hcju/isabelleps huggingface dataset:
zip_eq_Nil_iff: fixes xs :: "'b list"   and ys :: "'c list" shows "(zip xs ys = []) = (xs = [] \<or> ys = [])"
mem_samp_primes_excl_prime: fixes prime_set :: "nat set"   and x :: "nat"   and y :: "nat" assumes "rsa_base prime_set"   and "x \<in> set_spmf (sample_set_excl prime_set {y})" shows "prime x"
cdens_ctxt_invar_imp_integrable: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and \<rho> :: "nat \<Rightarrow> val" assumes "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"   and "\<rho> \<in> space (state_measure (set vs') \<Gamma>)" shows "integrable (state_measure (set vs) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (set vs) (set vs') (\<sigma>, \<rho>)) \<delta>))"
monofun_mem: fixes N :: "('g, 'b, 'h) Module_scheme"   and h :: "'a"   and H :: "'a set"   and x :: "'a"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "h \<in> H"   and "free_generator R M H"   and "x \<in> fgs R M {h}"   and "f \<in> H \<rightarrow> carrier N" shows "monofun R M N f H h x \<in> carrier N"
T_def: shows "T = V1 \<inter> U1 \<inter> S"
tweight_def: fixes s :: "thread" shows "tweight s \<equiv> case s of INIT \<Rightarrow> 0 | WAIT x \<Rightarrow> 3 | HOLD x \<Rightarrow> 2 | REL x \<Rightarrow> 1"
vwb_coregion: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "sym_lens S" shows "vwb_lens \<C>\<^bsub>S\<^esub>"
regular_fixpoint: fixes f :: "'a set \<Rightarrow> 'a set"   and I :: "'a set" assumes "regular f" shows "f (limit f I) = limit f I"
unitary11_gen_cis_inversion': fixes k :: "complex"   and M :: "complex \<times> complex \<times> complex \<times> complex"   and b :: "complex" assumes "k \<noteq> 0"   and "M = k *\<^sub>s\<^sub>m (- 1, 0, 0, 1) *\<^sub>m\<^sub>m (0, b, cnj b, 0)"   and "b \<noteq> 0" shows "\<exists>k' \<phi>. k' \<noteq> 0 \<and> M = k' *\<^sub>s\<^sub>m (cis \<phi>, 0, 0, 1) *\<^sub>m\<^sub>m (0, 1, 1, 0)"
nn_integral_count_space_finite: fixes A :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "finite A" shows "integral\<^sup>N (count_space A) f = sum f A"
set_intros_2: fixes y :: "'f"   and x22 :: "'f list"   and x21 :: "'f" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
normalize_idem: fixes p :: "'a list"   and q :: "'a list" shows "local.normalize (local.normalize p @ q) = local.normalize (p @ q)"
inj_on_imp_infinite_image: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "infinite A"   and "inj_on f A" shows "infinite (f ` A)"
Int_insert_left_if0: fixes a :: "'d"   and C :: "'d set"   and B :: "'d set" assumes "a \<notin> C" shows "insert a B \<inter> C = B \<inter> C"
nth_list_update_eq: fixes i :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "i < |xs|" shows "xs[i := x] ! i = x"
assms_2: shows "a t \<noteq> [stop_vc]"
Aleph_def: shows "Aleph \<equiv> transrec3 \<omega> (\<lambda>x. csucc) (\<lambda>i r. \<Squnion> (r ` elts i))"
inject: fixes a :: "'b"   and a' :: "'b" shows "(Def a = Def a') = (a = a')"
weight_spmf_def: fixes p :: "'c spmf" shows "weight_spmf p = Sigma_Algebra.measure (measure_spmf p) UNIV"
remove_key_def: fixes k0 :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b" shows "remove_key k0 f = Abs_poly_mapping (\<lambda>k. lookup f k when k \<noteq> k0)"
is_nextElem1: fixes vs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "distinct vs"   and "x \<in> set vs"   and "nextElem vs (hd vs) x = y" shows "is_nextElem vs x y"
singleton: fixes b :: "'a" assumes "b \<in> Basis" shows "(\<Sum>i\<in>Basis. e i * (k \<bullet> i) * (if i = b then 1 else 0)) = (\<Sum>i\<in>{b}. e i * (k \<bullet> i) * (if i = b then 1 else 0))"
iso_presheaves_of_rings_axioms_def: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and \<FF>' :: "'c set \<Rightarrow> 'e set"   and \<rho>' :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'e \<Rightarrow> 'e"   and b' :: "'e"   and add_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str' :: "'c set \<Rightarrow> 'e"   and one_str' :: "'c set \<Rightarrow> 'e"   and fam_morphisms :: "'c set \<Rightarrow> 'd \<Rightarrow> 'e" shows "iso_presheaves_of_rings_axioms X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms \<equiv> \<exists>\<psi>. morphism_presheaves_of_rings X is_open \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' \<FF> \<rho> b add_str mult_str zero_str one_str \<psi> \<and> (\<forall>U. is_open U \<longrightarrow> (\<forall>x\<in>\<FF>' U. (fam_morphisms U \<circ> \<psi> U) x = x) \<and> (\<forall>x\<in>\<FF> U. (\<psi> U \<circ> fam_morphisms U) x = x))"
imp_dfa_accepts: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list"   and bs :: "bool list list" assumes "wf_dfa M n"   and "wf_dfa N n"   and "list_all (is_alph n) bs" shows "dfa_accepts (imp_dfa M N) bs = (dfa_accepts M bs \<longrightarrow> dfa_accepts N bs)"
even_mult_exp_div_exp_iff: 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"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (divide (times a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)) = (n < m \<or> power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n = zero \<or> m \<le> n \<and> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (n - m))))"
compare_expansions_GT: fixes F :: "'a"   and G :: "'a"   and c :: "real"   and c' :: "real"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and g :: "real \<Rightarrow> real" assumes "compare_expansions F G = (GT, c, c')"   and "trimmed F"   and "(f expands_to F) basis"   and "(g expands_to G) basis"   and "basis_wf basis" shows "g \<in> o(f)"
Ya_ide: fixes a' :: "'c" assumes "C.ide a'" shows "Y a a' = S.mkIde (Hom.set (a', a))"
induct: fixes x1 :: "((Class \<times> Method) \<times> nat) list"   and x2 :: "Expr"   and x3 :: "nat"   and P :: "((Class \<times> Method) \<times> nat) list \<Rightarrow> Expr \<Rightarrow> nat \<Rightarrow> bool" assumes "(x1, x2, x3) \<in> TP_expr"   and "\<And>\<Sigma> e m n. \<lbrakk>(\<Sigma>, e, m) \<in> TP_expr; P \<Sigma> e m; m \<le> n\<rbrakk> \<Longrightarrow> P \<Sigma> e n"   and "\<And>\<Sigma> p n. (\<Sigma>, p, n) \<in> TP_prim \<Longrightarrow> P \<Sigma> (PrimE p) n"   and "\<And>\<Sigma> p k e m n x. \<lbrakk>(\<Sigma>, p, k) \<in> TP_prim; (\<Sigma>, e, m) \<in> TP_expr; P \<Sigma> e m; n = k + m\<rbrakk> \<Longrightarrow> P \<Sigma> (LetE x p e) n"   and "\<And>\<Sigma> e1 n e2 x. \<lbrakk>(\<Sigma>, e1, n) \<in> TP_expr; P \<Sigma> e1 n; (\<Sigma>, e2, n) \<in> TP_expr; P \<Sigma> e2 n\<rbrakk> \<Longrightarrow> P \<Sigma> (CondE x e1 e2) n"   and "\<And>\<Sigma> e1 n e2 x h t. \<lbrakk>(\<Sigma>, e1, n) \<in> TP_expr; P \<Sigma> e1 n; (\<Sigma>, e2, n) \<in> TP_expr; P \<Sigma> e2 n\<rbrakk> \<Longrightarrow> P \<Sigma> (MatchE x e1 h t e2) n" shows "P x1 x2 x3"
order_in_range_Suc: shows "(order G \<in> range Suc) = finite (carrier G)"
assoc_simp: 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 "\<a> f g h = \<a>\<^sub>B[f, g, h]"
coeff_length: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list"   and i :: "nat" assumes "ring R"   and "length p \<le> i" shows "ring.coeff R p i = \<zero>\<^bsub>R\<^esub>"
block_comp_incomplete_nempty: fixes bl :: "'a set" assumes "incomplete_block bl" shows "bl\<^sup>c \<noteq> {}"
inv_unique': fixes x :: "nat"   and y :: "nat" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "y = inv\<^bsub>G\<^esub> x"
Suc3_eq_add_3: fixes n :: "nat" shows "Suc (Suc (Suc n)) = 3 + n"
lowner_lub_is_positive: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "positive (matrix_seq.lowner_lub f)"
mk_coeff_Suc_closed: fixes a :: "'c" shows "mk_coeff vs (c \<circ> Suc) a \<in> carrier R"
embed_iff_compat_inj_on_ofilter: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e" assumes "Well_order r"   and "Well_order r'" shows "embed r r' f = (compat r r' f \<and> inj_on f (Field r) \<and> ofilter r' (f ` Field r))"
hindered_by: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<epsilon> :: "ennreal" assumes "hindrance_by \<Gamma> f \<epsilon>"   and "current \<Gamma> f"   and "wave \<Gamma> f" shows "hindered_by \<Gamma> \<epsilon>"
traces_def: fixes ioa :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set" shows "traces ioa \<equiv> {NF tr |tr. has_trace ioa tr}"
remove: fixes A :: "'c set"   and x :: "'c"   and g :: "'c \<Rightarrow> 'd" assumes "finite A"   and "x \<in> A" shows "prod g A = g x * prod g (A - {x})"
sing_gen: fixes w :: "'b list"   and z :: "'b list" assumes "w \<in> \<langle>{z}\<rangle>" shows "w \<in> z*"
which_node_dyn_standard_control_dependence_source: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n :: "'b"   and as'' :: "'a list"   and n' :: "'b"   and a' :: "'a" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFG.path sourcenode targetnode valid_edge Entry (as @ a # as') n"   and "CFG.path sourcenode targetnode valid_edge n as'' Exit"   and "sourcenode a = n'"   and "sourcenode a' = n'"   and "n \<notin> set (CFG.sourcenodes sourcenode (a # as'))"   and "valid_edge a'"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit n"   and "\<not> Postdomination.postdominate sourcenode targetnode valid_edge Exit n (targetnode a')"   and "\<forall>ax ax'. ax \<in> set as' \<and> sourcenode ax = sourcenode ax' \<and> valid_edge ax' \<longrightarrow> Postdomination.postdominate sourcenode targetnode valid_edge Exit n (targetnode ax')" shows "Postdomination.dyn_standard_control_dependence sourcenode targetnode valid_edge Exit n' n (a # as')"
rhoare_trans: fixes R1 :: "('e \<times> 'f) set"   and T1 :: "('e \<times> 'e) set"   and T2 :: "('f \<times> 'f) set"   and R2 :: "('f \<times> 'g) set"   and T3 :: "('g \<times> 'g) set" assumes "{R1} T1, T2 {> R1}"   and "{R2} T2, T3 {> R2}" shows "{R1 O R2} T1, T3 {> R1 O R2}"
covering_space_local_homeomorphism: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'c" assumes "covering_space c p S"   and "x \<in> c" obtains T :: "'c set" and u :: "'d set" and q :: "'d \<Rightarrow> 'c" where "x \<in> T" and "openin (top_of_set c) T" and "p x \<in> u" and "openin (top_of_set S) u" and "homeomorphism T u p q"
linear_conv_bounded_linear: fixes f :: "'b \<Rightarrow> 'c" shows "linear f = bounded_linear f"
simps_3: fixes locals' :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and globals :: "'a"   and locals :: "'b \<Rightarrow> 'c"   and more :: "'d" shows "locals_update locals' \<lparr>globals = globals, locals = locals, \<dots> = more\<rparr> = \<lparr>globals = globals, locals = locals' locals, \<dots> = more\<rparr>"
s0_secrecyI: fixes x :: "'a s0_state_scheme" assumes "synth (analz (ik x)) \<inter> secret x = {}" shows "x \<in> s0_secrecy"
eventually_nonneg_def: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "eventually_nonneg F f = (\<forall>\<^sub>F x in F. (0::'b) \<le> f x)"
infinite_resources_suffice: fixes f :: "C \<rightarrow> 'a"   and r :: "C" assumes "f\<cdot>r \<noteq> \<bottom>" shows "f\<cdot>C\<^sup>\<infinity> \<noteq> \<bottom>"
gen_model_jAction_eq: fixes M :: "('a, 'p, 'b) KripkeStructure"   and w :: "'b"   and M' :: "('a, 'p, 'b) KripkeStructure"   and w' :: "'b" assumes "gen_model M w = gen_model M' w"   and "w' \<in> worlds (gen_model M' w)"   and "kripke M"   and "kripke M'" shows "jAction M w' = jAction M' w'"
fps_dvd1_right_trivial_unit_factor: fixes f :: "'a fps" assumes "\<exists>g. 1 = g * f" shows "unit_factor f = f"
seqI: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.arr g"   and "\<F>\<^sub>SC.dom g = \<F>\<^sub>SC.cod f" shows "\<F>\<^sub>SC.seq g f"
s2'_def: shows "s2' \<equiv> interpret_floatarith s2 vs"
generalized_zipping_lemma2: fixes \<rho>1 :: "'e V_rec \<Rightarrow> 'e set" assumes "N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"   and "EventSystems.total ES1 (C\<^bsub>\<V>1\<^esub> \<inter> N\<^bsub>\<V>2\<^esub>)"   and "BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>" shows "\<forall>\<tau> lambda t1 t2. set \<tau> \<subseteq> E\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> set lambda \<subseteq> V\<^bsub>\<V>\<^esub> \<and> set t1 \<subseteq> E\<^bsub>ES1\<^esub> \<and> set t2 \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> @ t1 \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> @ t2 \<in> Tr\<^bsub>ES2\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> t1 \<upharpoonleft> C\<^bsub>\<V>1\<^esub> = [] \<and> t2 \<upharpoonleft> C\<^bsub>\<V>2\<^esub> = [] \<longrightarrow> (\<exists>t. \<tau> @ t \<in> Tr\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = lambda \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"
lookup8: fixes x :: "name"   and a :: "coname"   and \<Delta> :: "(coname \<times> ty) list"   and y :: "name" shows "lookupd x a (idc \<Delta> y) = Ax x a"
streams_mono: fixes s :: "'c stream"   and A :: "'c set"   and B :: "'c set" assumes "s \<in> streams A"   and "A \<subseteq> B" shows "s \<in> streams B"
simps_2: fixes k :: "'a" shows "atoms_of_lit (k\<inverse>) = k"
max_Var_floatarith_FDERIV_floatarith: fixes xs :: "nat list"   and d :: "floatarith list"   and fa :: "floatarith" assumes "length xs = length d" shows "max_Var_floatarith (FDERIV_floatarith fa xs d) \<le> max (max_Var_floatarith fa) (max_Var_floatariths d)"
intros_2: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "((f, gap_relabel_effect f l u), f, l) \<in> gap_algo_rel"
compactE: fixes S :: "'a set"   and \<T> :: "'a set set" assumes "compact S"   and "S \<subseteq> \<Union> \<T>"   and "\<And>B. B \<in> \<T> \<Longrightarrow> open B" obtains \<T>' :: "'a set set" where "\<T>' \<subseteq> \<T>" and "finite \<T>'" and "S \<subseteq> \<Union> \<T>'"
add_id_valid: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and i1 :: "'a"   and i2 :: "'a"   and e :: "'b" assumes "xs prefix of j"   and "Deliver (i1, Add i2 e) \<in> set xs" shows "i1 = i2"
replication_num_simple_def: fixes x :: "'a" shows "\<B> rep x = int (card {b \<in> design_support. x \<in> b})"
ValPj0: shows "False \<in> val[pJ,cJ]"
Hash_Enc: fixes X :: "msg"   and X' :: "msg"   and Y' :: "msg" shows "Hash X \<noteq> Enc X' Y'"
simps: fixes nsc :: "('i \<times> 'a ns_constraint) list"   and J :: "'i set"   and asi :: "('i, ('i \<times> 'a atom) list) mapping"   and tv :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and ui :: "'i list"   and s :: "'s" shows "invariant_nsc nsc J ((asi, tv, ui), s) = (case preprocess nsc of (t, as, tv', ui') \<Rightarrow> invariant_as_asi (set as) asi tv tv' ui ui' \<and> invariant_s t (set as \<inter> J \<times> UNIV) s \<and> J \<inter> set ui = {})"
mkIde_def: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set" shows "mkIde A = (if A \<subseteq> Univ \<and> |A| <o natLeq then inv_into (Collect ide) local.set A else null)"
exhaust_sel: fixes tm :: "('c, 'd) tm" obtains "tm = Hd (head tm)"   | "tm = App (fun tm) (arg tm)"
2: shows "w\<^sub>2 @ v\<^sub>2 =\<^sub>F w\<^sub>3"
bind_spmf_def: fixes x :: "'i spmf"   and f :: "'i \<Rightarrow> 'h spmf" shows "x \<bind> f = x \<bind> (\<lambda>a. case a of None \<Rightarrow> return_pmf None | Some a' \<Rightarrow> f a')"
release_not_unshared_no_write: fixes sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option"   and a :: "nat" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}"   and "non_volatile_writes_unshared \<S> sb"   and "release sb (dom \<S>) \<R> a \<noteq> Some False"   and "a \<in> dom (share sb \<S>)" shows "a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb"
wf_1: shows "prefs_from_table_wf agents alts xs"
lookup_mult_scalar_monomial_right_plus: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" shows "lookup (local.punit.mult_scalar p (monomial c v)) (local.punit.splus t v) = lookup p t * c"
cSUP_least: 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 A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and M :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "less_eq (Sup (f ` A)) M"
l_new_element_get_tag_name_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_tag_name type_wf get_tag_name get_tag_name_locs \<equiv> l_new_element type_wf \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_new_element_get_tag_name_axioms get_tag_name get_tag_name_locs"
Auth_fresh_not_AKcryptSK: fixes authK :: "nat"   and evs :: "event list"   and servK :: "nat" assumes "Key authK \<notin> used evs"   and "evs \<in> kerbV" shows "\<not> AKcryptSK authK servK evs"
fps_X_neq_fps_const: fixes c :: "'a" shows "fps_X \<noteq> fps_const c"
mu_repr: shows "mu_repr (IArray.of_fun (\<lambda>i. IArray.of_fun ((!!) (d\<mu>_impl fs_init !! i)) i) m) fs_init"
bdd_below_Ioi: fixes a :: "'a" shows "bdd_below {a<..}"
arith_simps_62: fixes a :: "'a" shows "(0::'a) * a = (0::'a)"
initial_value_def: fixes pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" shows "initial_value pf = constant_image (val_process Mkt pf 0)"
a: shows "k - 1 \<le> length (int_to_bv i) - 2"
spawn: fixes \<iota> :: "nat"   and V :: "('a, 'b) MWLsCom list"   and m :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool" shows "((spawn\<^bsub>\<iota>\<^esub> V, m), V, None, m) \<in> MWLs_semantics.MWLsSteps_det E BMap"
inessential_eq_continuous_logarithm_circle: fixes S :: "'b set"   and f :: "'b \<Rightarrow> complex" shows "(\<exists>a. homotopic_with_canon (\<lambda>h. True) S (sphere 0 1) f (\<lambda>t. a)) = (\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (\<i> * complex_of_real (g x))))"
distinct_69: fixes x7 :: "('a, 'b) form"   and x9 :: "('a, 'b) form" shows "Neg x7 \<noteq> Exists x9"
assms_15: shows "wf_tuple n V z \<and> (\<forall>(A, X)\<in>Q. restrict A z \<in> X) \<and> (\<forall>(A, X)\<in>Qn. restrict A z \<notin> X)"
wordinterval_element_set_eq: fixes el :: "'b word"   and rg :: "'b wordinterval" shows "wordinterval_element el rg = (el \<in> wordinterval_to_set rg)"
embed_measure_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c measure"   and B :: "'c measure" assumes "inj f" shows "(embed_measure A f = embed_measure B f) = (A = B)"
reduction_doesnt_break_things_invertibility_step1: fixes subsets :: "nat list list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots_copr p qs) \<subseteq> set signs"   and "satisfy_equation p qs subsets signs" shows "vec_space.rank (length signs) (reduce_mat_cols (M_mat signs subsets) (solve_for_lhs p qs subsets (M_mat signs subsets))) = length (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets)))"
simps: fixes i :: "nat" shows "gs2.gso i = map of_int_hom.vec_hom fs' ! i + gs.M.sumlist (map (\<lambda>j. - gs2.\<mu> i j \<cdot>\<^sub>v gs2.gso j) [0..<i])"
unat_eq_nat_uint: fixes w :: "'a word" shows "unat w = nat (uint w)"
delete_insert_index: fixes i :: "nat"   and i' :: "nat" shows "delete_index i (insert_index i i') = i'"
basis_reduction_swap_def: fixes m :: "nat"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "LLL_Impl.basis_reduction_swap m i state = (let di = d_state state i; dsi = d_state state (Suc i); dim1 = d_state state (i - 1); fi = fi_state state; fim1 = fim1_state state; dmu_i_im1 = dmu_ij_state state i (i - 1); fi' = fim1; fim1' = fi in case state of (f, dmus, djs) \<Rightarrow> (False, i - 1, dec_i (update_im1 (update_i f fi') fim1'), swap_mu m dmus i dmu_i_im1 dim1 di dsi, iarray_update djs i ((dsi * dim1 + dmu_i_im1 * dmu_i_im1) div di)))"
pseudo_proj_True_shift: fixes l :: "bool list"   and n :: "nat" assumes "length l = n" shows "pseudo_proj_True n (l @- sconst True) = l @- sconst True"
free_gen_condition_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and z :: "'e" shows "free_gen_condition f i a z = (\<forall>n. z \<notin> add_set f {a} n)"
map_simp: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.map f = f"
assoc_add_vecset: fixes A :: "'b vec set"   and n :: "nat"   and B :: "'b vec set"   and C :: "'b vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n"   and "C \<subseteq> carrier_vec n" shows "A + (B + C) = A + B + C"
mergesort_by_rel_permutes: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "mset (mergesort_by_rel R xs) = mset xs"