hcju/isabelleps huggingface dataset:
normalize_replace_primitive_matchexpr: fixes m :: "'b match_expr"   and \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'd)"   and replace_f :: "'d negation_type \<Rightarrow> 'b match_expr"   and a :: "action"   and p :: "'c" assumes "normalized_nnf_match m"   and "\<And>m. normalized_nnf_match m \<Longrightarrow> matches \<gamma> (replace_primitive_matchexpr disc_sel replace_f m) a p = matches \<gamma> m a p" shows "match_list \<gamma> (normalize_match (replace_primitive_matchexpr disc_sel replace_f m)) a p = matches \<gamma> m a p"
rtranclp_power: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" shows "P\<^sup>*\<^sup>* x y = (\<exists>n. (P ^^ n) x y)"
simps_5: fixes ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'a atom formula"   and va :: "'a atom formula" shows "wf_fmla_atom ty_ent (v \<^bold>\<and> va) = False"
consistent_sign_vectors_R_def: fixes qs :: "real poly list"   and S :: "real set" shows "consistent_sign_vectors_R qs S = consistent_sign_vec qs ` S"
VectorSpaceHomI: fixes T :: "'v \<Rightarrow> 'a"   and smult' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a" assumes "GroupHom V T"   and "\<And>r m. \<lbrakk>r \<in> UNIV; m \<in> V\<rbrakk> \<Longrightarrow> T (r \<cdot> m) = smult' r (T m)" shows "VectorSpaceHom (\<cdot>) V smult' T"
finite_funcset_iff: fixes I :: "'b set"   and S :: "'c set" shows "finite (I \<rightarrow>\<^sub>E S) = ((\<exists>a. S \<subseteq> {a}) \<or> I = {} \<or> finite I \<and> finite S)"
A_1: shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<alpha>* \<langle>l', Z'\<rangle>"
reciprocal_reciprocal: fixes P :: "'b poly"   and p :: "nat" assumes "degree P \<le> p" shows "reciprocal_poly p (reciprocal_poly p P) = P"
elims_2: fixes x :: "'b option"   and xa :: "'b option" assumes "x \<sqsubseteq> xa" obtains xb :: "'b" and y :: "'b" where "x = Some xb" and "xa = Some y__" and "xb \<sqsubseteq> y__"   | y :: "'b option" where "x = None" and "xa = y"
T_def: shows "\<T> = Abs (Abs (Var 0 \<degree> Var 1))"
ex_prop_imp: fixes Y :: "'b program set" assumes "ex_prop Y" shows "Y = UNIV guarantees Y"
intros_3: fixes c :: "nat" shows "valid_intv c (Greater c)"
degree_gt: fixes x :: "'d pdevs"   and j :: "nat" assumes "pdevs_apply x j \<noteq> (0::'d)" shows "j < degree x"
id2_2: shows "(\<lambda>x y. y) \<in> PrimRec2"
rstep_def: fixes l :: "'b"   and s :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "rstep l s i \<equiv> lstep l (s i) (s (Suc i))"
rTS_def: fixes h :: "nat list" shows "rTS h = (\<lambda>s. h, TS_step_d)"
multiset_of_partition_cards_eq_implies_permutes: fixes A :: "'c set"   and B :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd" assumes "finite A"   and "finite B"   and "f \<in> A \<rightarrow>\<^sub>E B"   and "f' \<in> A \<rightarrow>\<^sub>E B"   and "image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}})) = image_mset card (mset_set ((\<lambda>b. {x \<in> A. f' x = b}) ` B - {{}}))" obtains p\<^sub>A :: "'c \<Rightarrow> 'c" and p\<^sub>B :: "'d \<Rightarrow> 'd" where "p\<^sub>A permutes A" and "p\<^sub>B permutes B" and "\<forall>x\<in>A. f x = p\<^sub>B (f' (p\<^sub>A x))"
l_def: shows "l = Inf {u n / real n |n. 0 < n}"
func: shows "f \<circ> cv \<in> [U]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}"
pred_unique: fixes X :: "'a set"   and x :: "'a" assumes "derivation X"   and "x \<in> X" shows "{z \<in> X. z \<sqsubset> x} = {z. derivable z \<and> z \<sqsubset> x}"
t: fixes r :: "'d rexp"   and s :: "'d rexp"   and t :: "'d rexp" assumes "r \<approx> s"   and "s \<approx> t" shows "r \<approx> t"
less_x_plus_1: fixes x :: "'b word"   and y :: "'b word" assumes "x \<noteq> - 1" shows "(y < x + 1) = (y < x \<or> y = x)"
hfref_imp: fixes RR :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)"   and RR' :: "('a \<Rightarrow> 'b \<Rightarrow> assn) \<times> ('a \<Rightarrow> 'b \<Rightarrow> assn)"   and P :: "'a \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'c \<Rightarrow> assn" assumes "hrp_imp RR RR'" shows "[P]\<^sub>a RR \<rightarrow> S \<subseteq> [P]\<^sub>a RR' \<rightarrow> S"
dim_null_space_iarray: fixes A :: "(('d, 'e) vec, 'f) vec" shows "vec.dim (null_space A) = ncols_iarray (matrix_to_iarray A) - rank_iarray (matrix_to_iarray A)"
zmod_int: fixes a :: "nat"   and b :: "nat" shows "int (a mod b) = int a mod int b"
M2: shows "(\<lambda>x. - f x * indicat_real {g a..g b} x) \<in> borel_measurable borel"
assms_1: shows "\<forall>cfg. reachable c cfg \<longrightarrow> agreement cfg"
HInv3_L_sym: fixes s :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "HInv3_L s p q d" shows "HInv3_L s q p d"
neg_prod_lt: fixes c :: "'b"   and x :: "'b" assumes "c < (0::'b)" shows "c * x < (0::'b) \<equiv> (0::'b) < x"
simps_1: fixes v :: "bool"   and s :: "char list \<Rightarrow> int" shows "bval (Bc v) s = v"
dgrad_setD: fixes t :: "'c"   and d :: "'c \<Rightarrow> nat"   and m :: "nat" assumes "t \<in> dgrad_set d m" shows "d t \<le> m"
set_plus_def: fixes A :: "'a set"   and B :: "'a set" shows "A + B = {c. \<exists>a\<in>A. \<exists>b\<in>B. c = a + b}"
lub_cong: fixes f :: "nat \<Rightarrow> ordinal"   and g :: "nat \<Rightarrow> ordinal"   and F :: "ordinal \<Rightarrow> 'a" assumes "\<forall>n. f n < oLimit f"   and "\<forall>m. g m < oLimit g"   and "oLimit f = oLimit g"   and "\<forall>y<oLimit g. \<forall>x\<le>y. F x << F y" shows "lub (\<lambda>n. F (f n)) = lub (\<lambda>n. F (g n))"
one_neq_neg_one: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus one"
cmifi: shows "m\<inverse> O f\<inverse> \<subseteq> m\<inverse>"
axiom_def: shows "axiom \<equiv> \<lambda>\<phi>. \<forall>v. [\<phi> in v]"
add_comm: fixes x :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and y :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "x \<in> \<O> U"   and "y \<in> \<O> U" shows "add_sheaf_spec U x y = add_sheaf_spec U y x"
TOTAL_def: shows "TOTAL \<equiv> {U. \<exists>\<psi> s. learn_total \<psi> U s}"
g_finite: fixes i :: "nat" shows "value_flow \<Delta> (f_i i) \<noteq> \<top>"
simps_1: fixes p :: "'b" shows "Splay_Heap.partition p \<langle>\<rangle> = (\<langle>\<rangle>, \<langle>\<rangle>)"
inf_r_bl_def: fixes a :: "'a"   and b :: "'a" shows "a \<sqinter> b = a * (a r\<rightarrow> b)"
Quo1: fixes x :: "nat"   and y :: "nat" shows "x * Quo x y \<le> y"
generate_pow_nat: fixes a :: "'a" assumes "a \<in> carrier G"   and "ord a \<noteq> 0" shows "generate G {a} = {a [^] k |k. k \<in> UNIV}"
intro: fixes T :: "'a \<Rightarrow> 'a"   and M :: "'a set" assumes "T ` M \<subseteq> M" shows "RModuleEnd_axioms M T"
simps_1: fixes n :: "nat" shows "liftM_ctxt \<diamond> n = \<diamond>"
less_le_not_le: fixes x :: "'b word"   and y :: "'b word" shows "(x <s y) = (x \<le>s y \<and> \<not> y \<le>s x)"
hyps: fixes p :: "'b list" assumes "length p < length pa__"   and "closed_w p" shows "Ex cycle"
inverts_mat_def: fixes A :: "'c mat"   and B :: "'c mat" shows "inverts_mat A B \<equiv> A * B = 1\<^sub>m (dim_row A)"
FL_valid_Pred: fixes P :: "'state"   and f :: "'effect"   and \<phi> :: "'pred" shows "P \<Turnstile> FL_Formula.Pred f \<phi> = \<langle>f\<rangle>P \<turnstile> \<phi>"
Instantiate: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "[\<^bold>\<exists>x. \<phi> x in v]" obtains x :: "'b" where "[\<phi> x in v]"
getParts_same_fw_behaviour: fixes A :: "'a word set"   and rs :: "'a simple_rule list"   and s1 :: "'a word"   and s2 :: "'a word" assumes "A \<in> set (map wordinterval_to_set (getParts rs))"   and "s1 \<in> A"   and "s2 \<in> A" shows "same_fw_behaviour s1 s2 rs"
induced_image: shows "image.additive.induced ` kernel.additive.Partition = \<eta> ` R"
fempty_iff: fixes c :: "'c" shows "(c |\<in>| {||}) = False"
domino_singletons: fixes d :: "(nat \<times> nat) set" assumes "d \<in> domino" shows "(\<exists>i j. whites \<inter> d = {(i, j)}) \<and> (\<exists>m n. blacks \<inter> d = {(m, n)})"
sorted_stake: fixes s :: "'b stream"   and i :: "nat" assumes "ssorted s" shows "sorted (stake i s)"
con_comp_def: fixes P :: "'f process"   and Q :: "'g process"   and p :: "'f \<Rightarrow> 'e"   and q :: "'g \<Rightarrow> 'e" shows "P \<parallel> Q <p, q> \<equiv> Abs_process (con_comp_failures P Q p q, con_comp_divergences P Q p q)"
a1: shows "s' = snd (fst (div_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
nn: shows "\<integral>\<^sup>+ x. ennreal (f x * indicat_real {a..b} x) \<partial>lborel = ennreal (F b - F a)"
ball_one_point1: fixes A :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>A. x = a \<longrightarrow> P x) = (a \<in> A \<longrightarrow> P a)"
is_equivalence_pseudofunctor: shows "equivalence_pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C FG.map FG.cmp"
finite_is_class: fixes P :: "'a prog" shows "finite {C. is_class P C}"
trg_trg: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr \<mu>" shows "Span.trg (Span.trg \<mu>) = Span.trg \<mu>"
H_eq_zero: fixes x :: "'b" assumes "x \<notin> ball (0::'b) 2" shows "H x = 0"
orbit_subset_eq_affected: fixes a :: "'b"   and f :: "'b perm" assumes "a \<in> affected f" shows "orbit f a \<subseteq> affected f"
UBex: fixes f :: "atom fm"   and x :: "real"   and xs :: "real list" assumes "nqfree f"   and "R.I f (x # xs)"   and "\<not> R.I (inf\<^sub>+ f) xs"   and "x \<notin> EQ f xs" shows "\<exists>u\<in>UB f xs. x < u"
prv_cnjI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv \<phi>"   and "bprv \<chi>" shows "bprv (cnj \<phi> \<chi>)"
left_mult_zero_mat: fixes A :: "'a mat"   and n :: "nat"   and nc :: "nat"   and nr :: "nat" assumes "A \<in> carrier_mat n nc" shows "0\<^sub>m nr n * A = 0\<^sub>m nr nc"
simps: fixes t :: "'t"   and a1 :: "'x1 \<times> 'm1"   and a2 :: "'x1 \<times> 'm1" shows "r1.silent_move t a1 a2 = (\<exists>s s' tl. a1 = s \<and> a2 = s' \<and> t \<turnstile> s -1-tl\<rightarrow> s' \<and> \<tau>move1 s tl s')"
RcdT_canonical: fixes v :: "trm"   and fTs :: "(char list \<times> POPLmarkRecord.type) list" assumes "[] \<turnstile> v : RcdT fTs"   and "v \<in> value" shows "\<exists>fs. v = Rcd fs \<and> (\<forall>(l, t)\<in>set fs. t \<in> value)"
Compl_set_less_Compl: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c set" assumes "class.linorder less_eq less" shows "ord.set_less less_eq (- A) (- B) = ord.set_less less_eq B A"
simps_2: fixes m :: "nat"   and l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" assumes "HBT m" shows "HBT.hbt m \<langle>l, (a, n), r\<rangle> = (\<bar>int (height l) - int (height r)\<bar> \<le> int m \<and> n = max (height l) (height r) + 1 \<and> HBT.hbt m l \<and> HBT.hbt m r)"
le_iff_inf: fixes inf :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type"   and less_eq :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and less :: "'a::type \<Rightarrow> 'a::type \<Rightarrow> bool"   and x :: "'a::type"   and y :: "'a::type" assumes "class.semilattice_inf (inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) (less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (less::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool)" shows "(less_eq::'a::type \<Rightarrow> 'a::type \<Rightarrow> bool) (x::'a::type) (y::'a::type) = ((inf::'a::type \<Rightarrow> 'a::type \<Rightarrow> 'a::type) x y = x)"
sp_cf_def: fixes S :: "('e, 'f) Ring_scheme"   and a :: "'e"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "sp_cf S a c = (fst c, \<lambda>j. a \<cdot>\<^sub>r\<^bsub>S\<^esub> snd c j)"
mult_inv_character: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex"   and k :: "'c" assumes "character G \<chi>" shows "\<chi> k * inv_character \<chi> k = principal_char G k"
borel_measurable_cond_exp2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "real_cond_exp M F f \<in> borel_measurable M"
perm_pi_simp_1: fixes pi :: "(name \<times> name) list"   and x :: "'d" shows "pi \<bullet> rev pi \<bullet> x = x"
CFG_wf_axioms_def: fixes \<alpha>n :: "'d \<Rightarrow> 'e list"   and invar :: "'d \<Rightarrow> bool"   and inEdges' :: "'d \<Rightarrow> 'e \<Rightarrow> ('e \<times> 'f) list"   and Entry :: "'d \<Rightarrow> 'e"   and defs :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set"   and "uses" :: "'d \<Rightarrow> 'e \<Rightarrow> 'g set" shows "CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses \<equiv> \<forall>g. \<forall>m\<in>set (\<alpha>n g). \<forall>v\<in>uses g m. CFG_base.defAss' \<alpha>n invar inEdges' Entry defs g m v"
poss_in_map_clauseD: fixes AA :: "'b multiset"   and f :: "'c \<Rightarrow> 'b"   and C :: "'c literal multiset" assumes "poss AA \<subseteq># map_clause f C" shows "\<exists>AA0. poss AA0 \<subseteq># C \<and> AA = image_mset f AA0"
future_bounded_get_and: fixes \<phi> :: "formula" shows "list_all future_bounded (get_and_list \<phi>) = future_bounded \<phi>"
B: shows "bdd_below {u n / real n |n. 0 < n}"
list_of_dlist_def: shows "list_of_dlist \<equiv> (rep_dlist ---> id) remdups"
dropWhile_append1: fixes x :: "'b"   and xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "x \<in> set xs"   and "\<not> P x" shows "dropWhile P (xs @ ys) = dropWhile P xs @ ys"
h1: shows "\<forall>elem. List.member (subsets_smash (length qs1) subsets1 subsets2) elem \<longrightarrow> (\<exists>elem1 elem2. elem1 \<in> set subsets1 \<and> elem2 \<in> set subsets2 \<and> elem = elem1 @ map ((+) (length qs1)) elem2)"
ArrElem: fixes AT :: "Arraytype" shows "treach (ArrT AT) (at2jt AT)"
circuit_subset_carrier: fixes X :: "'a set" assumes "\<E>.circuit X" shows "X \<subseteq> \<E>"
ereal_mult_mono: fixes b :: "ereal"   and c :: "ereal"   and a :: "ereal"   and d :: "ereal" assumes "0 \<le> b"   and "0 \<le> c"   and "a \<le> b"   and "c \<le> d" shows "a * c \<le> b * d"
s_PrjPair2: fixes v\<^sub>1 :: "Syntax.term"   and v\<^sub>2 :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and va :: "mode" assumes "value v\<^sub>1"   and "value v\<^sub>2" shows "\<lless>\<pi>, Prj2 (Syntax.Pair v\<^sub>1 v\<^sub>2)\<ggreater> va\<rightarrow> \<lless>\<pi>, v\<^sub>2\<ggreater>"
CompHigh: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set"   and c :: "OBJ"   and d :: "OBJ" assumes "G \<rhd> c : HighSec"   and "G \<rhd> d : HighSec" shows "G \<rhd> Comp c d : HighSec"
wtPB_gC_\<Phi>: shows "GE.wtPB (gC ` \<Phi>)"
suboapp: shows "Subobjs P C ((C # Cs') @\<^sub>p Ds)"
basis_subspace_exists: fixes S :: "'a set" assumes "subspace S" obtains B :: "'a set" where "finite B" and "B \<subseteq> S" and "independent B" and "span B = S" and "card B = dim S"
del_vert_props_def: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and u :: "'c" shows "del_vert_props G M u \<equiv> digraph_map G M \<and> del_vert_props_axioms G u"
infinite: fixes f :: "'a \<Rightarrow> nat" assumes "infinite S"   and "bij_betw f S UNIV" shows "thesis"
Suc_3: shows "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"
simps_3: fixes tst :: "'test"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_siso (if tst then c1 else c2) = (compatTst tst \<and> SC_siso c1 \<and> SC_siso c2)"
map_add_disj: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and h\<^sub>2 :: "'c \<Rightarrow> 'd option" shows "h\<^sub>0 \<bottom> h\<^sub>1 ++ h\<^sub>2 = (h\<^sub>0 \<bottom> h\<^sub>1 \<and> h\<^sub>0 \<bottom> h\<^sub>2)"
topological_basis_fbint: shows "topological_basis {arith_prog_fb a b |a b. 0 < b}"
extended_Gromov_product_tendsto_PInf_a_b: fixes a :: "'b"   and u :: "'a \<Rightarrow> 'b Gromov_completion"   and v :: "'a \<Rightarrow> 'b Gromov_completion"   and F :: "'a filter"   and b :: "'b" assumes "((\<lambda>n. extended_Gromov_product_at a (u n) (v n)) \<longlongrightarrow> \<infinity>) F" shows "((\<lambda>n. extended_Gromov_product_at b (u n) (v n)) \<longlongrightarrow> \<infinity>) F"
lead_coeff_i_def: fixes ops :: "'b arith_ops_record"   and pp :: "'b list" shows "lead_coeff_i ops pp = (case pp of [] \<Rightarrow> arith_ops_record.zero ops | a # list \<Rightarrow> last pp)"
rev_finite_subset: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "A \<subseteq> B" shows "finite A"
sorted_spvec_add_spmat: fixes A :: "(nat \<times> (nat \<times> 'a) list) list"   and B :: "(nat \<times> (nat \<times> 'a) list) list" assumes "sorted_spvec A"   and "sorted_spvec B" shows "sorted_spvec (add_spmat A B)"
simps_4: fixes vb :: "'b"   and vc :: "'b"   and vd :: "'c"   and va :: "('b, 'c) Combinators list" shows "noDenyAll1 (AllowPortFromTo vb vc vd # va) = noDenyAll (AllowPortFromTo vb vc vd # va)"
add_strict_right_mono: fixes a :: "complex"   and b :: "complex"   and c :: "complex" assumes "a <\<^sub>\<complex> b" shows "a + c <\<^sub>\<complex> b + c"
nodes_target_elim: fixes q :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and initial :: "'a \<Rightarrow> bool" assumes "q \<in> transition_system_initial.nodes execute enabled initial" obtains r :: "'b list" and p :: "'a" where "initial p" and "transition_system.path execute enabled r p" and "q = fold execute r p"
IH: shows "\<Turnstile> {\<lambda>a. P a \<and> bval b a} c {P}"
SeqReds2: fixes e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v\<^sub>1 :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>2 :: "char list exp"   and e\<^sub>2' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2',s\<^sub>2,b\<^sub>2\<rangle>"
clock_coEC_relation_image: fixes s :: "'s"   and Y :: "'s odlist"   and a :: "'a" assumes "s \<in> toSet Y" shows "toSet (clock_knowledge envObs Y a s) = relations (clock_repMC (toSet Y)) a `` {s}"
balR_def: fixes A :: "('c \<times> nat) tree"   and a :: "'c"   and BC :: "('c \<times> nat) tree" shows "balR A a BC = (if ht BC = ht A + 2 then case BC of \<langle>B, (c, x), C\<rangle> \<Rightarrow> if ht B \<le> ht C then node (node A a B) c C else case B of \<langle>B\<^sub>1, (b, x), B\<^sub>2\<rangle> \<Rightarrow> node (node A a B\<^sub>1) b (node B\<^sub>2 c C) else node A a BC)"
discounted_adapted: fixes F :: "nat \<Rightarrow> 'b measure"   and X :: "nat \<Rightarrow> 'b \<Rightarrow> real"   and r :: "real" assumes "borel_adapt_stoch_proc F X" shows "borel_adapt_stoch_proc F (discounted_value r X)"
list_ex1_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_ex1 P xs = (\<exists>!x. x \<in> set xs \<and> P x)"
is_ladder_prop1: fixes u :: "nat" assumes "u < length L'" shows "ladder_n L' u \<le> length D - ladder_stepdown_diff L"
set_supp_mono: fixes L :: "'b list"   and L2 :: "'b list" assumes "set L \<subseteq> set L2" shows "supp L \<subseteq> supp L2"
cases: fixes a1 :: "'b LDCFG_node"   and a2 :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list"   and a3 :: "'b LDCFG_node" assumes "CFGExit_wf.path a1 a2 a3" obtains n :: "'b LDCFG_node" where "a1 = n__" and "a2 = []" and "a3 = n__" and "CFGExit_wf.valid_node n__"   | n'' :: "'b LDCFG_node" and as :: "('b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node) list" and n' :: "'b LDCFG_node" and a :: "'b LDCFG_node \<times> 'c edge_kind \<times> 'b LDCFG_node" and n :: "'b LDCFG_node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "CFGExit_wf.path n'' as n'" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a" and "src a = n" and "trg a = n''"
lfinite_lnths: fixes xs :: "'b llist"   and A :: "nat set" shows "lfinite (lselect A xs) = (lfinite xs \<or> finite A)"
assms: shows "Abstract_Linear_Poly.coeff (vec_to_lpoly v) i \<noteq> 0"
pos_bounded: shows "\<exists>K>0. \<forall>x. norm (h x) \<le> norm x * K"
little_Picard: fixes f :: "complex \<Rightarrow> complex"   and a :: "complex"   and b :: "complex" assumes "f holomorphic_on UNIV"   and "a \<noteq> b"   and "range f \<inter> {a, b} = {}" obtains c :: "complex" where "f = (\<lambda>x. c)"
hpi: shows "hpi \<equiv> map (\<lambda>(f, i). (hp f, i))"
arccos_le_mono: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> \<le> 1" shows "(arccos x \<le> arccos y) = (y \<le> x)"
IH: assumes "set Xs \<subseteq> carrier_vec n" shows "lincomb_list c Xs $ i = (\<Sum>j = 0..<length Xs. c j * Xs ! j $ i)"
set_pmf_bernoulli: fixes p :: "real" assumes "0 < p"   and "p < 1" shows "set_pmf (bernoulli_pmf p) = UNIV"
sorted_wrt_append: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "sorted_wrt P (xs @ ys) = (sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
freshAbs_def: fixes xs :: "'d"   and x :: "'e"   and A :: "('f, 'g, 'd, 'e, 'h) qAbs set" shows "freshAbs xs x A \<equiv> qFreshAbs xs x (pick A)"
is_valid_problem_strips_initial_of_dom: fixes \<Pi> :: "'b strips_problem" assumes "is_valid_problem_strips \<Pi>" shows "dom (\<Pi>\<^sub>I) = set (\<Pi>\<^sub>\<V>)"
simps_2: fixes x21 :: "tree"   and x22 :: "tree" shows "Leaf \<noteq> Branching x21 x22"
bind_cong_All: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and g :: "'c \<Rightarrow> 'd measure" assumes "\<forall>x\<in>space M. f x = g x" shows "M \<bind> f = M \<bind> g"
bounded_linear_compose: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'f \<Rightarrow> 'd" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f (g x))"
max_absorb2: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "max x y = y"
assms_6: fixes t :: "'c \<times> 'b \<times> 'c" assumes "t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0'" shows "(f t \<in> I) = (t \<in> I')"
poly_poly_as_poly2: fixes p :: "'c poly poly"   and x :: "'c"   and q :: "'c poly" shows "poly2 p x (poly q x) = poly (poly p q) x"
rref_first_position_zero_imp_column_0: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "reduced_row_echelon_form A"   and "A $ (0::'d) $ (0::'c) = (0::'b)" shows "column (0::'c) A = 0"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and T1 :: "'fsym trm"   and T2 :: "'fsym trm" shows "satA \<xi> (Eq T1 T2) = (int \<xi> T1 = int \<xi> T2)"
in_set_conv_decomp: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
pid_run_sim: fixes r :: "nat \<Rightarrow> (nat, local_state, global_state) Pid_Scheduler.pid_global_config" assumes "pid.is_run r" shows "lv.sa.is_run (pidgc_\<alpha> \<circ> r)"
D2_def: shows "D2 = (SIGMA u:{0<..<1}. {u<..<1})"
push_bit_of_0: fixes n :: "nat" shows "push_bit n (0::'a) = (0::'a)"
Un_absorb2: fixes B :: "'d set"   and A :: "'d set" assumes "B \<subseteq> A" shows "A \<union> B = A"