hcju/isabelleps huggingface dataset:
rel_G_neg_distr: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and uu :: "('z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af) itself"   and L1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and L1' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and L2 :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and L2' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') \<le> rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2'"
freeword_conv_freeletter_sum_list: fixes x :: "'a freeword" shows "x = sum_list (map (apply_sign Abs_freeletter) (freeword x))"
h: shows "|~ P \<and> N \<longrightarrow> \<circle>P \<or> \<circle>Q"
hyps_2: shows "input \<in> responses_\<I> \<I>' out"
terminates_onp_terminates_on_eq: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" shows "terminates_onp g = (\<lambda>x. x \<in> terminates_on g)"
1: fixes j' :: "'a" assumes "j' \<in> local.underS i" shows "(j', j) \<in> r"
simps_1: fixes v :: "bool"   and s :: "char list \<Rightarrow> int option" shows "pbval' (Bc v) s = Some v"
recurrence_series_infinite_transfer_operator: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M" shows "AE x in M. 0 < f x \<longrightarrow> (\<Sum>n. (nn_transfer_operator ^^ n) f x) = \<infinity>"
rec_inside_def: shows "rec_inside = CN rec_less [Z, rec_swap (rec_iter rec_pdec2)]"
put_env_def: fixes put :: "'i \<Rightarrow> 'h \<Rightarrow> 'h"   and s :: "'i"   and m :: "('g, 'h) envT" shows "put_env put s m = EnvT (\<lambda>r. put s (run_env m r))"
pw_smooth: shows "\<gamma>2 piecewise_C1_differentiable_on {0..1}"
span_superset: fixes S :: "'b set" shows "S \<subseteq> cspan S"
arg_cong2: fixes a :: "'a"   and b :: "'a"   and c :: "'b"   and d :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "a = b"   and "c = d" shows "f a c = f b d"
1: shows "span (local.comp f (pr1 (local.dom f) (local.dom g))) (local.comp g (pr0 (local.dom f) (local.dom g)))"
has_bot_ltx: shows "has_bottom (P .\<^bold>\<le> x)"
changes_R_smods_ext_def: fixes p :: "real poly"   and q :: "real poly" shows "changes_R_smods_ext p q = (let ps = smods_ext p q in changes_poly_neg_inf ps - changes_poly_pos_inf ps)"
nn_integral_mono_measure: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "sets M = sets N"   and "M \<le> N" shows "integral\<^sup>N M f \<le> integral\<^sup>N N f"
assms_3: shows "\<not> (l = \<infinity> \<and> m = \<infinity> \<or> l = - \<infinity> \<and> m = - \<infinity>)"
assms_6: shows "eq = eq_for_lvar (\<T> s) xs"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "Labeling c s t f l"   and "push_effect_locale_axioms c f l u v" shows "push_effect_locale c s t f l u v"
D_mono: fixes A :: "char list set option"   and A' :: "char list set option"   and e :: "expr" assumes "A \<sqsubseteq> A'"   and "\<D> e A" shows "\<D> e A'"
inverse_float_interval: fixes p :: "nat"   and X :: "float interval"   and Y :: "float interval" assumes "inverse_float_interval p X = Some Y" shows "inverse ` set_of (real_interval X) \<subseteq> set_of (real_interval Y)"
19: shows "irreflexive (e * (w \<sqinter> - prim_EP w v e)\<^sup>\<star> * prim_P w v e\<^sup>T\<^sup>\<star> * (w \<sqinter> - prim_EP w v e)\<^sup>\<star>)"
fetch_state_eq: fixes b :: "cell"   and r :: "nat"   and tp :: "(action \<times> nat) list"   and st :: "nat"   and nact :: "action"   and ns :: "nat" assumes "block_map b = scan r"   and "fetch tp st b = (nact, ns)"   and "st \<le> length tp div 2" shows "newstat (code tp) st r = ns"
find_base_vectors_gen_def: fixes uminus :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and A :: "'b mat" shows "find_base_vectors_gen uminus zero one A \<equiv> let pp = pivot_positions_gen zero A in Let (filter (\<lambda>j. j \<notin> set (map snd pp)) [0..<dim_col A]) (map (non_pivot_base_gen uminus zero one A pp))"
Let_supp: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "supp (Let \<Gamma> e) = supp e \<union> supp \<Gamma> - atom ` domA \<Gamma>"
untion_image_mset_Pair_distribute: fixes C :: "'b \<Rightarrow> 'c multiset"   and J :: "'b multiset"   and I :: "'b multiset" shows "(\<Sum>x\<in>#J - I. image_mset (Pair x) (C x)) = (\<Sum>x\<in>#J. image_mset (Pair x) (C x)) - (\<Sum>x\<in>#I. image_mset (Pair x) (C x))"
U: shows "splits_at (\<alpha>' @ \<delta>) (i + ix) a1 U a2 \<and> splits_at \<gamma> j b1 U a2 \<and> LeftDerivation a1 D b1"
is_val_def: fixes v :: "exp" shows "is_val v \<equiv> isval v \<and> FV v = {}"
check_extension_l_new_var_multiple_err_impl_def: fixes v :: "'a"   and p :: "'b" shows "check_extension_l_new_var_multiple_err_impl v p = ''Error while checking side conditions of extensions polynow, var is '' @ show v @ '' but it either appears at least once in the polynomial or another new variable is created '' @ show p @ '' but should not.''"
odd_power_less_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less a zero" shows "less (power.power one times a (Suc (2 * n))) zero"
pred_def: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and P :: "'d set" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "sifum_lang_no_dma.pred eval\<^sub>B P \<equiv> \<lambda>mem. \<forall>p\<in>P. eval\<^sub>B mem p"
antimono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'b" shows "antimono f = (\<forall>n. f (Suc n) \<le> f n)"
set_intros_2: fixes x2 :: "'e" shows "x2 \<in> set2_step (Skip x2)"
absorb_iff2: fixes a :: "'d"   and b :: "'d" shows "(a \<le> b) = (sup a b = b)"
wpw_lupd: fixes x :: "char list"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c :: "com"   and n :: "nat"   and l :: "char list \<Rightarrow> nat"   and u :: "nat" assumes "x \<notin> support Q" shows "wpw b c n Q (l(x := u)) = wpw b c n Q l"
abs_summable_on_reindex: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and A :: "'d set" assumes "(\<lambda>x. f (g x)) abs_summable_on A" shows "f abs_summable_on g ` A"
Domainp_mod_ring_rel64: shows "Domainp mod_ring_rel64 = (\<lambda>x. 0 \<le> x \<and> x < pp)"
test_galois1: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "test p"   and "test q"   and "test r" shows "(p \<le> q + r) = (p \<cdot> !q \<le> r)"
strategyproofness: shows "pmf (sds R18) c + pmf (sds R18) d \<le> pmf (sds R23) d + pmf (sds R23) c"
right_idem: fixes a :: "'d"   and b :: "'d" shows "sup (sup a b) b = sup a b"
Ide_Diagonalize_rep_ide: fixes a :: "'c local.term set" assumes "ide a" shows "Ide \<^bold>\<lfloor>rep a\<^bold>\<rfloor>"
hyp2: assumes "isTable E vs ps" shows "ExcessNotAtRec ps g = ListSum (ExcessNotAtRecList ps g) E"
is_rectangle_def: fixes x :: "'a" shows "is_rectangle x \<equiv> x ; 1 ; x \<le> x"
NoneI: fixes A :: "'a set" shows "None \<in> A\<^sub>\<bottom>"
unitor_coincidence_2: fixes a :: "'a" assumes "obj a" shows "\<r>[a] = \<i>[a]"
simps_5: fixes x21 :: "(Term_Class.term \<times> sterm) list"   and x22 :: "(name, value) fmap"   and x11 :: "name"   and x12 :: "value list" shows "Vabs x21 x22 \<noteq> Vconstr x11 x12"
at_most_once: fixes i :: "'a \<times> nat"   and s :: "'a \<times> nat"   and m :: "('a \<times> nat, 'b) message"   and j :: "'a \<times> nat" assumes "event_at i (Receive s m)"   and "event_at j (Receive s m)"   and "fst i = fst j" shows "i = j"
dims_Aw'_pow_1: shows "dim_row Aw' = last rs ^ N_half"
dgrad_sig_set_closed_monomial: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t"   and j :: "nat"   and c :: "'b" assumes "d (pp_of_term u) \<le> dgrad_max d"   and "component_of_term u < j" shows "monomial c u \<in> dgrad_sig_set' j d"
indRelTEQ_impl_TRel_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "rel_respects_binary_pred (indRelTEQ TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))" shows "rel_respects_binary_pred TRel (\<lambda>a. hasBarb a TWB)"
welltyped_subst: fixes u :: "expr"   and U :: "Unary_PCF.type"   and t :: "expr"   and xT :: "char list \<times> Unary_PCF.type" assumes "u ::: U"   and "t ::: snd xT" shows "subst xT t u ::: U"
x_3: shows "s \<sqsubseteq> \<delta> x"
inject: fixes x2 :: "'a"   and y2 :: "'a" shows "(Num x2 = Num y2) = (x2 = y2)"
f1: fixes x :: "atom" assumes "x \<in> set (ba xs)" shows "x \<sharp> f xs T"
gc1_def: shows "gc1 \<equiv> SOME gc1. 0 < gc1 \<and> (\<forall>\<^sub>F x in sequentially. g x \<le> gc1 * g' (real x))"
map_2: fixes f :: "'f \<Rightarrow> 'e"   and x21 :: "'f"   and x22 :: "'f list" shows "map f (x21 # x22) = f x21 # map f x22"
dom_simp: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<times> 'b" assumes "product_category C1 C2"   and "partial_magma.arr (product_category.comp C1 C2) f" shows "partial_magma.dom (product_category.comp C1 C2) f = (partial_magma.dom C1 (fst f), partial_magma.dom C2 (snd f))"
Re_linepath': fixes a :: "complex"   and b :: "complex"   and x :: "real" shows "Re (linepath a b x) = linepath (Re a) (Re b) x"
Bseq_iff1a: fixes X :: "nat \<Rightarrow> 'a" shows "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real (Suc N))"
SUCCI: shows "(succi, E) \<in> \<langle>Rv\<rangle>succg_rel"
assms_1: shows "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> \<turnstile>\<^sub>F \<Phi>"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> vint_mult \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> vint_mult \<in>\<^sub>\<circ> Vset \<alpha>" shows "vint_mult \<in>\<^sub>\<circ> Vset \<alpha>"
extension: fixes G :: "'b monoid"   and K :: "'b monoid"   and l :: "'b monoid list"   and g :: "'b \<Rightarrow> 'b"   and q :: "('b \<Rightarrow> 'b) list"   and H :: "'b monoid"   and h :: "'b \<Rightarrow> 'b" assumes "exact_seq (G # K # l, g # q)"   and "Group.group H"   and "h \<in> hom G H"   and "kernel G H h = g ` carrier K" shows "exact_seq (H # G # K # l, h # g # q)"
assms_2: shows "monotone (\<le>) (\<le>) f"
rpost_def: shows "rpost \<equiv> \<lambda>S (V, v) (V', r). (r \<longrightarrow> tgt \<in> local.reachable) \<and> (\<not> r \<longrightarrow> tgt \<notin> V' \<and> V \<subseteq> V' \<and> v \<in> V' \<and> V' \<subseteq> local.reachable \<and> E `` (V' - S) \<subseteq> V')"
is_lub_some_lub: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and u :: "'b"   and y :: "'b" assumes "single_valued r"   and "acyclic r"   and "(x, u) \<in> r\<^sup>*"   and "(y, u) \<in> r\<^sup>*" shows "is_lub (r\<^sup>*) x y (some_lub (r\<^sup>*) x y)"
primerec_all: fixes rt :: "recf"   and rf :: "recf"   and n :: "nat" assumes "primerec (rec_all rt rf) n" shows "primerec rt n \<and> primerec rf (Suc n)"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_diff a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "vars_mset s \<subseteq># vars_mset t" and "wt t = wt s" and "head t >\<^sub>h\<^sub>d head s"
rquot_mult: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<preceq>\<^sub>R y"   and "z = rquot y x" shows "x \<oplus> z = y"
worker_wrapper_id: fixes wrap :: "'c \<rightarrow> 'b"   and unwrap :: "'b \<rightarrow> 'c"   and computation :: "'b"   and body :: "'b \<rightarrow> 'b" assumes "wrap oo unwrap = ID"   and "computation = fix\<cdot>body" shows "computation = wrap\<cdot>(fix\<cdot>(unwrap oo body oo wrap))"
nat_mirror_inj_on: fixes l :: "nat"   and r :: "nat" shows "inj_on (\<lambda>x. nat_mirror x l r) {..l + r}"
prod_zero_i: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and i :: "'d" assumes "\<forall>k\<in>I. aGroup (A k)"   and "i \<in> I" shows "prod_zero I A i = \<zero>\<^bsub>A i\<^esub>"
lemVecPlusCommute: fixes u :: "'a Vector"   and v :: "'a Vector" shows "u \<oplus> v = v \<oplus> u"
map_add_Some_iff: fixes m :: "'d \<Rightarrow> 'c option"   and n :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and x :: "'c" shows "((m ++ n) k = Some x) = (n k = Some x \<or> n k = None \<and> m k = Some x)"
isPath_append_edge: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and p :: "(nat \<times> nat) list"   and v' :: "nat"   and v'' :: "nat" assumes "Graph.isPath c v p v'"   and "(v', v'') \<in> Graph.E c" shows "Graph.isPath c v (p @ [(v', v'')]) v''"
hd_map: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "xs \<noteq> []" shows "hd (map f xs) = f (hd xs)"
eq: fixes s :: "'a \<Rightarrow>\<^sub>0 nat"   and c :: "'b" shows "focus X (monomial c s * q) = focus X (monomial c s) * focus X q"
coprime_commute: fixes p :: "'a"   and q :: "'a" shows "comm_monoid_mult_class.coprime p q = comm_monoid_mult_class.coprime q p"
eq_f': shows "\<forall>i. f (X $ i) = (\<Sum>j\<in>UNIV. matrix' X Y f $ j $ i *s Y $ j)"
"1_2": shows "R \<in> {[x, y], [y, x]}"
distinct_6: fixes x3 :: "agent"   and x2 :: "agent" shows "privK x3 \<noteq> publK x2"
unif_rtrancl_preserves_unifiable: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset"   and E' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "(E, E') \<in> unif\<^sup>*"   and "unifiable (set_mset E)" shows "unifiable (set_mset E')"
distinct_189: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x15 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> throw x15"
left_hom_axioms: shows "left_hom (\<cdot>) (\<star>) b"
is_HOLD_def: fixes thread :: "thread" shows "is_HOLD thread = (\<exists>x3. thread = HOLD x3)"
lookup_times_monomial_right_plus: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and v :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" shows "lookup (p * monomial c v) (t + v) = lookup p t * c"
utp_top: fixes P :: "'a hrel" assumes "P is \<H>" shows "P \<sqsubseteq> \<^bold>\<top>"
transpose_columnvector: fixes v :: "('a, 'b) vec" shows "Finite_Cartesian_Product.transpose (columnvector v) = rowvector v"
not_summable_infsetsum_eq: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "\<not> f abs_summable_on A" shows "infsetsum f A = (0::'e)"
chamber_image_complement_closer: fixes D :: "'a set"   and B :: "'a set"   and Cs :: "'a set list" assumes "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "B \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "B \<noteq> D"   and "folding.gallery (B # Cs @ [f ` D])" shows "\<exists>Ds. folding.gallery (B # Ds @ [D]) \<and> order.greater (length Cs) (length Ds)"
antisym_imp_exe_antisym: fixes l :: "('b \<times> 'b) list" assumes "antisym (set l)" shows "exe_antisym l"
dist_s_A: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init" shows "distinct (BIT_Off.s_A acts qs init n)"
2: shows "\<forall>i. (mtop \<odot> f) (i, j) = top \<and> (\<forall>k. j \<noteq> k \<longrightarrow> (mtop \<odot> f) (i, k) = bot)"
simps: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t\<^sub>1 :: "term"   and t\<^sub>2 :: "'c"   and u :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.rewrite_step subst unapp unconst (t\<^sub>1, t\<^sub>2) u = map_option (subst t\<^sub>2) (term.match unapp unconst t\<^sub>1 u)"
intros: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<leadsto>[bisim] Q"   and "Q \<sim> P" shows "P \<sim> Q"
IH: fixes x2a :: "('fun, 'var) Term.term" assumes "x2a \<in> set T'"   and "Fun f T \<sqsubseteq> x2a"   and "wf\<^sub>t\<^sub>r\<^sub>m x2a"   and "\<Gamma> (Fun f T) = Fun f (map \<Gamma> T)" shows "f \<in> funs_term (\<Gamma> x2a)"
dist_midpoint_2: fixes b :: "'b"   and a :: "'b" shows "dist b (midpoint a b) = dist a b / 2"
override_on_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
execute_map_entry_1: fixes i :: "nat"   and h :: "heap"   and a :: "'a array"   and f :: "'a \<Rightarrow> 'a" assumes "i < Array.length h a" shows "execute (Array.map_entry i f a) h = Some (a, Array.update a i (f (Array.get h a ! i)) h)"
times_segment_def: shows "(\<oplus>) \<equiv> map_fun Rep_segment (map_fun Rep_segment Abs_segment) (\<lambda>x y. if snd x = fst y then (fst x, snd y) else x)"
cval_add_def: fixes u :: "'b \<Rightarrow> 'c"   and d :: "'c" shows "u \<oplus> d = (\<lambda>x. u x + d)"
encL: fixes S :: "'procS" shows "TargetTerm (\<lbrakk>S\<rbrakk>) \<R>\<lbrakk>\<cdot>\<rbrakk>L SourceTerm S"
\<sigma>: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) F G \<sigma>"
mndet_FD_subset: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> 'b process" assumes "A \<noteq> {}"   and "A \<subseteq> B" shows "mndet B P \<le> mndet A P"
RedBlock: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and u :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>{V:T; Val u},s\<rangle> \<rightarrow> \<langle>Val u,s\<rangle>"
minus_stream_def: fixes x :: "'a stream"   and y :: "'a stream" shows "x - y = pure (-) \<diamondop> x \<diamondop> y"
STStep_step_2: fixes T :: "'procT"   and P' :: "('procS, 'procT) Proc" shows "((\<exists>S S'. S \<in>S TargetTerm T \<and> S' \<in>S P' \<and> S \<longmapsto>Source S') \<or> (\<exists>Ta T'. Ta \<in>T TargetTerm T \<and> T' \<in>T P' \<and> Ta \<longmapsto>Target T')) = (\<exists>T'. T' \<in>T P' \<and> T \<longmapsto>Target T')"
intro: fixes U\<^sub>M\<^sub>_\<^sub>1 :: "'d set"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis1 :: "'d set"   and U\<^sub>M\<^sub>_\<^sub>2 :: "'f set"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f"   and basis2 :: "'f set" assumes "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>1 scale\<^sub>1 basis1"   and "finite_dimensional_vector_space_on U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>2 basis2" shows "finite_dimensional_vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 scale\<^sub>1 scale\<^sub>2 basis1 basis2"
elem_min_elems: fixes c :: "'a"   and B :: "'a set" assumes "c \<in> min_elems B" shows "c \<in> B"
P0: shows "\<zero>\<^bsub>residue_ring 1\<^esub> \<oplus>\<^bsub>residue_ring 1\<^esub> \<zero>\<^bsub>residue_ring 1\<^esub> = \<zero>\<^bsub>residue_ring 1\<^esub>"
S2: shows "comp_by_index index_of_id (pr_index_enumerator n m) = c_pair 4 (c_pair index_of_id (pr_index_enumerator n m))"
lspasl_magicl_eq: fixes h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool" shows "((h1,h2\<triangleright>h0) \<and> (A \<longrightarrow>* B) h2) = ((h1,h2\<triangleright>h0) \<and> \<not> A h1 \<and> (A \<longrightarrow>* B) h2 \<or> (h1,h2\<triangleright>h0) \<and> B h0 \<and> (A \<longrightarrow>* B) h2)"
simpleI: fixes \<mu> :: "('m \<times> 'w) set" assumes "match \<mu>"   and "\<And>m w. blocks m w \<mu> \<Longrightarrow> w \<notin> Range \<mu>" shows "simple \<mu>"
NextRecvUpd'_preserves_InvGlobPointstampsEq: fixes c0 :: "('b, 't, 'loc) Combined.configuration"   and q :: "'b"   and c1 :: "('b, 't, 'loc) Combined.configuration"   and p :: "'b" assumes "prop_safe (prop_config c0 q)"   and "InvGlobPointstampsEq c0"   and "cri.InvMsgInGlob (exchange_config c0)"   and "NextRecvUpd' c0 c1 p q" shows "InvGlobPointstampsEq c1"
A: shows "c_fst (c_fst key) mod 7 = 1"
terminates_fix_gen_raw: fixes g :: "'e \<Rightarrow> 'f \<Rightarrow> ('g, 'f) step" assumes "\<And>x. terminates (g x)" shows "terminates (fix_gen_raw g)"
rep_eq: fixes x :: "'c ltl set"   and xa :: "'c ltl_prop_equiv_quotient" shows "x \<up>\<Turnstile>\<^sub>P xa = x \<Turnstile>\<^sub>P Rep xa"
replacement: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B"   and "B \<subseteq> carrier V"   and "V.module.lin_indpt A"   and "A \<subseteq> V.module.span B" shows "\<exists>C. finite C \<and> C \<subseteq> carrier V \<and> C \<subseteq> V.module.span B \<and> C \<inter> A = {} \<and> int (card C) \<le> int (card B) - int (card A) \<and> V.module.span (A \<union> C) = V.module.span B"
wrapA_def: shows "wrapA \<equiv> Abs_cfun ((oo) a2n)"
DirProds_mult: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" shows "(\<otimes>\<^bsub>DirProds G I\<^esub>) = (\<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i)"
F_def: fixes z :: "complex" shows "F z = (if z = 0 then 0 else (cos (a * z) - sin (a * z) / (a * z)) / z)"
finite_PiE: fixes S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)" shows "finite (Pi\<^sub>E S T)"
is_valid_problem_sas_plus_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem" shows "is_valid_problem_sas_plus \<Psi> \<equiv> let ops = \<Psi>\<^sub>\<O>\<^sub>+; vs = \<Psi>\<^sub>\<V>\<^sub>+; I = \<Psi>\<^sub>I\<^sub>+; G = \<Psi>\<^sub>G\<^sub>+; D = range_of \<Psi> in list_all (\<lambda>v. D v \<noteq> None) vs \<and> list_all (is_valid_operator_sas_plus \<Psi>) ops \<and> (\<forall>v. (I v \<noteq> None) = ListMem v vs) \<and> (\<forall>v. I v \<noteq> None \<longrightarrow> ListMem (the (I v)) (the (D v))) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem v (\<Psi>\<^sub>\<V>\<^sub>+)) \<and> (\<forall>v. G v \<noteq> None \<longrightarrow> ListMem (the (G v)) (the (D v)))"
l10: shows "(e\<^sup>2 + d\<^sup>2) * x\<^sup>2 + (2 * e * b * d - 2 * a * e\<^sup>2 - 2 * d * f) * x + (a\<^sup>2 * e\<^sup>2 + f\<^sup>2 - 2 * e * b * f + b\<^sup>2 * e\<^sup>2 - e\<^sup>2 * c) = 0"
rref_upt_condition3: fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. i < i + (1::'f) \<and> \<not> is_zero_row_upt_k i k A \<and> \<not> is_zero_row_upt_k (i + (1::'f)) k A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'d)) < (LEAST n. A $ (i + (1::'f)) $ n \<noteq> (0::'d))"