hcju/isabelleps huggingface dataset:
InvariantConflictFlagCharacterizationAfterApplyBackjump_2: fixes state :: "State"   and F0 :: "Literal list list" assumes "InvariantConsistent (getM state)"   and "InvariantUniq (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantUniqC (getC state)"   and "getC state \<noteq> [opposite (getCl state)]"   and "InvariantNoDecisionsWhenConflict (butlast (getF state)) (getM state) (currentLevel (getM state))"   and "getF state \<noteq> []"   and "last (getF state) = getC state"   and "getConflictFlag state"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCllCharacterization (getCl state) (getCll state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "0 < currentLevel (getM state)"   and "isUIP (opposite (getCl state)) (getC state) (getM state)" shows "let state' = applyBackjump state in InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
del_height: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t"   and "bal t" shows "height (del k x t) = height t"
r_result1_total: shows "Partial_Recursive.total r_result1"
hom_0_iff: fixes x :: "'a poly poly" shows "(x_y * x = 0) = (x = 0)"
embedS_def: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
rcosets_assoc: fixes M1 :: "'a set"   and M2 :: "'a set"   and M3 :: "'a set" assumes "M1 \<in> rcosets H"   and "M2 \<in> rcosets H"   and "M3 \<in> rcosets H" shows "M1 <#> M2 <#> M3 = M1 <#> (M2 <#> M3)"
Ord_hadd: fixes x :: "hf"   and y :: "hf" assumes "Ord x"   and "Ord y" shows "Ord (x @+ y)"
WTrtLAss: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and e :: "char list exp"   and T' :: "ty" assumes "E V = \<lfloor>T\<rfloor>"   and "P,E,h \<turnstile> e : T'"   and "P \<turnstile> T' \<le> T" shows "P,E,h \<turnstile> V:=e : Void"
elims_2: fixes x :: "('fsym, 'tp) efsym" assumes "TE_wtFsym x" obtains f :: "'fsym" where "x = Oldf f" and "wtFsym f"   | \<sigma> :: "'tp" where "x = Tag \<sigma>__"   | \<sigma> :: "'tp" where "x = Wit \<sigma>" and "\<not> isRes \<sigma>"
dens: shows "has_density (distr M lborel (\<lambda>x. - inverse x)) lborel (\<lambda>x. f (- inverse x) * ennreal (inverse (x * x)))"
lhd_lmap_Sts: fixes f :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> 'b" shows "lhd (lmap f Sts) = f (lhd Sts)"
constrains_weaken_R: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A co A'"   and "A' \<subseteq> B'" shows "F \<in> A co B'"
1: shows "\<forall>x\<in>{\<parallel>diag_mat f *v x\<parallel> |x. \<parallel>x\<parallel> = 1}. x \<le> Max {\<bar>f i\<bar> |i. i \<in> UNIV}"
assms_3: shows "\<forall>i. (\<tau> ^^ i) {} \<subset> (\<tau> ^^ (i + 1)) {}"
supp_nprod: fixes x :: "'b"   and y :: "'c" shows "supp (nPair x y) = supp x \<union> supp y"
Vagree_antimon: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set"   and W :: "variable set" shows "Vagree \<nu> \<nu>' V \<and> W \<subseteq> V \<longrightarrow> Vagree \<nu> \<nu>' W"
equiintegrable_limit: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "range f equiintegrable_on cbox a b"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x" shows "g integrable_on cbox a b \<and> (\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> integral (cbox a b) g"
stone_phi_embed_triple_pair: fixes f :: "('c regular, 'c dense) phi"   and x :: "'c stone_phi_pair" shows "(if Rep_phi f = stone_phi then Rep_stone_phi_pair x else triple.pairs_uminus (Rep_phi f) (triple.pairs_uminus (Rep_phi f) (Rep_stone_phi_pair x))) \<in> triple.pairs (Rep_phi f)"
zero_divisor_smult_left: fixes a :: "'a"   and f :: "'a poly" assumes "zero_divisor a" shows "zero_divisor (smult a f)"
polypoly_polypoly': fixes p :: "poly"   and n0 :: "nat"   and x :: "'a"   and bs :: "'a list" assumes "isnpolyh p n0" shows "polypoly (x # bs) p = polypoly' bs p"
inj_into_vcard: fixes A :: "V" obtains f :: "V \<Rightarrow> V" where "f \<in> elts A \<rightarrow> elts (vcard A)" and "inj_on f (elts A)"
compile_load: fixes p1 :: "'prog1"   and p2 :: "'prog2"   and s2 :: "'state2" assumes "compile p1 = Some p2"   and "load2 p2 s2" shows "\<exists>s1 i. load1 p1 s1 \<and> match i s1 s2"
assms_8: shows "d \<le> (w \<sqinter> - d)\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
eWlsBinp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and ebinp :: "'bindex \<Rightarrow> 'd withERR option" assumes "eWlsBinp MOD delta ebinp" shows "\<exists>binp. igWlsBinp MOD delta binp \<and> ebinp = OKI binp"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) atm" assumes "F.satA x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = \<xi>'__" and "xa = Eq T1 T2" and "F.int \<xi>'__ T1 = F.int \<xi>'__ T2"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and p :: "'psym" and Tl :: "'fsym trm list" where "x = \<xi>" and "xa = Pr p Tl" and "intPF p (map (F.int \<xi>) Tl)"
mult_Sup_distrib: fixes Y :: "V set"   and x :: "V" assumes "small Y" shows "x * \<Squnion> Y = \<Squnion> ((*) x ` Y)"
dimension_direct_sum_space: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set"   and m :: "nat"   and F :: "'a set" assumes "subfield K R"   and "dimension n K E"   and "dimension m K F"   and "E \<inter> F = {\<zero>}" shows "dimension (n + m) K (E <+> F)"
complement_path_component_Union: fixes S :: "'b set"   and x :: "'b" shows "S - path_component_set S x = \<Union> ({path_component_set S y |y. y \<in> S} - {path_component_set S x})"
conj_comm: fixes P :: "'a upred"   and Q :: "'a upred" shows "(P \<and> Q) = (Q \<and> P)"
strict_order_iff: fixes b :: "'d multiset"   and a :: "'d multiset" shows "(b \<subset># a) = (a = a \<union># b \<and> a \<noteq> b)"
simps: fixes E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and n :: "nat" shows "commit_for_sc P (E, ws) n = (if enat n \<le> llength E then let (E', ws') = SOME (E', ws'). E' \<in> \<E> \<and> P \<turnstile> (E', ws') \<surd> \<and> enat n \<le> llength E' \<and> ltake (enat (n - 1)) E = ltake (enat (n - 1)) E' \<and> (0 < n \<longrightarrow> action_tid E' (n - 1) = action_tid E (n - 1) \<and> (if n - 1 \<in> read_actions E then sim_action else (=)) (action_obs E' (n - 1)) (action_obs E (n - 1)) \<and> (\<forall>i<n - 1. i \<in> read_actions E \<longrightarrow> ws' i = ws i)) \<and> (\<forall>r\<in>read_actions E'. n - 1 \<le> r \<longrightarrow> P,E' \<turnstile> ws' r \<le>hb r) in \<lparr>committed = {..<n}, justifying_exec = E', justifying_ws = ws', action_translation = id\<rparr> else \<lparr>committed = actions E, justifying_exec = E, justifying_ws = ws, action_translation = id\<rparr>)"
test_is_inj_fun: fixes x :: "'a" assumes "is_test x" shows "is_p_fun x \<and> is_inj x"
ZObisT_raw_coind: fixes indis :: "('a \<times> 'a) set"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis"   and "sym theta"   and "theta \<subseteq> PL_Indis.ZOretrT tval aval indis theta" shows "theta \<subseteq> PL_Indis.ZObisT tval aval indis"
err_le_unfold: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "Err.le r (OK a) (OK b) = r a b"
continuous_closedin_preimage: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on S f"   and "closed T" shows "closedin (top_of_set S) (S \<inter> f -` T)"
HSem_restr_cong_below: fixes h :: "('b \<times> 'a) list"   and \<rho> :: "'b \<Rightarrow> 'c"   and \<rho>' :: "'b \<Rightarrow> 'c" assumes "\<rho> f|` (- domA h) \<sqsubseteq> \<rho>' f|` (- domA h)" shows "\<lbrace> h \<rbrace>\<rho> \<sqsubseteq> \<lbrace> h \<rbrace>\<rho>'"
ptrm_alpha_equiv_fvs: fixes X :: "'b ptrm"   and Y :: "'b ptrm" assumes "X \<approx> Y" shows "ptrm_fvs X = ptrm_fvs Y"
polynomial_p_not_0: shows "polynomial_p \<noteq> 0"
real_sqrt_le_iff: fixes x :: "real"   and y :: "real" shows "(sqrt x \<le> sqrt y) = (x \<le> y)"
composition_rule: fixes x :: "'c"   and g1 :: "'a"   and g2 :: "'a" assumes "x \<in> E"   and "g1 \<in> carrier G"   and "g2 \<in> carrier G" shows "\<phi> (g1 \<otimes> g2) x = \<phi> g1 (\<phi> g2 x)"
is_functor_if_value_is_ide: assumes "C.ide a" shows "functor (\<cdot>\<^sub>J) (\<cdot>) \<alpha>.map"
sat_models_inter_inj: fixes P :: "'a ltln set" shows "inj_on (\<lambda>\<phi>. sat_models \<phi> \<inter> Pow P) {abs_ltln\<^sub>P \<phi> |\<phi>. prop_atoms \<phi> \<subseteq> P}"
assms_4: shows "(exe_wf_osig sa \<and> fst ` set (snd sa) = fst ` set tao \<and> (\<forall>type\<in>fst ` set (snd sa). \<forall>ars\<in>snd ` set (the (lookup (\<lambda>k. k = type) (snd sa))). the (lookup (\<lambda>k. k = type) tao) = length ars)) \<and> (\<forall>ty\<in>snd ` set cto. typ_ok_sig (map_of cto, map_of tao, translate_osig sa) ty)"
case_2: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case OK x2 of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = f2 x2"
noteq_irrefl: fixes r :: "'a ref" assumes "r =!= r" shows "False"
inv_solve_right: fixes a :: "'a \<Rightarrow> 'b"   and b :: "'a \<Rightarrow> 'b"   and c :: "'a \<Rightarrow> 'b" assumes "a \<in> carrier (DirProds G I)"   and "b \<in> carrier (DirProds G I)"   and "c \<in> carrier (DirProds G I)" shows "(a = b \<otimes>\<^bsub>DirProds G I\<^esub> inv\<^bsub>DirProds G I\<^esub> c) = (b = a \<otimes>\<^bsub>DirProds G I\<^esub> c)"
simps_2: fixes x :: "var\<^sub>C"   and m :: "Mode"   and mds :: "Mode \<Rightarrow> var\<^sub>C set" shows "C.update_modes (x -=\<^sub>m m) mds = mds(m := {y \<in> mds m. y \<noteq> x})"
accept_def: fixes w :: "nat \<Rightarrow> 'a set" shows "local.gba.accept w \<equiv> \<exists>r. gba.is_acc_run r \<and> (\<forall>i. local.gba.L (r i) (w i))"
bounded_def: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.metric_space dist uniformity open" shows "metric_space.bounded dist S = (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
tseytin_update_def: fixes fresh :: "'a set \<Rightarrow> 'a"   and \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula" assumes "freshstuff_sema fresh" shows "freshstuff_sema.tseytin_update fresh \<A> F \<equiv> Let (map (apsnd ((\<Turnstile>) \<A>)) (freshstuff.tseytin_assmt fresh F)) (foldl pair_fun_upd \<A>)"
Col_perm: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet A B C \<and> Tarski_neutral_dimensionless.Col Bet A C B \<and> Tarski_neutral_dimensionless.Col Bet B A C \<and> Tarski_neutral_dimensionless.Col Bet B C A \<and> Tarski_neutral_dimensionless.Col Bet C A B \<and> Tarski_neutral_dimensionless.Col Bet C B A"
prems_2: shows "f \<noteq> \<zero>\<^bsub>P\<^esub>"
coreflexive_transitive: fixes x :: "'a" assumes "x \<le> top" shows "x \<sqinter> x \<le> x"
delta_hedging_self_financing: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat" shows "self_financing Mkt (delta_hedging N der matur)"
finite_measure_Union': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> sets M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M (B - A)"
cases: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'a literal multiset" assumes "gd.ord_resolve Sts a1 a2 a3 a4 a5" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<Sum>\<^sub># (mset Cs) + D" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "\<forall>i<n. \<forall>A\<in>#AAs ! i. A = As ! i" and "gd.eligible Sts As (D + negs (mset As))" and "\<forall>i<n. gd.strictly_maximal_wrt (As ! i) (Cs ! i)" and "\<forall>i<n. S_Q Sts (CAs ! i) = {#}"
SubstTerm_def: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf" shows "SubstTerm v u x x' \<equiv> Term u \<and> (\<exists>s k. SeqStTerm v u x x' s k)"
distinct_adj_Cons_Cons: fixes x :: "'g"   and y :: "'g"   and xs :: "'g list" shows "distinct_adj (x # y # xs) = (x \<noteq> y \<and> distinct_adj (y # xs))"
dual_eq_iff: fixes x :: "'a dual"   and y :: "'a dual" shows "(x = y) = (undual x = undual y)"
rel_pmf_mono: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a pmf"   and g :: "'b pmf"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_pmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_pmf B f g"
satA_e: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and at :: "('a, 'b) atm" assumes "M_FullModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP" shows "U.Struct.satA intF intP \<xi> at = M.Struct.satA intF intP \<xi> at"
set_cases: fixes e :: "'b"   and a :: "'b flat_complete_lattice" assumes "e \<in> set_flat_complete_lattice a" obtains "a = flat_complete_lattice.Value e"
hunion_less_iff: fixes i :: "hf"   and j :: "hf"   and k :: "hf" assumes "Ord i"   and "Ord j" shows "(i \<squnion> j < k) = (i < k \<and> j < k)"
beta_reduction_left_fnE: fixes x :: "'a"   and T :: "PreSimplyTyped.type"   and A :: "'a trm"   and M' :: "'a trm" assumes "Fn x T A \<rightarrow>\<beta> M'" shows "\<exists>A'. A \<rightarrow>\<beta> A' \<and> M' = Fn x T A'"
sd_power_prop2: fixes x :: "'a"   and y :: "'b"   and i :: "nat" shows "sd_power (x, y) i = (x ^ i, \<Squnion> {\<alpha> (x ^ k) y |k. k < i})"
A6_Acc: shows "Acc level0 sA6 = {sA7, sA8, sA9}"
dim_zero_eq': fixes A :: "('b, 'c) vec set" assumes "A = {} \<or> A = {0}" shows "vec.dim A = 0"
prv_imp_com: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and \<phi> :: "'c"   and \<chi> :: "'c"   and \<psi> :: "'c" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> (imp \<chi> \<psi>))" shows "prv (imp \<chi> (imp \<phi> \<psi>))"
dickson_gradingD1: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and t :: "'a" assumes "dickson_grading d" shows "d (s + t) = max (d s) (d t)"
Cons_1: assumes "\<alpha> \<in># r|as|" shows "\<exists>\<sigma>1 \<sigma>3. as = \<sigma>1 @ [\<alpha>] @ \<sigma>3 \<and> \<alpha> \<notin> r \<down>l \<sigma>1"
cases: fixes x :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" obtains x :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" and xs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" where "xa__ = x # xs"   | "xa__ = []"
ccINF_greatest: fixes A :: "'c set"   and u :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> u \<le> f i" shows "u \<le> Inf (f ` A)"
l2polR_eq_el: fixes N :: "((int \<times> int) set set, int) Combinators list"   and p :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" assumes "N \<noteq> []" shows "C (list2policyR N) p = list2policy (map C N) p"
exits_single: fixes x :: "instr" shows "exits [x] = isuccs x 0 - {0}"
I: fixes v :: "nat \<Rightarrow> 'a" assumes "(\<lambda>n. to_Gromov_completion (v n)) \<longlonglongrightarrow> xi" shows "limsup (\<lambda>n. ereal (dist x (v n) - dist y (v n))) \<le> liminf (\<lambda>n. ereal (dist x (u n) - dist y (u n))) + ereal (2 * deltaG TYPE('a))"
not_lnull_conv: fixes xs :: "'a llist" shows "(\<not> lnull xs) = (\<exists>x xs'. xs = LCons x xs')"
hyps: fixes s :: "'a" assumes "(s, ts) \<in> seq ars" shows "(s, map (Pair True) ts) \<in> conv ars \<and> lst_conv (s, map (Pair True) ts) = lst (s, ts) \<and> labels (s, ts) = labels_conv (s, map (Pair True) ts)"
sup_is_lubI: fixes wf_md :: "'b prog \<Rightarrow> String.literal \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and T :: "ty"   and U :: "ty"   and V :: "ty" assumes "wf_prog wf_md P"   and "is_type P T"   and "is_type P U"   and "SemiType.sup P T U = OK V" shows "P \<turnstile> lub(T, U) = V"
s_Let_inv': fixes \<pi> :: "Syntax.term list"   and e\<^sub>1 :: "Syntax.term"   and x :: "Syntax.var"   and e\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Syntax.Let e\<^sub>1 x e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   and "atom x \<sharp> (e\<^sub>1, \<pi>)" obtains e\<^sub>1' :: "Syntax.term" where "e' = e\<^sub>2[e\<^sub>1 / x] \<and> value e\<^sub>1 \<and> \<pi> = \<pi>' \<or> \<lless>\<pi>, e\<^sub>1\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>1'\<ggreater> \<and> e' = Syntax.Let e\<^sub>1' x e\<^sub>2 \<and> \<not> value e\<^sub>1"
Agent_Tag: fixes X :: "agent"   and X' :: "tag" shows "Messages.Agent X \<noteq> Tag X'"
indicator_union_arith: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" shows "indicator (A \<union> B) x = indicator A x + indicator B x - indicator A x * indicator B x"
omit_set: fixes xs :: "nat set"   and ps :: "('b, 'c) fm list" shows "set (omit xs ps) \<subseteq> set ps"
set_subset_Univ: fixes a :: "'s" assumes "S.ide a" shows "S.set a \<subseteq> S.Univ"
le_2p_upper_bits: fixes p :: "'b word"   and n :: "nat" assumes "p \<le> 2 ^ n - 1"   and "n < LENGTH('b)" shows "\<forall>n'\<ge>n. n' < LENGTH('b) \<longrightarrow> \<not> p !! n'"
trm_naturality: fixes f :: "'a" assumes "arr f" shows "\<t>[cod f] \<cdot> f = \<t>[local.dom f]"
ennreal_SUP_const_minus: fixes I :: "'b set"   and c :: "ennreal"   and f :: "'b \<Rightarrow> ennreal" assumes "I \<noteq> {}"   and "c < top" shows "(INF x\<in>I. c - f x) = c - Sup (f ` I)"
null_rec_1: fixes x :: "'c"   and xs :: "'c list" shows "List.null (x # xs) = False"
bij_betw_open_intervals: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "a < b"   and "c < d" shows "\<exists>f. bij_betw f {a<..<b} {c<..<d}"
cases: fixes a :: "'d"   and r :: "('d \<times> 'e) set" assumes "a \<in> Domain r" obtains aa :: "'d" and b :: "'e" where "a = aa" and "(aa, b) \<in> r"
subset_trans: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" assumes "A \<subseteq> B"   and "B \<subseteq> C" shows "A \<subseteq> C"
m_upper: fixes j :: "nat"   and N :: "nat"   and i :: "nat"   and x :: "'a" assumes "j \<le> N"   and "i \<le> N"   and "x \<in> B j i" shows "j \<le> m N x"
R3_idem2: fixes x :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" shows "R3 (R3 x) = R3 x"
space_eq_univ: fixes n :: "nat" shows "space (M n) = UNIV"
iso_iff: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r" shows "iso r r' f = (embed r r' f \<and> f ` Field r = Field r')"
qconv_comp: fixes c\<^sub>1 :: "'g \<Rightarrow>\<^sub>U 'e"   and c\<^sub>2 :: "'f \<Rightarrow>\<^sub>U 'g"   and x :: "'c['d, 'f]" shows "qconv (c\<^sub>1 \<circ>\<^sub>C c\<^sub>2) x = qconv c\<^sub>1 (qconv c\<^sub>2 x)"
distinct_2: fixes x1 :: "nat" shows "(_Entry_) \<noteq> (_ x1 _)"
leadsETo_Trans: fixes F :: "'d program"   and A :: "'d set"   and CC :: "'d set set"   and B :: "'d set"   and C :: "'d set" assumes "F \<in> A leadsTo[CC] B"   and "F \<in> B leadsTo[CC] C" shows "F \<in> A leadsTo[CC] C"
PTrans: shows "P \<Longrightarrow>\<^sub>l\<^sup>^\<alpha> \<prec> P'"
multi_self_add_other_not_self: fixes M :: "'f multiset"   and x :: "'f" shows "(M = add_mset x M) = False"
Quo0: fixes y :: "nat" shows "Quo 0 y = 0"
eval_poly_as_sum: fixes h :: "'c \<Rightarrow> 'b"   and p :: "'c poly"   and x :: "'b" assumes "h (0::'c) = (0::'b)" shows "eval_poly h p x = (\<Sum>i\<le>degree p. x ^ i * h (coeff p i))"
T_def: fixes s :: "'a cfg" shows "MC.T s = distr (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K_cfg x)))) (stream_space (count_space UNIV)) (MC.walk s)"
lassocr\<^sub>C_def: shows "lassocr\<^sub>C = map_converter rsuml lsumr id id 1\<^sub>C"
well_com_mat_Ph: shows "well_com (Utrans_P vars1 mat_Ph)"
v11D_1: fixes r :: "V" assumes "v11 r" shows "vsv r"
ContractsWithSubstitutesAndLAD_axioms: shows "ContractsWithSubstitutesAndLAD Xd Xh Pd Ch"
assms_2: shows "invariantsHoldInState state F0 decisionVars"
l4_13: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Col A B C"   and "A B C Cong3 A' B' C'" shows "Col A' B' C'"
l_one: fixes x :: "int vec" assumes "x \<in> carrier_vec n" shows "0\<^sub>v n + x = x"
arrow_determines_cone_2: fixes d :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr"   and a :: "'a"   and x :: "('b \<Rightarrow> 'a, 'b \<Rightarrow> 'a) concrete_category.arr" assumes "J_C.ide d"   and "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) \<Delta>.map a d x" shows "J_C.MkArr (constant_functor.map J (\<cdot>) a) (J_C.Map d) (J_C.Map x) = x"
\<F>: fixes V :: "'a set" assumes "V \<in> \<F>" shows "\<exists>U. U \<in> \<U> \<and> V = U \<inter> S"
moebius_mu_eq_zero_iff: fixes n :: "nat" shows "(moebius_mu n = (0::'b)) = (\<not> squarefree n)"