hcju/isabelleps huggingface dataset:
Lambert_W'_less_iff: fixes x :: "real"   and y :: "real" assumes "x \<in> {- exp (- 1)..<0}"   and "y \<in> {- exp (- 1)..<0}" shows "(Lambert_W' x < Lambert_W' y) = (y < x)"
t: fixes x :: "'a"   and y :: "'a" assumes "t x y" shows "s x y"
widen_RefT: fixes G :: "prog"   and R :: "ref_ty"   and T :: "ty" assumes "G\<turnstile>RefT R\<preceq>T" shows "\<exists>t. T = RefT t"
subseteq_eqvt: fixes pi :: "(name \<times> name) list"   and X :: "'a set"   and Y :: "'a set" shows "pi \<bullet> (X \<subseteq> Y) = (pi \<bullet> X \<subseteq> pi \<bullet> Y)"
sup_is_type: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and A :: "ty"   and B :: "ty"   and T :: "ty" assumes "wf_prog wf_md P"   and "is_type P A"   and "is_type P B"   and "SemiType.sup P A B = OK T" shows "is_type P T"
assms_3: shows "no_matching_Goto \<gamma> p [r]"
Red_dom_lcl: fixes e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"
fresh_fmdrop: fixes x :: "atom"   and B :: "('a, 'b) fmap"   and y :: "'a" assumes "x \<sharp> B"   and "x \<sharp> y" shows "x \<sharp> fmdrop y B"
assms: shows "y * x \<le> x * x\<^sup>\<circ> * ((1::'a) \<squnion> y)"
is_interval_connected: fixes S :: "'a set" assumes "is_interval S" shows "connected S"
simps_4: fixes i :: "char"   and "is" :: "char list"   and p_i :: "char"   and p_is :: "char list" shows "internal_iface_name_match (i # is) (p_i # p_is) = (if i = CHR ''+'' \<and> is = [] then True else p_i = i \<and> internal_iface_name_match is p_is)"
map_le_mp: fixes m :: "'b \<Rightarrow> 'c option"   and m' :: "'b \<Rightarrow> 'c option"   and k :: "'b"   and v :: "'c" assumes "m \<subseteq>\<^sub>m m'"   and "m k = Some v" shows "m' k = Some v"
set_val_def: shows "set_val \<equiv> l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val"
lift_def: fixes f :: "'a \<Rightarrow> int"   and w :: "(bool \<times> 'a) list" shows "int.lift f w = int.m_concat (map (int.lift_gi f) w)"
Ogood_union: shows "Ogood = (\<Union>K. Ogood \<inter> {x \<in> space M. \<forall>n\<in>{1..Nf 1}. \<forall>l\<in>{1..n}. - (real K * real l) < u n x - u (n - l) x})"
commute: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "comm_semigroup f" shows "f a b = f b a"
pair: fixes A :: "'b ptrm"   and B :: "'b ptrm"   and C :: "'b ptrm"   and D :: "'b ptrm" assumes "A \<approx> B"   and "C \<approx> D" shows "PPair A C \<approx> PPair B D"
sum_wt_ParT_notWtFT_finished: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and s :: "'state" assumes "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "finished (cl ! n)" shows "sum (wt (ParT cl) s) {brnL cl n..<brnL cl n + brn (cl ! n)} = 0"
matrix_left_invertible_independent_columns: fixes A :: "(('b, 'd) vec, 'c) vec" shows "(\<exists>B. B ** A = mat (1::'b)) = (\<forall>c. (\<Sum>i\<in>UNIV. c i *s column i A) = 0 \<longrightarrow> (\<forall>i. c i = (0::'b)))"
vp_potent_zero: fixes v :: "'a \<Rightarrow> ant"   and n :: "ant" assumes "valuation K v"   and "0 \<le> n" shows "(n = \<infinity>) = (vp K v\<^bsup> Vr K v n\<^esup> = {\<zero>\<^bsub>Vr K v\<^esub>})"
induct: fixes P :: "('c, 'e, 'fun, 'op, 'opinl) Inca.instr \<Rightarrow> 'd option list \<Rightarrow> bool"   and a0 :: "('c, 'e, 'fun, 'op, 'opinl) Inca.instr"   and a1 :: "'d option list" assumes "\<And>x \<Sigma>. P (Inca.instr.IPush x) \<Sigma>"   and "\<And>uu_ \<Sigma>. P Inca.instr.IPop (uu_ # \<Sigma>)"   and "\<And>x \<Sigma>. P (Inca.instr.ILoad x) (None # \<Sigma>)"   and "\<And>x \<Sigma>. P (Inca.instr.IStore x) (None # None # \<Sigma>)"   and "\<And>op \<Sigma>. P (Inca.instr.IOp op) \<Sigma>"   and "\<And>opinl \<Sigma>. P (Inca.instr.IOpInl opinl) \<Sigma>"   and "\<And>f \<Sigma>. P (Inca.instr.ICall f) \<Sigma>"   and "P Inca.instr.IPop []"   and "\<And>v. P (Inca.instr.ILoad v) []"   and "\<And>v vc vb. P (Inca.instr.ILoad v) (Some vc # vb)"   and "\<And>v. P (Inca.instr.IStore v) []"   and "\<And>v vc vb. P (Inca.instr.IStore v) (Some vc # vb)"   and "\<And>v va. P (Inca.instr.IStore v) [va]"   and "\<And>v va ve vd. P (Inca.instr.IStore v) (va # Some ve # vd)"   and "\<And>v uw_. P (Inca.instr.ICJump v) uw_" shows "P a0 a1"
Gets_certificate_valid: fixes A :: "agent"   and SK :: "nat"   and lid :: "msg"   and xid :: "msg"   and cc :: "msg"   and cm :: "msg"   and C :: "agent"   and EK :: "nat"   and evs :: "event list" assumes "Gets A (sign SK \<lbrace>lid, xid, cc, cm, cert C EK onlyEnc (priSK RCA)\<rbrace>) \<in> set evs"   and "evs \<in> set_pur" shows "EK = pubEK C"
Int_atLeastLessThan: 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.atLeastLessThan less_eq less a b \<inter> ord.atLeastLessThan less_eq less c d = ord.atLeastLessThan less_eq less (ord.max less_eq a c) (ord.min less_eq b d)"
Max_eqI: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>y. y \<in> A \<Longrightarrow> less_eq y x"   and "x \<in> A" shows "linorder.Max less_eq A = x"
le: fixes x :: "real" assumes "1 \<le> x" shows "\<bar>sum_upto (divisor_sigma (- \<alpha>)) x - \<zeta> * x\<bar> \<le> ub x"
HFail_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HFail s s' p"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
induct: fixes x :: "('t, 'f) Language"   and P :: "('t, 'f) Language \<Rightarrow> bool" assumes "x \<in> Theory"   and "\<And>A. A \<in> aAxioms Ax \<Longrightarrow> P A"   and "\<And>A e B. Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e : B \<Longrightarrow> P Vx : A \<turnstile> e\<equiv>e : B"   and "\<And>A e1 e2 B. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> e2\<equiv>e1 : B"   and "\<And>A e1 e2 B e3. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B; Vx : A \<turnstile> e2\<equiv>e3 : B \<in> Theory; P Vx : A \<turnstile> e2\<equiv>e3 : B\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> e1\<equiv>e3 : B"   and "\<And>A e1 e2 B f C. \<lbrakk>Vx : A \<turnstile> e1\<equiv>e2 : B \<in> Theory; P Vx : A \<turnstile> e1\<equiv>e2 : B; f \<in> Sig aS\<^bsub>Ax\<^esub> : B \<rightarrow> C\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> f E@ e1\<equiv>f E@ e2 : C"   and "\<And>A e1 B e2 e3 C. \<lbrakk>Sig aS\<^bsub>Ax\<^esub> \<triangleright> Vx : A \<turnstile> e1 : B; Vx : B \<turnstile> e2\<equiv>e3 : C \<in> Theory; P Vx : B \<turnstile> e2\<equiv>e3 : C\<rbrakk> \<Longrightarrow> P Vx : A \<turnstile> sub e1 in e2\<equiv>sub e1 in e3 : C" shows "P x"
execute_change: fixes f :: "'a \<Rightarrow> 'a"   and r :: "'a ref"   and h :: "heap" shows "execute (Ref.change f r) h = Some (f (Ref.get h r), Ref.set r (f (Ref.get h r)) h)"
suffix_\<mu>_stable: fixes \<phi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<forall>\<^sub>\<infinity>i. \<mu>_stable \<phi> (suffix i w)"
prems: shows "\<forall>a|\<in>|e. case a of (a, b) \<Rightarrow> (case a of (from, to) \<Rightarrow> \<lambda>t. 0 < to) b"
to_nat_eq_card: shows "Rep a + 1 = int CARD('a)"
frac_eq_eq: fixes y :: "'a"   and z :: "'a"   and x :: "'a"   and w :: "'a" assumes "y \<noteq> (0::'a)"   and "z \<noteq> (0::'a)" shows "(x / y = w / z) = (x * z = w * y)"
IH: shows "map sgn (coeffs (reduce_root a p)) = map sgn (coeffs p)"
continuous_on_diff: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x - g x)"
unit_char_2: fixes a :: "'c" assumes "C.obj a" shows "D.iso (G.unit a)"
absp_def: fixes \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> 'a" shows "absp \<phi> \<equiv> \<phi> \<circ> reps"
hyps_1: shows "\<not> \<Gamma>\<turnstile> (c\<^sub>1, Fault x) \<rightarrow> \<dots>(\<infinity>)"
set_bd_2: fixes x :: "('l, 'm, 'k, 'n) F" shows "|F'set1 x| \<le>o F'bd"
frag_cmul_eq_0_iff: fixes k :: "int"   and c :: "'b \<Rightarrow>\<^sub>0 int" shows "(frag_cmul k c = 0) = (k = 0 \<or> c = 0)"
prems: fixes v :: "'a vec" assumes "v \<in> set (v__ # vs)" shows "v \<in> carrier_vec n"
comp_CONJ: fixes f :: "int \<Rightarrow> int"   and g :: "int \<Rightarrow> int" shows "CONJ (f \<circ> g) = CONJ f \<circ> CONJ g"
posterior_def: fixes t :: "transition"   and i :: "value list"   and r :: "nat \<Rightarrow>f value option" shows "posterior t i r = posterior_separate (Arity t) (Guards t) (Updates t) i r"
reduce_basis_cost_expanded: fixes Lg :: "nat" assumes "nat \<lceil>log (real_of_rat (4 * \<alpha> / (4 + \<alpha>))) (real N)\<rceil> \<le> Lg" shows "cost (reduce_basis_cost fs_init) \<le> 4 * Lg * m * m * m * n + 4 * Lg * m * m * m * m + 16 * Lg * m * m * m + 4 * Lg * m * m + 3 * m * m * m + 3 * m * m * n + 10 * m * m + 2 * n * m + 3 * m"
leg0_in_hom_2: shows "\<guillemotleft>s\<^sub>0 : s\<^sub>0 \<Rightarrow> s\<^sub>0\<guillemotright>"
safe_def: fixes s :: "v_state"   and r :: "nat"   and v :: "val" shows "safe s r v = (\<forall>r'<r. \<forall>Q w. quorum_for Q w (votes s r') \<longrightarrow> v = w)"
dgrad_p_set_leI: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>f. f \<in> F \<Longrightarrow> dgrad_p_set_le d {f} G" shows "dgrad_p_set_le d F G"
induct: fixes P :: "v sem_env \<Rightarrow> 'b state \<Rightarrow> exp0 list \<Rightarrow> bool"   and a0 :: "v sem_env"   and a1 :: "'b state"   and a2 :: "exp0 list" assumes "\<And>env s. P env s []"   and "\<And>env s e es. (\<And>x y x1. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e; y = Rval x1\<rbrakk> \<Longrightarrow> P env x es) \<Longrightarrow> P env s (e # es)" shows "P a0 a1 a2"
Tqm: shows "T \<in> quasi_measure_preserving M M"
compl_inf_bot: fixes x :: "'a upred" shows "(\<not> x \<and> x) = false"
vimage_sets_compl_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and M :: "'a measure" shows "(f -` A \<inter> space M \<in> sets M) = (f -` (- A) \<inter> space M \<in> sets M)"
axioms_5: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "is_dghm_axioms \<AA> \<BB> \<FF>"
gcd_ff_list_def: fixes X :: "'b fract list"   and g :: "'b fract" shows "gcd_ff_list X g = ((\<forall>x\<in>set X. divides_ff g x) \<and> (\<forall>d. (\<forall>x\<in>set X. divides_ff d x) \<longrightarrow> divides_ff d g))"
succs_2: shows "h' [succs]\<rightarrow>* h''"
Inf_to_Sup: shows "Inf = \<partial> \<circ> Sup \<circ> (`) \<partial>"
minus_minus: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "\<ominus>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> x) = x"
integral_has_vector_derivative_piecewise_continuous: fixes a :: "real"   and b :: "real"   and D :: "real set"   and f :: "real \<Rightarrow> 'a"   and x :: "real" assumes "piecewise_continuous_on a b D f"   and "x \<in> {a..b} - D" shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within {a..b} - D)"
SN_trancl_SN_conv: fixes R :: "('b \<times> 'b) set" shows "SN (R\<^sup>+) = SN R"
suffix_ConsD: fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "suffix (x # xs) ys" shows "suffix xs ys"
simps: fixes p :: "float poly"   and e :: "float interval" shows "tm_neg (TaylorModel p e) = TaylorModel (~\<^sub>p p) (- e)"
param_RECT: fixes B :: "('e \<Rightarrow> 'f nres) \<Rightarrow> 'e \<Rightarrow> 'f nres"   and B' :: "('g \<Rightarrow> 'h nres) \<Rightarrow> 'g \<Rightarrow> 'h nres"   and Ra :: "('e \<times> 'g) set"   and Rr :: "('f \<times> 'h) set" assumes "(B, B') \<in> (Ra \<rightarrow> \<langle>Rr\<rangle>nres_rel) \<rightarrow> Ra \<rightarrow> \<langle>Rr\<rangle>nres_rel"   and "trimono B" shows "(REC\<^sub>T B, REC\<^sub>T B') \<in> Ra \<rightarrow> \<langle>Rr\<rangle>nres_rel"
C1: shows "\<lfloor>\<lambda>w. \<forall>x. x actualizedAt w \<longrightarrow> Abstract x w \<longrightarrow> (\<exists>xa. (Concrete xa \<^bold>\<and> x dependsOn xa) w)\<rfloor>"
mpoly_Const_1: shows "Const (1::'b) = 1"
vec_Suc: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "vec (Suc n) f = vCons (f 0) (vec n (f \<circ> Suc))"
FVar: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn"   and C :: "qtname"   and Q :: "'b assn"   and e :: "expr"   and stat :: "bool"   and fn :: "vname"   and R :: "'b assn"   and accC :: "qtname" assumes "G,A\<turnstile>{Normal P} .Init C. {Q}"   and "G,A\<turnstile>{Q} e-\<succ> {\<lambda>Val:a:. fvar C stat fn a ..; R}" shows "G,A\<turnstile>{Normal P} {accC,C,stat}e..fn=\<succ> {R}"
AboveS_subset_Above: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "AboveS r A \<subseteq> Above r A"
simps_2: fixes U :: "complex mat" shows "ps_P.well_com (Utrans U) = (U \<in> carrier_mat ps_P.d ps_P.d \<and> unitary U)"
inject_1: fixes x11 :: "'c"   and x12 :: "'d"   and y11 :: "'c"   and y12 :: "'d" shows "(qVar x11 x12 = qVar y11 y12) = (x11 = y11 \<and> x12 = y12)"
rpf_loose_def: fixes rtbl :: "('h, 'i) routing_rule_scheme list"   and p :: "('h, 'j) simple_packet_scheme" shows "rpf_loose rtbl p \<equiv> \<exists>rr\<in>set rtbl. prefix_match_semantics (routing_match rr) (p_src p)"
assm_5: fixes x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set" assumes "(x1, K1) \<in> s"   and "(x2, K2) \<in> s"   and "(x1, K1) \<noteq> (x2, K2)" shows "interior K1 \<inter> interior K2 = {}"
isLub_least: fixes S :: "'a set"   and L :: "'a"   and z :: "'a" assumes "isLub S cl L"   and "z \<in> A"   and "\<forall>y\<in>S. (y, z) \<in> r" shows "(L, z) \<in> r"
nxt_singles_below_singles: fixes S :: "'b set"   and x :: "'b" shows "nxt (singles S) x \<sqsubseteq> singles S"
left_neutral: fixes a :: "'c \<times> 'd" shows "times_prod' one_prod a = a"
irreflp_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "irreflp R = (\<forall>a. \<not> R a a)"
symmetric: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> Congruence" shows "(b, a) \<in> Congruence"
sub_in_const: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "deg R g = 0" shows "deg R (Cring_Poly.compose R f g) = 0"
Diff_cancel: fixes A :: "'b set" shows "A\<setminus>A = {}"
resChainAlpha: fixes p :: "(name \<times> name) list"   and xvec :: "name list"   and P :: "('e, 'f, 'g) psi" assumes "(p \<bullet> xvec) \<sharp>* P"   and "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" shows "\<lparr>\<nu>*xvec\<rparr>P = \<lparr>\<nu>*p \<bullet> xvec\<rparr>p \<bullet> P"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<longmapsto> a[b] \<prec> P'"   and "Q \<longmapsto> a<x> \<prec> Q'" shows "P \<parallel> Q \<longmapsto> \<tau> \<prec> P' \<parallel> Q'[x::=b]"
conf_ClassD: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and v :: "val"   and C :: "char list" assumes "P,h \<turnstile> v :\<le> Class C" shows "v = Null \<or> (\<exists>a obj T. v = Addr a \<and> h a = \<lfloor>obj\<rfloor> \<and> obj_ty obj = T \<and> P \<turnstile> T \<le> Class C)"
dom_fun_upd: fixes f :: "'c \<Rightarrow> 'd option"   and x :: "'c"   and y :: "'d option" shows "dom (f(x := y)) = (if y = None then dom f - {x} else insert x (dom f))"
assms_4: fixes x :: "complex" assumes "x \<in> path_image (part_circlepath c r a b)" shows "cmod (f x) \<le> B"
sound_nneg: fixes P :: "'a \<Rightarrow> real" assumes "sound P" shows "nneg P"
b: shows "ordered_powerprod_lin.is_le_rel top_tail"
bounded_clinear_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "'a" assumes "bounded_cbilinear prod" shows "bounded_clinear (prod a)"
uadmit_fml_ntadjoint: fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<phi> :: "('sf + 'b, 'sc, 'sz) formula"   and U :: "('sz + 'sz) set"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and I :: "('sf, 'sc, 'sz) interp" assumes "FUadmitFO \<sigma> \<phi> U"   and "Vagree \<nu> \<omega> (- U)"   and "\<And>i. dsafe (\<sigma> i)"   and "fsafe \<phi>"   and "is_interp I" shows "fml_sem (adjointFO I \<sigma> \<nu>) \<phi> = fml_sem (adjointFO I \<sigma> \<omega>) \<phi>"
splits_wrtD_1: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "splits_wrt (ps, qs) T F" shows "cone_decomp T (ps @ qs)"
below_l: shows "\<forall>n. (\<Sum>i<2 * n. (- 1) ^ i * a i) \<le> l"
r_snoc: fixes es :: "nat"   and e :: "nat" shows "eval r_snoc [es, e] \<down>= e_snoc es e"
s'_t_equals_s_t: fixes A :: "'a"   and t :: "typ"   and S :: "nat \<Rightarrow> typ" shows "$ (\<lambda>n. if n \<in> free_tv A \<union> free_tv t then S n else TVar n) t = $ S t"
ground\<^sub>l\<^sub>s: shows "ground\<^sub>l\<^sub>s L"
poly_const_conv: fixes c :: "'a"   and x :: "'a"   and y :: "'a" shows "(poly [:c:] x = y) = (c = y)"
restrict_indets_dehomogenize: fixes p :: "('b option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "restrict_indets (dehomogenize None p) = restrict_indets p"
neg_2: shows "negligible {x \<in> T - S. f x \<noteq> (0::'a)}"
simps_2: fixes x21 :: "'c"   and x22 :: "'c"   and x23 :: "'d" shows "Null \<noteq> MkArr x21 x22 x23"
assert_all_def: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('d \<times> 'e atom) list" shows "assert_all t as \<equiv> let s = AssertAllState''.assert_all_state init_state assert_bound_code check_code t as in if \<U> s then Inl (the (\<U>\<^sub>c s)) else Inr (\<V> s)"
true_cls_empty: fixes I :: "'b set" shows "\<not> I \<TTurnstile> {#}"
bex1_equality: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "\<exists>!x. x \<in> A \<and> P x"   and "x \<in> A"   and "P x"   and "y \<in> A"   and "P y" shows "x = y"
comp_arr_dom: fixes f :: "'a2"   and a :: "'a2" assumes "C2.arr f"   and "C2.dom f = a" shows "f \<cdot>\<^sub>2 a = f"
vars2': fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<sigma> \<in> space M" shows "\<forall>x\<in>free_vars e'. val_type (\<sigma> x) = \<Gamma> x"
fdia_im: fixes R :: "('e \<times> 'f) set"   and f :: "'f \<Rightarrow> 'd"   and x :: "'e" shows " |R\<rangle> f x = \<Squnion>(f ` R `` {x})"
simps_3: fixes b :: "nat"   and t1 :: "trm"   and t2 :: "trm" shows "fvi b (formula.Eq t1 t2) = fvi_trm b t1 \<union> fvi_trm b t2"
fmdom_notD: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x |\<notin>| fmdom m" shows "m $$ x = None"
fst: shows "p \<approx>[\<R>] mix_pmf Up B W"
sorted_add_spmat_helper2: fixes a :: "nat"   and b :: "(nat \<times> 'a) list"   and arr :: "(nat \<times> (nat \<times> 'a) list) list"   and brr :: "(nat \<times> (nat \<times> 'a) list) list"   and ab :: "nat"   and bb :: "(nat \<times> 'a) list"   and list :: "(nat \<times> (nat \<times> 'a) list) list"   and aa :: "nat"   and ba :: "(nat \<times> 'a) list" assumes "add_spmat ((a, b) # arr) brr = (ab, bb) # list"   and "aa < a"   and "sorted_spvec ((aa, ba) # brr)" shows "aa < ab"
Block: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and e\<^sub>0 :: "expr"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and e\<^sub>1 :: "expr"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option" assumes "P,E(V \<mapsto> T) \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0(V := None))\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1)\<rangle>" shows "P,E \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V))\<rangle>"
k33: shows "K\<^bsub>3,3\<^esub> (with_proj G)"
simul_bis: fixes Retr :: "('b \<times> 'b) set \<Rightarrow> ('b \<times> 'b) set" assumes "Bisim.mono Retr" shows "simul Retr (bis Retr)"
not_less: fixes x :: "'c word"   and y :: "'c word" shows "(\<not> x <s y) = (y \<le>s x)"
lossless_\<A>: shows "rivest_commit.lossless (\<A> n)"
inj_def_var1: fixes x :: "'a" shows "is_inj x = (- 1' ; x \<le> - x)"
simps_7: fixes prec :: "nat"   and p :: "float interval poly"   and n :: "nat"   and I :: "float interval list"   and a :: "float interval list" shows "compute_bound_poly prec (Pw p n) I a = power_float_interval prec n (compute_bound_poly prec p I a)"
Arg2pi_times_of_real: fixes r :: "real"   and z :: "complex" assumes "0 < r" shows "Arg2pi (complex_of_real r * z) = Arg2pi z"
SIDEI_1: fixes P :: "bool" assumes "P" shows "PREFER_tag P"
DF_def: fixes A :: "'d set" shows "DF A \<equiv> \<mu> x. \<sqinter>xa\<in>A \<rightarrow>  x"
pell_power_def: fixes z :: "'a \<times> 'a"   and n :: "nat" shows "pell_power z n = ((\<lambda>z'. pell_mul z' z) ^^ n) (1::'a, 0::'a)"
assms_1: shows "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"
count_const_lt_components_alt: fixes hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list" shows "count_const_lt_components hs = card (const_lt_component ` fst ` set hs - {None})"
clm_sing: fixes sensors :: "cars \<Rightarrow> traffic \<Rightarrow> cars \<Rightarrow> real"   and c :: "cars" assumes "hmlsl sensors" shows "\<forall>ts v. \<nexists>va u. v=va--u \<and> (0 < \<parallel>ext va\<parallel> \<and> sensors.len sensors va ts c = ext va \<and> restrict va (clm ts) c = lan va \<and> |lan va| = 1) \<and> 0 < \<parallel>ext u\<parallel> \<and> sensors.len sensors u ts c = ext u \<and> restrict u (clm ts) c = lan u \<and> |lan u| = 1"
oth_class_taut_5_a: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<psi> \<^bold>\<equiv> \<^bold>\<not>(\<phi> \<^bold>& \<^bold>\<not>\<psi>) in v]"
map_of_aset_update: fixes S :: "('b \<times> 'c) set"   and k :: "'b"   and v :: "'c"   and v' :: "'c" assumes "unique_keys_set S"   and "(k, v) \<in> S" shows "map_of_aset (S - {(k, v)} \<union> {(k, v')}) =  map_of_aset S { k \<rightarrow> v' }"
complex_norm_square: fixes z :: "complex" shows "cor ((cmod z)\<^sup>2) = z * cnj z"
BRANCH_inject: fixes c :: "'f"   and rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set"   and c' :: "'f"   and rs' :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "(BRANCH c rs = BRANCH c' rs') = (c = c' \<and> rs = rs')"
fps_power_eqD': fixes f :: "'a fps"   and m :: "nat"   and g :: "'a fps" assumes "f ^ m = g ^ m"   and "fps_nth f (subdegree f) = fps_nth g (subdegree g)"   and "0 < m" shows "f = g"
partition_set: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and yes :: "'a list"   and no :: "'a list" assumes "partition P xs = (yes, no)" shows "set yes \<union> set no = set xs"
assms_9: shows "q \<cdot> l \<le> l \<cdot> q"