And>v va i. P (Add v va) i"   and "\<And>v va i. P (num.Mul v va) i"   and "\<And>v i. P (num.Var v) i" shows "P a0 a1"
Fun_similar'_step: fixes s :: "Value \<Rightarrow> CValue \<Rightarrow> bool"   and f :: "Value \<rightarrow> Value"   and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" assumes "\<And>x y. s x (y\<cdot>C\<^sup>\<infinity>) \<Longrightarrow> s (f\<cdot>x) (g\<cdot>y\<cdot>C\<^sup>\<infinity>)" shows "similar'_step s (Fn\<cdot>f) (CFn\<cdot>g)"
Diagonalize_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Dom t\<^bold>\<rfloor> = Dom \<^bold>\<lfloor>t\<^bold>\<rfloor>"
l_to_tree_order_wf_get_root_node_wf_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_to_tree_order_wf_get_root_node_wf type_wf known_ptr known_ptrs to_tree_order get_root_node heap_is_wellformed \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_to_tree_order_wf_get_root_node_wf_axioms type_wf known_ptrs to_tree_order get_root_node heap_is_wellformed"
lnth_lappend1: fixes n :: "nat"   and xs :: "'a llist"   and ys :: "'a llist" assumes "enat n < llength xs" shows "lnth (xs @\<^sub>l ys) n = lnth xs n"
cone_carrier: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "m.cone A \<subseteq> carrier_vec n"
strict_preference_iff_strict_utility: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> \<P> outcomes"   and "q \<in> \<P> outcomes" shows "(p \<succeq> q \<and> (q, p) \<notin> relation) = (measure_pmf.expectation q u < measure_pmf.expectation p u)"
unreachStaticE: fixes s :: "Store"   and f :: "AttId"   and x :: "Value"   and P :: "bool" assumes "s\<turnstile> staticLoc f reachable_from x" shows "P"
get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: shows "get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<equiv> l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r get_root_node get_disconnected_nodes"
nrename_credu: fixes M :: "trm"   and x :: "name"   and y :: "name"   and M' :: "trm" assumes "M[x\<turnstile>n>y] \<longrightarrow>\<^sub>c M'" shows "\<exists>M0. M0[x\<turnstile>n>y] = M' \<and> M \<longrightarrow>\<^sub>c M0"
execn: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"
has_least_method_has_method: 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 C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" assumes "P \<turnstile> C has least M = mthd via Cs" shows "P \<turnstile> C has M = mthd via Cs"
unit_l_inv: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a" assumes "Dmonoid prod one"   and "Dmonoid.unit prod one x" shows "prod (Dmonoid.inv prod one x) x = one"
Min_pre_ranks_pre_oldest_tokens: fixes p :: "'b"   and n :: "nat"   and q :: "'b" assumes "configuration p (Suc n) \<noteq> {}"   and "configuration q (Suc n) \<noteq> {}"   and "\<not> sink q"   and "\<not> sink p" shows "(Min (pre_ranks (\<lambda>q. state_rank q n) (w n) p) < Min (pre_ranks (\<lambda>q. state_rank q n) (w n) q)) = (Min (pre_oldest_tokens p n) < Min (pre_oldest_tokens q n))"
simps_5: fixes a :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Cos a) vs = cos (interpret_floatarith a vs)"
d_surjective: fixes x :: "'a" assumes "test x" shows "\<exists>y. d y = x"
execution_simulates_def: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset" shows "execution_simulates e1 e2 = (\<exists>f. \<forall>t. execution_simulation f e1 0 <> e2 0 <> t)"
cospan_\<nu>\<pi>: shows "C.cospan trg.leg0 f.leg1"
gauss_sum_nonzero_noncoprime_necessary_condition_3: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and k :: "nat"   and a :: "nat" assumes "dcharacter n \<chi>"   and "dcharacter.gauss_sum n \<chi> k \<noteq> 0"   and "\<not> coprime k n"   and "0 < k"   and "coprime a n"   and "[a = 1] (mod n div gcd k n)" shows "\<chi> a = 1"
config_n_fst_init_length2: fixes s0 :: "'b list"   and qs :: "'b list" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand BIT (fst BIT s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs). length (fst (snd x)) = length s0"
strong_induct: fixes P :: "'a \<Rightarrow> ty \<Rightarrow> bool"   and z :: "'a"   and ty :: "ty" assumes "\<And>z. P z TBase"   and "\<And>z. P z TUnit"   and "\<And>ty1 ty2 z. \<lbrakk>\<And>z. P z ty1; \<And>z. P z ty2\<rbrakk> \<Longrightarrow> P z (ty1\<rightarrow>ty2)" shows "P z ty"
insertion_replace_coeff: fixes f :: "nat \<Rightarrow> 'a"   and pp :: "'a mpoly mpoly" shows "insertion f (replace_coeff (insertion f) pp) = insertion f (reduce_nested_mpoly pp)"
simps: fixes h :: "heap"   and as :: "nat set" shows "in_range (h, as) = (\<forall>a\<in>as. a < lim h)"
val_of_vec_def: fixes v :: "rat vec"   and x :: "nat" shows "val_of_vec v x = v $ x"
bottom_lower: fixes x :: "'a" assumes "x \<in> carrier (fpl L f)" shows "\<bottom>\<^bsub>fpl L f\<^esub> \<sqsubseteq>\<^bsub>fpl L f\<^esub> x"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" assumes "indRelp a1 a2" obtains S :: "'procS" where "S__ \<in>S a1" and "\<lbrakk>S__\<rbrakk> \<in>T a2"   | S :: "'procS" where "\<lbrakk>S\<rbrakk> \<in>T a1" and "S \<in>S a2"
opp_chambers_subset_opp_half_apartment: shows "order.greater_eq folding.Y (folding.\<C> - f \<turnstile> folding.\<C>)"
a_set_val_locs_def: fixes character_data_ptr :: "(_) character_data_ptr" shows "CD.a_set_val_locs character_data_ptr \<equiv> all_args (put_M\<^sub>C\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>D\<^sub>a\<^sub>t\<^sub>a character_data_ptr val_update)"
False: shows "\<not> (k = 0 \<or> fst (interp_ops ops) = [])"
Un_insert_left: fixes a :: "'c"   and B :: "'c set"   and C :: "'c set" shows "insert a B \<union> C = insert a (B \<union> C)"
assms_3: shows "v \<in> iso_verts hom ` verts G"
atleast_free_SucD: fixes h :: "'a \<Rightarrow> 'b option"   and n :: "nat"   and a :: "'a"   and b :: "'b" assumes "atleast_free h (Suc n)" shows "atleast_free (h(a \<mapsto> b)) n"
rbt_comp_bulkload: fixes c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "comparator c" shows "rbt_comp_bulkload c = ord.rbt_bulkload (lt_of_comp c)"
Un_upper2: fixes B :: "'a set"   and A :: "'a set" shows "B \<subseteq> A \<union> B"
density_discrete: fixes A :: "'c set"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "countable A"   and "sets N = Pow A"   and "\<And>x. 0 \<le> f x"   and "\<And>x. x \<in> A \<Longrightarrow> f x = emeasure N {x}" shows "density (count_space A) f = N"
t: fixes a :: "nat"   and w :: "int"   and b :: "nat" assumes "(a, w, b) \<in> edges \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr> - edges \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr>" shows "\<not> forest (add_edge a w b \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E'\<rparr>)"
range_build: fixes x :: "'c \<Longrightarrow>\<^sub>\<triangle> 'd" assumes "wb_prism x" shows "range build\<^bsub>x\<^esub> = dom match\<^bsub>x\<^esub>"
strict_boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < inf_class.inf b c" obtains "a < b" and "a < c"
splay_Leaf_iff: fixes a :: "'b"   and t :: "'b tree" shows "(splay a t = \<langle>\<rangle>) = (t = \<langle>\<rangle>)"
matrix_def: fixes f :: "('d, 'e) vec \<Rightarrow> ('d, 'f) vec" shows "matrix f = (\<chi>i j. f (axis j (1::'d)) $ i)"
permutation_of_list_unique: fixes xs :: "('b \<times> 'b) list"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "list_permutes xs A"   and "(x, y) \<in> set xs" shows "permutation_of_list xs x = y"
mcont2mcont: fixes lub' :: "'a set \<Rightarrow> 'a"   and ord' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and lub :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'a" assumes "mcont lub' ord' Sup (\<le>) f"   and "mcont lub ord lub' ord' t" shows "mcont lub ord Sup (\<le>) (\<lambda>x. f (t x))"
quietOutput: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "quiet P" shows "False"
dbl_dec_simps_2: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one zero = uminus one"
cross3_simps_13: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - b - c = a - (b + c)"
gen_entries_simps_1: shows "RBT_Impl.gen_entries [] rbt.Empty = []"
supp_neg_eq_supp: fixes f :: "'a \<Rightarrow> 'b" shows "supp (- f) = supp f"
type_definition_sat: shows "type_definition nat_of Abs_sat {..LENGTH('b)}"
nosplit_simps1_1: shows "nosplit [] = [] @ [1::'a]"
f_expand_Cons: fixes k :: "nat"   and x :: "'d message_af"   and xs :: "'d message_af list" assumes "0 < k" shows "(x # xs) \<odot> k = x # NoMsg\<^bsup>k - Suc 0\<^esup> @ xs \<odot> k"
mod_add_self2: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a b) b = modulo a b"
iIN_0_iTILL_conv: fixes n :: "nat" shows "[0\<dots>,n] = [\<dots>n]"
simps_2: fixes n :: "nat" shows "numbound0 (Bound n) = (0 < n)"
that: fixes b :: "'n" assumes "to_nat a = Bij_Nat.to_nat b" shows "thesis"
ipresIGWlsAll_termMOD: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'sort, 'h, 'i, 'd, 'e, 'j) model_scheme" shows "ipresIGWlsAll h hA termMOD MOD = ipresWlsAll h hA MOD"
simps_7: fixes cl :: "('d, 'e, 'f) cmd list" shows "proper (ParT cl) = (cl \<noteq> [] \<and> Ball (set cl) proper)"
collapse_5: fixes event :: "('d, 'e, 'f) event" assumes "isRecvMarker event" shows "RecvMarker (getId event) (occurs_on event) (partner event) = event"
step_Normal_elim_cases_5: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Seq c1 c2, Normal s) \<rightarrow> u" obtains c\<^sub>1' :: "('b, 'a, 'c) com" and s' :: "('b, 'c) xstate" where "u = (Seq c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, Normal s) \<rightarrow> (c\<^sub>1', s')"   | "c1 = Skip" and "u = (c2, Normal s)"   | "u = (Throw, Normal s)" and "c1 = Throw"
mono_neutral_cong_right: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and T :: "'c set"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b" assumes "abel_semigroup_set f z"   and "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = z"   and "\<And>x. x \<in> S \<Longrightarrow> g x = h x" shows "abel_semigroup_set.F f z g T = abel_semigroup_set.F f z h S"
\<PP>_def: shows "\<PP> = \<Q> (length Doc)"
set_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = {x21} \<union> set x22"
asprod_0: fixes z :: "int"   and x :: "ant" assumes "z \<noteq> 0"   and "z *\<^sub>a x = 0" shows "x = 0"
nat_subst\<^sub>E_distr: fixes \<alpha> :: "'a \<Rightarrow> 'a"   and \<beta> :: "'b \<Rightarrow> 'b"   and e :: "('a, 'b, nat) expr"   and x :: "nat"   and e' :: "('a, 'b, nat) expr" shows "\<R>\<^sub>E \<alpha> \<beta> (nat_subst\<^sub>E e x e') = nat_subst\<^sub>E (\<R>\<^sub>E \<alpha> \<beta> e) x (\<R>\<^sub>E \<alpha> \<beta> e')"
is_unit_fract_poly_iff: fixes p :: "'a poly" shows "(p dvd 1) = (is_unit (map_poly to_fract p) \<and> content p = (1::'a))"
that: fixes s :: "'c \<Rightarrow>\<^sub>0 nat" assumes "s \<in> keys p"   and "t \<in> keys (subst_pp f s)" shows "thesis"
listrel_mono: fixes r :: "('g \<times> 'h) set"   and s :: "('g \<times> 'h) set" assumes "r \<subseteq> s" shows "listrel r \<subseteq> listrel s"
axioms_2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('d \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'d \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_reader_state return bind ask get put" shows "monad_state return bind get put"
bind_FAIL: fixes f :: "'c \<Rightarrow> 'b nres" shows "FAIL \<bind> f = FAIL"
distinct_15: fixes x2 :: "char list"   and x51 :: "'b fm"   and x52 :: "'b fm" shows "Pro x2 \<noteq> (x51 \<^bold>\<longrightarrow> x52)"
map2_val_compatI: fixes f :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list" assumes "\<And>zs. oalist_inv_raw zs \<Longrightarrow> oalist_inv_raw (f zs)"   and "\<And>zs. oalist_inv_raw zs \<Longrightarrow> fst ` set (f zs) \<subseteq> fst ` set zs" shows "map2_val_compat f"
nonmeets3: fixes u :: "'a"   and r :: "'a" shows "\<not> (u \<parallel> r \<and> (\<exists>p. u \<parallel> p \<and> p \<parallel> r))"
fmdip_cons_trueI: fixes cld :: "cld"   and meth_defs :: "meth_def list"   and meth :: "char list"   and meth_def :: "meth_def"   and ctx :: "ctx"   and ctxcld_list :: "(ctx \<times> cld) list" assumes "class_methods cld meth_defs"   and "find_meth_def_in_list meth_defs meth (Some meth_def)" shows "find_meth_def_in_path ([(ctx, cld)] @ ctxcld_list) meth (Some (ctx, meth_def))"
v_def: shows "v \<equiv> \<lambda>f. sign_changes (coeffs f)"
pick_one_lt: fixes xs :: "real list" assumes "list_neq xs (mean xs) \<noteq> []" shows "\<exists>e. e \<in> set (list_neq xs (mean xs)) \<and> e < mean xs"
gcd_def: fixes m :: "nat"   and n :: "nat" shows "Arith2.gcd m n = (SOME x. cd x m n \<and> (\<forall>y. cd y m n \<longrightarrow> y \<le> x))"
or: shows "(\<sigma>, s) \<in> oreachable A (act I) U"
finite_enumerate_Diff_singleton: fixes S :: "'a set"   and i :: "nat"   and x :: "'a" assumes "finite S"   and "i < card S"   and "enum S i < x" shows "enum (S - {x}) i = enum S i"
bind_pure_returns_result_I: fixes f :: "('m, 'n, 'o) prog"   and h :: "'m"   and x :: "'o"   and g :: "'o \<Rightarrow> ('m, 'n, 'p) prog"   and y :: "'p" assumes "pure f h"   and "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
le: fixes x :: "'a" assumes "x \<in> S" shows "dist (g a) a \<le> dist (g x) x"
rel_dist_to_stop_def: fixes v :: "real"   and a :: "real" shows "rel_dist_to_stop v a \<equiv> - v\<^sup>2 / (2 * a)"
union_ac_1: fixes M :: "'a multiset"   and N :: "'a multiset"   and K :: "'a multiset" shows "M + N + K = M + (N + K)"
matrix_compose: fixes f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and g :: "(real, 'b) vec \<Rightarrow> (real, 'c) vec" assumes "linear f"   and "linear g" shows "matrix (g \<circ> f) = matrix g ** matrix f"
AEf2: shows "AE x in M. real_cond_exp M Invariants f2 x = 0"
scale_sum_right: fixes a :: "'a"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "a *s sum f A = (\<Sum>x\<in>A. a *s f x)"
e1: shows "- a = lincomb (\<lambda>x. - (1::'a) * aa x) A"
ide_dom: fixes f :: "'a arrow_of_spans_data" assumes "V.arr f" shows "V.ide (V.dom f)"
AL_delete3: fixes L :: "('b \<times> 'a) list"   and a :: "'b" assumes "L\<down>a = None" shows "delete L a = L"
Mp_pow: fixes f :: "int poly" shows "pl.Mp f =m f"
mat_rel_eq: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "(R ===> R ===> (=)) (=) (=)" shows "(mat_rel R ===> mat_rel R ===> (=)) (=) (=)"
mpt_factorE_1: fixes proj :: "'a \<Rightarrow> 'c"   and M2 :: "'c measure"   and T2 :: "'c \<Rightarrow> 'c" assumes "mpt_factor proj M2 T2" shows "proj \<in> measure_preserving M M2"
IH: fixes y :: "nat"   and a :: "real"   and b :: "real" assumes "y < i"   and "0 \<le> a \<and> a \<le> b \<or> odd y \<and> a \<le> b \<or> even y \<and> a \<le> 0 \<and> b \<le> a" shows "power_down prec a y \<le> power_down prec b y"
vfinite_vinsertD: fixes x :: "V"   and F :: "V" assumes "vfinite (vinsert x F)" shows "vfinite F"
filterlim_at_top_ge: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and c :: "'b" shows "filterlim f at_top F = (\<forall>Z\<ge>c. \<forall>\<^sub>F x in F. Z \<le> f x)"
list_all2_Cons2: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and y :: "'e"   and ys :: "'e list" shows "list_all2 P xs (y # ys) = (\<exists>z zs. xs = z # zs \<and> P z y \<and> list_all2 P zs ys)"
preserves_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "VVV.arr f" shows "local.dom (\<alpha>.map f) = HoHV (VVV.dom f)"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and a :: "('fsym, 'psym) atm" shows "satL \<xi> (Neg a) = (\<not> satA \<xi> a)"
separating_hyperplane_closed_0: fixes S :: "'b set" assumes "convex S"   and "closed S"   and "(0::'b) \<notin> S" shows "\<exists>a b. a \<noteq> (0::'b) \<and> 0 < b \<and> (\<forall>x\<in>S. b < a \<bullet> x)"
assms_5: fixes i :: "'a" assumes "i \<in> I" shows "f i \<le>\<^sub>o\<^sub>w c"
queue_invar_def: fixes q :: "('c, 'd) BinomialTree list" shows "queue_invar q \<equiv> \<forall>t\<in>set q. tree_invar t"
that: fixes s :: "real" assumes "0 \<le> s"   and "s < t"   and "flow0 x ` {0..s} \<subseteq> M"   and "flow0 x s \<in> frontier M"   and "\<exists>\<^sub>F s' in at_right s. flow0 x s' \<notin> M" shows "thesis"
downclosed_downset_set: fixes X :: "'a set" shows "downclosed_set (\<Down> X)"
build_split: fixes w :: "nat \<Rightarrow> 'a" shows "w = w 0 ## suffix 1 w"
compact_cbox: fixes a :: "'c"   and b :: "'c" shows "compact (cbox a b)"
pentagon: fixes a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "ide a"   and "ide b"   and "ide c"   and "ide d" shows "(a \<otimes> \<a>[b, c, d]) \<cdot> \<a>[a, b \<otimes> c, d] \<cdot> (\<a>[a, b, c] \<otimes> d) = \<a>[a, b, c \<otimes> d] \<cdot> \<a>[a \<otimes> b, c, d]"
RMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O>s :: "nat set list"   and i :: "nat"   and \<R>s :: "(nat \<Rightarrow> bool option) list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool" assumes "cond (\<theta>(t \<mapsto> m a))"   and "a \<in> \<O> \<or> a \<in> dom \<S>"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> a \<notin> \<O>s ! j \<union> dom (\<R>s ! j)"   and "A \<subseteq> dom \<S> \<union> \<O>"   and "L \<subseteq> A"   and "R \<subseteq> \<O>"   and "A \<inter> R = _emptyset"   and "\<forall>j<length \<O>s. i \<noteq> j \<longrightarrow> A \<inter> (\<O>s ! j \<union> dom (\<R>s ! j)) = _emptyset"   and "a \<notin> read_only \<S>" shows "\<O>s,\<R>s,i\<turnstile> (RMW a t (D, f) cond ret A L R W # is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
rem_closed: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "UP_root_rem f a \<in> carrier P"
length_def: fixes h :: "heap"   and a :: "'b array" shows "Array.length h a = length (Array.get h a)"
assm: shows "y \<sqinter> nc \<noteq> (0::'a)"
ordIso_Well_order_simp: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r =o r'" shows "Well_order r \<and> Well_order r'"
cases: fixes x :: "tm \<times> tm \<times> tm \<times> tm" obtains t :: "name" and v :: "tm" and i :: "tm" and y :: "tm" and y' :: "tm" and t' :: "name" and u :: "name" and u' :: "name" where "atom t \<sharp> (v, i, y, y', t', u, u')" and "atom t' \<sharp> (v, i, y, y', u, u')" and "atom u \<sharp> (v, i, y, y', u')" and "atom u' \<sharp> (v, i, y, y')" and "x = (v, i, y, y')"
Block_None_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and V :: "nat"   and T :: "ty" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "\<tau>red1gr uf P t h ({V:T=None; e}, xs) ({V:T=None; e'}, xs')"
Units_one_side_I_1: fixes a :: "'a mat"   and p :: "'a mat" assumes "a \<in> carrier (ring_mat TYPE('a) n ())"   and "p \<in> Units (ring_mat TYPE('a) n ())"   and "p \<otimes>\<^bsub>ring_mat TYPE('a) n ()\<^esub> a = \<one>\<^bsub>ring_mat TYPE('a) n ()\<^esub>" shows "a \<in> Units (ring_mat TYPE('a) n ())"
Derive: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "Derivation a D b" shows "Derive a D = b"
inj_on_f_imageI: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'c set set" assumes "inj_on f S"   and "\<And>t. t \<in> T \<Longrightarrow> t \<subseteq> S" shows "inj_on ((`) f) T"
reusables_neq_Nil_if_well_shaped_and_overlapping: fixes t :: "'b tree"   and w :: "nat \<times> nat" assumes "well_shaped t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "fst w \<le> r t" shows "reusables t w \<noteq> []"
bconf_SCall: fixes es :: "'a exp list"   and P :: "'b prog"   and sh :: "sheap"   and C :: "char list"   and M :: "char list"   and b :: "bool" assumes "map_vals_of es = None" shows "(P,sh \<turnstile>\<^sub>b (C\<bullet>\<^sub>sM(es),b) \<surd>) = (P,sh \<turnstile>\<^sub>b (es,b) \<surd>)"