hcju/isabelleps huggingface dataset:
bdy: shows "\<forall>s. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' s) bdy {t. return s t \<in> R s t},{t. return s t \<in> A}"
pwd_owner_v_def: fixes n :: "nat" shows "pwd_owner_v n \<equiv> stage_owner_v n (pwd_spy_iv n)"
ldistinct_lprefix: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "ldistinct xs"   and "lprefix ys xs" shows "ldistinct ys"
check_addition_l_def: fixes spec :: "(char list list \<times> int) list"   and A :: "(nat, (char list list \<times> int) list) fmap"   and \<V> :: "char list set"   and p :: "nat"   and q :: "nat"   and i :: "nat"   and r :: "(char list list \<times> int) list" shows "check_addition_l spec A \<V> p q i r = (let b = p \<in># dom_m A \<and> q \<in># dom_m A \<and> i \<notin># dom_m A \<and> vars_llist r \<subseteq> \<V> in if \<not> b then RETURN (error_msg i ((if p \<notin># dom_m A then error_msg_notin_dom p else []) @ (if q \<notin># dom_m A then error_msg_notin_dom p else []) @ (if i \<in># dom_m A then error_msg_reused_dom p else []))) else ASSERT (p \<in># dom_m A) \<bind> (\<lambda>_. let p = the (fmlookup A p) in ASSERT (q \<in># dom_m A) \<bind> (\<lambda>_. let q = the (fmlookup A q) in add_poly_l (p, q) \<bind> (\<lambda>pq. weak_equality_l pq r \<bind> (\<lambda>b. weak_equality_l r spec \<bind> (\<lambda>b'. if b then if b' then RETURN CFOUND else RETURN CSUCCESS else check_not_equal_dom_err p q pq r \<bind> (\<lambda>c. RETURN (error_msg i c))))))))"
topspace_subtopology_subset: fixes S :: "'a set"   and X :: "'a topology" assumes "S \<subseteq> topspace X" shows "topspace (subtopology X S) = S"
induct: fixes P :: "typ list \<Rightarrow> Core.term \<Rightarrow> bool"   and a0 :: "typ list"   and a1 :: "Core.term" assumes "\<And>uu_ uv_ T. P uu_ (Ct uv_ T)"   and "\<And>Ts i. P Ts (Bv i)"   and "\<And>uw_ ux_ T. P uw_ (Fv ux_ T)"   and "\<And>Ts T body. P (T # Ts) body \<Longrightarrow> P Ts (Abs T body)"   and "\<And>Ts t u. \<lbrakk>P Ts u; \<And>a. typ_of1 Ts u = Some a \<Longrightarrow> P Ts t\<rbrakk> \<Longrightarrow> P Ts (t $ u)" shows "P a0 a1"
WhileTrue: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int option"   and c :: "com"   and s' :: "(char list \<Rightarrow> int option) option"   and s'' :: "(char list \<Rightarrow> int option) option" assumes "Def_Init_Exp.bval b s = Some True"   and "(c, Some s) \<Rightarrow> s'"   and "(WHILE b DO c, s') \<Rightarrow> s''" shows "(WHILE b DO c, Some s) \<Rightarrow> s''"
factorial_condition_one: fixes G :: "('c, 'd) monoid_scheme" shows "(divisor_chain_condition_monoid G \<and> primeness_condition_monoid G) = factorial_monoid G"
coprime_0_2: fixes p :: "'a" shows "coprime (0::'a) p = (p dvd (1::'a))"
step_converter: fixes step :: "nat \<Rightarrow> atom list \<Rightarrow> atom fm list \<Rightarrow> atom fm"   and var :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and \<Gamma> :: "real list" assumes "\<And>xs var L F \<Gamma>. length xs = var \<Longrightarrow> (\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval (step var L F) (xs @ x # \<Gamma>))" shows "(\<exists>xs. length xs = var + 1 \<and> eval (list_conj (map fm.Atom L @ F)) (xs @ \<Gamma>)) = (\<exists>xs. length xs = var + 1 \<and> eval (step var L F) (xs @ \<Gamma>))"
alpha_Tree_eqvt': fixes t1 :: "('b, 'c, 'd) Tree"   and t2 :: "('b, 'c, 'd) Tree"   and p :: "perm" shows "(t1 =\<^sub>\<alpha> t2) = (p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2)"
is_pole_def: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c" shows "is_pole f a = filterlim f at_infinity (at a)"
normalize_alt_simps_12: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and op :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and as :: "'a expr list"   and call :: "'a call_expr" shows "((\<Gamma>, \<tau>) \<turnstile>\<^sub>C Operation op as \<Rrightarrow> call) = (\<exists>params\<^sub>2. call = Operation op params\<^sub>2 \<and> \<Gamma> \<turnstile>\<^sub>L as \<Rrightarrow> params\<^sub>2)"
C1_differentiable_compose: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> 'a" assumes "f C1_differentiable_on S"   and "g C1_differentiable_on f ` S"   and "\<And>x. finite (S \<inter> f -` {x})" shows "g \<circ> f C1_differentiable_on S"
smult_1_left: fixes p :: "'a poly" shows "Polynomial.smult (1::'a) p = p"
path_independent_on_Chernoff_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "path_independent_on A f"   and "f_range_on A f" shows "Chernoff_on A f"
HomFtorContraFtor: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "Ftor (Hom\<^bsub>C\<^esub>[\<midarrow>,X]) : Op C \<longrightarrow> SET"
Sigma: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "finite A"   and "\<forall>x\<in>A. finite (B x)" shows "(\<Sum>x\<in>A. sum (g x) (B x)) = (\<Sum>(x, y)\<in>Sigma A B. g x y)"
fundadjset_eq_S_image: fixes D :: "'a set" assumes "D \<in> adjacentset C0 - {C0}" shows "\<exists>s\<in>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}. D = s `\<rightarrow> C0"
abc_Hoare_haltI: fixes P :: "nat list \<Rightarrow> bool"   and p :: "abc_inst list"   and Q :: "nat list \<Rightarrow> bool" assumes "\<And>lm. P lm \<Longrightarrow> \<exists>n. abc_final (abc_steps_l (0, lm) p n) p \<and> Q abc_holds_for abc_steps_l (0, lm) p n" shows "{P} p {Q}"
cmp\<^sub>D\<^sub>N_in_hom_2: fixes \<mu>\<nu> :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "arr (fst \<mu>\<nu>)"   and "arr (snd \<mu>\<nu>)"   and "src (fst \<mu>\<nu>) = trg (snd \<mu>\<nu>)" shows "\<guillemotleft>cmp\<^sub>D\<^sub>N \<mu>\<nu> : DN (local.dom (fst \<mu>\<nu>)) \<star>\<^sub>B DN (local.dom (snd \<mu>\<nu>)) \<Rightarrow>\<^sub>B DN (cod (fst \<mu>\<nu>) \<star> cod (snd \<mu>\<nu>))\<guillemotright>"
higher_deriv_comp_lemma: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex"   and z :: "complex"   and t :: "complex set"   and g :: "complex \<Rightarrow> complex"   and i :: "nat"   and n :: "nat" assumes "open s"   and "f holomorphic_on s"   and "z \<in> s"   and "open t"   and "g holomorphic_on t"   and "f ` s \<subseteq> t"   and "i \<le> n"   and "deriv f z = 1"   and "\<And>i. \<lbrakk>1 < i; i \<le> n\<rbrakk> \<Longrightarrow> (deriv ^^ i) f z = 0" shows "(deriv ^^ i) (g \<circ> f) z = (deriv ^^ i) g (f z)"
card_subset_eq: fixes B :: "'b set"   and A :: "'b set" assumes "finite B"   and "A \<subseteq> B"   and "card A = card B" shows "A = B"
bot_unique: fixes a :: "'a" shows "(a \<le> bot) = (a = bot)"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "indRelLp a1 a2 = (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2)"
doctor_optimal_matchI: fixes ds :: "'b set"   and Y :: "'a set" assumes "stable_on ds Y"   and "\<And>X x. \<lbrakk>x \<in> X; stable_on ds X\<rbrakk> \<Longrightarrow> \<exists>y\<in>Y. (x, y) \<in> Pd (Xd x)" shows "doctor_optimal_match ds Y"
code_status_status_rel_def: shows "code_status_status_rel = {(CFOUND, FOUND), (CSUCCESS, SUCCESS)} \<union> {(CFAILED a, FAILED) |a. True}"
If_eq_bottom_iff: fixes x :: "'a"   and y :: "'a"   and b :: "tr" shows "(If b then x else y = \<bottom>) = (b = \<bottom> \<or> b = TT \<and> x = \<bottom> \<or> b = FF \<and> y = \<bottom>)"
prems_1: shows "\<forall>op\<in>set (op # \<psi>). are_operator_effects_consistent op op"
cd_closed: fixes n' :: "'node"   and S :: "'node set"   and n :: "'node" assumes "n' \<in> PDG_BS S"   and "n controls\<^sub>s n'" shows "n \<in> PDG_BS S"
EF: shows "is_sentence [A] \<and> is_sentence (u @ [B] @ v) \<and> LeftDerivation [A] D (u @ [B] @ v) \<and> 0 < length [A] \<and> length u < length (u @ [B] @ v) \<and> [A] ! 0 = (u @ [B] @ v) ! length u \<and> D = E @ derivation_shift F 0 (Suc (length u)) \<and> LeftDerivation (take 0 [A]) E (take (length u) (u @ [B] @ v)) \<and> LeftDerivation (drop (Suc 0) [A]) F (drop (Suc (length u)) (u @ [B] @ v))"
typeof_lit_eq_Void: fixes v :: "'a val" shows "(typeof v = \<lfloor>Void\<rfloor>) = (v = Unit)"
L3_5iv: fixes b :: "'b set"   and c1 :: "'b com"   and c2 :: "'b com"   and X :: "'b set" shows "SEM (Cond b c1 c2) X = SEM c1 (X \<inter> b) \<union> SEM c2 (X \<inter> - b)"
Int_greaterThanLessThan: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "ord.greaterThanLessThan less a b \<inter> ord.greaterThanLessThan less c d = ord.greaterThanLessThan less (ord.max less_eq a c) (ord.min less_eq b d)"
equiv_test_def: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and b1 :: "'d"   and b2 :: "'d" shows "equiv_test ifex_of b1 b2 = (let t1 = ifex_of b1; t2 = ifex_of b2 in Trueif = normif Mapping.empty t1 t2 (normif Mapping.empty t2 Falseif Trueif))"
norm_divide: fixes a :: "'d"   and b :: "'d" shows "norm (a / b) = norm a / norm b"
vec_plus: fixes nr :: "nat"   and u :: "'c list"   and v :: "'c list"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "vec nr u"   and "vec nr v" shows "vec nr (vec_plusI pl u v)"
drefinesD2: fixes \<phi> :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> bool"   and A :: "bool \<Rightarrow> ('d \<Rightarrow> real) \<Rightarrow> 'd \<Rightarrow> real"   and B :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'d \<Rightarrow> real"   and Q :: "'d \<Rightarrow> real"   and s :: "'c" assumes "drefines \<phi> G A B"   and "unitary P"   and "unitary Q"   and "P \<tturnstile> wp A Q"   and "G s" shows "(P \<circ> \<phi>) s \<le> wp B (Q \<circ> \<phi>) s"
fin_eep: shows "finite ((ee \<circ> p) ` k)"
4: shows "\<forall>i. T (v (Y i)) \<subseteq> T (u (Y i))"
select_convs_1: fixes carrier :: "'e set"   and mult :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and more :: "'f" shows "(\<otimes>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub>) = mult"
sym_group_is_group: fixes n :: "nat" shows "Group.group (sym_group n)"
case_prodD': fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'c"   and b :: "'d"   and c :: "'e" assumes "(case (a, b) of (c, d) \<Rightarrow> R c d) c" shows "R a b c"
vector_space_on_axioms: shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1"
integrable_straddle_interval: fixes a :: "'a"   and b :: "'a"   and f :: "'a \<Rightarrow> real" assumes "\<And>e. 0 < e \<Longrightarrow> \<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and> \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>cbox a b. g x \<le> f x \<and> f x \<le> h x)" shows "f integrable_on cbox a b"
not_eq_extremum: fixes a :: "'b list" shows "(a \<noteq> []) = strict_prefix [] a"
Inf_from_mono: fixes N :: "'a set"   and N' :: "'a set"   and Inf :: "'a inference set" assumes "N \<subseteq> N'" shows "inference_system.Inf_from Inf N \<subseteq> inference_system.Inf_from Inf N'"
assms: shows "\<langle>a, b\<rangle> \<^bold>\<in> A * B"
connected_edge: fixes v :: "Vertices"   and w :: "Vertices" assumes "v \<rightarrow>\<^bsub>T2\<^esub> w" shows "connected v w"
strip_while_append_rec: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "P x" shows "strip_while P (xs @ [x]) = strip_while P xs"
simps_1: fixes hosts :: "'b set"   and flows_fix :: "('b \<times> 'b) set"   and flows_state :: "('b \<times> 'b) set"   and more :: "'c" shows "hosts \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = hosts"
igSubstIPresIGWlsSTR_imp_igSubstIPresIGWls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstIPresIGWlsSTR Delta MOD" shows "FixSyn.igSubstIPresIGWls Delta MOD"
PDG_cdep_edge: fixes n :: "'node"   and n' :: "'node" assumes "n controls\<^sub>s n'" shows "PDG.PDG_edge sourcenode targetnode valid_edge Def Use standard_control_dependence n None n'"
cs_not_not_started_stable: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and cid :: "nat"   and p :: "'p"   and q :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "snd (cs c cid) \<noteq> NotStarted"   and "channel cid = Some (p, q)" shows "snd (cs c' cid) \<noteq> NotStarted"
the_model_def: fixes C :: "'b set"   and Rs :: "(('b Standard_Constant, nat) labeled_graph \<times> ('b Standard_Constant, nat) labeled_graph) set" shows "the_model C Rs \<equiv> let L = fst ` \<Union> ((edges \<circ> snd) ` Rs) \<union> {S_Bot, S_Top, S_Idt} \<union> S_Const ` C; Rules = Rs \<union> standard_rules C L; sel = non_constructive_selector Rules in the_lcg sel Rules (0, {})"
normalize_poly_old_def: fixes f :: "'b poly" shows "normalization_semidom_class.normalize f = smult (inverse_class.inverse (unit_factor_class.unit_factor (lead_coeff f))) f"
prv_exu_imp_exi: fixes \<phi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "x \<in> var" shows "prv (imp (exu x \<phi>) (exi x \<phi>))"
transpose_vector: fixes x :: "('d, 'f) vec"   and A :: "(('d, 'e) vec, 'f) vec" shows "x v* A = Finite_Cartesian_Product.transpose A *v x"
mod_factor_div: fixes x :: "nat"   and m :: "nat"   and k :: "nat"   and y :: "nat" assumes "x mod (m * k) = y * k mod (m * k)" shows "x div k * k = x"
\<phi>_\<psi>: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "A.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>B G x\<guillemotright>" shows "\<phi> y (\<psi> x g) = g"
image_set_diff: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'b set" assumes "inj f" shows "f ` (A - B) = f ` A - f ` B"
in_arcsD2: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs G" shows "v \<in> pverts G"
Terminal_def: fixes a :: "'f" shows "Terminal a = Abs_resumption (TERMINAL a)"
pj_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" shows "pj R I = Pj (b_ag R) I"
statically_complete: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and B :: "'a"   and N :: "'a set" assumes "statically_complete_calculus Bot Inf entails Red_I Red_F"   and "B \<in> Bot"   and "calculus.saturated Inf Red_I N"   and "entails N {B}" shows "\<exists>B'\<in>Bot. B' \<in> N"
h1: shows "closure (rel_interior S) \<subseteq> closure S"
x_U_prop2: fixes x :: "('a \<times> 'a set) set" shows "x \<cdot> NC = mrel_c_lattice.d (x \<inter> NC) \<cdot> NC \<union> x \<cdot> {}"
run_catch_option: fixes return :: "'b option \<Rightarrow> 'a"   and bind :: "'a \<Rightarrow> ('b option \<Rightarrow> 'a) \<Rightarrow> 'a"   and m1 :: "('b, 'a) optionT"   and m2 :: "('b, 'a) optionT" shows "run_option (catch_option return bind m1 m2) = bind (run_option m1) (\<lambda>x. if x = None then run_option m2 else return x)"
lunit_tensor': fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<l>\<^sub>1[a \<otimes>\<^sub>1 b] \<cdot> \<a>\<^sub>1[\<I>\<^sub>1, a, b] = \<l>\<^sub>1[a] \<otimes>\<^sub>1 b"
elims_2: fixes x :: "'a simple_rule list" assumes "has_default_policy x" obtains m :: "'a simple_match" and uu_ :: "simple_action" where "x = [SimpleRule m uu_]" and "m = simple_match_any"   | uv_ :: "'a simple_rule" and v :: "'a simple_rule" and va :: "'a simple_rule list" where "x = uv_ # v # va" and "has_default_policy (v # va)"
sccs_are_sccs: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "\<forall>scc\<in>sccs s. is_scc (g_E G) scc"
invar_rel_br: fixes b :: "'h \<Rightarrow> 'g set"   and a' :: "'f \<Rightarrow> 'h"   and I' :: "'f \<Rightarrow> bool"   and a :: "'e \<Rightarrow> 'g set"   and I :: "'e \<Rightarrow> bool" shows "\<langle>br a' I', br a I\<rangle>invar_rel b = br (\<lambda>(x, s). a x) (\<lambda>(x, s). I x \<and> I' s \<and> a x \<subseteq> b (a' s))"
mult_neg_numeral_simps_2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and m :: "num"   and n :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "times (uminus (numeral.numeral one plus m)) (numeral.numeral one plus n) = uminus (numeral.numeral one plus (m * n))"
prime_factorization_1: shows "prime_factorization (1::'a) = {#}"
idealprod_whole_l: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set" assumes "Ring R"   and "ideal R I" shows "carrier R \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> I = I"
lang_pow_add: fixes n :: "nat"   and m :: "nat"   and A :: "'b list set" shows "A ^^ (n + m) = A ^^ n @@ A ^^ m"
stream_of_llist_llist_of_stream: fixes xs :: "'a stream" shows "stream_of_llist (llist_of_stream xs) = xs"
deflation: fixes y :: "'b" assumes "y \<in> carrier \<Y>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> y) \<sqsubseteq>\<^bsub>\<Y>\<^bsub>G\<^esub>\<^esub> y"
AE_in_set_eq_1: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M"   and "A \<in> sets M" shows "(AE x in M. x \<in> A) = (Sigma_Algebra.measure M A = 1)"
between_hyp2_S: fixes p :: "proj2"   and r :: "proj2"   and k :: "real" assumes "p \<in> hyp2 \<union> S"   and "r \<in> hyp2 \<union> S"   and "0 \<le> k"   and "k \<le> 1" shows "proj2_pt (k *\<^sub>R cart2_pt r + (1 - k) *\<^sub>R cart2_pt p) \<in> hyp2 \<union> S"
fmap_error_def: shows "fmap = error_map\<cdot>ID"
fixp_induct_uc: fixes U :: "'e \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'e \<Rightarrow> 'e"   and C :: "('c \<Rightarrow> 'd) \<Rightarrow> 'e"   and f :: "'e"   and P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool" assumes "\<And>x. lfp.mono_body (\<lambda>f. U (F (C f)) x)"   and "f \<equiv> C (lfp.fixp_fun (\<lambda>f. U (F (C f))))"   and "\<And>f. U (C f) = f"   and "lfp.admissible P"   and "P (\<lambda>_. \<Squnion> {})"   and "\<And>f. P (U f) \<Longrightarrow> P (U (F f))" shows "P (U f)"
g: shows "g \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M UNIV (\<lambda>i. M)"
simps_2: fixes A :: "'b mat"   and I :: "nat" shows "swap_col_to_front A (Suc I) = swap_col_to_front (swapcols I (Suc I) A) I"
intros_1: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and h :: "'c \<Rightarrow> 'd" assumes "bo x y" shows "blinding_of_blindable h bo (Unblinded x) (Unblinded y)"
ans1: fixes x :: "'a" shows "ad x \<cdot> x = (zero_class.zero::'a)"
simps_1: fixes var :: "nat"   and a :: "atom"   and \<Gamma> :: "real list" shows "convert_fm var (fm.Atom a) \<Gamma> = map_option AtomUni (convert_atom var a \<Gamma>)"
ground_var_denott: fixes t :: "fterm"   and E :: "char list \<Rightarrow> 'b"   and F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and E' :: "char list \<Rightarrow> 'b" assumes "ground\<^sub>t t" shows "eval\<^sub>t E F t = eval\<^sub>t E' F t"
SHOmsgVectors_get_msgs: fixes A :: "('d, 'f, 'e, 'g) CHOAlgorithm_scheme"   and r :: "nat"   and p :: "'d"   and cfg :: "'d \<Rightarrow> 'f"   and HOp :: "'d set"   and SHOp :: "'d set" shows "SHOmsgVectors A r p cfg HOp SHOp = get_msgs (sendMsg A r) cfg (K HOp) (K SHOp) p"
R1_def: fixes msgs :: "'a \<times> 'a"   and \<sigma> :: "bool" shows "R1 msgs \<sigma> = (let (m0, m1) = msgs in sample_uniform (order \<G>) \<bind> (\<lambda>a. sample_uniform (order \<G>) \<bind> (\<lambda>b. let c\<^sub>\<sigma> = a * b in sample_uniform (order \<G>) \<bind> (\<lambda>c\<^sub>\<sigma>'. return_spmf (msgs, \<^bold>g [^] a, \<^bold>g [^] b, if \<sigma> then \<^bold>g [^] c\<^sub>\<sigma>' else \<^bold>g [^] c\<^sub>\<sigma>, if \<sigma> then \<^bold>g [^] c\<^sub>\<sigma> else \<^bold>g [^] c\<^sub>\<sigma>')))))"
known_ptrs_known_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "ShadowRootClass.known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "ShadowRootClass.known_ptr ptr"
\<psi>': shows "adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g' \<phi> \<psi>'"
row_mat1: fixes i :: "nat"   and nr :: "nat"   and ze :: "'c"   and on :: "'c" assumes "i < nr" shows "row (mat1I ze on nr) i = vec1I ze on nr i"
B_theorem_3_1: fixes F :: "nat \<Rightarrow> 'b fract poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b fract"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b fract"   and G1 :: "'b poly"   and G2 :: "'b poly"   and i :: "nat" assumes "subresultant_prs_locale2 F n \<delta> f k \<beta> G1 G2"   and "3 \<le> i"   and "i \<le> k" shows "subresultant_prs_locale.h \<delta> f i = subresultant_prs_locale.\<Theta> n \<delta> f \<beta> i * f i"
aform_rel_internal: fixes R :: "('a \<times> 'b) set" shows "aform_rel R = br Affine top O \<langle>R\<rangle>set_rel"
A4: shows "Esigma (3\<^sup>2 * 7) = Esigma (3\<^sup>2) * Esigma 7"
imirror_iMin: fixes I :: "nat set" assumes "finite I" shows "iMin (imirror I) = iMin I"
islimpt_def: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (\<forall>T. x \<in> T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x))"
Seq_decomp_star: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p\<^sub>1 :: "('d, 'e, 'f) com"   and p\<^sub>2 :: "('d, 'e, 'f) com"   and s :: "'d"   and p' :: "('d, 'e, 'f) com"   and s' :: "'d" assumes "\<Gamma> \<turnstile> (Seq p\<^sub>1 p\<^sub>2, Normal s) \<rightarrow>\<^sup>* (p', Normal s')"   and "final (p', Normal s')" shows "\<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Throw, Normal s') \<and> p' = Throw \<or> (\<exists>t. \<Gamma> \<turnstile> (p\<^sub>1, Normal s) \<rightarrow>\<^sup>* (Skip, Normal t) \<and> \<Gamma> \<turnstile> (p\<^sub>2, Normal t) \<rightarrow>\<^sup>* (p', Normal s'))"
h_nonneg: fixes n :: "nat"   and xy :: "'v \<times> 'v" shows "0 \<le> h n xy"
module_hom_axioms: shows "module_hom (*s) (*s) f"
components_gb_red_aux_subset: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "component_of_term ` Keys (set (gb_red_aux bs ps)) \<subseteq> component_of_term ` Keys (args_to_set ([], bs, ps))"
E_linear_sum_allg: fixes D :: "'c pmf"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> real"   and A :: "'d set" assumes "finite (set_pmf D)" shows "E (map_pmf (\<lambda>x. \<Sum>i\<in>A. f i x) D) = (\<Sum>i\<in>A. E (map_pmf (f i) D))"
select_convs_2: fixes lens_get :: "'b \<Rightarrow> 'c"   and lens_put :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and more :: "'d" shows "put\<^bsub>\<lparr>lens_get = lens_get, lens_put = lens_put, \<dots> = more\<rparr>\<^esub> = lens_put"
simps_1: fixes ops :: "'b arith_ops_record"   and q :: "'b list"   and r :: "'b list"   and d :: "'b list"   and n :: "nat" shows "divmod_poly_one_main_i ops q r d (Suc n) = (let a = hd r; qqq = cCons_i ops a q; rr = tl (if a = arith_ops_record.zero ops then r else minus_poly_rev_list_i ops r (map (arith_ops_record.times ops a) d)) in divmod_poly_one_main_i ops qqq rr d n)"
lemma_2_10_30: fixes b :: "'b"   and a :: "'b" shows "((b r\<rightarrow> a) l\<rightarrow> a) r\<rightarrow> a = b r\<rightarrow> a"
usubst_lookup_upd: fixes x :: "'e \<Longrightarrow> 'f"   and \<sigma> :: "'g \<Rightarrow> 'f"   and v :: "('e, 'g) uexpr" assumes "weak_lens x" shows "\<langle>\<sigma>(x \<mapsto>\<^sub>s v)\<rangle>\<^sub>s x = v"
subst_apply_fv_unfold: fixes t :: "('b, 'c) Term.term"   and \<delta> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "fv (t \<cdot> \<delta>) = fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` fv t)"
bin_eq_iff: fixes x :: "int"   and y :: "int" shows "(x = y) = (\<forall>n. bit x n = bit y n)"