hcju/isabelleps huggingface dataset:
cases: fixes x :: "('c, 'd) tm" obtains uu_ :: "('c, 'd) hd" where "x = Hd uu_"   | s :: "('c, 'd) tm" and t :: "('c, 'd) tm" where "x = App s t"
rational_preference_axioms_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" shows "rational_preference_axioms carrier relation \<equiv> total_on carrier relation"
chamber_retraction2: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set" assumes "ChamberComplexRetraction X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X" shows "f ` C = C"
wens_single_Un_eq: fixes act :: "('b \<times> 'b) set"   and B :: "'b set" assumes "single_valued act" shows "wens_single act B \<union> wp act (wens_single act B) = wens_single act B"
Red_I_\<G>_def: fixes N :: "('f \<times> 'l) set" shows "Red_I_\<G> N = {\<iota> \<in> Inf_FL. \<G>_I_L \<iota> \<noteq> None \<and> the (\<G>_I_L \<iota>) \<subseteq> Red_I_G (\<Union> (\<G>_F_L ` N)) \<or> \<G>_I_L \<iota> = None \<and> \<G>_F_L (concl_of \<iota>) \<subseteq> \<Union> (\<G>_F_L ` N) \<union> Red_F_G (\<Union> (\<G>_F_L ` N))}"
wf_while_option_Some: fixes P :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and s :: "'a" assumes "wf {(t, s). (P s \<and> b s) \<and> t = c s}"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "P s" shows "\<exists>t. while_option b c s = Some t"
match_pres_def: fixes pres :: "('c \<times> 'd) list"   and s :: "'c \<Rightarrow> 'd option" shows "Solve_SASP.match_pres pres s \<equiv> \<forall>pre\<in>set pres. Solve_SASP.match_pre pre s"
compare_Integer_def: shows "compare = (\<Lambda> (MkI\<cdot>x) (MkI\<cdot>y). if x < y then LT else if y < x then GT else EQ)"
unit_div_eq_0_iff: 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 b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "(divide a b = zero) = (a = zero)"
Rhovar: shows "|Rho| <o |UNIV| \<or> finite Rho"
\<A>_split_1: shows "\<A> = \<A>1 \<union> \<A>2"
K_record_comp: fixes c :: "'c"   and f :: "'b \<Rightarrow> 'd" shows "(\<lambda>x. c) \<circ> f = (\<lambda>x. c)"
smap_onto_PosetComplex: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b set" assumes "ComplexLikePoset less_eq less P" shows "ordering.poset_simplex_map less_eq less P ` P = ordering.PosetComplex less_eq less P"
Re_cos: fixes z :: "complex" shows "Re (cos z) = cos (Re z) * (exp (Im z) + exp (- Im z)) / 2"
terminal_arr_unique: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f' :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "par f f'"   and "terminal_arr f"   and "terminal_arr f'" shows "f = f'"
has_integral_restrict_UNIV: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'c" shows "((\<lambda>x. if x \<in> s then f x else (0::'c)) has_integral i) UNIV = (f has_integral i) s"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "norm x = onorm ((*\<^sub>V) x)"
sq_mtx_unique_sol_autonomous_affine: fixes U :: "(real, 'b) vec \<Rightarrow> real set"   and s :: "(real, 'b) vec"   and t :: "real"   and Y\<^sub>1 :: "real \<Rightarrow> (real, 'b) vec"   and A :: "'b sq_mtx"   and B :: "(real, 'b) vec"   and t\<^sub>0 :: "real"   and Y\<^sub>2 :: "real \<Rightarrow> (real, 'b) vec" assumes "is_interval (U s)"   and "t \<in> U s"   and "Y\<^sub>1 \<in> Sols (\<lambda>t s. A *\<^sub>V s + B) U UNIV t\<^sub>0 s"   and "Y\<^sub>2 \<in> Sols (\<lambda>t s. A *\<^sub>V s + B) U UNIV t\<^sub>0 s" shows "Y\<^sub>1 t = Y\<^sub>2 t"
one_coreflexive_def: shows "1 \<equiv> Abs_coreflexive (1::'b)"
\<psi>_in_hom: fixes f :: "'a" assumes "C.ide f" shows "\<guillemotleft>\<psi> f : F f \<Rightarrow>\<^sub>D \<sigma>.map\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D G f \<star>\<^sub>D \<tau>\<^sub>0 (src\<^sub>C f)\<guillemotright>"
FORWEAK_LIST_transfer_nfoldli: fixes xs :: "'f list"   and c :: "'e \<Rightarrow> 'f \<Rightarrow> 'e nres"   and a :: "'e" shows "nfoldli xs (\<lambda>_. True) (\<lambda>x a. c a x) a \<le> WHILE (\<lambda>(a, xs). xs \<noteq> []) (\<lambda>(a, xs). RETURN (hd xs, tl xs) \<bind> (\<lambda>(x, xs). c a x \<bind> (\<lambda>a. RETURN (a, xs)))) (a, xs) \<bind> (\<lambda>(a, uu_). RETURN a)"
arr_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
intT_proj: 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 infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP" shows "intT \<sigma> (ModelIkPolMcalc2C.proj intT \<sigma> a)"
open_Collect_disj: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}"   and "open {x. Q x}" shows "open {x. P x \<or> Q x}"
swap_in_arcs: fixes x :: "'b" shows "((a \<rightleftharpoons>\<^sub>F b) x \<in> arcs G) = (x \<in> arcs G)"
erH_add: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and p :: "'a"   and q :: "'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "p \<in> carrier R"   and "q \<in> carrier R" shows "erH R S X A B Y h (p \<plusminus> q) = erH R S X A B Y h p \<plusminus>\<^bsub>A\<^esub> erH R S X A B Y h q"
const_unconst: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and u :: "'b"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "unconst u = Some name" shows "u = const name"
ext_singleton_def: fixes ext :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" shows "ext_singleton ext \<equiv> ext ext \<and> ext_singleton_axioms ext"
det_row_0: fixes k :: "nat"   and n :: "nat"   and c :: "nat \<Rightarrow> 'b vec" assumes "k < n"   and "c \<in> {0..<n} \<rightarrow> carrier_vec n" shows "det (mat\<^sub>r n n (\<lambda>i. if i = k then 0\<^sub>v n else c i)) = (0::'b)"
NextRecvUpd'_preserves_prop_invs: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b"   and q :: "'b" assumes "cri.InvMsgInGlob (exchange_config c0)"   and "inv_init_imp_prop_safe c0"   and "\<forall>p. prop_invs (prop_config c0 p)"   and "InvGlobPointstampsEq c0"   and "NextRecvUpd' c0 c1 p q" shows "\<forall>p. prop_invs (prop_config c1 p)"
rep_eq: fixes x :: "'b"   and xa :: "'b poly" shows "coeff (pCons x xa) = case_nat x (coeff xa)"
Dirichlet_strong: fixes h :: "nat" assumes "coprime h n" shows "(\<lambda>x. (\<Sum>x\<in>{p. prime p \<and> [p = h] (mod n) \<and> real p \<le> x}. ln (real x) / real x) - ln x / real (totient n)) \<in> O(\<lambda>_. 1)"
bin_lcp_def: shows "symcode.\<alpha> = u\<^sub>1 \<cdot> u\<^sub>0 \<and>\<^sub>p u\<^sub>0 \<cdot> u\<^sub>1"
measure_preservingI: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and N :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "\<And>A. A \<in> sets N \<Longrightarrow> emeasure M (f -` A \<inter> space M) = emeasure N A" shows "f \<in> measure_preserving M N"
hyps_3: fixes a :: "coname"   and M :: "trm" assumes "<a>:M \<in> \<parallel><B>\<parallel>" shows "SNa M"
psubst_subst_propagate: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and u :: "trm"   and t :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><t[x::=u]> = \<theta><t>[x::=\<theta><u>]"
assoc: fixes a :: "'d list"   and b :: "'d list"   and c :: "'d list" shows "(a @ b) @ c = a @ b @ c"
map_component_id: fixes x :: "'t" shows "map_component (\<lambda>k. k) x = x"
strict_order_iff: fixes b :: "'b"   and a :: "'b" shows "(b < a) = (a = ord.max (\<le>) a b \<and> a \<noteq> b)"
A_def: shows "A = 2 * c * (real n / real (n - 1)) / b ^ (n - 1)"
set_mset_filter: fixes P :: "'k \<Rightarrow> bool"   and M :: "'k multiset" shows "set_mset (filter_mset P M) = {a. a \<in># M \<and> P a}"
chain: shows "chain_on P M {x \<in> A. \<not> Q x}"
summable: fixes z :: "real" assumes "0 \<le> z"   and "z < 1" shows "summable (\<lambda>n. f n * z ^ n)"
right_inverse: fixes a :: "'a" shows "a + - a = (0::'a)"
HUnion_hinsert: fixes a :: "hf"   and A :: "hf" shows "\<Squnion>(A \<triangleleft> a) = a \<squnion> \<Squnion>A"
RSD_pareto_eqclass_insert: fixes agents' :: "'agent set"   and A :: "'alt set"   and i :: "'agent" assumes "RSD_pareto_eqclass agents' alts R A"   and "finite alts"   and "i \<in> agents"   and "agents' \<subseteq> agents" shows "RSD_pareto_eqclass (insert i agents') alts R (Max_wrt_among (R i) A)"
split_cong: fixes q :: "'f \<times> 'g"   and f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and g :: "'f \<Rightarrow> 'g \<Rightarrow> 'h"   and p :: "'f \<times> 'g" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
permute_Arity_def: fixes p :: "perm"   and a :: "Arity" shows "p \<bullet> a = a"
orelse_thms_3: fixes y :: "tr" shows "(\<bottom> orelse y) = \<bottom>"
exec: shows "steps (s, l, r) (ci ly (start_of ly as) ins, start_of ly as - 1) stp = (s', l', r')"
subterm_vars: fixes u :: "'b trm"   and C :: "'b literal set" assumes "u \<in> subterms_of_cl C" shows "vars_of u \<subseteq> vars_of_cl C"
sets_diff_measure: fixes M :: "'c measure"   and N :: "'c measure" shows "sets (diff_measure M N) = sets M"
OP_def: fixes x :: "'e" shows "OP x \<equiv> x"
chainTransitionAppend_2: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
laplace_integrand_absolutely_integrable: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> complex"   and B :: "real \<Rightarrow> real"   and s :: "complex" assumes "exponential_order M c f"   and "\<And>k. AE x\<in>{0..k} in lebesgue. cmod (f x) \<le> B k"   and "\<And>k. f integrable_on {0..k}"   and "c < Re s" shows "complex_set_integrable lebesgue {0..} (laplace_integrand f s)"
NEST_END: fixes i :: "'b" assumes "\<I> i" shows "NEST (END i)"
rep_eq: fixes x :: "'a coreflexive"   and xa :: "'a coreflexive" shows "Rep_coreflexive (x \<sqinter> xa) = Rep_coreflexive x \<sqinter> Rep_coreflexive xa"
N_def: shows "N = bernoulli_stream ((1 + r - d) / (u - d))"
Concrete_def: shows "E! \<equiv> make\<Pi>\<^sub>1 (\<lambda>u s w. case u of \<omega>\<upsilon> x \<Rightarrow> ConcreteInWorld x w | \<sigma>\<upsilon> \<sigma> \<Rightarrow> False)"
set_mult_mono: fixes R :: "('d, 'e) Ring_scheme"   and A1 :: "'d set"   and A2 :: "'d set"   and B :: "'d set" assumes "Ring R"   and "A1 \<subseteq> carrier R"   and "A2 \<subseteq> carrier R"   and "A1 \<subseteq> A2"   and "B \<subseteq> carrier R" shows "set_mult R A1 B \<subseteq> set_mult R A2 B"
singleActionChain_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and u :: "name" assumes "P \<longmapsto> a<x> \<prec> P'" shows "P \<Longrightarrow>\<^sub>lu in P'\<rightarrow>a<x> \<prec> P'[x::=u]"
a1: shows "s' = snd (fst (load_sub3 instr curr_win rd asi address s)) \<and> get_S (cpu_reg_val PSR s) = 0"
linear_eq_stdbasis: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "linear f"   and "linear g"   and "\<And>b. b \<in> Basis \<Longrightarrow> f b = g b" shows "f = g"
intro: fixes U :: "'a set"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "minus_ow U minus"   and "uminus_ow U uminus"   and "monoid_add_ow U plus zero"   and "group_add_ow_axioms U minus plus zero uminus" shows "group_add_ow U minus plus zero uminus"
mono: fixes x1 :: "'a acom"   and x2 :: "'a acom" assumes "P x1"   and "P x2"   and "x1 \<le> x2" shows "f x1 \<le> f x2"
intros_3: fixes l :: "int"   and d :: "int"   and u :: "int" assumes "- l \<le> d"   and "d < u" shows "valid_intv' l u (Intv' d)"
simps_10: fixes x5 :: "char list" shows "Call x5 \<noteq> Accept"
val_neg_nonzero: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "v x < 0" shows "x \<noteq> \<zero>"
beta_preserves_fresh: fixes t :: "lam"   and s :: "lam"   and a :: "name" assumes " t \<longrightarrow>\<^sub>\<beta> s"   and "a \<sharp> t" shows "a \<sharp> s"
SUP_UNION: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd set"   and A :: "'e set" shows "Sup (f ` \<Union> (g ` A)) = (SUP y\<in>A. Sup (f ` g y))"
adjoint_linear: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f" shows "linear (adjoint f)"
minimal_exists: fixes A :: "'g set"   and P :: "'g set \<Rightarrow> bool" assumes "finite A"   and "P A" shows "\<exists>B\<subseteq>A. minimal B P"
prems: shows "simplicial_chain (Suc p) (standard_simplex s) c"
invert_side_of_triangle_2: fixes h :: "'a"   and f :: "'a"   and g :: "'a" assumes "C.arr h"   and "f \<cdot> g = h"   and "C.iso g" shows "C.seq h (C.inv g) \<and> f = h \<cdot> C.inv g"
smult_content_normalize_primitive_part: fixes p :: "'b poly" shows "smult (content p) (normalize (primitive_part p)) = normalize p"
Delay_def: fixes a2 :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and d :: "nat"   and a1 :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list" shows "Delay a2 i1 d a1 i2 \<equiv> \<forall>t. (t < d \<longrightarrow> a1 t = [] \<and> i2 t = []) \<and> (d \<le> t \<longrightarrow> a1 t = a2 (t - d) \<and> i2 t = i1 (t - d))"
card_Suc_k': shows "card {i. i < dim_col A \<and> i \<in> insert b J} = Suc k"
aezfun_finite_supp: fixes a :: "('b, 'a) aezfun" shows "finite (supp (aezfun a))"
inv_on_def: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'b" shows "inv_on f A x \<equiv> SOME y. y \<in> A \<and> f y = x"
aligned\<^sub>d_mono_hmset: fixes e :: "nat"   and M :: "hmultiset"   and M' :: "hmultiset" assumes "local.aligned\<^sub>d e M"   and "hmsetmset M' \<subseteq># hmsetmset M" shows "local.aligned\<^sub>d e M'"
d_subid: fixes x :: "'a" assumes "d x = x" shows "x \<le> 1\<^sub>\<sigma>"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (\<ll> f) = L (S.dom f)"
A: shows "q \<rightarrow> l qs \<in> \<delta>"
lnull_lSup: fixes A :: "'b llist set" shows "lnull (lSup A) = (\<forall>x\<in>A. lnull x)"
lookup_correct: fixes m :: "'m1"   and k :: "'V" assumes "m1.invar m" shows "m1.lookup k m = m1.\<alpha> m k"
actions_def: fixes asig :: "'d set \<times> 'd set \<times> 'd set" shows "actions asig \<equiv> inputs asig \<union> outputs asig \<union> internals asig"
dist_triangle_le: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and z :: "'b"   and y :: "'b"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "dist x z + dist y z \<le> e" shows "dist x y \<le> e"
intro: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "reduced_calculus_axioms Inf Red_I Red_F" shows "reduced_calculus Bot Inf entails Red_I Red_F"
cspan_singleton: fixes x :: "'d" shows "cspan {x} = {\<alpha> *\<^sub>C x |\<alpha>. True}"
le_oSucE: fixes x :: "ordinal"   and y :: "ordinal" assumes "x \<le> oSuc y" obtains "x \<le> y"   | "x = oSuc y"
M1: shows "mat_of_cblinfun M \<in> carrier_mat nB nA"
B_consistent_prfOf_iff_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "B.consistent"   and "prf \<in> proof"   and "\<phi> \<in> fmla" shows "Fvars \<phi> = {} \<longrightarrow> prfOf prf \<phi> = bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
module_hom_on_axioms_def: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'e set"   and scale\<^sub>1 :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and scale\<^sub>2 :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e" shows "module_hom_on_axioms U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 f \<equiv> f ` U\<^sub>M\<^sub>_\<^sub>1 \<subseteq> U\<^sub>M\<^sub>_\<^sub>2 \<and> (\<forall>b1 b2. b1 \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> b2 \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> f (b1 + b2) = f b1 + f b2) \<and> (\<forall>r b. b \<in> U\<^sub>M\<^sub>_\<^sub>1 \<longrightarrow> f (scale\<^sub>1 r b) = scale\<^sub>2 r (f b))"
fv_subst_subset: fixes x :: "'c"   and t :: "('d, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "x \<in> fv t" shows "fv (\<theta> x) \<subseteq> fv (t \<cdot> \<theta>)"
norm_triangle_le: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and x :: "'d"   and y :: "'d"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (plus x y) \<le> e"
2: shows "\<not> prv (exi x (neg (cnj fls (eql (Var x) (Var x)))))"
projs_3: fixes dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat"   and pre :: "nat set" shows "\<pi>\<^sub>4 (dsn, dsk, flag, hops, nhip, pre) = flag"
\<alpha>\<sigma>_surj: shows "surj \<alpha>\<sigma>"
pred_Field_succ: fixes i :: "'a" assumes "isSucc i" shows "pred i \<in> Field r \<and> local.aboveS (pred i) \<noteq> {} \<and> succ (pred i) = i"
A_invariant: fixes a :: "'a"   and b :: "'a" assumes "PA a"   and "A a b" shows "QA b"
Lim_cong_within: fixes a :: "'d"   and b :: "'d"   and x :: "'e"   and y :: "'e"   and S :: "'d set"   and T :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "a = b"   and "x = y"   and "S = T"   and "\<And>x. \<lbrakk>x \<noteq> b; x \<in> T\<rbrakk> \<Longrightarrow> f x = g x" shows "(f \<longlongrightarrow> x) (at a within S) = (g \<longlongrightarrow> y) (at b within T)"
single_resolve: fixes n :: "nat"   and x :: "nat" assumes "0 < n"   and "x + n \<le> x" shows "False"
Zn_iso: fixes G :: "('c, 'd) monoid_scheme"   and gen :: "'c"   and n :: "nat" assumes "Finite_And_Cyclic_Groups.cyclic_group G gen"   and "order G = n" shows "G \<cong> Z n"
length_eq: shows "IArray.length (A - 0) = IArray.length A"
elims: fixes x :: "Preliminaries.var \<Rightarrow> 'c trm"   and xa :: "('c, 'd) lit"   and y :: "('c, 'd) lit" assumes "substL x xa = y" obtains \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm" and at :: "('c, 'd) atm" where "x = \<pi>__" and "xa = Pos at__" and "y = Pos (substA \<pi>__ at__)"   | \<pi> :: "Preliminaries.var \<Rightarrow> 'c trm" and at :: "('c, 'd) atm" where "x = \<pi>" and "xa = Neg at" and "y = Neg (substA \<pi> at)"
assms: fixes x :: "'a"   and y :: "'b" assumes "x |\<in>| S"   and "fmlookup m x = Some y" shows "P y y"
l2_inv5_trans: shows "{l2_inv5 \<inter> l2_inv2 \<inter> l2_inv3} TS.trans l2 {> l2_inv5}"
sgn_3: fixes x :: "real_alg_3" shows "real_of_rat (sgn_3 x) = sgn (real_of_3 x)"
assms_5: shows "\<forall>x. x \<in> ins PQ \<longrightarrow> \<not> exprChannel x m"
Number_Hash: fixes X :: "nat"   and X' :: "msg" shows "Number X \<noteq> Hash X'"
sets_2: shows "Y \<in> sets (Pi\<^sub>M J M)"
monom_Suc: fixes a :: "'c"   and n :: "nat" shows "monom a (Suc n) = pCons (0::'c) (monom a n)"
DC_refines_AC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "(a \<Sqinter> b) \<sqsubseteq> (a \<Squnion> b)"
simps_1: fixes int :: "int"   and int' :: "int" shows "(eint int = eint int') = (int = int')"
scalar_product_def: fixes id :: "'a"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zer :: "'a"   and g :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inver :: "'a \<Rightarrow> 'a"   and v :: "'a list"   and w :: "'a list" assumes "plus_mult id f zer g inver" shows "plus_mult.scalar_product f zer g v w = scalar_prodI zer g f v w"
Refl: fixes H :: "fmla set"   and t :: "trm" shows "H \<turnstile> t EQ t"
assoc: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" shows "signed.max (signed.max a b) c = signed.max a (signed.max b c)"
prems_2: fixes x :: "'a node \<times> 'a node set" assumes "True" shows "f x \<le> SPEC (\<lambda>r. expand_assm_exist \<xi> x \<longrightarrow> expand_rslt_exist \<xi> x r)"
sig_red_single_lt_rep_list: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single sing_reg top_tail p q f t" shows "punit.lt (rep_list q) \<preceq> punit.lt (rep_list p)"
catch_return: fixes x :: "'a"   and m :: "'m" shows "catch (return x) m = return x"
tendsto_componentwise: fixes b :: "'f \<Rightarrow> 'e \<Rightarrow>\<^sub>L 'd"   and a :: "'e \<Rightarrow>\<^sub>L 'd"   and F :: "'f filter" assumes "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> \<Longrightarrow> ((\<lambda>n. blinfun_apply (b n) j \<bullet> i) \<longlongrightarrow> blinfun_apply a j \<bullet> i) F" shows "(b \<longlongrightarrow> a) F"
lsl_prelocalic_var: fixes x :: "'b"   and y :: "'b" shows "\<nu> x \<cdot> y \<le> \<nu> x \<sqinter> \<nu> y"
ide_mkarr_Ide: fixes a :: "'c local.term" assumes "Ide a" shows "ide (mkarr a)"
irreducible_factor_singleton: fixes a :: "'a"   and F :: "'a multiset" assumes "irreducible a" shows "mset_factors F a = (F = {#a#})"
lower_minus: fixes A :: "'b interval"   and B :: "'b interval" shows "lower (A - B) = lower A - upper B"
Dom_F: shows "(\<AA> \<times>\<^sub>D\<^sub>G\<^sub>3 \<BB> \<times>\<^sub>D\<^sub>G\<^sub>3 \<CC>)\<lparr>Dom\<rparr> \<lparr>f, f', f''\<rparr>\<^sub>\<bullet> = A"
bot_apply: fixes x :: "'e" shows "\<bottom> x = \<bottom>"
assms_3: shows "H \<turnstile> LstSeqP f2 k2 y2"
invertible_iarray_def: fixes A :: "'b iarray iarray" shows "invertible_iarray A = (rank_iarray A = nrows_iarray A)"
simps_1: fixes x21 :: "'a"   and x22 :: "'a llist"   and y21 :: "'a"   and y22 :: "'a llist" shows "(LCons x21 x22 = LCons y21 y22) = (x21 = y21 \<and> x22 = y22)"
sum_mset_image_mset_singleton: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b multiset" shows "(\<Sum>x\<in>#A. {#f x#}) = image_mset f A"
k: shows "int (card (set_mset (\<Sum>bl\<in>#remove1_mset blv \<B>. mset_set (bl \<inter> blv)))) = \<k>"