hcju/isabelleps huggingface dataset:
postQ_transl: fixes C :: "Nielson_VCG.acom" shows "Nielson_VCGi.postQ (transl C) = Nielson_VCG.postQ C"
l_get_dom_component_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" shows "l_get_dom_component_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr \<equiv> known_ptr = DocumentClass.known_ptr"
restrict_PiE: fixes f :: "'b \<Rightarrow> 'c"   and I :: "'b set"   and S :: "'b \<Rightarrow> 'c set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
intro: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "\<And>gt xs x. ext gt (xs @ [x]) xs" shows "ext_snoc_axioms ext"
assms_2: fixes res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource"   and a :: "'a" assumes "X res res'"   and "a \<in> A" shows "rel_spmf (rel_prod (=) (\<lambda>res res'. X res res' \<or> A \<turnstile>\<^sub>R res \<sim> res')) (run_resource res a) (run_resource res' a)"
step_6_correct: fixes p :: "prog" assumes "spec\<^sub>6 p" shows "spec\<^sub>5 p"
is_rbt_rbt_unionwk: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t1 :: "('a, 'b) RBT_Impl.rbt"   and t2 :: "('a, 'b) RBT_Impl.rbt"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.linorder less_eq less"   and "ord.is_rbt less t1"   and "ord.is_rbt less t2" shows "ord.is_rbt less (ord.rbt_union_with_key less f t1 t2)"
1: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>l\<^sub>s\<^sub>t A')"
s_1: fixes i :: "nat"   and n :: "nat" shows "i < s n i"
f3: shows "(dim_row (mk_poly A) = dim_col (mk_poly A)) = False"
ap_congR: fixes R :: "'a itrm \<Rightarrow> 'a itrm \<Rightarrow> bool"   and x :: "'a itrm"   and x' :: "'a itrm"   and f :: "'a itrm" assumes "itrm_cong R x x'" shows "itrm_cong R (f \<diamondop> x) (f \<diamondop> x')"
T_del_ub: fixes x :: "'a"   and t :: "'a tree" shows "T_del x t \<le> height t + 1"
assms: shows "\<exists>\<sigma>. v |\<notin>| fmdom \<sigma> \<and> \<sigma>1 = [ {|v|} |=> Top ] ++\<^sub>f \<sigma> \<and> \<sigma>2 = [ {|v|} |=> Bot ] ++\<^sub>f \<sigma>"
simps: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and k :: "nat"   and t :: "'c btree" assumes "split split" shows "invar_inorder k t = (bal t \<and> root_order k t)"
inEdge_to_edge: fixes e :: "'node \<times> 'edgeD \<times> 'node"   and g :: "'g"   and n :: "'node" assumes "e \<in> set (inEdges g n)"   and "invar g" obtains eD :: "'edgeD" and n' :: "'node" where "(n', eD, n) \<in> \<alpha>e g"
power_dvd_iff_le_multiplicity: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and p :: "'b"   and n :: "nat" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "dvd.dvd times (power.power one times p n) x = (n \<le> factorial_semiring.multiplicity times one p x)"
SDF_nodictator_witness: fixes x :: "'c"   and y :: "'c"   and A :: "'c set"   and i :: "'d"   and j :: "'d"   and Is :: "'d set" assumes "hasw [x, y] A"   and "hasw [i, j] Is" obtains P :: "'d \<Rightarrow> ('c \<times> 'c) set" where "profile A Is P" and "x \<^bsub>(P i)\<^esub>\<prec> y" and "y \<^bsub>(P j)\<^esub>\<prec> x"
Zero_ring1: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R"   and "\<not> zeroring R" shows "1\<^sub>r\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
assms_1: shows "d (pp_of_term v) \<le> m"
inputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "a<b>.P \<longmapsto> Rs" obtains x :: "name" and aa :: "name" and u :: "name" and P :: "pi" where "a = aa \<and> [b].Pa__ = [x].P" and "Rs = aa<u> \<prec> P[x::=u]" and "x \<noteq> aa" and "x \<noteq> u"
RHS: shows "\<forall>a b. {x \<in> S. f x \<in> cbox a b} \<in> sets (lebesgue_on S)"
sympD: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and a :: "'c" assumes "symp r"   and "r b a" shows "r a b"
language_def: shows "Power.language \<equiv> {xs. Power.nextl (dfa.init Power_dfa) xs \<in> dfa.final Power_dfa}"
lookup_tenv: fixes \<rho> :: "(nat \<times> val) list"   and x :: "nat"   and v :: "val" assumes "lookup \<rho> x = Some v" shows "lookup (Tenv \<rho>) x = Some (T v)"
substitute_T_cong': fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and t :: "'b ttree"   and T' :: "'b set" assumes "xs \<in> paths (substitute f T t)"   and "\<And>x. (x \<in> T) = (x \<in> T') \<or> f x = TTree.empty" shows "xs \<in> paths (substitute f T' t)"
of_bl_def: shows "of_bl \<equiv> map_fun id Word.Word bl_to_bin"
lran_bwd_simp: fixes a :: "int \<Rightarrow> 'b"   and l :: "int"   and h :: "int" shows "lran a l h = (if l < h then lran a l (h - 1) @ [a (h - 1)] else [])"
subspace_add: fixes S :: "('c \<Rightarrow>\<^sub>0 'b) set"   and x :: "'c \<Rightarrow>\<^sub>0 'b"   and y :: "'c \<Rightarrow>\<^sub>0 'b" assumes "phull.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
ffold_rec_exp: fixes k :: "'e"   and x :: "'e \<rightharpoonup>\<^sub>f 'f"   and ky :: "'e \<times> nat"   and f :: "'f \<Rightarrow> nat" assumes "k |\<in>| fmdom x"   and "ky = (k, the (fmlookup (fmmap f x) k))" shows "ffold tcf 0 (fset_of_fmap (fmmap f x)) = tcf ky (ffold tcf 0 (fset_of_fmap (fmmap f x) |-| {|ky|}))"
axiom_instance: fixes \<phi> :: "\<o>"   and v :: "i" assumes "Axioms.axiom \<phi>" shows "[\<phi> in v]"
parallelE: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<parallel> ys" obtains "\<not> prefix xs ys \<and> \<not> prefix ys xs"
ide_cod: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f" shows "J_C.ide (J_C.cod f)"
integrable_on_all_intervals_UNIV: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> real" assumes "\<And>a b. f integrable_on cbox a b"   and "\<And>x. norm (f x) \<le> g x"   and "g integrable_on UNIV" shows "f integrable_on UNIV"
game_1_def: fixes \<A> :: "(bool, '\<beta>, '\<gamma>) gpv" shows "prf'.game_1 \<A> = exec_gpv prf.random_oracle \<A> Map.empty \<bind> (\<lambda>(b, uu_). return_spmf b)"
bigoI_tendsto_norm: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and c :: "real"   and F :: "'b filter" assumes "((\<lambda>x. norm (f x / g x)) \<longlongrightarrow> c) F"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "f \<in> O[F](g)"
open_translation_subtract: fixes S :: "'a set"   and a :: "'a" assumes "open S" shows "open ((\<lambda>x. x - a) ` S)"
assms_3: shows "\<forall>j. j \<noteq> i \<longrightarrow> S (\<sigma> j) (\<sigma>' j)"
vlrestriction_atI: fixes r :: "V"   and a :: "V"   and A :: "V"   and b :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
map_resource_def: fixes a :: "'k \<Rightarrow> 'i"   and b :: "'j \<Rightarrow> 'l" shows "map_resource a b \<equiv> corec_resource (\<lambda>res. map_spmf (map_prod b Inr) \<circ> run_resource res \<circ> a)"
merge_single: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'm" shows "merge (single x) f = f x"
RS1n_nMT: fixes p :: "('a, 'b) Combinators list" assumes "p \<noteq> []" shows "removeShadowRules1_alternative p \<noteq> []"
stock_filtration_eq: fixes N :: "bool stream measure"   and q :: "real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1" shows "G n = stoch_proc_filt N geom_proc borel n"
map_fun_apply: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'd"   and x :: "'b" shows "(f ---> g) h x = g (h (f x))"
star_decompose_1: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> = (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star>"
proper_designI: assumes "int (size \<B>\<^sup>R) \<noteq> 0" shows "proper_design (bl\<^sup>c) \<B>\<^sup>R"
faceq: fixes h :: "'a set" assumes "h \<in> F" shows "a h \<noteq> (0::'a) \<and> h = {x. a h \<bullet> x \<le> b h}"
plus_fps_def: fixes f :: "('c, 'd) fps"   and g :: "('c, 'd) fps" shows "f + g = Abs_fps (\<lambda>n. f $ n + g $ n)"
borel_measurable_abs: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. \<bar>f x\<bar>) \<in> borel_measurable M"
tendsto_compose: fixes g :: "'b \<Rightarrow> 'c"   and l :: "'b"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "g \<midarrow>l\<rightarrow> g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
assms: shows "\<forall>i\<in>frees t. n \<le> i"
exhaust: fixes y :: "latin_type" obtains "y = Row"   | "y = Col"   | "y = Num"
props_3: shows "\<forall>(k, \<gamma>)\<in>subdiv. \<gamma> C1_differentiable_on {0..1}"
LLL_many_reconstruction: fixes p :: "int"   and pl :: "int"   and f :: "int poly"   and us :: "int poly list"   and fs :: "int poly list"   and F :: "int poly"   and l :: "nat" assumes "LLL_implementation.LLL_many_reconstruction p pl f us = fs"   and "degree f \<noteq> 0"   and "poly_mod.unique_factorization_m pl f (lead_coeff f, mset us)"   and "f dvd F"   and "\<And>ui. ui \<in> set us \<Longrightarrow> poly_mod.Mp pl ui = ui"   and "F \<noteq> 0"   and "comm_monoid_mult_class.coprime (lead_coeff F) p"   and "poly_mod.square_free_m p F"   and "1 < pl"   and "pl = p ^ l"   and "prime p"   and "2 ^ (5 * (degree F div 2) * (degree F div 2)) * \<parallel>F\<parallel>\<^sup>2 ^ (2 * (degree F div 2)) < pl\<^sup>2" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
inf_bot_left: fixes x :: "'h multiset" shows "{#} \<inter># x = {#}"
multiplicity_power_nat: fixes p :: "nat"   and n :: "nat"   and k :: "nat" assumes "prime p"   and "0 < n" shows "multiplicity p (n ^ k) = k * multiplicity p n"
nres_monad1: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a nres" shows "RETURN x \<bind> f = f x"
hyps_2: assumes "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow>\<notin>Fault ` F" shows "strip F \<Gamma>\<turnstile>c1 \<down> Normal s"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'c" shows "poly_mapping.lookup (x - xa) = (\<lambda>k. poly_mapping.lookup x k - poly_mapping.lookup xa k)"
drop_Nil: fixes n :: "nat" shows "[] \<up> n = []"
vsingletonI: fixes x :: "V" shows "x \<in>\<^sub>\<circ> set {x}"
exists_factorization_prod_list: fixes P :: "'b poly list" assumes "0 < degree (prod_list P)"   and "\<And>u. u \<in> set P \<Longrightarrow> 0 < degree u \<and> monic u"   and "square_free (prod_list P)" shows "\<exists>Q. prod_list Q = prod_list P \<and> length P \<le> length Q \<and> (\<forall>u. u \<in> set Q \<longrightarrow> irreducible u \<and> monic u)"
eval_poly_mult: fixes p :: "'a list"   and q :: "'a list"   and a :: "'a" assumes "set p \<subseteq> carrier R"   and "set q \<subseteq> carrier R"   and "a \<in> carrier R" shows "local.eval (poly_mult p q) a = local.eval p a \<otimes> local.eval q a"
size_less_than_median_of_medians_strong: fixes d :: "nat"   and med :: "'b list \<Rightarrow> 'b"   and x :: "'b"   and xs :: "'b list" assumes "0 < d"   and "\<And>xs. \<lbrakk>xs \<noteq> []; length xs \<le> d\<rbrakk> \<Longrightarrow> is_median (med xs) xs"   and "is_median x (map med (chop d xs))" shows "size {#y \<in># mset xs. y < x#} \<le> length (chop d xs) * (d div 2) + length (chop d xs) div 2 * ((d + 1) div 2)"
straight_path_diffrentiable_y: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and y2 :: "real"   and y1 :: "real"   and b :: "real" assumes "\<gamma> = (\<lambda>x. (y2 + y1 * x, b))" shows "\<forall>x. \<gamma> differentiable at x"
allE_id: fixes y :: "'var"   and \<phi> :: "'fmla" assumes "y \<in> var"   and "\<phi> \<in> fmla"   and "bprv (all y \<phi>)" shows "bprv \<phi>"
less_eq_multiset_plus_left: fixes N :: "'b multiset"   and M :: "'b multiset" shows "N \<le> M + N"
invert_side_of_triangle_2: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "arr h"   and "f \<cdot> g = h"   and "local.iso g" shows "seq h (local.inv g) \<and> f = h \<cdot> local.inv g"
val_axiom5: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>x. x \<in> carrier K \<and> v x \<noteq> \<infinity> \<and> v x \<noteq> 0"
reachableNodes_append_edge: fixes u :: "nat"   and s :: "nat"   and v :: "nat" assumes "u \<in> cf.reachableNodes s"   and "(u, v) \<in> cf.E" shows "v \<in> cf.reachableNodes s"
that: fixes N :: "nat" assumes "\<And>n0 n1. N < n0 \<and> N < n1 \<Longrightarrow> s n0 n = s n1 n" shows "thesis"
rel_cong: fixes x :: "'b option"   and ya :: "'b option"   and y :: "'c option"   and xa :: "'c option"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "x = ya"   and "y = xa"   and "\<And>z yb. \<lbrakk>z \<in> set_option ya; yb \<in> set_option xa\<rbrakk> \<Longrightarrow> R z yb = Ra z yb" shows "rel_option R x y = rel_option Ra ya xa"
fmlookup_inverse: fixes x :: "('a, 'b) fmap" shows "map_of_fmap (fmlookup x) = x"
inverse_pow_pow: fixes a :: "'a"   and r :: "nat" assumes "a \<in> carrier G" shows "inv (a [^] r) = inv a [^] r"
less_TC_imp_VWO: fixes x :: "V"   and y :: "V" assumes "x \<sqsubset> y" shows "(x, y) \<in> VWO"
lists_length_Suc_eq: fixes A :: "'g set"   and n :: "nat" shows "{xs. set xs \<subseteq> A \<and> length xs = Suc n} = (\<lambda>(xs, n). n # xs) ` ({xs. set xs \<subseteq> A \<and> length xs = n} \<times> A)"
mkSPRDetAuto_def: shows "mkSPRDetAuto \<equiv> \<lambda>agents jkbp envInit envAction envTrans envVal envObsC envObs. mkAlgAuto acts_MapOps trans_MapOps (SPRViewDet.spr_simObs envObsC) (SPRViewDet.spr_simInit envInit envObsC envObs) (SPRViewDet.spr_simTrans agents jkbp envAction envTrans envVal envObsC envObs) (SPRViewDet.spr_simAction jkbp envVal envObs) (\<lambda>a. map (SPRViewDet.spr_simInit envInit envObsC envObs a \<circ> envObs a) envInit)"
size_mset_mono: fixes A :: "'b multiset"   and B :: "'b multiset" assumes "A \<subseteq># B" shows "size A \<le> size B"
card_eq_UNIV: fixes S :: "'a set" shows "(card S = card UNIV) = (S = UNIV)"
TokensAt_def: fixes k :: "nat"   and I :: "item set" shows "TokensAt k I = {uu_. \<exists>t s x l. uu_ = (t, s) \<and> x \<in> bin I k \<and> next_symbol x = Some t \<and> is_terminal t \<and> l \<in> Lex t Doc k \<and> s = take l (drop k Doc)}"
igAbsCongSSTR_def: fixes MOD :: "('h, 'i, 'j, 'k, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igAbsCongSSTR MOD \<equiv> \<forall>xs x x' y X X'. igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSwap MOD xs y x X = igSwap MOD xs y x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
has_derivative_zero_unique_strong_convex: fixes S :: "'a set"   and K :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and c :: "'a"   and y :: "'b"   and x :: "'a" assumes "convex S"   and "finite K"   and "continuous_on S f"   and "c \<in> S"   and "f c = y"   and "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0::'b)) (at x within S)"   and "x \<in> S" shows "f x = y"
lconsNlist_inv: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and xs :: "('c, 'd) Node list" assumes "is_leveln_ftree n t"   and "is_measured_ftree t"   and "\<forall>x\<in>set xs. is_leveln_node n x \<and> is_measured_node x" shows "is_leveln_ftree n (lconsNlist xs t) \<and> is_measured_ftree (lconsNlist xs t)"
bit_0: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a 0 = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
1: shows "(p - q)\<^sup>2 = rat_poly_times (p - q) (p - q)"
simps: fixes m :: "nat"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wadjust_loop_start m rs (l, r) = (\<exists>ln rn ml mr. l = Oc \<up> ml @ Bk # Oc \<up> Suc m \<and> r = Oc # Bk \<up> ln @ Bk # Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr)"
order_iff: fixes b :: "'a"   and a :: "'a" shows "order.greater_eq a b = (a = sup a b)"
tuple_prj: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.cospan f g"   and "C.seq \<p>\<^sub>1[f, g] h" shows "\<langle>\<p>\<^sub>1[f, g] \<cdot> h \<lbrakk>f, g\<rbrakk> \<p>\<^sub>0[f, g] \<cdot> h\<rangle> = h"
blinfun_eqI: fixes x :: "'d \<Rightarrow>\<^sub>L 'e"   and y :: "'d \<Rightarrow>\<^sub>L 'e" assumes "\<And>i. blinfun_apply x i = blinfun_apply y i" shows "x = y"
word_and_mask_shiftl: fixes x :: "'b word"   and n :: "nat"   and m :: "nat" shows "x AND (mask n << m) = (x >> m) AND mask n << m"
real_is_int_rep: fixes x :: "real" assumes "real_is_int x" shows "\<exists>!a. real_of_int a = x"
unique_authKeys: fixes A :: "agent"   and Ka :: "nat"   and authK :: "nat"   and Ta :: "msg"   and X :: "msg"   and evs :: "event list"   and A' :: "agent"   and Ka' :: "nat"   and Ta' :: "msg"   and X' :: "msg" assumes "Says Kas A \<lbrace>Crypt Ka \<lbrace>Key authK, Agent Tgs, Ta\<rbrace>, X\<rbrace> \<in> set evs"   and "Says Kas A' \<lbrace>Crypt Ka' \<lbrace>Key authK, Agent Tgs, Ta'\<rbrace>, X'\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "A = A' \<and> Ka = Ka' \<and> Ta = Ta' \<and> X = X'"
fpxs_const_uminus: fixes x :: "'b" shows "fpxs_const (- x) = - fpxs_const x"
sep_substate_disj_add': fixes x :: "'a"   and y :: "'a" assumes "x ## y" shows "x \<preceq> y + x"
simps: fixes V :: "nat set"   and s :: "(nat set \<times> 'b option list set) set" shows "New_max.projectQuery V s = New_max.projectTable V ` s"
sublist_MEM: fixes h :: "'c"   and l1 :: "'c list"   and l2 :: "'c list" assumes "subseq (h # l1) l2" shows "ListMem h l2"
distinct_41: fixes x61 :: "msg"   and x62 :: "msg"   and x71 :: "nat"   and x72 :: "msg" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Crypt x71 x72"
onorm_f: shows "\<parallel>f\<parallel> = (SUP x\<in>ball (0::'a) 1. \<parallel>f *\<^sub>v x\<parallel>)"
isCont_rGamma: fixes z :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "isCont f z" shows "isCont (\<lambda>x. rGamma (f x)) z"
M: shows "M \<in> count_space UNIV \<Otimes>\<^sub>M count_space UNIV \<rightarrow>\<^sub>M subprob_algebra N"
arc_simple_path: fixes g :: "real \<Rightarrow> 'a" shows "arc g = (simple_path g \<and> pathfinish g \<noteq> pathstart g)"
deg_fun_zero: shows "deg_fun 0 = (0::'c)"
subset_co_segment: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "({a--b} \<subseteq> {c<--<d}) = (a \<in> {c<--<d} \<and> b \<in> {c<--<d})"
has_derivative_continuous_on: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)" shows "continuous_on s f"
mset_fset_fset_mset: fixes N :: "'c multiset" shows "mset_fset (fset_mset N) = remdups_mset N"
span_zero: fixes S :: "'c set" shows "(0::'c) \<in> vs2.span S"
simps_3: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm"   and v :: "(real, 'd) vec" shows "frechet I (Plus t1 t2) v = (\<lambda>v'. frechet I t1 v v' + frechet I t2 v v')"
diff_ev_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "??.Jordan_Normal_Form_Existence.diff_ev A i j \<equiv> i < j \<longrightarrow> A $$ (i, i) \<noteq> A $$ (j, j) \<longrightarrow> A $$ (i, j) = (0::'b)"
l_heap_is_wellformed_axioms: shows "l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes"
assms_2: shows "\<forall>\<^sub>F x in F. h1 x = h2 x"
simps_4: fixes i :: "int"   and c :: "int"   and e :: "Cooper.num"   and d :: "int" shows "d_\<delta> (NDvd i (CN 0 c e)) d = (i dvd d)"
absorb1: fixes b :: "'a word"   and a :: "'a word" assumes "b \<le>s a" shows "signed.max a b = a"
slp_of_fa_bin_def: fixes Binop :: "floatarith \<Rightarrow> floatarith \<Rightarrow> floatarith"   and a :: "floatarith"   and b :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M2 :: "(floatarith, nat) mapping"   and slp2 :: "floatarith list" shows "slp_of_fa_bin Binop a b M slp M2 slp2 = (case Mapping.lookup M (Binop a b) of None \<Rightarrow> (Mapping.update (Binop a b) (length slp2) M2, slp2 @ [Binop (Var (slp_index_lookup slp2 M2 a)) (Var (slp_index_lookup slp2 M2 b))]) | Some i \<Rightarrow> (Mapping.update (Binop a b) (length slp) M, slp @ [Var (slp_index slp i)]))"
I: fixes t :: "'a" shows "dist x z - Busemann_function_at (to_Gromov_completion t) z x \<le> max (dist x y - Busemann_function_at (to_Gromov_completion t) y x) (dist y z - Busemann_function_at (to_Gromov_completion t) z y - 2 * Busemann_function_at (to_Gromov_completion t) y x) + 2 * deltaG TYPE('a)"
pick_middlep: fixes P :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'b"   and c :: "'f" assumes "(P OO Q) a c" shows "P a (BNF_Def.pick_middlep P Q a c) \<and> Q (BNF_Def.pick_middlep P Q a c) c"
c: shows "\<forall>\<^sub>F n in F. continuous_on A (f n)"
lin_dep_iff_in_span: fixes S :: "'c set"   and v :: "'c" assumes "S \<subseteq> carrier V"   and "lin_indpt S"   and "v \<in> carrier V"   and "v \<notin> S" shows "(v \<in> span S) = lin_dep (S \<union> {v})"
rEqualsF: fixes \<theta>\<^sub>1 :: "trm"   and \<nu> :: "fin_string \<Rightarrow> real"   and r1 :: "real"   and \<theta>\<^sub>2 :: "trm"   and r2 :: "real" assumes "[\<theta>\<^sub>1]\<nu> \<down> r1"   and "[\<theta>\<^sub>2]\<nu> \<down> r2"   and "r1 \<noteq> r2" shows "[Equals \<theta>\<^sub>1 \<theta>\<^sub>2]\<nu> \<downharpoonright> False"
must_sync_def2: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1" shows "t \<turnstile> \<langle>x, m\<rangle> \<wrong>1 = (\<exists>ta x' m' s. r1.r_syntax t x m ta x' m' \<and> r1.actions_ok s t ta)"
last_appendR: fixes ys :: "'c list"   and xs :: "'c list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
n_add_op_def: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y \<equiv> !(!x \<cdot> !y)"
sp_one: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and e :: "'d edge"   and v2 :: "'d" shows "Graph.subpath g v1 [e] v2 = (src e = v1 \<and> e \<in> edges g \<and> tgt e = v2)"
disc_2: fixes x21 :: "'d"   and x22 :: "'e \<Rightarrow> ('d, 'e, 'f) resumption" shows "\<not> is_Done (Pause x21 x22)"
gcsrp_func1: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "gcsrp_map G H \<in> set_rcs G H \<rightarrow> gcsrp_map G H ` set_rcs G H"
IH: shows "\<sigma> l i j = d l * (\<Sum>k<l. \<mu> i k * \<mu> j k * \<beta> fs k)"
eval: shows "prg Env\<turnstile>s0 \<midarrow>es\<doteq>\<succ>vs\<rightarrow> s1"
decisive2: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and A :: "'a set"   and swf :: "('b \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> ('a \<times> 'a) set"   and Is :: "'b set"   and j :: "'b" assumes "hasw [x, y, z] A"   and "iia swf A Is"   and "SWF swf A Is universal_domain"   and "weak_pareto swf A Is universal_domain"   and "semidecisive swf A Is {j} x y" shows "decisive swf A Is {j} z y"
conga_line: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C"   and "A' \<noteq> B'"   and "B' \<noteq> C'"   and "Bet A B C"   and "Bet A' B' C'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"