hcju/isabelleps huggingface dataset:
run_asset_iii_msg: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "case run_asset_iii n s of (s', S, A, B, C, D) \<Rightarrow> {Asset n} \<times> {Crypt (Auth_ShaKey n) (PriKey S), \<lbrace>Num 2, PubKey B\<rbrace>, \<lbrace>Num 4, PubKey D\<rbrace>} \<union> {Owner n} \<times> {\<lbrace>Num 1, PubKey A\<rbrace>, \<lbrace>Num 3, PubKey C\<rbrace>} \<subseteq> s' \<and> (Asset n, PubKey B) \<notin> s'"
Suc_ile_eq: fixes m :: "nat"   and n :: "enat" shows "(enat (Suc m) \<le> n) = (enat m < n)"
APP_def: fixes f :: "'k \<Rightarrow> 'j"   and a :: "'k" shows "f $ a \<equiv> f a"
rev_appendSubobj: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ rev Cs')"   and "Cs \<noteq> []" shows "Subobjs P C Cs"
subset_imageE: fixes B :: "'b set"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "order.greater_eq (f ` A) B" obtains C :: "'c set" where "order.greater_eq A C" and "B = f ` C"
Singl: fixes s :: "('St, 'U, 'C) state" shows "wffp [s]"
mk_next_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (mk_next \<phi>) = atoms_ltln \<phi>"
tendsto_eventually: fixes "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and net :: "'b filter" assumes "class.topological_space open"   and "\<forall>\<^sub>F x in net. f x = l" shows "filterlim f (topological_space.nhds open l) net"
eval_poly_vars: fixes p :: "('c monom \<times> 'd) list"   and \<alpha> :: "'c \<Rightarrow> 'd"   and \<beta> :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> poly_vars p \<Longrightarrow> \<alpha> x = \<beta> x" shows "eval_poly \<alpha> p = eval_poly \<beta> p"
set_intros_1: fixes x1 :: "'c"   and x2 :: "'c stream" shows "x1 \<in> sset (x1 ## x2)"
threshold_properties_2: fixes \<phi> :: "'b ltl"   and w :: "nat \<Rightarrow> 'b set"   and \<G> :: "'b ltl set"   and i :: "nat"   and j :: "nat" assumes "threshold \<phi> w \<G> = Some i"   and "i \<le> j" shows "\<PP> \<phi> \<G> w j"
l_get_parent_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_get_parent_axioms type_wf known_ptrs get_parent get_parent_locs get_child_nodes \<equiv> ((\<forall>node_ptr h h'. reads get_parent_locs (get_parent node_ptr) h h') \<and> (\<forall>h node_ptr. type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> node_ptr |\<in>| node_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_parent node_ptr) \<and> (\<forall>h node_ptr. h \<turnstile> ok get_parent node_ptr \<longrightarrow> node_ptr |\<in>| node_ptr_kinds h)) \<and> ((\<forall>node_ptr h. pure (get_parent node_ptr) h) \<and> (\<forall>h child_node parent. h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent \<longrightarrow> parent |\<in>| object_ptr_kinds h)) \<and> (\<forall>h child ptr thesis. h \<turnstile> get_parent child \<rightarrow>\<^sub>r Some ptr \<longrightarrow> (\<forall>children. h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children \<longrightarrow> child \<in> set children \<longrightarrow> thesis) \<longrightarrow> thesis) \<and> (\<forall>ptr. preserved (get_M\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t ptr RObject.nothing) \<in> get_parent_locs)"
reachable_alt_def: shows "reachable \<equiv> \<lambda>x. \<exists>p. path G x p a"
notm: shows "a * t P s + b * t Q s \<le> t (\<lambda>s. a * P s + b * Q s) s"
cupcake_v_update_preserve: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and f :: "(char list, char list, v) namespace \<Rightarrow> (char list, char list, v) namespace" assumes "cakeml_static_env static_cenv"   and "cakeml_static_env.is_cupcake_all_env static_cenv env"   and "cakeml_static_env.is_cupcake_ns static_cenv (f (sem_env.v env))" shows "cakeml_static_env.is_cupcake_all_env static_cenv (update_v f env)"
ereal_add_nonneg_eq_0_iff: fixes a :: "ereal"   and b :: "ereal" assumes "0 \<le> a"   and "0 \<le> b" shows "(a + b = 0) = (a = 0 \<and> b = 0)"
assms_2: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j"   and "j < n" shows "b j \<le> b i"
size_16: fixes x51 :: "'b ltln"   and x52 :: "'b ltln" shows "size (x51 and\<^sub>n x52) = size x51 + size x52 + Suc 0"
HEndPhase1_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4 s"   and "Inv2a s"   and "Inv2a s'"   and "Inv2c s"   and "\<not> maxBalInp s' (bal (dblock s' p)) (inp (dblock s' p))" shows "\<exists>D\<in>MajoritySet. \<exists>q. \<forall>d\<in>D. bal (dblock s' p) < mbal (disk s' d q) \<and> \<not> hasRead s' p d q"
upd_set_fold: fixes A :: "'a set"   and m :: "('a, 'b) mapping"   and f :: "'a \<Rightarrow> 'b" assumes "finite A" shows "upd_set m f A = Finite_Set.fold (\<lambda>a. Mapping.update a (f a)) m A"
lookup_def: fixes \<Gamma> :: "(ty \<times> nat) list \<times> nat"   and n :: "nat" shows "lookup \<Gamma> n \<equiv> if n < length (fst \<Gamma>) then let k = snd \<Gamma> - snd (fst \<Gamma> ! n) in Some (shift k 0 (fst (fst \<Gamma> ! n))) else None"
circ_right_unfold_sub: fixes x :: "'a" shows "Rf (x\<^sup>\<star>) x \<le> x\<^sup>\<star>"
mod_self: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_modulo divide plus minus zero times one modulo" shows "modulo a a = zero"
inline1_id_oracle: fixes gpv :: "('d, 'f, 'g) gpv"   and s :: "'e" shows "inline1 id_oracle gpv s = map_spmf (\<lambda>generat. case generat of Pure x \<Rightarrow> Inl (x, s) | IO out c \<Rightarrow> Inr (out, \<lambda>x. Generative_Probabilistic_Value.Done (x, s), c)) (the_gpv gpv)"
negative_len_shortest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and xs :: "nat list"   and n :: "nat"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and i :: "nat" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "length xs = n"   and "less (linordered_ab_monoid_add.len plus m i i xs) neutral" shows "\<exists>j ys. distinct (j # ys) \<and> less (linordered_ab_monoid_add.len plus m j j ys) neutral \<and> j \<in> set (i # xs) \<and> set ys \<subseteq> set xs"
dsafe_Plus_simps: fixes a :: "('d, 'e) trm"   and b :: "('d, 'e) trm" shows "dsafe (Plus a b) = (dsafe a \<and> dsafe b)"
pred_succ_order: fixes m' :: "('a, 'b) insert_message" assumes "m' \<in> insert_messages M \<union> {m}" shows "a (P m') < a \<lbrakk>I m'\<rbrakk> \<and> a \<lbrakk>I m'\<rbrakk> < a (S m')"
bezout_matrix_works2: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a \<noteq> b"   and "A $h a $h j \<noteq> (0::'b) \<or> A $h b $h j \<noteq> (0::'b)" shows "(bezout_matrix A a b j bezout ** A) $h b $h j = (0::'b)"
id_right: fixes CC :: "('d, 'e, 'f) category_scheme"   and f :: "'e" assumes "category CC"   and "f \<in> Ar\<^bsub>CC\<^esub>" shows "f \<bullet>\<^bsub>CC\<^esub> Id\<^bsub>CC\<^esub> (Dom\<^bsub>CC\<^esub> f) = f"
v_2: shows "f_b `` vertices (graph_union A B) \<subseteq> vertices G"
lessE: fixes i :: "nat"   and k :: "nat" assumes "i < k" obtains "k = Suc i"   | j :: "nat" where "i < j" and "k = Suc j"
right_unique_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
rbt_insert_is_rbt: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and t :: "('b, 'c) rbt"   and k :: "'b"   and v :: "'c" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_insert less k v t)"
MGFnD': fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and t :: "Term.term" assumes "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{(\<lambda>Y' s' s. s' = s \<and> True) \<and>. G\<turnstile>init\<le>n} t\<succ> {(\<lambda>Y' s' s. G\<turnstile>s \<midarrow>t\<succ>\<rightarrow> (Y', s') \<and> True) \<and>. G\<turnstile>init\<le>n}"
der_f: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
simps: fixes uu :: "(String.literal \<times> String.literal) set"   and ars :: "String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option" shows "tcsigs (uu, ars) = ars"
bounded_f: shows "bounded ((\<lambda>xa. f (h * xa + t, x (h * xa + t))) ` {0..1})"
cn: shows "card ((\<lambda>i. cis (real i * 2 * pi / real n)) ` {0..<n}) = n"
ide_char: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2" shows "partial_magma.ide (product_category.comp C1 C2) f = (partial_magma.ide C1 (fst f) \<and> partial_magma.ide C2 (snd f))"
that: fixes x :: "'a" assumes "x \<in># A" shows "thesis"
precompact_neighborhoodE: fixes x :: "'a" assumes "x \<in> carrier" obtains C :: "'a set" where "x \<in> C" and "open C" and "compact (closure C)" and "closure C \<subseteq> carrier"
card_inversion: fixes xs :: "bool list" assumes "xs \<in> inversion ` dc_crypto" shows "card {dc \<in> dc_crypto. inversion dc = xs} = 2 * n"
renaming_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and V :: "'b set" shows "renaming \<sigma> V = (\<forall>x\<in>V. is_a_variable (Var x \<lhd> \<sigma>) \<and> (\<forall>x y. x \<in> V \<longrightarrow> y \<in> V \<longrightarrow> x \<noteq> y \<longrightarrow> Var x \<lhd> \<sigma> \<noteq> Var y \<lhd> \<sigma>))"
finite_before: fixes x :: "'c"   and xs :: "'c list" shows "finite (before x xs)"
MakeCatId: fixes X :: "'g"   and C :: "('g, 'h, 'i) Category_scheme" assumes "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X = id\<^bsub>MakeCat C\<^esub> X"
simps: fixes a1 :: "'a scheduler"   and a2 :: "'a scheduler" shows "eq_scheduler a1 a2 = (\<exists>sc1 sc2 D. a1 = sc1 \<and> a2 = sc2 \<and> action_sch sc1 = D \<and> action_sch sc2 = D \<and> (\<forall>s\<in>set_pmf D. eq_scheduler (cont_sch sc1 s) (cont_sch sc2 s)))"
cinner_eq_zero_iff: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "(cinner x x = 0) = (x = zero)"
filter_M_subset: fixes h :: "'d"   and P :: "'f \<Rightarrow> ('d, 'e, bool) prog"   and xs :: "'f list"   and ys :: "'f list" assumes "h \<turnstile> filter_M P xs \<rightarrow>\<^sub>r ys" shows "set ys \<subseteq> set xs"
gen_igbg_impl_rel_sv: fixes Racc :: "('a \<times> ('b \<Rightarrow> nat set)) set"   and Rm :: "('c \<times> 'd) set" assumes "single_valued Racc"   and "single_valued Rm" shows "single_valued (\<langle>Rm, Racc\<rangle>gen_igbg_impl_rel_eext)"
graph_homomorphismI: fixes s :: "('f, 'e) labeled_graph"   and h :: "('e \<times> 'g) set"   and t :: "('f, 'g) labeled_graph" assumes "vertices s = Domain h"   and "h `` vertices s \<subseteq> vertices t"   and "univalent h"   and "edge_preserving h (edges s) (edges t)"   and "graph s"   and "graph t" shows "graph_homomorphism s t h"
ChamberSubcomplexD_complex: fixes X :: "'b set set"   and Y :: "'b set set" assumes "ChamberComplex X"   and "ChamberComplex.ChamberSubcomplex X Y" shows "ChamberComplex Y"
orthogonal: fixes i :: "nat"   and j :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)"   and "j < length (map of_int_hom.vec_hom fs)"   and "i \<noteq> j" shows "fs.gs.gso i \<bullet> fs.gs.gso j = 0"
in_sum: fixes N1 :: "'c set"   and N2 :: "'c set" assumes "LinearCombinations.submodule K N1 V"   and "LinearCombinations.submodule K N2 V" shows "N1 \<subseteq> subspace_sum N1 N2"
eucl_of_list_matrix_vector_mult_eq_sum_nth_Basis_list: fixes M :: "real list"   and v :: "real list" assumes "length M = CARD('a) * CARD('b)"   and "length v = CARD('a)" shows "eucl_of_list M *v eucl_of_list v = (\<Sum>i<CARD('b). (\<Sum>j<CARD('a). M ! (i * CARD('a) + j) * v ! j) *\<^sub>R Basis_list ! i)"
Amortized_axioms: shows "Amortized init nxt inv T \<Phi> U"
norm_25: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "((\<lambda>s. True) \<U> P) = \<diamond>P"
Fun_def: fixes f :: "'s" shows "Fun f = restrict (some_img \<circ> (\<cdot>) f \<circ> inv_into (hom unity (local.dom f)) some_img) (Dom f)"
isLubD1: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "isLub R S x" shows "S *<= x"
hyps: fixes ps :: "('d \<times> 'c \<times> 'b) list" shows "set (fold (aps flag gs bs) hs ps) = Pair flag ` (set hs \<times> (set gs \<union> set bs)) \<union> set ps"
iffD2: fixes P :: "bool"   and Q :: "bool" assumes "P = Q"   and "Q" shows "P"
elims: fixes x :: "('a \<times> 'c) list \<times> 'o"   and y :: "('a \<times> 'c) list \<times> 'o" assumes "sort_oalist_raw x = y" obtains xs :: "('a \<times> 'c) list" and ko :: "'o" where "x = (xs, ko)" and "y = (sort_oalist ko xs, ko)"
id_eq_prop_prop_7: fixes v :: "i"   and p :: "\<o>" shows "[p \<^bold>= p in v]"
HIGH_SKIP: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" shows "G \<rhd> Skip : twiddle"
rep_eq: fixes x :: "'b boolean_subset" shows "Rep_boolean_subset (- x) = - Rep_boolean_subset x"
inter_overappr_impl: fixes X :: "'b set"   and Y :: "'b set" shows "split_intersecting X Y \<bind> (\<lambda>(X, uu_). RETURN X) \<le> inter_overappr X Y"
simps: fixes x :: "'b nonzero"   and a :: "'b nonzero" shows "proj_rel x a = (\<exists>c. a = c *\<^sub>R x \<and> c \<noteq> 0)"
rep_eq: fixes x :: "float"   and xa :: "float" shows "(x \<le> xa) = (real_of_float x \<le> real_of_float xa)"
integrable_integral: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'b set" assumes "f integrable_on i" shows "(f has_integral integral i f) i"
take_eq_Nil: fixes n :: "nat"   and xs :: "'b list" shows "(take n xs = []) = (n = 0 \<or> xs = [])"
gbinomial_mult_fact: fixes k :: "nat"   and a :: "'a" shows "fact k * (a gchoose k) = (\<Prod>i = 0..<k. a - of_nat i)"
Polygamma_real_strict_mono: fixes x :: "real"   and y :: "real"   and n :: "nat" assumes "0 < x"   and "x < y"   and "even n" shows "Polygamma n x < Polygamma n y"
rtrancl_refl: fixes a :: "'b"   and r :: "('b \<times> 'b) set" shows "(a, a) \<in> r\<^sup>*"
r_into_trancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "r a b" shows "r\<^sup>+\<^sup>+ a b"
older_seniors_not_self_referential: fixes x :: "nat"   and n :: "nat" shows "senior x n \<notin> older_seniors x n"
sub_unit_closed: shows "zero_local_ring_at \<in> \<ww>"
size_mset_set: fixes A :: "'b set" shows "size (mset_set A) = card A"
process_charn: fixes P :: "'c process" shows "([], {}) \<in> F P \<and> (\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s) \<and> (\<forall>s t. (s @ t, {}) \<notin> F P \<or> (s, {}) \<in> F P) \<and> (\<forall>s X Y. (s, Y) \<notin> F P \<or> \<not> X \<subseteq> Y \<or> (s, X) \<in> F P) \<and> (\<forall>s X Y. (s, X) \<in> F P \<and> (\<forall>c. c \<in> Y \<longrightarrow> (s @ [c], {}) \<notin> F P) \<longrightarrow> (s, X \<union> Y) \<in> F P) \<and> (\<forall>s X. (s @ [tick], {}) \<in> F P \<longrightarrow> (s, X - {tick}) \<in> F P) \<and> (\<forall>s t. s \<notin> D P \<or> \<not> tickFree s \<or> \<not> front_tickFree t \<or> s @ t \<in> D P) \<and> (\<forall>s X. s \<notin> D P \<or> (s, X) \<in> F P) \<and> (\<forall>s. s @ [tick] \<notin> D P \<or> s \<in> D P)"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<cdot> f = n \<and> f \<cdot> n = n"
And_def: fixes \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula" shows "MFOTL.And \<phi> \<psi> = formula.Neg (formula.Or (formula.Neg \<phi>) (formula.Neg \<psi>))"
CFGExit_axioms: shows "CFGExit src trg knd lve NewEntry NewExit"
a: shows "finite_measure (distr M borel (\<lambda>x n. f ((T ^^ n) x)))"
fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q_def: fixes x :: "('d, 'c) strand_step" shows "fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q x \<equiv> case x of \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<Rightarrow> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X | _ \<Rightarrow> {}"
subst_atm_mset_lists_nth: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and AAs :: "'b multiset list"   and \<sigma>s :: "'c list"   and i :: "nat" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "length AAs = length \<sigma>s"   and "i < length AAs" shows "substitution_ops.subst_atm_mset_lists subst_atm AAs \<sigma>s ! i = substitution_ops.subst_atm_mset subst_atm (AAs ! i) (\<sigma>s ! i)"
prod_simps_1: fixes f0 :: "'a"   and f1 :: "'a" assumes "arr f0"   and "arr f1" shows "arr (local.prod f0 f1)"
trm: shows "\<lambda>s. 1 \<tturnstile> t (\<lambda>s. 1)"
iso_ll_def: fixes A :: "nat list set"   and B :: "nat list set" shows "iso_ll A B \<equiv> iso (Restr (lenlex less_than) A) (Restr (lenlex less_than) B)"
sum_gp0: fixes x :: "'a"   and n :: "nat" shows "sum ((^) x) {..n} = (if x = (1::'a) then of_nat (n + 1) else ((1::'a) - x ^ Suc n) / ((1::'a) - x))"
wf_acyclic: fixes r :: "('a \<times> 'a) set" assumes "wf r" shows "acyclic r"
assms_3: shows "set some_ops \<subseteq> set all_ops"
eval_G': fixes s :: "complex" shows "eval_fps G s = g (1 + s)"
bot_or_not_bot: fixes x :: "Value"   and y :: "CValue" assumes "x \<triangleleft>\<triangleright> y" shows "(x = \<bottom>) = (y = \<bottom>)"
\<omega>consistentStd1_def: shows "\<omega>consistentStd1 \<equiv> \<forall>\<phi>\<in>fmla. \<forall>x\<in>var. Fvars \<phi> = {x} \<longrightarrow> (\<forall>n\<in>num. prv (neg (subst \<phi> n x))) \<longrightarrow> \<not> prv (exi x \<phi>)"
hcomp_null_2: fixes \<mu> :: "'a" shows "\<mu> \<star> null = null"
summable_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<forall>\<^sub>F x in sequentially. f x = g x" shows "summable f = summable g"
coprime_1_right: 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 a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a one"
strategyproofness: shows "pmf (sds R25) c + pmf (sds R25) a \<le> pmf (sds R1) c + pmf (sds R1) a"
restrict: fixes a :: "'a" assumes "a \<in> carrier R" shows "((\<lambda>x. the_elem (f ` x)) \<circ> (+>) I) a = f a"
subst_Var_notin_img: fixes x :: "'a"   and s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "x \<notin> range_vars s"   and "t \<cdot> s = Var x" shows "t = Var x"
inverse_t': shows "[int (t * u0) + int s * (int t * fst (bezw t (order \<G>))) = int (t * u0 + s * 1)] (mod int (order \<G>))"
prems: shows "\<I> \<turnstile>g gpv \<surd>"
Id_on_iff: fixes x :: "'g"   and y :: "'g"   and A :: "'g set" shows "((x, y) \<in> Id_on A) = (x = y \<and> x \<in> A)"
option_rule: fixes v :: "'a option"   and S1 :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and f2 :: "'a \<Rightarrow> 'b nres" assumes "v = None \<Longrightarrow> S1 \<le> SPEC \<Phi>"   and "\<And>x. v = Some x \<Longrightarrow> f2 x \<le> SPEC \<Phi>" shows "(case v of None \<Rightarrow> S1 | Some x \<Rightarrow> f2 x) \<le> SPEC \<Phi>"
mon_env_cil: fixes w :: "('a, 'b) label list el_step list"   and w1 :: "('a, 'b) label list el_step list"   and \<alpha> :: "('a, 'b) label list el_step \<Rightarrow> 'c set \<times> 'c set"   and w2 :: "('a, 'b) label list el_step list"   and fg :: "('e, 'a, 'f, 'd, 'g) flowgraph_rec_scheme" assumes "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" shows "mon_env fg w = mon_env fg w1 \<union> mon_env fg w2"
subspace_def: fixes S :: "'b set" shows "csubspace S = ((0::'b) \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x + y \<in> S) \<and> (\<forall>c. \<forall>x\<in>S. c *\<^sub>C x \<in> S))"
word_rep: fixes bw :: "bword" shows "\<exists>r. Rep_bword bw \<equiv>\<^sub>E r"
endo_comp: fixes g :: "'a \<Rightarrow> 'a" assumes "ChamberComplexEndomorphism X g" shows "ChamberComplexEndomorphism X (g \<circ> f)"
f_shrink_last_nth: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "n < length xs div k" shows "xs \<div>\<^sub>l k ! n = xs ! (n * k + k - Suc 0)"
IH: shows "(\<Sum>k = 1..<2 ^ n. f (2 * 2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ Suc k))"
contractible_imp_holomorphic_log: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "contractible S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on S" and "\<And>z. z \<in> S \<Longrightarrow> f z = exp (g z)"
reject: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "Semantics.matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
norm_assertion_simps_14: fixes x :: "assn"   and y :: "assn"   and z :: "assn" shows "(x \<and>\<^sub>A y) \<and>\<^sub>A z = x \<and>\<^sub>A y \<and>\<^sub>A z"
th_pr_rec: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "g \<in> PrimRec1"   and "h \<in> PrimRec3"   and "\<forall>x. f 0 x = g x"   and "\<forall>x y. f (Suc y) x = h y (f y x) x" shows "f \<in> PrimRec2"
loxodromic_attracting_fixed_point_3: fixes f :: "'b \<Rightarrow> 'b" assumes "loxodromic_isometry f" shows "additive_strength f (attracting_fixed_point f) < 0"
Pareto_iff: fixes x :: "'b pmf"   and y :: "'b pmf" shows "Pareto (SD \<circ> R') x y = (\<forall>i\<in>agents'. (SD \<circ> R') i x y)"
fun_unequal_in_support: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set"   and C :: "'e set"   and z :: "'e" assumes "f \<noteq> g"   and "f \<in> Func A B"   and "g \<in> Func A C" shows "(support z A f \<union> support z A g) \<inter> {a. f a \<noteq> g a} \<noteq> {}"
insert_not_empty: fixes a :: "'c"   and A :: "'c set" shows "{a} \<union> A \<noteq> {}"
hs_dfs_initial_def: fixes \<Sigma>i :: "('a, unit) HashMap.hashmap" shows "hs_dfs_initial \<Sigma>i \<equiv> (\<Sigma>i, hs.to_list \<Sigma>i)"
match_4: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "B.VV.seq g f"   and "B.VV.seq h g" shows "B.VV.seq h (B.VV.comp g f)"