hcju/isabelleps huggingface dataset:
less_le_not_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less" shows "less x y = strict less_eq x y"
dotprod_eq_0_iff: fixes x :: "nat list"   and a :: "nat list" assumes "length x = length a"   and "0 \<notin> set a" shows "(x \<bullet> a = 0) = (\<forall>e\<in>set x. e = 0)"
intros_1: fixes Rel :: "pi \<Rightarrow> pi \<Rightarrow> bool"   and P :: "pi"   and Q :: "pi" assumes "Rel P Q" shows "bangRelp Rel (!P) (!Q)"
mult_less_one_less_self: fixes x :: "real"   and i :: "real" assumes "0 < x"   and "i < 1" shows "i * x < x"
goodXux: shows "qGood (X #[[u \<and> x]]_xs)"
span_columns_divide_by_norm: fixes A :: "(('a, 'c) vec, 'b) vec" shows "span (columns A) = span (columns (divide_by_norm A))"
halloc_no_jump: fixes G :: "prog"   and s0 :: "state"   and oi :: "obj_tag"   and a :: "loc"   and s1 :: "state"   and j :: "jump" assumes "G\<turnstile>s0 \<midarrow>halloc oi\<succ>a\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)" shows "fst s1 \<noteq> Some (Jump j)"
countable_reachable: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "countable ((SIGMA x:UNIV. set_pmf (K x))\<^sup>* `` {s})"
PQ1: shows "four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2 * four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2 = 1\<^sub>m (n + m)"
2: shows "SubRel.ces (tgt re) res rv2 (subs prb)"
is_singleton_conv_Ex1: fixes A :: "'a set" shows "is_singleton A = (\<exists>!x. x \<in> A)"
starlike_negligible_bounded_gmeasurable: fixes S :: "'b set" assumes "S \<in> sets lebesgue"   and "bounded S"   and "\<And>c x. \<lbrakk>c *\<^sub>R x \<in> S; 0 \<le> c; x \<in> S\<rbrakk> \<Longrightarrow> c = 1" shows "S \<in> null_sets lebesgue"
ball_translation: fixes a :: "'c"   and c :: "'c"   and r :: "real" shows "ball (a + c) r = (+) a ` ball c r"
beta_cases_2: fixes r :: "dB"   and s :: "dB" assumes "Abs r \<rightarrow>\<^sub>\<beta> s" obtains t :: "dB" where "s = Abs t" and "r \<rightarrow>\<^sub>\<beta> t"
rv_edge_slice_kinds: fixes a :: "'edge"   and n :: "'node"   and n'' :: "'node"   and S :: "'node set"   and s :: "'state"   and s' :: "'state"   and as :: "'edge list"   and asx :: "'edge list" assumes "valid_edge a"   and "sourcenode a = n"   and "targetnode a = n''"   and "\<forall>V\<in>rv S n. state_val s V = state_val s' V"   and "preds (slice_kinds S (a # as)) s"   and "preds (slice_kinds S (a # asx)) s'" shows "\<forall>V\<in>rv S n''. state_val (transfer (slice_kind S a) s) V = state_val (transfer (slice_kind S a) s') V"
hyps_16: fixes thesis :: "bool" assumes "\<And>t' xa. \<lbrakk>t' \<in> .[X]; xa \<in> X; V \<subseteq> U; 0 \<notin> (\<lambda>s. s - t') ` (\<lambda>f. f - monomial 1 x) ` S; s = t' + (monomial 1 x + t) + monomial 1 xa; (monomial (1::'a) s, V) \<in> set (fst (split (t' + (monomial 1 x + t)) V ((\<lambda>s. s - t') ` (\<lambda>f. f - monomial 1 x) ` S))); set (snd (split (t' + (monomial 1 x + t)) V ((\<lambda>s. s - t') ` (\<lambda>f. f - monomial 1 x) ` S))) \<subseteq> set (snd (ps1, qs1))\<rbrakk> \<Longrightarrow> thesis"   and "0 \<notin> (\<lambda>f. f - monomial 1 x) ` S"   and "(monomial (1::'a) s, V) \<in> set (fst (ps1, qs1))" shows "thesis"
y_z_diff: shows "(f ^^ (funpow_dist1 f x z - funpow_dist1 f x y)) y = z"
altc_sample_state_def: fixes altc_sample :: "'b \<Rightarrow> ('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'c \<Rightarrow> ('s, 'm) stateT" shows "local.altc_sample_state altc_sample p f = StateT (\<lambda>s. altc_sample p (\<lambda>x. run_state (f x) s))"
Derives1_suffix: fixes \<delta> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and a :: "symbol list"   and b :: "symbol list"   and \<alpha> :: "symbol list"   and N :: "symbol"   and \<beta> :: "symbol list" assumes "Derives1 \<delta> i r (a @ b)"   and "splits_at \<delta> i \<alpha> N \<beta>" shows "is_suffix \<beta> b \<or> is_suffix b \<beta>"
push_down_configuration_token_run: fixes x :: "nat"   and q :: "'b"   and n :: "nat"   and y :: "nat" assumes "x \<in> configuration q n"   and "y \<in> configuration q n" shows "x \<le> n \<and> y \<le> n \<and> token_run x n = token_run y n"
face_of_face: fixes T :: "'a set"   and S :: "'a set"   and f :: "'a set" assumes "T face_of S" shows "(f face_of T) = (f face_of S \<and> f \<subseteq> T)"
digraph_isomorphism_invI: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism" assumes "wf_digraph G"   and "pre_digraph.digraph_isomorphism G hom" shows "pre_digraph.digraph_isomorphism (app_iso hom G) (pre_digraph.inv_iso G hom)"
module_vec_simps_2: fixes ty :: "'b itself"   and n :: "nat" shows "\<zero>\<^bsub>module_vec ty n\<^esub> = 0\<^sub>v n"
update_nil_none: fixes env :: "('f, 'g, 'h) env" shows "update [] None env = env"
risk_neutral_prob_def: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure"   and Mkt :: "('b, 'c) discrete_market"   and r :: "real"   and risk_free_asset :: "'c"   and N :: "'b measure" assumes "rfr_disc_equity_market M F Mkt r risk_free_asset" shows "rfr_disc_equity_market.risk_neutral_prob F Mkt r N = (prob_space N \<and> (\<forall>asset\<in>stocks Mkt. martingale N F (discounted_value r (prices Mkt asset))))"
f: shows "f \<in> {p. p permutes S} \<rightarrow> carrier R"
sgn_div_norm: fixes x :: "'c" shows "sgn x = x /\<^sub>R norm x"
tabL: shows "table (args_n args) (args_L args) rel1"
handshake_phase_rel_def: fixes hp :: "hs_phase"   and in_sync :: "bool"   and ph :: "gc_phase" shows "handshake_phase_rel hp in_sync ph = (case hp of hp_Idle \<Rightarrow> ph = ph_Idle | hp_IdleInit \<Rightarrow> ph = ph_Idle \<or> in_sync \<and> ph = ph_Init | hp_InitMark \<Rightarrow> ph = ph_Init \<or> in_sync \<and> ph = ph_Mark | hp_Mark \<Rightarrow> ph = ph_Mark | hp_IdleMarkSweep \<Rightarrow> ph = ph_Mark \<or> in_sync \<and> ph \<in> {ph_Idle, ph_Sweep})"
res_def: fixes \<eta> :: "nat" shows "MAC.RO.res \<eta> = RES (MAC.RO.rnd_oracle \<eta> \<oplus>\<^sub>O MAC.RO.rnd_oracle \<eta>) Map.empty"
Rep_G_order: fixes a :: "'b G"   and b :: "'b G" shows "(a \<le> b) = (Rep_G a \<le> Rep_G b)"
disjoint_eq_subset_Compl: fixes A :: "'c set"   and B :: "'c set" shows "(A \<inter> B = {}) = (A \<subseteq> - B)"
holomorphic_factor_order_of_zero: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and \<xi> :: "complex"   and n :: "nat" assumes "f holomorphic_on S"   and "open S"   and "\<xi> \<in> S"   and "0 < n"   and "(deriv ^^ n) f \<xi> \<noteq> 0"   and "\<And>i. \<lbrakk>0 < i; i < n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f \<xi> = 0" obtains g :: "complex \<Rightarrow> complex" and r :: "real" where "0 < r" and "g holomorphic_on ball \<xi> r" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> f w - f \<xi> = (w - \<xi>) ^ n * g w" and "\<And>w. w \<in> ball \<xi> r \<Longrightarrow> g w \<noteq> 0"
dim_gen_eigenspace: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and ev :: "'a"   and k :: "nat" assumes "jordan_nf A n_as" shows "dim_gen_eigenspace A ev k = sum_list (map (min k) (map fst (filter (\<lambda>(n, e). e = ev) n_as)))"
is_acc_type_def: fixes G :: "prog"   and P :: "pname"   and T :: "ty" shows "is_acc_type G P T = (is_type G T \<and> G \<turnstile> T accessible_in P)"
seqE_cons_Undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule"   and rs :: "'c rule list"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t" obtains ti :: "state" where "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], Undecided\<rangle> \<Rightarrow> ti" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, ti\<rangle> \<Rightarrow> t" and "no_matching_Goto \<gamma> p [r]"   | m :: "'c match_expr" and chain :: "char list" and rs' :: "'c rule list" where "r = Rule m (Goto chain)" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m (Goto chain)], Undecided\<rangle> \<Rightarrow> t" and "matches \<gamma> m p" and "\<Gamma> chain = Some rs'"
count_le_replicate_mset_subset_eq: fixes n :: "nat"   and M :: "'b multiset"   and x :: "'b" shows "(n \<le> count M x) = (replicate_mset n x \<subseteq># M)"
a: shows "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk> \<rightarrow> IMor g"
SourcesDef: fixes i :: "AbstrLevelsID"   and C :: "CSet" shows "Sources i C = DSources i C \<union> \<Union> (Sources i ` DSources i C)"
ex: shows "\<exists>L. subspace L \<and> affine_parallel S L"
RedFAssNull: fixes E :: "char list \<Rightarrow> ty option"   and F :: "char list"   and Cs :: "char list list"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>null\<bullet>F{Cs} := Val v,s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
port_range: fixes s :: "'c"   and e :: "'d"   and port :: "16 word" shows "s = (0::'c) \<and> e = (65535::'d) \<longrightarrow> port \<le> 65535"
surjective: fixes r :: "('c, 'd) partial_object_scheme" shows "r = \<lparr>carrier = carrier r, \<dots> = partial_object.more r\<rparr>"
PhiPlus_is_Phi_Suc: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "n < length qs" shows "BIT_Off.PhiPlus acts qs init n = BIT_Off.Phi acts qs init (Suc n)"
rel_compp: fixes R1 :: "'a \<Rightarrow> 'f \<Rightarrow> bool"   and S1 :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'g \<Rightarrow> bool"   and S2 :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_var (R1 OO S1) (R2 OO S2) = rel_var R1 R2 OO rel_var S1 S2"
hyps_3: obtains w :: "'action list" where "path w p" and "target w p = q" and "set w \<inter> cuts = {}" and "w \<noteq> []"
crit_def: shows "crit = (\<lambda>U s. s \<in> S \<and> (\<forall>u\<in>U. \<not> P u s))"
compress_normalize_primitive_hasdisc_negated: fixes disc2 :: "'c \<Rightarrow> bool"   and neg :: "bool"   and m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and f :: "'d negation_type list \<Rightarrow> ('d list \<times> 'd list) option"   and m' :: "'c match_expr" assumes "\<not> has_disc_negated disc2 neg m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc2 (C a)"   and "normalized_nnf_match m"   and "compress_normalize_primitive (disc, sel) C f m = Some m'" shows "normalized_nnf_match m' \<and> \<not> has_disc_negated disc2 neg m'"
assms: shows "split_block (four_block_mat A B C D) (dim_row A) (dim_col A) = (U, X, Y, V)"
Rep_myopt_inverse: fixes x :: "'d myopt" shows "Abs_myopt (Rep_myopt x) = x"
lemma3_5_d: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and \<tau> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>' :: "'a \<times> ('b \<times> 'a) list"   and \<tau>' :: "'a \<times> ('b \<times> 'a) list"   and \<upsilon> :: "'a \<times> ('b \<times> 'a) list"   and \<sigma>'' :: "'a \<times> ('b \<times> 'a) list"   and \<upsilon>' :: "'a \<times> ('b \<times> 'a) list" assumes "diagram ars (\<tau>, \<sigma>, \<sigma>', \<tau>')"   and "diagram ars (\<upsilon>, \<sigma>', \<sigma>'', \<upsilon>')" shows "diagram ars ((fst \<tau>, snd \<tau> @ snd \<upsilon>), \<sigma>, \<sigma>'', fst \<tau>', snd \<tau>' @ snd \<upsilon>')"
a: shows "homotopic_with_canon (\<lambda>x. True) S S id (\<lambda>x. a)"
ultra_filter_prime: fixes 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 F :: "'c set" assumes "class.distrib_lattice inf less_eq less sup"   and "(ord.filter less_eq F \<and> F \<noteq> UNIV) \<and> (\<forall>G. (ord.filter less_eq G \<and> G \<noteq> UNIV) \<and> F \<subseteq> G \<longrightarrow> F = G)" shows "(ord.filter less_eq F \<and> F \<noteq> UNIV) \<and> (\<forall>x y. sup x y \<in> F \<longrightarrow> x \<in> F \<or> y \<in> F)"
r_s_def: shows "r_s \<equiv> SOME r_s. recfn 1 r_s \<and> Partial_Recursive.total r_s \<and> s = (\<lambda>x. eval r_s [x])"
inj_on_set_zip_map: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and x1 :: "'a"   and y1 :: "'c"   and xs :: "'a list"   and ys :: "'c list" assumes "inj_on f X"   and "(f x1, y1) \<in> set (zip (map f xs) ys)"   and "set xs \<subseteq> X"   and "x1 \<in> X"   and "length xs = length ys" shows "(x1, y1) \<in> set (zip xs ys)"
restrict_spmf_trivial': fixes p :: "'h spmf"   and A :: "'h set" assumes "set_spmf p \<subseteq> A" shows "p \<upharpoonleft> A = p"
exhaust: fixes y :: "complex" obtains x1 :: "real" and x2 :: "real" where "y = Complex x1 x2"
left_unique_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "left_unique R = (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y)"
delete_id: fixes k :: "'a"   and al :: "('a \<times> 'b) list" assumes "k \<notin> fst ` set al" shows "delete k al = al"
bottom_lower: fixes x :: "'a hrel" assumes "x \<in> carrier thy_order" shows "le thy_order \<^bold>\<bottom> x"
mins_nonempty: fixes i :: "int"   and j :: "int" assumes "i < j" shows "mins i j \<noteq> {}"
order_refl: fixes x :: "('c, 'd) Term.term" shows "x \<sqsubseteq> x"
start_heap_data_def: shows "start_heap_data = foldl create_initial_object (empty_heap, [], True) initialization_list"
filter_eq_Cons_iff: fixes P :: "'c \<Rightarrow> bool"   and ys :: "'c list"   and x :: "'c"   and xs :: "'c list" shows "(filter P ys = x # xs) = (\<exists>us vs. ys = us @ x # vs \<and> (\<forall>u\<in>set us. \<not> P u) \<and> P x \<and> xs = filter P vs)"
hom_prod_mset: fixes hom :: "'a \<Rightarrow> 'b"   and X :: "'a multiset" assumes "comm_monoid_mult_hom hom" shows "hom (\<Prod>\<^sub># X) = \<Prod>\<^sub># (image_mset hom X)"
span_explicit: fixes b :: "'a set" shows "cspan b = {\<Sum>a\<in>t. r a *\<^sub>C a |t r. finite t \<and> t \<subseteq> b}"
antilinear: fixes f :: "'a \<Rightarrow> 'b" assumes "bounded_antilinear f" shows "antilinear f"
normal_Q: shows "Group.normal (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>"
le_inf_def: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2 \<equiv> \<forall>v\<^sub>1. v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1 \<longrightarrow> v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
simps_1: fixes prec :: "nat"   and f :: "float interval"   and I :: "float interval list"   and a :: "float interval list" shows "compute_bound_poly prec (f)\<^sub>p I a = f"
graph_path_def: fixes \<alpha>e :: "'d \<Rightarrow> ('e \<times> 'f \<times> 'e) set"   and \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list" shows "graph_path \<alpha>e \<alpha>n invar inEdges' \<equiv> graph (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar \<and> graph_nodes_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g. foldri (\<alpha>n g)) \<and> graph_pred_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g n. foldri (inEdges' g n))"
subresultant_zero_lt: fixes k :: "nat"   and f :: "'b poly"   and g :: "'b poly"   and J :: "nat" assumes "k < degree f + (degree g - J)"   and "J \<le> degree f"   and "J \<le> degree g"   and "J < k" shows "coeff (subresultant J f g) k = (0::'b)"
exhaust: fixes y :: "res_step" obtains x1 :: "res_crash" where "y = RSCrash x1"   | x21 :: "nat" and x22 :: "v list" where "y = RSBreak x21 x22"   | x3 :: "v list" where "y = RSReturn x3"   | x4 :: "e list" where "y = RSNormal x4"
rel_nabla_prop: fixes R :: "('a \<times> 'a) set" shows "(rel_nabla R = {}) = (\<forall>P. P \<subseteq> relfdia R P \<longrightarrow> P = {})"
assms_1: fixes v :: "i" shows "[\<lparr>A!,x\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<diamond>\<lparr>E!,x\<rparr> in v]"
arcs_subdivideD: fixes x :: "'b \<times> 'b"   and G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b"   and w :: "'b" assumes "x \<in> parcs (subdivide G e w)"   and "fst x \<noteq> w"   and "snd x \<noteq> w" shows "x \<in> parcs G"
SeqStTermP_imp_VarP: fixes v :: "tm"   and i :: "tm"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqStTermP v i t u s k} \<turnstile> VarP v"
id3: shows "subresultant j (F 1) (F 2) = Polynomial.smult (inverse (\<Prod>l\<leftarrow>[3..<i]. \<alpha> l ^ (n (l - 1) - j)) * (\<Prod>l\<leftarrow>[3..<i]. \<beta> l ^ (n (l - 1) - j) * f (l - 1) ^ (\<delta> (l - 2) + \<delta> (l - 1)) * (- 1) ^ ((n (l - 2) - j) * (n (l - 1) - j)))) (subresultant j (F (i - 2)) (F (i - 1)))"
ccApprox_strict: shows "ccApprox \<bottom> = \<bottom>"
defect_less: fixes v :: "real \<Rightarrow> real"   and v' :: "real \<Rightarrow> real"   and t0 :: "real"   and t1 :: "real"   and w :: "real \<Rightarrow> real"   and w' :: "real \<Rightarrow> real"   and e :: "real"   and f :: "real \<Rightarrow> real \<Rightarrow> real" assumes "(v has_vderiv_on v') {t0<..t1}"   and "(w has_vderiv_on w') {t0<..t1}"   and "t0 < t1"   and "0 < e"   and "t0 + e \<le> t1"   and "\<And>t. \<lbrakk>t0 < t; t < t0 + e\<rbrakk> \<Longrightarrow> v t < w t"   and "\<And>t. \<lbrakk>t0 < t; t \<le> t1\<rbrakk> \<Longrightarrow> v' t - f t (v t) < w' t - f t (w t)" shows "\<forall>t\<in>{t0<..t1}. v t < w t"
intros_2: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "swapidseq n p"   and "a \<noteq> b" shows "swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
octo_of_int_def: fixes r :: "int" shows "octo_of_int r = real_of_int r *\<^sub>R 1"
eq_neutr_iff: fixes a :: "nat"   and b :: "nat" shows "(gcd a b = 0) = (a = 0 \<and> b = 0)"
assms_6: fixes a :: "floatarith" shows "subterms (Unop a) = insert (Unop a) (subterms a)"
fwi_mono: fixes i :: "nat"   and n :: "nat"   and j :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i' :: "nat"   and j' :: "nat" assumes "i \<le> n"   and "j \<le> n" shows "fwi m n k i' j' i j \<le> m i j"
witnessed_omit_branch: fixes p :: "('b, 'c) fm"   and a :: "'c"   and xs :: "(nat \<times> nat) set"   and branch :: "(('b, 'c) fm list \<times> 'c) list" assumes "witnessed p a (omit_branch xs branch)" shows "witnessed p a branch"
red1_max_vars: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "max_vars e' \<le> max_vars e"
less_eq_st_rep_iff: fixes r1 :: "(char list \<times> 'a) list"   and r2 :: "(char list \<times> 'a) list" shows "less_eq_st_rep r1 r2 = (\<forall>x. fun_rep r1 x \<le> fun_rep r2 x)"
hyp2_axiom9: shows "\<forall>p q a b c. p \<noteq> q \<and> a p \<congruent>\<^sub>K a q \<and> b p \<congruent>\<^sub>K b q \<and> c p \<congruent>\<^sub>K c q \<longrightarrow> B\<^sub>K a b c \<or> B\<^sub>K b c a \<or> B\<^sub>K c a b"
S3_excl: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> S3 rmhist p \<longrightarrow> S3 rmhist p \<and> \<not> S1 rmhist p \<and> \<not> S2 rmhist p"
Assign': fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and x :: "char list"   and a :: "aexp"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<forall>s. eSuc (Q (s[a/x])) \<le> P s" shows "\<turnstile>\<^sub>2 {P} x ::= a {Q}"
rbt_lookup_keys: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt" assumes "class.linorder less_eq less"   and "ord.rbt_sorted less t" shows "dom (ord.rbt_lookup less t) = set (RBT_Impl.keys t)"
match_list_normalize_match: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> [m] a p = match_list \<gamma> (normalize_match m) a p"
empty_correct: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "(q = []) = (SkewBinomialHeapStruc.queue_to_multiset q = {#})"
density_context_axioms: shows "density_context V V' \<Gamma> (\<lambda>\<sigma>. \<delta> \<sigma> * (if P \<sigma> then 1 else 0))"
cBase: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'" shows "Prop P P'"
assms_4: shows "flow0 x t \<noteq> flow0 x t'"
semicontract_Tn: fixes n :: "nat"   and x :: "'a"   and y :: "'a" shows "dist ((T ^^ n) x) ((T ^^ n) y) \<le> dist x y"
le_ASSERTI: fixes \<Phi> :: "bool"   and M :: "'b nres"   and M' :: "'b nres" assumes "\<Phi> \<Longrightarrow> M \<le> M'" shows "M \<le> ASSERT \<Phi> \<bind> (\<lambda>_. M')"
length_sim_delta: fixes s :: "'s" assumes "wf_state s" shows "length (sim_delta s) = length (\<sigma> n)"
commute: fixes a :: "'c"   and b :: "'c" shows "a OR b = b OR a"
UN_I: fixes a :: "'a"   and A :: "'a set"   and b :: "'d"   and B :: "'a \<Rightarrow> 'd set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
DirProds_group_imp_groups: fixes G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme"   and I :: "'d set"   and i :: "'d" assumes "Group.group (DirProds G I)"   and "i \<in> I" shows "Group.group (G i)"
span_image: fixes S :: "'b set" shows "vs2.span (f ` S) = f ` vs1.span S"
ltln_expand_StrongRelease: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = \<xi> \<Turnstile>\<^sub>n \<psi> and\<^sub>n \<phi> or\<^sub>n X\<^sub>n (\<phi> M\<^sub>n \<psi>)"
freshStarPair_2: fixes xvec :: "name list"   and x :: "'d"   and y :: "'e" shows "xvec \<sharp>* (x, y) = (xvec \<sharp>* x \<and> xvec \<sharp>* y)"
minus_QDelta_def: fixes qd1 :: "QDelta"   and qd2 :: "QDelta" shows "qd1 - qd2 = QDelta (qdfst qd1 - qdfst qd2) (qdsnd qd1 - qdsnd qd2)"
I_intP_def: fixes p :: "'psym"   and al :: "univ list" shows "I_intP p al \<equiv> case polC p of Cext \<Rightarrow> intP p (map2 proj (parOf p) al) | Text \<Rightarrow> if list_all2 intT (parOf p) al then intP p al else True | Fext \<Rightarrow> if list_all2 intT (parOf p) al then intP p al else False"
offset_poly_eq_0_lemma: fixes c :: "'a"   and p :: "'a poly"   and a :: "'a" assumes "smult c p + pCons a p = 0" shows "p = 0"
emeasure_real: fixes s :: "'s"   and A :: "'s stream set" shows "\<exists>r\<ge>0. emeasure (T s) A = ennreal r"
le: shows "(\<Sum>n. norm (fps_nth f (Suc n)) * \<delta> ^ Suc n) \<le> 1"
path_appendE: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and l1 :: "'a"   and l3 :: "'a"   and xs :: "('a \<times> 'b \<times> 'a) list"   and ys :: "('a \<times> 'b \<times> 'a) list" assumes "graph weights"   and "graph.path weights l1 l3 (xs @ ys)" shows "\<exists>l2. graph.path weights l2 l3 ys \<and> graph.path weights l1 l2 xs"
assms_2: shows "c\<^sub>2 \<sqsubseteq> c\<^sub>1"
g_bounds: fixes x :: "real" assumes "1 \<le> x" shows "g x \<in> {c..c + 1 / (12 * x)}"
copr_nonz: shows "coprime_r (cast_rat_list fs) \<noteq> 0"
idx_of_impl_def: fixes SBIP :: "'b list \<times> nat list \<times> ('b \<Rightarrow> node_state option) \<times> (nat \<times> 'b set) list"   and v :: "'b" shows "GS.idx_of_impl SBIP v \<equiv> ASSERT (\<exists>i. GS.I SBIP v = Some (STACK i)) \<bind> (\<lambda>_. let j = GS.S_idx_of SBIP v in ASSERT (j < length (GS.S SBIP)) \<bind> (\<lambda>_. Let (GS.find_seg_impl SBIP j) RETURN))"
abs_in_float_intervalI: fixes a :: "real"   and A :: "float interval" assumes "a \<in>\<^sub>r A" shows "\<bar>a\<bar> \<in>\<^sub>r abs_interval A"