hcju/isabelleps huggingface dataset:
vcgaDefiniteness: fixes \<Omega> :: "'c list"   and N :: "integer set"   and b :: "integer \<times> 'c set \<Rightarrow> 'd"   and r :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "card (Outside' {seller} ` (argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) r) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) b (allAllocations (N \<union> {seller}) (set \<Omega>)))) = 1"
change_eval_neq: fixes a :: "real"   and b :: "real"   and c :: "real"   and x :: "real"   and y :: "real" assumes "aEvalUni (NeqUni (a, b, c)) x \<noteq> aEvalUni (NeqUni (a, b, c)) y \<and> x < y" shows "\<exists>w\<ge>x. w \<le> y \<and> a * w\<^sup>2 + b * w + c = 0"
empty_subst: shows "\<exists>s. (\<forall>f. subst s f = f) \<and> subst_lconsts s = {}"
vimage_Collect_eq: fixes f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> bool" shows "f -` Collect P = {y. P (f y)}"
singular_relboundary_imp_relcycle: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relboundary p X S c" shows "singular_relcycle p X S c"
Union_path_components_of: fixes X :: "'c topology" shows "\<Union> (path_components_of X) = topspace X"
elims: fixes x :: "nat"   and xa :: "bool list"   and y :: "bool list" assumes "flip x xa = y" obtains uu_ :: "nat" where "x = uu_" and "xa = []" and "y = []"   | xb :: "bool" and xs :: "bool list" where "x = 0" and "xa = xb__ # xs__" and "y = (\<not> xb__) # xs__"   | n :: "nat" and xb :: "bool" and xs :: "bool list" where "x = Suc n" and "xa = xb # xs" and "y = xb # flip n xs"
Suc_2: shows "ndiff (Suc n) f {0<..}"
orderI: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a = a \<inter># b" shows "a \<subseteq># b"
r_inv_ex: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "\<exists>y\<in>carrier \<G>. x \<otimes> y = \<one>"
exhaust: fixes y :: "mconstraint" obtains "y = MEq"   | "y = MLess"   | "y = MLessEq"
xivp_1: shows " D X = (\<lambda>x. f x (X x)) on U (X t\<^sub>0)"
opt': fixes F :: "atom fm"   and L :: "real list" shows "eval (push_forall (nnf (unpower 0 (groupQuantifiers (clearQuantifiers F))))) L = eval F L"
legal_rules_finite: fixes Q :: "'a set" assumes "finite Q" shows "finite (legal_rules Q)"
fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p_NegCheck_1: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> = fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F \<union> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G - set X"
smult_dvd_cancel: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" assumes "smult a p dvd q" shows "p dvd q"
append_same_eq: fixes ys :: "'c list"   and xs :: "'c list"   and zs :: "'c list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
induct: fixes P :: "'b \<Rightarrow> nat \<Rightarrow> 'b mat \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "nat"   and a2 :: "'b mat" assumes "\<And>ev A. P ev 0 A"   and "\<And>ev i A. (\<not> A $$ (i, i) \<noteq> ev \<Longrightarrow> P ev i A) \<Longrightarrow> P ev (Suc i) A" shows "P a0 a1 a2"
extremum_unique: fixes a :: "'a" shows "(a \<le> \<bottom>) = (a = \<bottom>)"
simps: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE" shows "semBV I ODE = Inl ` ODE_vars I ODE \<union> Inr ` ODE_vars I ODE"
rows_carrier: fixes A :: "'a mat" shows "set (rows A) \<subseteq> carrier_vec (dim_col A)"
fmfilter_false: fixes m :: "('a, 'b) fmap"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> \<not> P x" shows "fmfilter P m = fmempty"
greaterThan_eq_iff: fixes x :: "'d"   and y :: "'d" shows "({x<..} = {y<..}) = (x = y)"
triangle_right: shows "(g \<star>\<^sub>C \<epsilon>) \<cdot>\<^sub>C \<a>\<^sub>C[g, f, g] \<cdot>\<^sub>C (\<eta> \<star>\<^sub>C g) = \<r>\<^sub>C\<^sup>-\<^sup>1[g] \<cdot>\<^sub>C \<l>\<^sub>C[g]"
LAcc: fixes A :: "triple set"   and P :: "vassn"   and x :: "vname" shows "A \<turnstile>\<^sub>e {\<lambda>s. P (s<x>) s} LAcc x {P}"
simps_1: fixes p :: "fm" shows "not (NOT (NOT p)) = not p"
col_k_rw: shows "column k (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column k A - proj_onto (column k A) {column i A |i. i < k}"
dist_at1: fixes vs :: "'c list"   and a :: "'c list"   and r :: "'c"   and b :: "'c list"   and c :: "'c list"   and d :: "'c list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "a = c"
insert_select2: fixes l1 :: "'c"   and l2 :: "'c"   and f :: "'c \<Rightarrow> 'd option"   and t :: "'d" assumes "l1 \<noteq> l2" shows "(f(l1 \<mapsto> t)) l2 = f l2"
simps_4: fixes f :: "'d"   and \<alpha> :: "'c"   and t :: "('a, 'b, 'c, 'd) FL_Formula.Tree" shows "L_transform_Tree (FL_Formula.Tree.tAct f \<alpha> t) = Formula.Act\<^sub>\<alpha> (Eff f) (Formula.Act\<^sub>\<alpha> (L_action.Act \<alpha>) (L_transform_Tree t))"
data_dependence_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and V :: "'c"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.data_dependence sourcenode targetnode kind valid_edge get_proc procs Def Use ParamDefs ParamUses n V n' \<equiv> \<exists>as. SDG.SDG_Def sourcenode targetnode kind valid_edge get_proc procs Def ParamDefs V n \<and> SDG.SDG_Use sourcenode targetnode kind valid_edge get_proc procs Use ParamUses V n' \<and> CFG.intra_path sourcenode targetnode kind valid_edge (parent_node n) as (parent_node n') \<and> (\<forall>n''. SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n'' \<and> parent_node n'' \<in> set (CFG.sourcenodes sourcenode (tl as)) \<longrightarrow> \<not> SDG.SDG_Def sourcenode targetnode kind valid_edge get_proc procs Def ParamDefs V n'')"
simps: fixes A :: "'a list"   and inc :: "nat"   and x :: "'a" shows "array_grow (Array A) inc x = Array (A @ replicate inc x)"
mod_add_cong: fixes a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "a mod c = a' mod c"   and "b mod c = b' mod c" shows "(a + b) mod c = (a' + b') mod c"
simps_1: shows "ext0.Bot + ext0.Bot = ext0.Val (0::'b)"
LeftDerivationIntrosAt_def: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and index :: "nat" shows "LeftDerivationIntrosAt a D L index = (let \<alpha> = ladder_\<alpha> a D L index; i = ladder_i L index; j = ladder_j L index; ix = ladder_ix L index; \<gamma> = ladder_\<gamma> a D L index; n = ladder_n L (index - 1); m = ladder_n L index; e = D ! n; E = drop (Suc n) (take m D) in i = fst e \<and> LeftDerivationIntro \<alpha> i (snd e) ix E j \<gamma>)"
face\<M>: fixes C :: "'a set"   and F :: "'a set" assumes "C \<in> \<M>"   and "F face_of C" shows "F \<in> \<M>"
compute_tm_by_comp_range: fixes f :: "floatarith"   and a :: "float interval list"   and I :: "float interval list"   and x :: "real"   and e :: "nat \<Rightarrow> real"   and tg :: "taylor_model"   and prec :: "nat"   and ord :: "nat"   and c :: "float interval \<Rightarrow> bool"   and t :: "taylor_model" assumes "max_Var_floatarith f \<le> 1"   and "a all_subset I"   and "x \<in>\<^sub>i range_tm e tg"   and "compute_tm_by_comp prec ord I a f (Some tg) c = Some t"   and "\<And>x. \<lbrakk>x \<in>\<^sub>r compute_bound_tm prec I a tg; c (compute_bound_tm prec I a tg)\<rbrakk> \<Longrightarrow> isDERIV 0 f [x]"   and "num_params (tm_poly tg) \<le> length I"   and "develops_at_within e a I" shows "interpret_floatarith f [x] \<in>\<^sub>i range_tm e t"
zero_less_numeral: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (numeral.numeral one plus n)"
vsv_axioms_def: fixes r :: "V" shows "vsv_axioms r \<equiv> \<forall>a b c. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<longrightarrow> \<langle>a, c\<rangle> \<in>\<^sub>\<circ> r \<longrightarrow> b = c"
add_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c \<le> d" shows "a + c \<le> b + d"
Ord_contains_Transset: fixes i :: "V"   and j :: "V" assumes "Ord i"   and "j \<in> elts i" shows "Transset j"
or_def: fixes P :: "bool"   and Q :: "bool" shows "P \<or> Q \<equiv> \<forall>R. (P \<longrightarrow> R) \<longrightarrow> (Q \<longrightarrow> R) \<longrightarrow> R"
in_Union_mset_iff: fixes x :: "'b"   and MM :: "'b multiset multiset" shows "(x \<in># \<Sum>\<^sub># MM) = (\<exists>M. M \<in># MM \<and> x \<in># M)"
tLength_tZip: fixes xs :: "'b Trace"   and ys :: "'c Trace"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "tLength xs = tLength ys" shows "tLength (tZip f xs ys) = tLength xs"
Suc_pred: fixes n :: "nat" assumes "order.greater n 0" shows "Suc (n - Suc 0) = n"
simps: fixes a :: "trm" shows "(a \<in> value) = ((\<exists>T t. a = (\<lambda>:T. t)) \<or> (\<exists>T t. a = (\<lambda><:T. t)) \<or> (\<exists>fs. a = Rcd fs \<and> (\<forall>(l, t)\<in>set fs. t \<in> value)))"
P_prop: fixes x' :: "'a" assumes "P x"   and "P x'" shows "x' = x"
assms_2: shows "x @ y \<le>\<^sub>v w @ v"
s1: shows "simple_ruleset (packet_assume_new (rs::'i common_primitive rule list))"
rep_eq: fixes x :: "'state"   and xa :: "('b, 'pred, 'act, 'effect) Tree\<^sub>\<alpha>" shows "FL_valid_Tree\<^sub>\<alpha> x xa = FL_valid_Tree x (rep_Tree\<^sub>\<alpha> xa)"
conj_subset_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "((A::'b set) \<subseteq> {x::'b. (P::'b \<Rightarrow> bool) x \<and> (Q::'b \<Rightarrow> bool) x}) = (A \<subseteq> {x::'b. P x} \<and> A \<subseteq> {x::'b. Q x})"
ParamDefs_return_target_length: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option"   and ins :: "'c list"   and outs :: "'c list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses"   and "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'"   and "(p, ins, outs) \<in> set procs" shows "length (ParamDefs (targetnode a)) = length outs"
axioms_1: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "full_embedding_functor A B F" shows "embedding_functor A B F"
Theorem_10: fixes x :: "'a"   and ds :: "'b set"   and d :: "'b" assumes "x \<in> Singleton_for_d.match (gfp_F ds)" shows "\<exists>y\<in>Singleton_for_d.match (Singleton_for_d.gfp_F ds d ds). (x, y) \<in> Pd_singletons_for_ds (Singleton_for_d.match (gfp_F ds)) {d} (Xd x)"
and_def: fixes \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm" shows "and \<phi>\<^sub>1 \<phi>\<^sub>2 = (if \<phi>\<^sub>1 = TrueF then \<phi>\<^sub>2 else if \<phi>\<^sub>2 = TrueF then \<phi>\<^sub>1 else if \<phi>\<^sub>1 = FalseF \<or> \<phi>\<^sub>2 = FalseF then FalseF else And \<phi>\<^sub>1 \<phi>\<^sub>2)"
confTs_map: fixes P :: "'a prog"   and h :: "heap"   and vs :: "val list"   and Ts :: "ty list" shows "(P,h \<turnstile> vs [:\<le>\<^sub>\<top>] map OK Ts) = (P,h \<turnstile> vs [:\<le>] Ts)"
iMOD_set_UN_set_eq: shows "iMOD_set = iMOD_UN_set"
type_definition_prod: shows "type_definition Rep_prod Abs_prod Product_Type.prod"
DynPDG_cdep_edge_CFG_path_2: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>c\<^sub>d n'" shows "as \<noteq> []"
borel_def: fixes "open" :: "'b set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.borel open = sigma UNIV {S. open S}"
rb_aux_term_def: shows "rb_aux_term \<equiv> rb_aux_term2 \<inter> {(x, y). rb_aux_inv x \<and> rb_aux_inv y}"
set_mono_sublist: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le>f ys" shows "set xs \<subseteq> set ys"
seteq_qle_def: fixes qle :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set"   and N :: "'a set" assumes "quasi_order qle" shows "quasi_order.seteq_qle qle M N \<equiv> quasi_order.subseteq_qle qle M N \<and> quasi_order.subseteq_qle qle N M"
MPair_parts: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> parts H" obtains "X \<in> parts H" and "Y \<in> parts H"
eint_uminus_eq: fixes a :: "eint" shows "a + - a = a - a"
orthog_proj_exists: fixes M :: "'b set"   and h :: "'b" assumes "closed_csubspace M" shows "\<exists>k. h - k \<in> orthogonal_complement M \<and> k \<in> M"
sub1_triangulizes_column: fixes A :: "'a mat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub1 (A $$ (l, l)) (dim_row A - Suc l) l (r, A) = (r', A')"   and "local.triangular_to l A"   and "0 < dim_row A"   and "l < dim_row A"   and "l < dim_col A" shows "local.triangular_column l A'"
NatTransComp'_def: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "\<eta>1 \<bullet>1 \<eta>2 = \<lparr>NTDom = NTDom \<eta>1, NTCod = NTCod \<eta>2, NatTransMap = \<lambda>X. \<eta>1 $$ X ;;\<^bsub>NTCatCod \<eta>1\<^esub> (\<eta>2 $$ X)\<rparr>"
min_elems_charn: fixes t :: "'b list"   and A :: "'b list set" assumes "t \<in> A" shows "\<exists>t' r. t = t' @ r \<and> t' \<in> min_elems A"
augFlow_val: fixes p :: "(nat \<times> nat) list" assumes "isAugmentingPath p" shows "Flow.val cf s (augmentingFlow p) = resCap p"
split_paired_Ex: fixes P :: "'f \<times> 'g \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>a b. P (a, b))"
induct: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a list"   and x2 :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "lexp order x1 x2"   and "\<And>x y xs ys. \<lbrakk>order x y; length xs = length ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>xs ys x. \<lbrakk>lexp order xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (x # ys)" shows "P x1 x2"
remove_cycles_id: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<notin> set xs" shows "remove_cycles xs x ys = rev ys @ xs"
zsgn_def: fixes i :: "Quotient_Int.int" shows "sgn i = (if i = 0 then 0 else if 0 < i then 1 else - 1)"
multiple_point_rep_num: fixes n :: "nat"   and x :: "'a" shows "d.multiple_blocks n rep x = multiple_blocks n rep x * int n"
I: fixes s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> s'"   and "P s"   and "bval b s" shows "P s'"
force_simps_1: fixes g :: "('e, 'f) generator"   and s :: "'f" assumes "generator g s = Done" shows "force g s = None"
cinner_adj_right: fixes x :: "'c"   and G :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and y :: "'d" shows "x \<bullet>\<^sub>C (G* *\<^sub>V y) = (G *\<^sub>V x) \<bullet>\<^sub>C y"
nn_integral_count_space_indicator: fixes X :: "'c set"   and f :: "'c \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = set_nn_integral (count_space UNIV) X f"
mertens_bounded: shows "(\<lambda>x. primes_M x - ln x) \<in> O(\<lambda>_. 1)"
open_countable_basis_ex: fixes X :: "'a set" assumes "p X" shows "\<exists>B'\<subseteq>B. X = \<Union> B'"
measurable_embed_measure2: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure" assumes "inj f" shows "f \<in> M \<rightarrow>\<^sub>M embed_measure M f"
word_exp_length_eq_0: shows "2 ^ LENGTH('k) = 0"
finite_supp_map_bset_tau_steps: fixes x :: "('idx, 'pred, 'act) formula" shows "finite (supp (map_bset (tau_steps x) (Abs_bset UNIV)))"
simps_5: fixes I :: "('a, 'b, 'c) interp"   and r :: "real"   and v :: "(real, 'c) vec" shows "sterm_sem I (Const r) v = r"
index_map_vec_2: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b vec" shows "dim_vec (map_vec f v) = dim_vec v"
inj: shows "inj_on ((`) \<pi>) {x. x \<subseteq> b1 \<and> x \<subseteq> b2 \<and> card x = n}"
closed_cdi_Un: fixes \<Omega> :: "'c set"   and M :: "'c set set"   and A :: "'c set"   and B :: "'c set" assumes "closed_cdi \<Omega> M"   and "{} \<in> M"   and "A \<in> M"   and "B \<in> M"   and "A \<inter> B = {}" shows "A \<union> B \<in> M"
snd_rep_nat_term_eqI: fixes ps :: "bool"   and u :: "'a"   and v :: "'a" assumes "ps \<or> is_scalar TYPE('a)"   and "ps \<Longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v)" shows "snd (rep_nat_term u) = snd (rep_nat_term v)"
cpred_def: shows "cpred \<equiv> \<lambda>c (pc, s, stk) (pc', s', stk'). pc' = pc + size (ccomp c) \<and> (c, s) \<Rightarrow> s' \<and> stk' = stk"
isomorphisms_UNIV_UNIV: assumes "DIM('b) = DIM('c)" obtains f :: "'b \<Rightarrow> 'c" and g :: "'c \<Rightarrow> 'b" where "linear f" and "linear g" and "\<And>x. norm (f x) = norm x" and "\<And>y. norm (g y) = norm y" and "\<And>x. g (f x) = x" and "\<And>y. f (g y) = y"
R2_def: fixes h :: "'a"   and w :: "nat \<times> nat"   and c :: "nat" shows "R2 h w c = (let (x1, x2) = w in sample_uniform (order \<G>) \<bind> (\<lambda>r1. sample_uniform (order \<G>) \<bind> (\<lambda>r2. let z1 = (c * x1 + r1) mod order \<G>; z2 = (c * x2 + r2) mod order \<G> in return_spmf (\<^bold>g [^] r1 \<otimes> g' [^] r2, c, z1, z2))))"
memset: shows "pseudo_proj_True n -` {w} \<in> events"
preserves_assoc_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "C.ide f"   and "C.ide g"   and "C.ide h"   and "src\<^sub>C f = trg\<^sub>C g"   and "src\<^sub>C g = trg\<^sub>C h" shows "F \<a>\<^sub>C[f, g, h] = \<Phi> (f, g \<star>\<^sub>C h) \<cdot>\<^sub>D (F f \<star>\<^sub>D \<Phi> (g, h)) \<cdot>\<^sub>D \<a>\<^sub>D[F f, F g, F h] \<cdot>\<^sub>D (D.inv (\<Phi> (f, g)) \<star>\<^sub>D F h) \<cdot>\<^sub>D D.inv (\<Phi> (f \<star>\<^sub>C g, h))"
rcoset_rel_def: fixes H :: "'g set" shows "rcoset_rel H \<equiv> {(g, g'). g \<in> G \<and> g' \<in> G \<and> g - g' \<in> H}"
InvInv_1: fixes f :: "'b" assumes "ciso f" shows "ciso Cinv f"
natural_transformation_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'b \<Rightarrow> 'd"   and G :: "'b \<Rightarrow> 'd"   and \<tau> :: "'b \<Rightarrow> 'd" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
estep_def: fixes e :: "nat" shows "estep e \<equiv> if e2stack e = 0 then prod_encode (0, e2rv e) else if e2i e = 0 then prod_encode (e2tail e, 1) else if e2i e = 1 then prod_encode (e2tail e, Suc (Suc (e_hd (e2xs e)))) else if encode_kind (e2i e) = 2 then prod_encode (e2tail e, Suc (e_nth (e2xs e) (pdec22 (e2i e)))) else if encode_kind (e2i e) = 3 then estep_Cn e else if encode_kind (e2i e) = 4 then estep_Pr e else if encode_kind (e2i e) = 5 then estep_Mn e else 0"
vinsert_3: fixes r :: "V" assumes "F = \<D>\<^sub>\<circ> r"   and "Limit \<alpha>"   and "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "vsv r" shows "\<R>\<^sub>\<circ> r \<in>\<^sub>\<circ> Vset \<alpha>"
Rep_ratfps_inject: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "(Rep_ratfps x = Rep_ratfps y) = (x = y)"
max_pos_imp_Interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and C :: "'b literal multiset"   and N :: "'b literal multiset set"   and A :: "'b" assumes "ground_resolution_with_selection S"   and "C \<in> N"   and "C \<noteq> {#}"   and "Max_mset C = Pos A"   and "S C = {#}" shows "ground_resolution_with_selection.Interp S N C \<Turnstile> C"
maddux_3_11: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> - y = x"
dcs: fixes x :: "'a" assumes "x \<in> set (map (\<lambda>(x, y). x - y) xys)" shows "d \<le> x"
sup_top_right: fixes x :: "'a" shows "x \<squnion> \<top> = \<top>"
enabled_conj1: fixes F :: "state \<times> state \<Rightarrow> bool"   and G :: "state \<times> state \<Rightarrow> bool" shows "\<turnstile> Enabled (F \<and> G) \<longrightarrow> Enabled F"
invertible: fixes u :: "'b" assumes "u \<in> A" shows "A.additive.invertible u"
components_are_iso: fixes a :: "'c" assumes "Cop.ide a" shows "S.iso (\<Phi> a)"
prems_3: shows "(\<sigma>', coll'') \<in> csmall_nstep P \<sigma> n'"
dg_0_components_2: shows "dg_0\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
measurable_inequality_set_4: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{x \<in> space M. g x < f x} \<in> sets M"
inner_add_left: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "(x + y) \<bullet> z = x \<bullet> z + y \<bullet> z"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
reachableNodes_def: fixes u :: "nat" shows "cf.reachableNodes u \<equiv> {v. cf.connected u v}"
app\<^sub>i_mono: fixes p :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>' :: "ty list \<times> ty err list"   and i :: "instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty" assumes "wf_prog p P"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')" shows "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>)"
real_is_int_add: fixes a :: "real"   and b :: "real" assumes "real_is_int a"   and "real_is_int b" shows "real_is_int (a + b)"
Rep_1: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "Rep one = 1"
cond: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s" shows "f s = f s'"
less_trans: fixes x :: "'a list"   and y :: "'a list"   and z :: "'a list" assumes "x <lex y"   and "y <lex z" shows "x <lex z"
igSwapIGAbsSTR_imp_igSwapIGAbs: fixes MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'i, 'j, 'k) model_scheme" assumes "igSwapIGAbsSTR MOD" shows "igSwapIGAbs MOD"
add_strict_increasing2: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset" assumes "{#} \<subseteq># a"   and "b \<subset># c" shows "b \<subset># a + c"