hcju/isabelleps huggingface dataset:
maximum_modulus_frontier: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and B :: "real"   and \<xi> :: "complex" assumes "f holomorphic_on interior S"   and "continuous_on (closure S) f"   and "bounded S"   and "\<And>z. z \<in> frontier S \<Longrightarrow> cmod (f z) \<le> B"   and "\<xi> \<in> S" shows "cmod (f \<xi>) \<le> B"
PLemmas_4: fixes x :: "((int \<times> int) set set, int) Combinators"   and y :: "((int \<times> int) set set, int) Combinators" shows "C (x \<oplus> y) = C y \<Oplus> C x"
assms_2: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" assumes "P cfg" shows "state cfg \<in> S"
rcos_ring_hom: shows "(+>) I \<in> ring_hom R (R Quot I)"
G: shows "({x} \<in> refusals P xs) \<noteq> ({x} \<inter> next_events P xs = {})"
Not_eq_iff: fixes x1 :: "('b, 'c, 'd) formula"   and x2 :: "('b, 'c, 'd) formula" shows "(Formula.Not x1 = Formula.Not x2) = (x1 = x2)"
assms_2: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" shows "P f [x, y]"
assms_1: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
assms_2: shows "typeof_addr h a = \<lfloor>Class_type C\<rfloor>"
update_convs_2: fixes ypos' :: "nat \<Rightarrow> nat"   and xpos :: "nat"   and ypos :: "nat"   and more :: "'a" shows "ypos_update ypos' \<lparr>xpos = xpos, ypos = ypos, \<dots> = more\<rparr> = \<lparr>xpos = xpos, ypos = ypos' ypos, \<dots> = more\<rparr>"
prems: shows "init_exp_of e0 = \<lfloor>unit\<rfloor>"
equiv_aux: fixes g :: "'g"   and ns :: "'node list"   and s :: "'var \<Rightarrow> 'node option"   and s' :: "'val \<Rightarrow> 'node option"   and ms :: "'node list"   and m :: "'node"   and v :: "'val" assumes "g \<turnstile> ns\<Down>s"   and "g \<turnstile> ns\<Down>\<^sub>s s'"   and "g \<turnstile> last ns-ms\<rightarrow>m"   and "v \<in> allUses g m"   and "\<forall>n\<in>set (tl ms). var g v \<notin> var g ` allDefs g n" shows "s (var g v) = s' v"
set_sel_2: fixes a :: "'b llist"   and x :: "'b" assumes "a \<noteq> LNil"   and "x \<in> lset (ltl a)" shows "x \<in> lset a"
w_def: shows "w = (\<lambda>n. if WtFT cl = 1 then if n = pickFT cl then 1 else 0 else if finished (cl ! n) then 0 else 1 / real (length cl) / (1 - WtFT cl) * sum (wt (cl ! n) s) {..<brn (cl ! n)})"
ofilter_underS_Field: fixes A :: "'a set" shows "local.ofilter A = ((\<exists>a\<in>Field r. A = local.underS a) \<or> A = Field r)"
Int_Union_Union: fixes B :: "'c set set"   and A :: "'c set" shows "\<Union> B \<inter> A = (\<Union>C\<in>B. C \<inter> A)"
encR: fixes S :: "'procS" shows "indRelp (SourceTerm S) (TargetTerm (\<lbrakk>S\<rbrakk>))"
simps_10: fixes vs :: "real list" shows "interpret_floatarith Pi vs = pi"
config'_set: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and S :: "'a list \<times> 'b"   and qs :: "'a list" shows "set (fst (Partial_Cost_Model.config' A S qs)) = set (fst S)"
FW_valid_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'a DBMEntry" assumes "valid_dbm M" shows "valid_dbm (FW M n)"
inst_fm_semantics: fixes E :: "nat \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) fm"   and t :: "'b tm"   and m :: "nat" shows "\<lbrakk>E, F, G\<rbrakk> (p\<langle>t/m\<rangle>) = \<lbrakk>E\<langle>m:\<lparr>E, F\<rparr> t\<rangle>, F, G\<rbrakk> p"
map_val_pair_cong: fixes xs :: "('a \<times> 'b) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<And>k v. (k, v) \<in> set xs \<Longrightarrow> f k v = g k v" shows "map_val_pair f xs = map_val_pair g xs"
execute_serial_plan_is_execute_parallel_plan_ii: fixes \<pi> :: "'a strips_operator list"   and G :: "'a \<Rightarrow> bool option"   and I :: "'a \<Rightarrow> bool option" assumes "\<forall>op\<in>set \<pi>. are_operator_effects_consistent op op"   and "G \<subseteq>\<^sub>m execute_serial_plan I \<pi>" shows "G \<subseteq>\<^sub>m execute_parallel_plan I (List_Supplement.embed \<pi>)"
Nil_is_append_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "([] = xs @ ys) = (xs = [] \<and> ys = [])"
Par2B: fixes Q :: "pi"   and a :: "subject"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> Q'"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>acc. P acc []"   and "\<And>acc x xs. P (x # acc) xs \<Longrightarrow> P acc (x # xs)" shows "P a0 a1"
assms_5: shows "h \<turnstile> create_document \<rightarrow>\<^sub>h h'"
digraph_map_axioms: shows "digraph_map (with_proj (list_digraph G_list)) (to_map' (snd G_list) xss)"
assms_1: shows "Lower_Semicontinuous.convex_on s f"
left_zero_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n + v = v"
edka_refine: shows "edka \<le> \<Down> Id edka_partial"
b_take_i: shows "b1 @ [fst q] = take (Suc i) b"
get_ancestors_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
enumerate_eq_zip: fixes n :: "nat"   and xs :: "'d list" shows "List.enumerate n xs = zip [n..<n + length xs] xs"
factor_theorem: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a"   and g :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "a \<in> carrier R"   and "g = UP_root_div f a"   and "to_fun f a = \<zero>" shows "f = X_poly_minus R a \<otimes>\<^bsub>P\<^esub> g"
lucas_lehmer_to_real_simps1_1: fixes a :: "int"   and b :: "int" shows "lucas_lehmer_to_real (a, b) = real_of_int a + real_of_int b * sqrt 3"
fresh_def: fixes xs :: "'a set"   and x :: "'a" shows "infinite_class.fresh xs x \<equiv> if x \<notin> xs \<or> infinite xs then x else SOME y. y \<notin> xs"
1: shows "has_bochner_integral (lebesgue_on {a..a + c}) f (integral\<^sup>L (lebesgue_on {a..a + c}) f)"
chambers_1: shows "C \<in> folding.\<C> - f \<turnstile> folding.\<C>"
refl_on_Un: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and B :: "'c set"   and s :: "('c \<times> 'c) set" assumes "refl_on A r"   and "refl_on B s" shows "refl_on (A \<union> B) (r \<union> s)"
par_neq2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Par C D" shows "C \<noteq> D"
range_to_nat: shows "range Bij_Nat.to_nat = {0..<CARD('d)}"
is_valid_problem_strips_operator_variable_sets_3: fixes \<Pi> :: "'a strips_problem"   and op :: "'a strips_operator" assumes "is_valid_problem_strips \<Pi>"   and "op \<in> set (\<Pi>\<^sub>\<O>)" shows "set (delete_effects_of op) \<subseteq> set (\<Pi>\<^sub>\<V>)"
Max_mono: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and N :: "'b set" assumes "class.linorder less_eq less"   and "M \<subseteq> N"   and "M \<noteq> {}"   and "finite N" shows "less_eq (linorder.Max less_eq M) (linorder.Max less_eq N)"
in_list_induct: fixes P :: "'a list \<Rightarrow> bool"   and xs :: "'a list" assumes "P []"   and "\<And>x xs. \<lbrakk>x \<in> set xs; P xs\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P xs"
has_sgnx_times: fixes f :: "real \<Rightarrow> real"   and c1 :: "real"   and F :: "real filter"   and g :: "real \<Rightarrow> real"   and c2 :: "real" assumes "(f has_sgnx c1) F"   and "(g has_sgnx c2) F" shows "((\<lambda>x. f x * g x) has_sgnx c1 * c2) F"
dg_is_arrD_1: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<DD>\<^esub> b" shows "f \<in>\<^sub>\<circ> \<DD>\<lparr>Arr\<rparr>"
cnf2bhv_mono_strict: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lActive active c t \<le> n"   and "n < n'" shows "dynamic_component.cnf2bhv tCMP active c t n < dynamic_component.cnf2bhv tCMP active c t n'"
rel_inject: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e" shows "rel_id R (return_id x) (return_id y) = R x y"
induct: fixes x :: "('sf, 'sz) ODE"   and P :: "('sf, 'sz) ODE \<Rightarrow> bool" assumes "ORadmit x"   and "\<And>x \<theta>. P (OSing x \<theta>)"   and "\<And>ODE1 ODE2. \<lbrakk>ORadmit ODE1; P ODE1; ORadmit ODE2; P ODE2\<rbrakk> \<Longrightarrow> P (OProd ODE1 ODE2)" shows "P x"
bres_canc1: fixes x :: "'b"   and y :: "'b" shows "x \<cdot> (x \<rightarrow> y) \<le> y"
weakly_bisimilar_transp: shows "transp (\<approx>\<cdot>\<^sub>S)"
Call: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and b :: "('a, 'b, 'c) com"   and s :: "'a" assumes "\<Gamma> p = Some b" shows "\<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow> (b, Normal s)"
assms_2: shows "F \<in> set_offending_flows G nP"
horizontal_identity_is_ide: fixes \<mu> :: "'a" assumes "H.ide \<mu>" shows "ide \<mu>"
hyperUn_assoc: fixes A :: "'f set option"   and B :: "'f set option"   and C :: "'f set option" shows "A \<squnion> B \<squnion> C = A \<squnion> (B \<squnion> C)"
EquivI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<phi> in v] = [\<psi> in v]" shows "[\<phi> \<^bold>\<equiv> \<psi> in v]"
f_Exec_Stream_Init_tl_eq_f_Exec_Stream: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and c :: "'d" shows "tl (f_Exec_Comp_Stream_Init trans_fun xs c) = f_Exec_Comp_Stream trans_fun xs c"
polyneg_normh: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p n" shows "isnpolyh (~\<^sub>p p) n"
osp__ncop2: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B C OSP P Q" shows "\<not> Coplanar A B C Q"
weak_unit_is_horizontal_identity: fixes a :: "'a" assumes "weak_unit a" shows "H.ide a"
monic_degree_m: fixes m :: "int"   and f :: "int poly" assumes "poly_mod_2 m"   and "monic f" shows "degree (poly_mod.Mp m f) = degree f"
llength_lnull: fixes xs :: "'b llist" assumes "lnull xs" shows "llength xs = 0"
tObsC_absI: fixes t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace"   and u :: "('a, 'es, 'as) BEState"   and v :: "('a, 'es, 'as) BEState" assumes "t' \<in> SPR.jkbpC"   and "tObsC t' = tObsC t"   and "u = tFirst t'"   and "v = tLast t'" shows "(u, v) \<in> tObsC_abs t"
A12_NotSource_level1: shows "isNotDSource level1 sA12"
Equivalence_def: shows "{(x, y). stk2.rel x y} = {(a, b). \<exists>A\<in>e2.Partition. a \<in> A \<and> b \<in> A}"
elims_2: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> 'i" assumes "P1 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "'i" where "x = (s1, s2)" and "\<forall>p. status s1 p \<in> {Pending, Aborted} \<longrightarrow> fst (pending s1 p) = p"
has_contour_integral_shiftpath_eq: fixes g :: "real \<Rightarrow> complex"   and a :: "real"   and f :: "complex \<Rightarrow> complex"   and i :: "complex" assumes "valid_path g"   and "pathfinish g = pathstart g"   and "a \<in> {0..1}" shows "(f has_contour_integral i) (shiftpath a g) = (f has_contour_integral i) g"
degree_uminus: fixes A :: "'a mpoly"   and x' :: "nat" shows "MPoly_Type.degree (- A) x' = MPoly_Type.degree A x'"
main_4: assumes "c = round (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)" shows "\<bar>gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs') i j\<bar> \<le> 1 / 2"
r_amalgamate: fixes i :: "nat"   and j :: "nat" shows "eval r_amalgamate [i, j] \<down>= amalgamate i j"
cases: fixes x :: "'b tree" obtains hsx :: "'b tree" and xa :: "'b" and hsy :: "'b tree" and y :: "'b" and hs :: "'b tree" where "x = \<langle>hsx, xa, \<langle>hsy, y, hs\<rangle>\<rangle>"   | "x = \<langle>\<rangle>"   | v :: "'b tree" and va :: "'b" where "x = \<langle>v, va, \<langle>\<rangle>\<rangle>"
expr_sem_eq_on_vars: fixes e :: "expr"   and \<sigma>\<^sub>1 :: "nat \<Rightarrow> val"   and \<sigma>\<^sub>2 :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<sigma>\<^sub>1 x = \<sigma>\<^sub>2 x" shows "expr_sem \<sigma>\<^sub>1 e = expr_sem \<sigma>\<^sub>2 e"
combine_infinity_invariant_1: fixes i :: "enat"   and x :: "'a ltln"   and j :: "enat"   and y :: "'a ltln"   and z :: "'a ltln"   and k :: "enat" assumes "(i = \<infinity>) = is_constant x"   and "(j = \<infinity>) = is_constant y"   and "combine mk_and (x, i) (y, j) = (z, k)" shows "(k = \<infinity>) = is_constant z"
append_subset_1: fixes xs :: "'c list"   and ys :: "'c list"   and zs :: "'c list" assumes "set xs = set (ys @ zs)" shows "set ys \<subseteq> set xs"
bot_def: fixes c :: "com" shows "bot c = annotate Map.empty c"
reds_preserves_sconf: fixes es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "char list exp list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"   and "P,E,hp s \<turnstile> es [:] Ts"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
arr_char: fixes f :: "'a" shows "hom.arr f = \<guillemotleft>f : src (src f) \<rightarrow> src e\<guillemotright>"
ih: shows "set (labels (u, xs)) \<subseteq> (ars\<^sup>+)\<inverse> \<down>s {\<beta>}"
rel_pmf_iff_equivp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "equivp R" shows "rel_pmf R p q = (\<forall>C\<in>UNIV // {(x, y). R x y}. measure_pmf.prob p C = measure_pmf.prob q C)"
accepting_pair\<^sub>R_def: fixes \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and P :: "('d \<times> 'e \<times> 'd) set \<times> ('d \<times> 'e \<times> 'd) set"   and w :: "nat \<Rightarrow> 'e" shows "accepting_pair\<^sub>R \<delta> q\<^sub>0 P w \<equiv> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> fst P = {} \<and> limit (run\<^sub>t \<delta> q\<^sub>0 w) \<inter> snd P \<noteq> {}"
Sup_less_def: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'b \<Rightarrow> 'a"   and w :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.Sup_less Sup x w = Sup {y. \<exists>v<w. y = x v}"
funpow_face_cycle_succ_no_arc: fixes a :: "'b"   and n :: "nat" assumes "a \<notin> arcs G" shows "(face_cycle_succ ^^ n) a = a"
degree_add_le: fixes p :: "'a poly"   and n :: "nat"   and q :: "'a poly" assumes "degree p \<le> n"   and "degree q \<le> n" shows "degree (p + q) \<le> n"
lambda_one: shows "(\<lambda>x. x) = (*) (1::'a)"
ncoll: fixes x :: "'b" assumes "x \<in> X" shows "ccw (0::'a) (f x) a"
simps_2: fixes x21 :: "char list"   and x22 :: "'a list"   and y21 :: "char list"   and y22 :: "'a list" shows "(Neg x21 x22 = Neg y21 y22) = (x21 = y21 \<and> x22 = y22)"
monotonicity: fixes \<Gamma> :: "(name \<times> ty) list"   and \<theta> :: "(name \<times> trm) list"   and e :: "trm"   and T :: "ty"   and x :: "name" assumes "\<theta> Vcloses \<Gamma>"   and "e \<in> V T"   and "valid ((x, T) # \<Gamma>)" shows "(x, e) # \<theta> Vcloses (x, T) # \<Gamma>"
tr_impl_while_body_def: shows "tr_impl_while_body \<equiv> \<lambda>s. gds_get_pending gds s \<bind> (\<lambda>(u, Vs, s). case Vs of None \<Rightarrow> gds_finish gds u s | Some v \<Rightarrow> if gds_is_discovered gds v s then if gds_is_finished gds v s then gds_cross_edge gds u v s else gds_back_edge gds u v s else gds_discover gds u v s)"
irbd_rbd_inv2_iff: fixes \<phi> :: "'c set \<Rightarrow> 'd set" shows "((bd\<^sub>\<R> \<circ> bd\<^sup>-\<^sub>\<R>) \<phi> = \<phi>) = Sup_pres \<phi>"
from_hma\<^sub>v_def: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v = Matrix.vec CARD('d) (\<lambda>i. v $h mod_type_class.from_nat i)"
adj_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a" shows "f \<stileturn> g = (\<forall>x y. (f x \<le> y) = (x \<le> g y))"
enabled_opr'_correct: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and \<pi> :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "s \<in> valid_states"   and "lookup_operator name = Some \<pi>" shows "enabled_opr' \<pi> s = enabled name s"
regular_UNION: fixes r :: "('a \<times> 'a) set"   and A :: "'b set"   and F :: "'b \<Rightarrow> 'c set" assumes "regular r"   and "|A| <o r"   and "\<And>a. a \<in> A \<Longrightarrow> |F a| <o r" shows "|\<Union> (F ` A)| <o r"
execute_opr'_correct: fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and \<pi> :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "s \<in> valid_states"   and "lookup_operator name = Some \<pi>" shows "execute_opr' \<pi> s = execute name s"
assms_2: shows "Prod_any f \<noteq> (0::'a)"
distinct_set_conv_list: fixes xs :: "'b list"   and g :: "'b \<Rightarrow> 'c" assumes "distinct xs" shows "prod g (set xs) = prod_list (map g xs)"
R2: fixes a :: "name"   and \<Psi>' :: "'b"   and b :: "name"   and Q :: "('a, 'b, 'c) psi" assumes "a \<sharp> \<Psi>'"   and "b \<sharp> \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>a\<rparr>(\<lparr>\<nu>b\<rparr>Q), \<lparr>\<nu>b\<rparr>(\<lparr>\<nu>a\<rparr>Q)) \<in> Rel"
keys_focus: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "keys (focus X p) = (\<lambda>t. except t (- X)) ` keys p"
simps_5: fixes t :: "'c local.term" shows "Cod \<^bold>\<l>\<^bold>[t\<^bold>] = Cod t"
dim_image_eq: fixes f :: "'c \<Rightarrow> 'b \<times> 'c"   and S :: "'c set" assumes "Vector_Spaces.linear (*b) local.scale f"   and "inj_on f (vs2.span S)" shows "p.dim (f ` S) = vs2.dim S"
IH: assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len m i i xs"   and "k \<le> n" shows "canonical_subs n {0..k} (fw m n k) \<and> (\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..k} \<longrightarrow> (0::'a) \<le> len (fw m n k) i i xs)"
rep_eq: fixes x :: "(char list \<times> 'a list) prefix" shows "msteps_stateless x = msteps0_stateless (Rep_prefix x)"
Inf_pair_ow_def: fixes U\<^sub>1 :: "'a set"   and Inf\<^sub>1 :: "'a set \<Rightarrow> 'a"   and U\<^sub>2 :: "'b set"   and Inf\<^sub>2 :: "'b set \<Rightarrow> 'b" shows "Inf_pair_ow U\<^sub>1 Inf\<^sub>1 U\<^sub>2 Inf\<^sub>2 \<equiv> Inf_ow U\<^sub>1 Inf\<^sub>1 \<and> Inf_ow U\<^sub>2 Inf\<^sub>2"
id_is_mor_pr_rngs: shows "morphism_presheaves_of_rings S is_open id.im_sheaf id.im_sheaf_morphisms b id.add_im_sheaf id.mult_im_sheaf id.zero_im_sheaf id.one_im_sheaf id.im_sheaf id.im_sheaf_morphisms b id.add_im_sheaf id.mult_im_sheaf id.zero_im_sheaf id.one_im_sheaf (\<lambda>U. identity (id.im_sheaf U))"
Mem_HFun_Sigma_OrdP: fixes t :: "tm"   and u :: "tm"   and f :: "tm" shows "{HPair t u IN f, HFun_Sigma f} \<turnstile> OrdP t"
asym: fixes b :: "'c"   and a :: "'c" assumes "b < a"   and "a < b" shows "False"
indep_vars: shows "local.P.indep_vars (\<lambda>_. M) (\<lambda>i f. f i) UNIV"
map_length_ci: fixes xs :: "abc_inst list"   and x :: "abc_inst" shows "map (length \<circ> (\<lambda>(xa, y). ci (layout_of xs @ [length_of x]) xa y)) (tpairs_of xs) = map (length \<circ> (\<lambda>(x, y). ci (layout_of xs) x y)) (tpairs_of xs)"
get_invar_def: fixes a :: "'d \<Rightarrow> 'c set"   and X :: "'c set" shows "get_invar a X = SPEC (\<lambda>(Y, invar). Y = X \<and> Y \<subseteq> a invar)"
NormalizeI_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P Language.normalize c Q,A"
R_def: fixes j :: "nat" shows "R j = (if j = n 2 then sdiv_poly (Polynomial.smult (lead_coeff G2 ^ \<delta> 1) G2) (lead_coeff G2) else subresultant j G1 G2)"
R\<^sub>C_of_simp: fixes dma\<^sub>A :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> Sec"   and dma\<^sub>C :: "('e \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> Sec"   and \<C>_vars\<^sub>A :: "'c \<Rightarrow> 'c set"   and \<C>_vars\<^sub>C :: "'e \<Rightarrow> 'e set"   and eval\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and eval\<^sub>C :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and var\<^sub>C_of :: "'c \<Rightarrow> 'e"   and \<R>\<^sub>A :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) set"   and \<R> :: "((('f \<times> (Mode \<Rightarrow> 'c set)) \<times> ('c \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and P :: "((('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) \<times> ('g \<times> (Mode \<Rightarrow> 'e set)) \<times> ('e \<Rightarrow> 'd)) set"   and c\<^sub>1\<^sub>C :: "'g"   and mds\<^sub>C :: "Mode \<Rightarrow> 'e set"   and mem\<^sub>1\<^sub>C :: "'e \<Rightarrow> 'd"   and c\<^sub>2\<^sub>C :: "'g"   and mem\<^sub>2\<^sub>C :: "'e \<Rightarrow> 'd" assumes "sifum_refinement dma\<^sub>A dma\<^sub>C \<C>_vars\<^sub>A \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of"   and "sifum_refinement.secure_refinement dma\<^sub>C \<C>_vars\<^sub>C eval\<^sub>A eval\<^sub>C var\<^sub>C_of \<R>\<^sub>A \<R> P" shows "((((c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C), mem\<^sub>2\<^sub>C) \<in> sifum_refinement.R\<^sub>C_of dma\<^sub>C \<C>_vars\<^sub>C \<R>\<^sub>A \<R> P) = ((\<exists>c\<^sub>1\<^sub>A c\<^sub>2\<^sub>A. (((c\<^sub>1\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>1\<^sub>C), (c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C) \<in> \<R> \<and> (((c\<^sub>2\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>2\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C), mem\<^sub>2\<^sub>C) \<in> \<R> \<and> (((c\<^sub>1\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>A, sifum_refinement.mds\<^sub>A_of var\<^sub>C_of mds\<^sub>C), sifum_refinement.mem\<^sub>A_of var\<^sub>C_of mem\<^sub>2\<^sub>C) \<in> \<R>\<^sub>A) \<and> sifum_security_init.low_mds_eq dma\<^sub>C \<C>_vars\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C mem\<^sub>2\<^sub>C \<and> (((c\<^sub>1\<^sub>C, mds\<^sub>C), mem\<^sub>1\<^sub>C), (c\<^sub>2\<^sub>C, mds\<^sub>C), mem\<^sub>2\<^sub>C) \<in> P)"