hcju/isabelleps huggingface dataset:
that: fixes n :: "nat" assumes "p ^^ n = id"   and "m < n" shows "thesis"
args_def: shows "args = (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})"
tt_min: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "lookup p v \<noteq> (0::'c)" shows "punit.tt p \<preceq> v"
InvariantConsistent_def: fixes M :: "(Literal \<times> bool) list" shows "InvariantConsistent M \<equiv> consistent (elements M)"
norm_mult_less: fixes x :: "'b"   and r :: "real"   and y :: "'b"   and s :: "real" assumes "norm x < r"   and "norm y < s" shows "norm (x * y) < r * s"
arr_compI: fixes f :: "'a"   and g :: "'a" assumes "seq f g" shows "arr (f \<cdot>\<^sub>B g)"
fact6: assumes "\<exists>v1 v2. fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') \<noteq> 0" shows "False"
take_take: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list" shows "take n (take m xs) = take (min n m) xs"
chamber_distance_le: fixes X :: "'b set set"   and C :: "'b set"   and Cs :: "'b set list"   and D :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])" shows "order.greater_eq (Suc (length Cs)) (ChamberComplex.chamber_distance X C D)"
piecewise_C1_differentiable_compose: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "f piecewise_C1_differentiable_on S"   and "g piecewise_C1_differentiable_on f ` S"   and "\<And>x. finite (S \<inter> f -` {x})" shows "g \<circ> f piecewise_C1_differentiable_on S"
tObsC_initial_3: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and envInit :: "('b, 'e, 'f) BEState list"   and envAction :: "('b, 'e, 'f) BEState \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> ('b, 'e, 'f) BEState \<Rightarrow> ('b, 'e, 'f) BEState"   and envVal :: "('b, 'e, 'f) BEState \<Rightarrow> 'c \<Rightarrow> bool"   and agents :: "'b odlist"   and envObsC :: "'e \<Rightarrow> 'h"   and t :: "('b, 'e, 'f) BEState Trace"   and cobs :: "'h" assumes "FiniteDetBroadcastEnvironment jkbp envInit envAction envTrans envVal agents envObsC" shows "(FiniteDetBroadcastEnvironment.tObsC envObsC t = tInit cobs) = (\<exists>s. t = tInit s \<and> envObsC (es s) = cobs)"
prems: shows "f \<in> insert a A \<rightarrow> carrier_vec n"
actionPar1Dest: fixes \<alpha> :: "'d action"   and P :: "('d, 'e, 'f) psi"   and \<beta> :: "'d action"   and Q :: "('d, 'e, 'f) psi"   and R :: "('d, 'e, 'f) psi" assumes "\<alpha> \<prec> P = \<beta> \<prec> Q \<parallel> R"   and "bn \<alpha> \<sharp>* bn \<beta>" obtains T :: "('d, 'e, 'f) psi" and p :: "(name \<times> name) list" where "set p \<subseteq> set (bn \<alpha>) \<times> set (bn \<beta>)" and "P = T \<parallel> (p \<bullet> R)" and "\<alpha> \<prec> T = \<beta> \<prec> Q"
simps: fixes a :: "('St, 'U, 'C) state list" shows "wffp a = ((\<exists>s. a = [s]) \<or> (\<exists>s' s sl. a = s # s' # sl \<and> s' \<in> \<delta> s \<and> wffp (s' # sl)))"
dist_fs_zero1: fixes z :: "complex_homo" shows "dist_fs z z = 0"
total_preorder_on_axioms: shows "total_preorder_on alts (R i)"
coeff_minus: fixes p :: "'b poly"   and n :: "nat" shows "coeff (- p) n = - coeff p n"
permutation_face_cycle_succ: shows "permutation face_cycle_succ"
diag_B': shows "isDiagonal (fst (snd (diagonal_to_Smith_aux_PQ [0..<k] bezout (P, P ** A ** Q, Q))))"
projection_concatenation_commute: fixes l1 :: "'b list"   and l2 :: "'b list"   and X :: "'b set" shows "(l1 @ l2) \<upharpoonleft> X = l1 \<upharpoonleft> X @ l2 \<upharpoonleft> X"
simps_2: fixes f :: "'d \<Rightarrow> 'd"   and x :: "'d" shows "stl (siterate f x) = siterate f (f x)"
weak_preservation_of_barbs_and_closures_2: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "rel_weakly_preserves_barbs (Rel\<^sup>+) CWB"
assms_2: fixes s' :: "('a, 'c) Term.term" assumes "s' \<sqsubseteq> s" shows "t \<noteq> s' \<cdot> \<theta>"
reds_preserves_sconf: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es',s',b'\<rangle>"   and "P,E,hp s,shp s \<turnstile> es [:] Ts"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
size_single: fixes b :: "'b" shows "size (\<LM> b  \<RM>) = 1"
iT_arith_iAll_conv_3: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<oplus>- k. P t) = (\<box> t I \<down>\<ge> k. P (t - k))"
closed_diff_open: fixes A :: "'a set"   and B :: "'a set" assumes "A closed"   and "B open" shows "A - B closed"
not_matches_removeAll: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and m :: "'b match_expr"   and a :: "action"   and p :: "'c"   and rs :: "'b rule list" assumes "\<not> matches \<gamma> m a p" shows "approximating_bigstep_fun \<gamma> p (removeAll (Rule m a) rs) Undecided = approximating_bigstep_fun \<gamma> p rs Undecided"
f_antecedent_def: fixes r :: "'rule" shows "f_antecedent r = fset_from_list (antecedent r)"
simps_5: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
matchC_LC_def: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" shows "matchC_LC theta c d \<equiv> \<forall>s t. s \<approx> t \<longrightarrow> (\<exists>I0 P F. mC_ZOC theta c d s t I0 P F)"
iso_presheaves_of_rings_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 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> 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' fam_morphisms \<and> 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"
CatExpCod: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" assumes "f \<in> mor\<^bsub>CatExp A B\<^esub>" shows "cod\<^bsub>CatExp A B\<^esub> f = NTCod f"
hcomp_in_vhom: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a"   and \<nu> :: "'a"   and h :: "'a"   and k :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>B g\<guillemotright>"   and "\<guillemotleft>\<nu> : h \<Rightarrow>\<^sub>B k\<guillemotright>"   and "src\<^sub>B h = trg\<^sub>B f" shows "\<guillemotleft>\<nu> \<star>\<^sub>B \<mu> : h \<star>\<^sub>B f \<Rightarrow>\<^sub>B k \<star>\<^sub>B g\<guillemotright>"
normalize_ports_generic_preserves_normalized_n_primitive: fixes m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and disc2 :: "'b common_primitive \<Rightarrow> bool"   and m' :: "'b common_primitive match_expr"   and sel2 :: "'b common_primitive \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> bool" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc (Prot a)"   and "\<forall>a. \<not> disc2 (C a)"   and "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"   and "m' \<in> set (normalize_ports_generic (normalize_positive_ports_step (disc, sel) C) (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one) m)"   and "normalized_n_primitive (disc2, sel2) f m" shows "normalized_n_primitive (disc2, sel2) f m'"
assms_1: shows "\<not> hb (i, Create i e1) (ir, Delete is e2)"
simps_3: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list"   and s :: "'state_t" assumes "\<not> interrupt (Suc n)"   and "thread_empty (execs (current s))" shows "run_total (Suc n) s execs = run_total n s execs"
sup_lf: fixes M :: "('c \<times> real) set set"   and E :: "'c set"   and p :: "'c \<Rightarrow> real"   and F :: "'c set"   and f :: "'c \<Rightarrow> real"   and c :: "('c \<times> real) set set"   and H :: "'c set"   and h :: "'c \<Rightarrow> real" assumes "M = norm_pres_extensions E p F f"   and "c \<in> chains M"   and "graph H h = \<Union> c" shows "linearform H h"
agent_abs: fixes t :: "'s Trace"   and ec :: "'s odlist \<times> 's odlist"   and a :: "'a" assumes "t \<in> Clock.jkbpC"   and "clock_simAbs ec = Clock.sim_equiv_class a t" shows "toSet (snd ec) = agent_abs a t"
st_proj1_inv_ne_top: fixes xa :: "'b" shows "st_proj1_inv xa \<noteq> top_sphere"
l9_18_R2: fixes X :: "'p"   and Y :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" assumes "Col X Y P"   and "Col A B P"   and "Bet A P B"   and "\<not> Col X Y A"   and "\<not> Col X Y B" shows "X Y TS A B"
not_None_eq: fixes x :: "'h option" shows "(x \<noteq> None) = (\<exists>y. x = Some y)"
rell_G_compp_Grp: fixes L1 :: "'j \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'k \<Rightarrow> 'r \<Rightarrow> bool" shows "rell_G L1 L2 = (Grp {x. set1_G x \<subseteq> {(x, y). L1 x y} \<and> set2_G x \<subseteq> {(x, y). L2 x y}} (mapl_G fst fst))\<inverse>\<inverse> OO Grp {x. set1_G x \<subseteq> {(x, y). L1 x y} \<and> set2_G x \<subseteq> {(x, y). L2 x y}} (mapl_G snd snd)"
smc_smcf_id_is_semifunctor: shows "dghm_id \<BB> : \<BB> \<mapsto>\<mapsto>\<^sub>S\<^sub>M\<^sub>C\<^bsub>\<alpha>\<^esub> \<BB>"
absolutely_integrable_cos_product: fixes f :: "real \<Rightarrow> real"   and k :: "real" assumes "f absolutely_integrable_on {- pi..pi}" shows "(\<lambda>x. cos (k * x) * f x) absolutely_integrable_on {- pi..pi}"
assms_3: assumes "P \<Longrightarrow>\<alpha> \<prec> P'" shows "Prop \<alpha> P'"
Act_eq_iff_perm: fixes f1 :: "'e"   and \<alpha>1 :: "'d"   and x1 :: "('b, 'c, 'd, 'e) formula"   and f2 :: "'e"   and \<alpha>2 :: "'d"   and x2 :: "('b, 'c, 'd, 'e) formula" shows "(Act f1 \<alpha>1 x1 = Act f2 \<alpha>2 x2) = (f1 = f2 \<and> (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2))"
comp_exi2: fixes l' :: "'a literal" shows "\<exists>l. l' = l\<^sup>c"
is_trail_split: fixes v :: "'v"   and p1 :: "('v \<times> 'w \<times> 'v) list"   and p2 :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v (p1 @ p2) v'" shows "\<exists>u. is_trail v p1 u \<and> is_trail u p2 v'"
restrict_lub_spmf: fixes Y :: "'h spmf set"   and A :: "'h set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y" shows "lub_spmf Y \<upharpoonleft> A = lub_spmf ((\<lambda>p. p \<upharpoonleft> A) ` Y)"
dghm_ArrMap_vdomain: shows "\<D>\<^sub>\<circ> (\<GG>\<lparr>ArrMap\<rparr>) = \<BB>\<lparr>Arr\<rparr>"
min_less_self_conv_2: fixes a :: "'a"   and b :: "'a" shows "(min a b < b) = (a < b)"
extreme_bound_quasi_const: fixes C :: "'a set"   and x :: "'a" assumes "C \<subseteq> A"   and "x \<in> A"   and "C \<noteq> {}"   and "\<forall>y\<in>C. y \<sim> x" shows "extreme_bound A (\<sqsubseteq>) C x"
fh: shows "\<forall>x\<in>X \<inter> K. f x = h x"
RModule_axioms: shows "RModule ActingGroup.group_ring (\<cdot>) V"
wfrec_unique_fixpoint: fixes r :: "('a \<times> 'a) set"   and H :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'b" assumes "wf r"   and "adm_wf r H"   and "f = H f" shows "f = wfrec r H"
assms: shows "f $$ fls_subdegree f * x = (1::'a)"
distinct_5: fixes x3 :: "'a" shows "Match_type_error \<noteq> Match x3"
assoc: fixes a :: "'e list"   and b :: "'e list"   and c :: "'e list" shows "(a @ b) @ c = a @ b @ c"
order_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
empty_not_add_mset: fixes a :: "'b"   and A :: "'b multiset" shows "{#} \<noteq> a, A"
1: shows "\<guillemotleft>\<iota> : \<I> \<otimes> \<I> \<rightarrow> \<I>\<guillemotright>"
ID_cEnum: fixes cEnum :: "('a list \<times> (('a \<Rightarrow> bool) \<Rightarrow> bool) \<times> (('a \<Rightarrow> bool) \<Rightarrow> bool)) option"   and enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.cenum cEnum"   and "ID cEnum = Some (enum, enum_all, enum_ex)" shows "UNIV = set enum \<and> enum_all = Ball UNIV \<and> enum_ex = Bex UNIV"
mat_extension_id: shows "ps_P.mat_extension (1\<^sub>m ps_P.d1) = 1\<^sub>m ps_P.d"
power_numeral_even: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and w :: "num" assumes "class.monoid_mult one times" shows "power.power one times z (numeral (num.Bit0 w)) = (let w = power.power one times z (numeral w) in times w w)"
isoI: fixes G :: "'b set"   and T :: "'b \<Rightarrow> 'c" assumes "GroupHom G T"   and "\<And>k. \<lbrakk>k \<in> G; T k = (0::'c)\<rbrakk> \<Longrightarrow> k = (0::'b)" shows "GroupIso G T"
fd: fixes u :: "char list" shows "simple_fw fw (p\<lparr>p_oiface := u\<rparr>) = Decision FinalDeny"
h3: shows "t mod cycleLength (nC i) mem schedule (nC i)"
total: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
Ball_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
test_mult_comp: fixes x :: "'a" shows "n x \<cdot> n n x = (0::'a)"
isCont_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "\<forall>\<^sub>F x in nhds x. f x = g x" shows "isCont f x = isCont g x"
comp_associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x * y * z = x * (y * z)"
rep_eq: fixes x :: "8 word" shows "Rep_uint8 (Abs_uint8' x) = x"
MWLsSteps_det_cases_3: fixes c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "\<langle>c1;c2,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>p,m'\<rangle>" obtains "p = Some c2" and "\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>None,m'\<rangle>"   | c1' :: "('exp, 'id) MWLsCom" where "p = Some (c1';c2)" and "\<langle>c1,m\<rangle> \<rightarrow>\<lhd>\<alpha>\<rhd> \<langle>Some c1',m'\<rangle>"
convergent_subseq_res: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and l :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a = acc_point s" shows "convergent_subseq s l l = residue (p ^ l) (acc_point s l)"
steps: shows "\<forall>i. (g i, g (Suc i)) \<in> Rw\<^sup>* O R O Rw\<^sup>*"
map_us_def: fixes f :: "'c \<Rightarrow> 'd" shows "map_us f = us \<circ> f \<circ> IInf"
build_relp_def: fixes \<alpha> :: "'b \<Rightarrow> 'e"   and I :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'e" shows "brp \<alpha> I x y \<equiv> y = \<alpha> x \<and> I x"
ncols_transpose: fixes A :: "'a matrix" shows "ncols (transpose_matrix A) = nrows A"
imp_neg_I: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IMP B"   and "H \<turnstile> A IMP neg B" shows "H \<turnstile> neg A"
inj_onD: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "inj_on f A"   and "f x = f y"   and "x \<in> A"   and "y \<in> A" shows "x = y"
compact_empty: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.compact open {}"
fmrestrict_set_empty: fixes A :: "'c set" shows "fmrestrict_set A {$$} = {$$}"
f: shows "monotone (\<lambda>x y. y \<le> x) (\<lambda>x y. y \<le> x) f"
left_tree_initial_edge: fixes s :: "'a"   and t :: "'a" assumes "s \<rightarrow> t" shows "t \<notin> left_tree s t"
orecvmsgEI: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and a :: "'b seq_action"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> bool" assumes "orecvmsg P \<sigma> a"   and "\<And>\<sigma> a. P \<sigma> a \<Longrightarrow> Q \<sigma> a" shows "orecvmsg Q \<sigma> a"
lm01: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "argmax f A = A \<inter> f -` {Max (f ` A)}"
closed_Collect_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "closed {x. f x = g x}"
wf_envs_def: fixes \<sigma> :: "('a \<times> 'b) trace"   and j :: "nat"   and P :: "'a \<Rightarrow> nat option"   and P' :: "'a \<Rightarrow> nat option"   and V :: "'a \<Rightarrow> (nat \<Rightarrow> 'b set) option"   and db :: "('a, 'b set list) mapping" shows "wf_envs \<sigma> j P P' V db = (dom V = dom P \<and> Mapping.keys db = dom P \<union> {p. p \<in> fst ` \<Gamma> \<sigma> j} \<and> rel_mapping (\<le>) P P' \<and> pred_mapping (\<lambda>i. i \<le> j) P \<and> pred_mapping (\<lambda>i. i \<le> Suc j) P' \<and> (\<forall>p\<in>Mapping.keys db - dom P. the (Mapping.lookup db p) = [{ts. (p, ts) \<in> \<Gamma> \<sigma> j}]) \<and> (\<forall>p\<in>dom P. list_all2 (\<lambda>i X. X = the (V p) i) [the (P p)..<the (P' p)] (the (Mapping.lookup db p))))"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "Ferrack.num" shows "ismaxcoeff (CN n c t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> ismaxcoeff t x)"
uncurry0_def: fixes c :: "'a" shows "uncurry0 c \<equiv> \<lambda>_. c"
simps_4: fixes n :: "nat"   and execs :: "'dom_t \<Rightarrow> 'action_t list list"   and s :: "'state_t" assumes "\<not> interrupt (Suc n)"   and "thread_empty (execs (current s))" shows "run (Suc n) (Some s) execs = run n (Some s) execs"
vNM_utility_def: fixes outcomes :: "'b set"   and relation :: "('b pmf \<times> 'b pmf) set"   and u :: "'b \<Rightarrow> real" shows "vNM_utility outcomes relation u \<equiv> relation \<subseteq> \<P> outcomes \<times> \<P> outcomes \<and> (\<forall>p q. p \<in> \<P> outcomes \<longrightarrow> q \<in> \<P> outcomes \<longrightarrow> p \<succeq>[relation] q = (measure_pmf.expectation q u \<le> measure_pmf.expectation p u))"
sameDom_lift2: fixes f :: "'c \<Rightarrow> 'b"   and inp :: "'a \<Rightarrow> 'c option" shows "sameDom (lift f inp) inp"
leg0_simps_3: shows "trg f = src (g \<star> f\<^sup>*)"
fset_of_list_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "fset_of_list (map f xs) = f |`| fset_of_list xs"
single_is_union: fixes a :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" shows "(\<LM> a  \<RM> = M + N) = (\<LM> a  \<RM> = M \<and> N = \<Empt> \<or> M = \<Empt> \<and> \<LM> a  \<RM> = N)"
wfG': shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
rank_initial: fixes x :: "nat" shows "\<exists>i. \<UU>.rank x x = Some i"
blinfun_apply_cases: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> {f. bounded_linear f}" obtains x :: "'c \<Rightarrow>\<^sub>L 'd" where "y = blinfun_apply x"
non_neg_vec_def: fixes A :: "('d, 'e) Finite_Cartesian_Product.vec" shows "non_neg_vec A \<equiv> \<forall>i. (0::'d) \<le> A $h i"
deg_comp: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> order" shows "deg_comp cmp = (\<lambda>x y. case comparator_of (deg_pp (fst (rep_nat_term x))) (deg_pp (fst (rep_nat_term y))) of Eq \<Rightarrow> cmp x y | Lt \<Rightarrow> Lt | Gt \<Rightarrow> Gt)"
lm086: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "Graph f ,, x = f x"
distinct_263: fixes x51 :: "tf"   and x52 :: "b_e list"   and x23 :: "v" shows "Block x51 x52 \<noteq> C x23"
bdia_whilei: fixes p :: "'a"   and i :: "'a"   and t :: "'a"   and q :: "'a"   and x :: "'a" assumes "r p \<le> r i"   and "r i \<cdot> ad t \<le> r q"   and "\<langle>x| (d t \<cdot> r i) \<le> r i" shows "\<langle>while t inv i do x od| p \<le> r q"
prime_filter: 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 bot :: "'c"   and F :: "'c set" assumes "class.distrib_lattice_bot inf less_eq less sup bot"   and "ord.filter less_eq F \<and> F \<noteq> UNIV" shows "\<exists>G. ((ord.filter less_eq G \<and> G \<noteq> UNIV) \<and> (\<forall>x y. sup x y \<in> G \<longrightarrow> x \<in> G \<or> y \<in> G)) \<and> F \<subseteq> G"
append_child_preserves_known_ptrs: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> append_child ptr child \<rightarrow>\<^sub>h h'" shows "known_ptrs h'"
scc_root_transfer: fixes s' :: "'v tarjan_state"   and r :: "'v"   and scc :: "'v set" assumes "DFS_invar G tarjan_params s'"   and "r \<in> dom (discovered s)"   and "dom (discovered s) \<subseteq> dom (discovered s')"   and "\<forall>x\<in>dom (discovered s). \<delta> s x = \<delta> s' x"   and "\<forall>x\<in>dom (discovered s') - dom (discovered s). counter s \<le> \<delta> s' x"   and "tree_edges s \<subseteq> tree_edges s'" shows "scc_root s r scc = scc_root s' r scc"
a: shows "liminf (\<lambda>n. u n + v n) + liminf (\<lambda>n. - u n) \<le> liminf (\<lambda>n. u n + v n + - u n)"
prj_in_hom_3: shows "\<guillemotleft>\<mu>_\<nu>_\<pi>\<rho>.Prj\<^sub>0 : HH\<mu>\<nu>H\<pi>\<rho>.chine \<rightarrow>\<^sub>C \<pi>\<rho>.apex\<guillemotright>"
PhiPlus_def: fixes n :: "nat" shows "\<Phi>\<^sup>+ n = (let nextconfig = config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) (take n qs) \<bind> (\<lambda>(s, is). BIT_step (s, is) (qs ! n) \<bind> (\<lambda>(a, nis). return_pmf (step s (qs ! n) a, nis))) in E (map_pmf (\<phi> (Suc n)) nextconfig))"
app_iso_inv: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "app_iso (H.inv_iso hom) (app_iso hom (app_iso iso_f G)) = app_iso iso_f G"
simps: fixes q :: "'b"   and n :: "nat" shows "state_rank q n = (if configuration q n \<noteq> {} \<and> \<not> sink q then Some (card (senior_states q n)) else None)"
f1: shows "\<not> ((fst instr = load_store_type STH \<or> fst instr = load_store_type STHA) \<and> ucast (get_addr (snd instr) (snd (fst (get_curr_win () s)))) \<noteq> 0)"