hcju/isabelleps huggingface dataset:
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "qtname \<Rightarrow> 'a"   and f2 :: "ty \<Rightarrow> int \<Rightarrow> 'a"   and obj_tag :: "obj_tag" shows "P (case obj_tag of CInst x \<Rightarrow> f1 x | Arr x xa \<Rightarrow> f2 x xa) = (\<not> ((\<exists>x1. obj_tag = CInst x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x21 x22. obj_tag = Arr x21 x22 \<and> \<not> P (f2 x21 x22))))"
select_index_suc_odd: fixes k :: "nat"   and n :: "nat"   and i :: "nat" assumes "k \<le> 2 ^ n - 1"   and "select_index n i k" shows "select_index (Suc n) i (2 * k + 1)"
size_Suc_Diff1: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "Suc (size (M - {#x#})) = size M"
P1_adv_real_ideal_model_def: fixes \<G> :: "'d cyclic_group"   and D :: "'e \<times> 'd \<Rightarrow> bool spmf"   and M :: "'d \<times> 'd"   and \<sigma> :: "bool"   and \<A> :: "('d \<times> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> (('d \<times> 'd \<times> 'd) \<times> 'g) spmf) \<times> ('d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<Rightarrow> 'g \<Rightarrow> ((('d \<times> 'd) \<times> 'd \<times> 'd) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and z :: "'f" assumes "ot_base \<G>" shows "ot_base.P1_adv_real_ideal_model \<G> D M \<sigma> \<A> z = \<bar>spmf (ot_base.P1_real_model \<G> M \<sigma> z \<A> \<bind> D) True - spmf (ot_base.P1_ideal_model \<G> M \<sigma> z \<A> \<bind> D) True\<bar>"
Edges_between_edges: fixes a :: "nat"   and b :: "nat"   and u :: "nat"   and f :: "face"   and v :: "nat"   and vs :: "nat list" assumes "(a, b) \<in> Edges (u # between (vertices f) u v @ [v])"   and "pre_split_face f u v vs" shows "(a, b) \<in> \<E> f"
not_integrable_contour_integral: fixes f :: "complex \<Rightarrow> complex"   and g :: "real \<Rightarrow> complex" assumes "\<not> f contour_integrable_on g" shows "contour_integral g f = 0"
lens_equivI: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "X \<subseteq>\<^sub>L Y"   and "Y \<subseteq>\<^sub>L X" shows "X \<approx>\<^sub>L Y"
half_gt_zero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (divide a (numeral.numeral one plus (num.Bit0 num.One)))"
le_sup_iff: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
bvars\<^sub>s\<^sub>s\<^sub>t_unlabel_Cons: fixes l :: "'f"   and b :: "('e, 'd) stateful_strand_step"   and A :: "('f \<times> ('e, 'd) stateful_strand_step) list" shows "bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, b) # A) = set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p b) \<union> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
simps_1: fixes N :: "nat" shows "freeleft (NONCE N) = NONCE N"
norm_memb_eq_rep_ARR: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "free_monoidal_category.ARR C f"   and "t \<in> f" shows "free_monoidal_category.norm C t = free_monoidal_category.rep C f"
uncurry_def: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
ptensor_encode1_encode2: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and dims :: "nat list" assumes "partial_state2 vars1 vars2" shows "partial_state.encode1 (partial_state2.dims0 dims vars1 vars2) (partial_state2.vars1' vars1 vars2) = partial_state.encode2 (partial_state2.dims0 dims vars1 vars2) (partial_state2.vars2' vars1 vars2)"
fds_diverges: fixes s :: "'a"   and f :: "'a fds" assumes "ereal (s \<bullet> (1::'a)) < conv_abscissa f" shows "\<not> fds_converges f s"
pmc_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>\<gamma>\<^esub> \<phi> = Cn (K \<div>\<^bsub>\<gamma>\<^esub> \<phi>)"
drop_guard_add_update_direct_subsumption_def: fixes t' :: "transition"   and t :: "transition"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s' :: "nat" shows "drop_guard_add_update_direct_subsumption t' t e s' = (case input_stored_in_reg t' t e of None \<Rightarrow> False | Some (i, r) \<Rightarrow> if no_illegal_updates t r then initially_undefined_context_check (tm e) r s' else False)"
simps_4: fixes F :: "'a formula" shows "to_mini_formula (\<^bold>\<not> F) = to_mini_formula F \<^bold>\<rightarrow> \<bottom>"
that: fixes A :: "'a set"   and c :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "finite A"   and "p = smult c (\<Prod>a\<in>A. [:- a, 1::'a:] ^ m a)"   and "\<And>a. (m a = 0) = (a \<notin> A)"   and "(c = (0::'a)) = (p = 0)"   and "\<And>z. (poly p z = (0::'a)) = (c = (0::'a) \<or> z \<in> A)" shows "thesis"
tlookup_some_inst: shows "sortedTree fst t \<and> tlookup fst i t = Some ia \<longrightarrow> ia \<in> setOf t \<and> fst ia = i"
norm_inverse: fixes a :: "'b" shows "norm (inverse a) = inverse (norm a)"
Int_stable_prod_algebra: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "Int_stable (prod_algebra I M)"
firstocc: shows "\<forall>j<l. cs ! j \<noteq> y"
path_snoc: fixes g :: "'g"   and ns :: "'node list"   and n :: "'node"   and m :: "'node" assumes "local.path g (ns @ [n])"   and "n \<in> set (predecessors g m)" shows "local.path g (ns @ [n, m])"
r_less_def: shows "r_less \<equiv> Cn 2 r_ifle [recf.Id 2 1, recf.Id 2 0, r_constn 1 1, r_constn 1 0]"
t_sum_a_sum: fixes ot :: "'op rose_tree" assumes "local.wf ot" shows "cost_sum ot = acost_sum ot - \<Phi> (state ot)"
not_present_alloc: fixes h :: "heap"   and v :: "'b" shows "\<not> Ref.present h (fst (Ref.alloc v h))"
get_delayed_pool_def: fixes state :: "'b sparc_state" shows "get_delayed_pool state \<equiv> dwrite state"
v'_def: shows "v' = \<lparr>ext = ext v, lan = lan v4 \<squnion> lan v2, own = own v\<rparr>"
gt_sub_Ge': shows "{(x, y). gt y x} \<subseteq> Ge'"
\<sigma>_vaccpt: shows "\<forall>k \<mu> \<eta>. \<mu> U\<^sub>r \<eta> \<in> old (\<sigma> k) \<longrightarrow> (\<exists>j. (\<forall>i<j. {\<mu>, \<mu> U\<^sub>r \<eta>} \<subseteq> old (\<sigma> (k + i))) \<and> \<eta> \<in> old (\<sigma> (k + j)))"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "'a" assumes "S.ide a"   and "S.ide a'"   and "\<guillemotleft>g : R a' \<Rightarrow>\<^sub>S R a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<Rightarrow>\<^sub>S a\<guillemotright> \<and> R f = g"
msb_word_def: fixes a :: "'b word" shows "msb a = (bin_sign (signed_take_bit (LENGTH('b) - 1) (uint a)) = - 1)"
atLeastLessThan_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b"   and u :: "'b" shows "ord.atLeastLessThan less_eq less l u \<equiv> ord.atLeast less_eq l \<inter> ord.lessThan less u"
space_distr: fixes M :: "'e measure"   and N :: "'d measure"   and f :: "'e \<Rightarrow> 'd" shows "space (distr M N f) = space N"
less_imp_le: fixes x :: "'c word"   and y :: "'c word" assumes "x <s y" shows "x \<le>s y"
accI_pos: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "0 < K1.p x y n" shows "(x, y) \<in> K1.acc"
x_axis_line_integral_is_only_horizontal: shows "one_chain_line_integral F {i} (two_chain_boundary two_chain) = one_chain_line_integral F {i} (two_chain_horizontal_boundary two_chain)"
Keys_mat_to_polys: fixes ts :: "'d list"   and A :: "'e mat" shows "Keys (set (mat_to_polys ts A)) \<subseteq> set ts"
Insert: fixes xs :: "'a list"   and x :: "'a" assumes "equal_base.list_distinct ceq' xs \<or> ID CEQ('a) = None"   and "\<not> equal_base.list_member ceq' xs x"   and "P xs" shows "P (equal_base.list_insert ceq' x xs)"
simps_14: fixes C' :: "char list"   and e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp" shows "not_init C' (e\<^sub>1;; e\<^sub>2) = (not_init C' e\<^sub>1 \<and> not_init C' e\<^sub>2)"
sum_product: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'a"   and B :: "'c set" shows "sum f A * sum g B = (\<Sum>i\<in>A. \<Sum>j\<in>B. f i * g j)"
scc_of_in_sccs_verts: fixes u :: "'a" assumes "u \<in> verts G" shows "scc_of u \<in> sccs_verts"
prems_5: shows "\<forall>c l. l \<noteq> literal \<longrightarrow> (c \<in> set (getWatchList state l)) = (Some l = getWatch1 state c \<or> Some l = getWatch2 state c)"
assms_4: shows "set (zip ys YS) \<subseteq> set (zip xs XS)"
not_Some_eq: fixes x :: "\<langle>'d\<rangle>\<^sub>\<bottom>" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
vertex_def: fixes G :: "('k, 'l) graph_scheme"   and x :: "'k" shows "vertex G x = (Domainp (edge G) x \<or> Rangep (edge G) x)"
Sigma_Diff_distrib1: fixes I :: "'c set"   and J :: "'c set"   and C :: "'c \<Rightarrow> 'd set" shows "Sigma (I - J) C = Sigma I C - Sigma J C"
involved_def: fixes oa :: "'action_t option" shows "involved oa \<equiv> case oa of None \<Rightarrow> {} | Some a \<Rightarrow> kinvolved a"
update_convs_1: fixes ok' :: "bool \<Rightarrow> bool"   and ok :: "bool"   and more :: "'b" shows "ok_update ok' \<lparr>ok = ok, \<dots> = more\<rparr> = \<lparr>ok = ok' ok, \<dots> = more\<rparr>"
v: shows "a pm < a \<lbrakk>I (s ! l)\<rbrakk> \<and> a \<lbrakk>I (s ! l)\<rbrakk> < a sm"
card_Un_disjnt: fixes A :: "'b set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "disjnt A B" shows "card (A \<union> B) = card A + card B"
E5: shows "(\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. (\<Sum>i | i < length qs \<and> qs ! i = y. ALG' A qs init i x) + (\<Sum>i | i < length qs \<and> qs ! i = x. ALG' A qs init i y)) = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. \<Sum>i | i < length qs \<and> (qs ! i = y \<or> qs ! i = x). ALG' A qs init i y + ALG' A qs init i x)"
mult_eq_def: fixes r :: "('b \<times> 'b) set" shows "mult_eq r = (mult1 r)\<^sup>*"
CallNull: 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 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 es :: "expr list"   and vs :: "val list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>2\<rangle>"
typeof_unboxed_inversion_3: fixes x :: "('a, 'b, 'c) unboxed" assumes "typeof x = Some Ubx2" shows "\<exists>b. x = OpUbx2 b"
countable_null_set: fixes M :: "'a measure"   and A :: "'a set" assumes "\<And>x. x \<in> space M \<Longrightarrow> {x} \<in> null_sets M"   and "countable A"   and "A \<subseteq> space M" shows "A \<in> null_sets M"
sum_inverse_squares_tail_bound_1: fixes d :: "real" assumes "0 < d" shows "summable (\<lambda>n. 1 / (real (Suc n) + d)\<^sup>2)"
IH: fixes V :: "'b set" assumes "\<forall>l. wf\<^sub>l\<^sub>s\<^sub>t V (proj l A)" shows "wf\<^sub>l\<^sub>s\<^sub>t V A"
arr_inv: fixes f :: "'a" assumes "local.iso f" shows "arr (local.inv f)"
simps_8: fixes v :: "'b poly"   and va :: "nat" shows "degreen (Pw v va) = (\<lambda>m. 0)"
1: shows "a k \<in> carrier (residue_ring (p ^ k))"
2: fixes k :: "nat" shows "set_borel_measurable lebesgue S (f k)"
smap_smap: fixes f :: "'b \<rightarrow> 'a"   and g :: "'c \<rightarrow> 'b"   and xs :: "'c Stream" shows "smap\<cdot>f\<cdot>(smap\<cdot>g\<cdot>xs) = smap\<cdot>(f oo g)\<cdot>xs"
delay_bisimulation_diverge_axioms: shows "delay_bisimulation_diverge trsys1 trsys2 bisim tlsim \<tau>move1 \<tau>move2"
induct: fixes P :: "int \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "int"   and a1 :: "int" assumes "\<And>i j. (i \<le> j \<Longrightarrow> P (i + 1) j) \<Longrightarrow> P i j" shows "P a0 a1"
prems_8: shows "getSATFlag stateA = getSATFlag stateB"
step_wf: shows "wf {(dwa_step WA s, s) |s. s \<in> dwa_invar WA \<and> dwa_cond WA s}"
regions_partition: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and u :: "'d \<Rightarrow> real" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "\<forall>x\<in>X. 0 \<le> u x" shows "\<exists>!R. R \<in> \<R> \<and> u \<in> R"
valid_index_dimsE: fixes "is" :: "nat list"   and A :: "'b tensor" assumes "is \<lhd> dims A" obtains "is = []" and "dims A = []"   | isa :: "nat list" and ds :: "nat list" and i :: "nat" and d :: "nat" where "is = i # isa" and "dims A = d # ds" and "isa \<lhd> ds" and "i < d"
assms_1: shows "liftAll (\<lambda>v. phi v \<longrightarrow> chi v) inp"
1: shows "x \<sqinter> y * (e \<sqinter> f) * (- e \<sqinter> f\<^sup>T) = bot"
ccField_cc_delete: fixes x :: "Vars.var"   and S :: "CoCalls" shows "ccField (cc_delete x S) \<subseteq> ccField S - {x}"
f: shows "f \<in> Hom\<^bsub>set_cat U\<^esub> A B"
is_embedding: fixes f :: "'b"   and f' :: "'b" assumes "F.B.arr f"   and "F.B.arr f'"   and "G f = G f'" shows "f = f'"
subseq_diagonal_rest: fixes k :: "nat" shows "strict_mono (\<lambda>x. nat.fold_reduce k x (k + x))"
rep_eq: fixes x :: "('a, 'b) finfun"   and xa :: "'a"   and xb :: "'b" shows "finfun_apply (finfun_update x xa xb) = (finfun_apply x)(xa := xb)"
matchers_Un: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) set"   and P' :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) set" shows "matchers (P \<union> P') = matchers P \<inter> matchers P'"
auxiliary1_9: fixes d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and s :: "nat"   and m :: "nat"   and n :: "nat" assumes "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "0 < s"   and "m \<le> n"   and "s \<le> m"   and "\<forall>n\<ge>m. real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1))) < d (n + 1) * (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))" shows "real_of_int (a (n + 1)) powr (1 / real_of_int (2 ^ (n + 1))) < prod d {m + 1..n + 1} * (MAX j\<in>{s..m}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))"
sigma_finite_measure_axioms: shows "sigma_finite_measure (count_space \<Omega>)"
supp_\<beta>'': shows "{q. \<beta>'' q \<noteq> 0} \<subseteq> (\<Union>X\<in>Roots_ms. prime_factors (Q X))"
A: shows "finite (pverts G) \<and> card (pverts G) = n \<and> parcs G = {(u, v). (u, v) \<in> pverts G \<times> pverts G \<and> u \<noteq> v}"
is_const_const: fixes name :: "name" shows "is_const (Const name)"
topspace_subtopology_subset: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "topspace (subtopology X S) = S"
mset_subset_eq_mono_add: fixes A :: "'a multiset"   and B :: "'a multiset"   and C :: "'a multiset"   and D :: "'a multiset" assumes "A \<subseteq># B"   and "C \<subseteq># D" shows "A + C \<subseteq># B + D"
prod_tuple: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "span f g"   and "seq h f"   and "seq k g" shows "local.prod h k \<cdot> tuple f g = tuple (h \<cdot> f) (k \<cdot> g)"
hyps_2: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! reduce p)"
iMODb_cut_ge: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and t :: "nat" shows "[ r, mod m, c ] \<down>\<ge> t = (if t \<le> r then [ r, mod m, c ] else if r + m * c < t then {} else [ t + m - Suc ((t - Suc r) mod m), mod m, c - (t + m - Suc r) div m ])"
simps_2: fixes V :: "('a, unit) RBT.rbt"   and v1 :: "'a"   and v2 :: "'a"   and Es :: "('a \<times> 'a) list" shows "wf_list_graph_impl_rs V ((v1, v2) # Es) = (rbt_contains v1 V \<and> rbt_contains v2 V \<and> wf_list_graph_impl_rs V Es)"
E_wf_2: fixes G :: "'c graph" assumes "wf_graph G" shows "snd ` edges G \<subseteq> nodes G"
assms: fixes u :: "real" shows "\<forall>\<^sub>F n in \<U>. u < norm (X n)"
trivial_def: fixes g :: "'g"   and v :: "'val" shows "uninst_code.ssa.trivial uses phis g v \<equiv> \<exists>v'\<in>uninst_code.ssa.allVars uses phis g. uninst_code.ssa.isTrivialPhi phis g v v'"
legacy_Complex_simps_1: fixes a :: "real"   and b :: "real" shows "(Complex a b = 0) = (a = 0 \<and> b = 0)"
sel_4: fixes x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "the_check \<langle>x61: x62 \<in> x63\<rangle> = x61"
nontriv_solution_pell_nat_mul1: fixes z1 :: "nat \<times> nat"   and z2 :: "nat \<times> nat" assumes "solution z1"   and "nontriv_solution z2" shows "nontriv_solution (pell_mul z1 z2)"
one_greatest2: fixes x :: "'a" shows "n x \<oplus> (1::'a) = (1::'a)"
X: shows "{t. t \<le> n \<and> token_run t n = token_run x n} \<noteq> {}"
lset_eq_lsetp: fixes xs :: "'c llist" shows "lset xs = {x. Coinductive_List.lsetp xs x}"
space_state_measure_shift_iff: fixes \<sigma> :: "nat \<Rightarrow> val"   and V :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "(\<sigma> \<in> space (state_measure (shift_var_set V) (case_nat t \<Gamma>))) = (\<exists>x \<sigma>'. x \<in> type_universe t \<and> \<sigma>' \<in> space (state_measure V \<Gamma>) \<and> \<sigma> = case_nat x \<sigma>')"
axioms_4: 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" assumes "valid_ownership_and_sharing \<S> ts" shows "no_outstanding_write_to_read_only_memory \<S> ts"
simps_1: fixes P :: "'c \<Rightarrow> ('a, 'b, bool) prog" shows "filter_M P [] = return []"
augment_edge_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and e :: "nat \<times> nat"   and cap :: "'a" shows "Graph.augment_edge c e cap \<equiv> c(e := c e - cap, prod.swap e := c (prod.swap e) + cap)"
degeneralize_nodes_card: fixes A :: "('e, 'd) dgbta" shows "card (DBTA.nodes (degeneralize A)) \<le> max 1 (length (dgbta.accepting A)) * card (DGBTA.nodes A)"
is_offending_flows_min_set_def: fixes f :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "is_offending_flows_min_set f G nP \<equiv> is_offending_flows f G nP \<and> (\<forall>(e1, e2)\<in>f. \<not> sinvar (add_edge e1 e2 (delete_edges G f)) nP)"
sorted_list_of_set_0_LEAST: fixes I :: "'a set" assumes "finite I"   and "I \<noteq> {}" shows "sorted_list_of_set I ! 0 = (LEAST n. n \<in> I)"
is_root_if_no_Empty: fixes h :: "'b heap" assumes "no_Empty h" shows "is_root h"
resV_netV: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and cf :: "nat \<times> nat \<Rightarrow> 'a" assumes "RPreGraph c s t cf" shows "Graph.V cf = Graph.V c"
not_primitiveE: fixes f :: "'b poly" assumes "\<not> primitive f" obtains x :: "'b" where "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y" and "\<not> x dvd (1::'b)"
integrity: fixes n :: "nat"   and p :: "'proc"   and v :: "bool" assumes "dec (rho n p) = Some v" shows "v \<in> range vals"
zeta_neg_of_nat: fixes n :: "nat" shows "zeta (- of_nat n) = - complex_of_real (bernoulli' (Suc n)) / of_nat (Suc n)"
pos_poly_pCons: fixes a :: "'b"   and p :: "'b poly" shows "pos_poly (pCons a p) = (pos_poly p \<or> p = 0 \<and> (0::'b) < a)"
expand: fixes prod :: "'g \<times> 'h"   and prod' :: "'g \<times> 'h" assumes "fst prod = fst prod' \<and> snd prod = snd prod'" shows "prod = prod'"
assign_post_def: fixes P :: "'BExp set"   and x :: "'Var"   and e :: "'AExp" shows "assign_post P x e \<equiv> if x \<in> aexp_vars e then P |` (- {x}) else P |` (- {x}) \<union> {bexp_assign x e}"
that_2: shows "\<I>2 \<le> \<I>2'"
fps_X_power_mult_nth: fixes k :: "nat"   and f :: "'a fps"   and n :: "nat" shows "(fps_X ^ k * f) $ n = (if n < k then 0::'a else f $ (n - k))"
star_inductr_var: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and y :: "'c"   and x :: "'c" assumes "class.kleene_algebra_zerol plus times one zero less_eq less star"   and "less_eq (times y x) y" shows "less_eq (times y (star x)) y"
map_cong: fixes x :: "'c dual"   and ya :: "'c dual"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "x = ya"   and "\<And>z. z \<in> set_dual ya \<Longrightarrow> f z = g z" shows "map_dual f x = map_dual g ya"
Integral_unique: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and k1 :: "real"   and k2 :: "real" assumes "a \<le> b"   and "Integral (a, b) f k1"   and "Integral (a, b) f k2" shows "k1 = k2"
right_neutral: fixes f :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and z :: "'a::type"   and a :: "'a::type" assumes "monoid (f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (z::'a::type)" shows "(f::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (a::'a::type) (z::'a::type) = a"
insert_rga_commutes: fixes i2 :: "'a option"   and e1 :: "'a"   and i1 :: "'a option"   and e2 :: "'a"   and xs :: "'a list" assumes "i2 \<noteq> Some e1"   and "i1 \<noteq> Some e2" shows "insert_rga (insert_rga xs (e1, i1)) (e2, i2) = insert_rga (insert_rga xs (e2, i2)) (e1, i1)"
ii: shows "(mset_lit L, mset_lit M) \<in> mult trm_ord"
periodic: fixes a :: "nat" shows "principal_dchar n (a + n) = principal_dchar n a"
circuit_elim: fixes C\<^sub>1 :: "'a set"   and C\<^sub>2 :: "'a set"   and x :: "'a" assumes "circuit C\<^sub>1"   and "circuit C\<^sub>2"   and "C\<^sub>1 \<noteq> C\<^sub>2"   and "x \<in> C\<^sub>1 \<inter> C\<^sub>2" shows "\<exists>C\<^sub>3\<subseteq>C\<^sub>1 \<union> C\<^sub>2 - {x}. circuit C\<^sub>3"
has_laplace_unique: fixes f :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex"   and M :: "complex" assumes "(f has_laplace L) s"   and "(f has_laplace M) s" shows "L = M"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "F.A.seq g f" shows "F (A g f) = B (F g) (F f)"
le_list_map: fixes xs :: "'a list"   and ys :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<preceq> ys" shows "map f xs \<preceq> map f ys"
eq_norm: fixes x :: "nat" shows "unat (word_of_nat x) = x mod 2 ^ LENGTH('b)"
cond_case_prod_eta: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<times> 'c \<Rightarrow> 'd" assumes "\<And>x y. f x y = g (x, y)" shows "(\<lambda>(x, y). f x y) = g"
fwi_step': fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i' :: "nat"   and n :: "nat"   and j' :: "nat"   and i :: "nat"   and j :: "nat" assumes "(0::'a) \<le> m k k"   and "i' \<le> n"   and "j' \<le> n"   and "k \<le> n"   and "i \<le> i'"   and "j \<le> j'" shows "fwi m n k i' j' i j = min (m i j) (m i k + m k j)"
below_halfspace_def: fixes sctn :: "'b sctn" shows "below_halfspace sctn = Collect (le_halfspace sctn)"