hcju/isabelleps huggingface dataset:
domA_not_fresh: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list" assumes "x \<in> domA \<Gamma>" shows "\<not> atom x \<sharp> \<Gamma>"
fps_ln_def: fixes c :: "'a" shows "fps_ln c = fps_const ((1::'a) / c) * Abs_fps (\<lambda>n. if n = 0 then 0::'a else (- (1::'a)) ^ (n - 1) / of_nat n)"
supset_glbound_in_of_lcoset_shift: fixes P :: "'b set set"   and X :: "'b set"   and Y :: "'b set"   and B :: "'b set"   and a :: "'b" assumes "dual_order.glbound_in_of P X Y B" shows "dual_order.glbound_in_of ((+o) a ` P) (a +o X) (a +o Y) (a +o B)"
split_relation_access: fixes as :: "'b list"   and ls :: "'b list"   and rs :: "'b list"   and i :: "nat"   and r :: "'b"   and rrs :: "'b list" assumes "split_relation as (ls, rs) i"   and "rs = r # rrs" shows "as ! i = r"
conj_zero_right: fixes x :: "'a" shows "x AND (0::'a) = (0::'a)"
cases: fixes x :: "atom" obtains uu_ :: "nat" where "x = Less uu_ 0"   | v :: "nat" where "x = Less 0 (Suc v__)"   | i :: "nat" and j :: "nat" where "x = Less (Suc i__) (Suc j__)"   | "x = Eq 0 0"   | v :: "nat" where "x = Eq 0 (Suc va__)"   | v :: "nat" where "x = Eq (Suc v) 0"   | i :: "nat" and j :: "nat" where "x = Eq (Suc i) (Suc j)"
sp_in_extends: fixes e :: "'c edge"   and g :: "('c, 'd) rgraph_scheme"   and g' :: "('c, 'd) rgraph_scheme"   and v1 :: "'c"   and es :: "'c edge list"   and v2 :: "'c" assumes "extends g e g'"   and "Graph.subpath g v1 es v2" shows "Graph.subpath g' v1 es v2"
dim_row_transfer_rule: fixes A :: "'c mat"   and A' :: "(('c, 'd) Finite_Cartesian_Product.vec, 'e) Finite_Cartesian_Product.vec" assumes "HMA_M A A'" shows "dim_row A = CARD('e)"
sep_disj_prod_def: fixes m1 :: "'e \<times> 'f"   and m2 :: "'e \<times> 'f" shows "m1 ## m2 \<equiv> fst m1 ## fst m2 \<and> snd m1 ## snd m2"
span_superset: fixes S :: "'c set" shows "S \<subseteq> vs2.span S"
simps_3: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and v :: "'a list"   and i :: "nat"   and \<phi> :: "'a MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Neg \<phi>) = (\<not> MFOTL.sat \<sigma> v i \<phi>)"
paths_nxt_eq: fixes xs :: "'b list"   and t :: "'b ttree"   and x :: "'b" shows "(xs \<in> paths (nxt t x)) = (xs = [] \<or> x # xs \<in> paths t)"
cvalids_def: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and D :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set" shows "(C |\<Turnstile> D) = (|\<Turnstile> C \<longrightarrow> |\<Turnstile> D)"
minus_p_integer_def: fixes p :: "integer"   and x :: "integer"   and y :: "integer" shows "minus_p_integer p x y \<equiv> if y \<le> x then x - y else x + p - y"
honest_never_says_current_timestamp_in_auth: fixes evs :: "event list"   and T :: "nat"   and X :: "msg" assumes "CT evs = T"   and "Number T \<in> parts {X}"   and "evs \<in> kerbIV" shows "\<forall>A B Y. A \<notin> bad \<longrightarrow> Says A B \<lbrace>Y, X\<rbrace> \<notin> set evs"
det_R: shows "det (snd (QR_decomposition A)) = (\<Prod>i\<in>UNIV. snd (QR_decomposition A) $ i $ i)"
intros_5: fixes X :: "msg"   and H :: "msg set"   and Y :: "msg" assumes "X \<in> synth H"   and "Y \<in> synth H" shows "\<lbrace>X, Y\<rbrace> \<in> synth H"
triangle_paths_3: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiPrimitive \<E> \<P>"   and "MinkowskiPrimitive.kinematic_triangle \<E> \<P> a b c" shows "\<exists>Q. Q \<in> \<P> \<and> b \<in> Q \<and> c \<in> Q \<and> b \<noteq> c"
1: fixes s' :: "'b set" assumes "s' \<in> set ss" shows "local.subspace s'"
wf_residual_networkD_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" assumes "countable_network \<Delta>"   and "countable_network.wf_residual_network \<Delta>"   and "edge \<Delta> x y" shows "\<not> edge \<Delta> y x"
Gramian_matrix_times_\<kappa>: fixes i :: "nat"   and l :: "nat" assumes "i < length fs"   and "l \<le> i" shows "Gramian_matrix fs l *\<^sub>v vec l (\<kappa> i l) = vec l (\<lambda>j. - (fs ! i \<bullet> fs ! j))"
comp_sub: fixes l :: "fterm literal"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "l\<^sup>c \<cdot>\<^sub>l \<sigma> = (l \<cdot>\<^sub>l \<sigma>)\<^sup>c"
2: shows "set_mset (r|\<upsilon>'| -s r \<down>l (\<sigma> @ \<tau>') \<inter>s r \<down>l \<tau>) \<subseteq> r \<down>m (r|\<sigma>| + r|\<tau>|) - r \<down>m r|\<sigma> @ \<tau>'|"
one: shows "dirichlet_prod f g 1 = (1::'a)"
Crypt_valid_pubSK: fixes i :: "nat"   and SKi :: "nat"   and evs :: "event list" assumes "Crypt (priSK RCA) \<lbrace>Agent (CA i), Key SKi, onlySig\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> set_mr" shows "SKi = pubSK (CA i)"
h1c: shows "\<forall>i. MPoly_Type.degree (Const (real (i mod 2))) var = 0"
cod_dom: fixes f :: "'d" assumes "D.arr f" shows "D.cod (D.dom f) = D.dom f"
Crypt_notin_image_Key: fixes K :: "nat"   and X :: "msg"   and A :: "nat set" shows "Crypt K X \<notin> Key ` A"
FunctorMFunctor: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme" assumes "Functor F" shows "FunctorM F"
inj_on_inverseI: fixes A :: "'k set"   and g :: "'l \<Rightarrow> 'k"   and f :: "'k \<Rightarrow> 'l" assumes "\<And>x. x \<in> A \<Longrightarrow> g (f x) = x" shows "inj_on f A"
diff_single_eq_union: fixes x :: "'a"   and M :: "'a multiset"   and N :: "'a multiset" assumes "x \<in># M" shows "(remove1_mset x M = N) = (M = add_mset x N)"
hyps_1: shows "cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> inj_on Xd (CH fp)"
lvar_def: fixes vn :: "lname"   and s :: "st" shows "lvar vn s = (the (locals s vn), \<lambda>v. supd lupd(vn\<mapsto>v))"
BPd'_BCh_mechanism_domain: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh BPd' BCh"
Abs1_eq_iff_fresh_3: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and x :: "'b"   and y :: "'b" assumes "atom c \<sharp> (a, b, x, y)" shows "([[atom a]]lst. x = [[atom b]]lst. y) = ((a \<leftrightarrow> c) \<bullet> x = (b \<leftrightarrow> c) \<bullet> y)"
sigma_finite_measure_axioms: fixes M :: "'c pmf" shows "sigma_finite_measure (measure_pmf M)"
freeword_funlift_uminus: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b freeword" shows "freeword_funlift f (- g) = - freeword_funlift f g"
HEndPhase1_HInv4a_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4a s q"   and "p \<noteq> q" shows "HInv4a s' q"
3: shows "\<not> choose_arc h \<le> - choose_arc h"
assms_1: shows "LLL_impl_inv state i (fs_state state)"
bdd_below_Ioo: fixes a :: "'a"   and b :: "'a" shows "bdd_below {a<..<b}"
xs_x: shows "v = xs @\<^sub>v [x]\<^sub>v"
finite_list: fixes A :: "'f set" assumes "finite A" shows "\<exists>xs. set xs = A"
lookup_default_0_map_values: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d"   and m :: "('d, 'e) mapping" assumes "f x (0::'e) = (0::'c)" shows "Mapping.lookup_default (0::'c) (Mapping.map_values f m) x = f x (Mapping.lookup_default (0::'e) m x)"
i_expand_gr0_i_expand_Suc: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'd message_af" assumes "0 < k" shows "f \<odot> k = f \<odot>\<^bsub>iSuc\<^esub> (k - Suc 0)"
column_def: fixes j :: "'c"   and A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "column j A = (\<chi>i. A $h i $h j)"
3: shows "(punit.lt (rep_list a) - punit.lt (rep_list g)) \<oplus> lt g = lt a"
sublist_single: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "WB_Sort.sublist xs i i = [xs ! i]"
no_infinite_subset_chain2: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "mono \<tau>"   and "\<forall>i. (\<tau> ^^ (i + 1)) UNIV \<subset> (\<tau> ^^ i) UNIV" shows "False"
ord_eq_le_trans: fixes a :: "'b"   and b :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "'b" assumes "a = b"   and "less_eq b c" shows "less_eq a c"
verit_minus_simplify_4: fixes b :: "'c" shows "- (- b) = b"
elims: fixes x :: "'b sort_key extended \<times> 'b sort_key extended"   and y :: "'b sort_key extended \<times> 'b sort_key extended" assumes "derive_left x = y" obtains l :: "'b sort_key extended" and u :: "'b sort_key extended" where "x = (l, u)" and "y = (derive_pos (\<gamma> l sort_dir.Right) l, derive_pos (\<gamma> l sort_dir.Right) u)"
Tensor_in_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "arr f"   and "arr g"   and "t \<in> f"   and "u \<in> g" shows "t \<^bold>\<otimes> u \<in> f \<otimes> g"
span_finiteE: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "finite B"   and "p \<in> pmdl B" obtains q :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c" where "p = (\<Sum>b\<in>B. q b \<odot> b)"
N2H_def: fixes b :: "nat"   and n :: "nat" shows "N2H b n = inv_into (hbase b) (H2N b) n"
new_Addr_def: fixes h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" shows "new_Addr h \<equiv> if \<exists>a. h a = None then \<lfloor>LEAST a. h a = None\<rfloor> else None"
sum_coprime_dvd_cong: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "coprime a b"   and "\<And>r s. \<lbrakk>coprime r s; r dvd a; s dvd b\<rbrakk> \<Longrightarrow> f r s = g r s" shows "(\<Sum>r | r dvd a. \<Sum>s | s dvd b. f r s) = (\<Sum>r | r dvd a. \<Sum>s | s dvd b. g r s)"
poincare_between_nonstrict_2: fixes u :: "complex_homo"   and v :: "complex_homo" shows "poincare_between u v v"
pseudominimal_inD1: fixes P :: "'b set"   and x :: "'b" assumes "codomain.pseudominimal_in P x" shows "x \<in> P"
empty_union: fixes A :: "'s" shows "empty \<^bold>\<union> A = A"
coeff_minus: fixes p :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and q :: "'a mpoly" shows "MPoly_Type.coeff p m - MPoly_Type.coeff q m = MPoly_Type.coeff (p - q) m"
divide_inf_right: fixes z :: "complex_homo" assumes "z \<noteq> \<infinity>\<^sub>h" shows "z :\<^sub>h \<infinity>\<^sub>h = 0\<^sub>h"
streamValue12: fixes d :: "nat"   and k :: "nat"   and lose :: "nat \<Rightarrow> 'a"   and t :: "nat"   and x :: "'a" assumes "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = x" shows "\<forall>j\<le>Suc d. lose (t + 2 + k + j) = x"
independent_replacement: fixes u :: "'a"   and Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (u # Us)"   and "local.independent K Vs"   and "Span K (u # Us) \<subseteq> Span K Vs" shows "\<exists>v\<in>set Vs. local.independent K (v # Us)"
globally_consistent_loweq: fixes A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set"   and x :: "'Var"   and v :: "'Val"   and v' :: "'Val" assumes "globally_consistent A mds"   and "A x = Some (v, v')"   and "dma x = Low" shows "v = v'"
run_asset_iv_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_asset_iv n s \<equiv> stage_asset_iv n (run_owner_iv n s)"
B: fixes x :: "hf"   and x' :: "hf" assumes "B x x'" shows "P x x'"
Sup_to_Inf: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and dual :: "'a \<Rightarrow> 'a" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Sup = dual \<circ> Inf \<circ> \<P> dual"
NSCauchyI: fixes X :: "nat \<Rightarrow> 'a" assumes "\<And>M N. \<lbrakk>M \<in> HNatInfinite; N \<in> HNatInfinite\<rbrakk> \<Longrightarrow> (*f* X) M \<approx> (*f* X) N" shows "NSCauchy X"
uminus_eq: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and implies :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.heyting_algebra uminus inf less_eq less sup bot top implies" shows "uminus x = implies x bot"
return_heap_E: fixes h :: "'a"   and h' :: "'a"   and h'' :: "'a" shows "h \<turnstile> return_heap h' \<rightarrow>\<^sub>h h'' = (h'' = h')"
NT_ND: fixes c :: "'b event list"   and P :: "'b process" assumes "c \<notin> T P" shows "c \<notin> D P"
mult_left_less_imp_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring plus minus less_eq less zero times"   and "less (times c a) (times c b)"   and "less_eq zero c" shows "less a b"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" shows "(k \<notin> Poly_Mapping.keys f) = (poly_mapping.lookup f k = (0::'c))"
poly_mult_zero_iff: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'a" shows "(poly (p * q) x = (0::'a)) = (poly p x = (0::'a) \<or> poly q x = (0::'a))"
p2_eq_0: fixes n :: "nat" shows "(2 ^ n = 0) = (LENGTH('b) \<le> n)"
rel_G_witness1: 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 tytok :: "('z \<times> ('z \<times> 'aa) \<times> 'aa \<times> 'ab \<times> ('ab \<times> 'ac) \<times> 'ac \<times> 'ad) itself"   and L1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and x :: "('z, 'ab, 'n, 'q, 't, 'w, 'ad) G"   and y :: "('aa, 'ac, 'p, 's, 'v, 'y, 'ad) G" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok"   and "rel_G L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_G (\<lambda>x (x', y). x' = x \<and> L1 x y) (\<lambda>x (x', y). x' = x \<and> L2 x y) Co1 Co2 Contra1 Contra2 x (rel_G_witness L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y))"
sets_eq_N: shows "sets (measure_pmf (pair_pmf A B) \<bind> M) = sets N"
set_bd_2: fixes x :: "('l, 'm, 'k) F3" shows "|F3set2 x| \<le>o bd_F3"
dist_commute: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "dist x y = dist y x"
fiter_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c" shows "fiter f g = gfp (fiter_fun f g)"
monotone_snd: fixes orda :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ordb :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "monotone (rel_prod orda ordb) ordb snd"
cases: fixes x :: "'a Literal" obtains A :: "'a" where "x = Pos A__"   | A :: "'a" where "x = Neg A"
a_host_shadow_root_rel_finite: fixes h :: "(_) heap" shows "finite (local.a_host_shadow_root_rel h)"
hcomp_in_hhom: fixes \<mu> :: "'b"   and a :: "'b"   and b :: "'b"   and \<nu> :: "'b"   and c :: "'b" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "\<guillemotleft>\<nu> \<star>\<^sub>D \<mu> : a \<rightarrow>\<^sub>D c\<guillemotright>"
A: shows "\<not> class.modular_lattice (\<sqinter>) (\<le>) (<) (\<squnion>) \<or> (\<exists>a b c. M5_lattice a b c)"
addpre_simps_4: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>5 (addpre v pre) = \<pi>\<^sub>5 v"
distinct_1: fixes x11 :: "'e trm"   and x12 :: "'e trm"   and x21 :: "'f"   and x22 :: "'e trm list" shows "Eq x11 x12 \<noteq> Pr x21 x22"
comp_assoc: fixes h :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" shows "(h \<odot> g) \<odot> f = h \<odot> g \<odot> f"
finite_union_of_complement: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set" shows "(finite union_of P) S = (finite intersection_of (\<lambda>S. P (- S))) (- S)"
unit_simps_2: fixes a :: "'a" assumes "obj a" shows "src \<i>[a] = a"
assms: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>)"
projectTable_idle: fixes n :: "nat"   and A :: "nat set"   and X :: "'b option list set"   and I :: "nat set" assumes "table n A X"   and "A \<subseteq> I" shows "New_max.projectTable I (A, X) = (A, X)"
of_nat_power_le_of_nat_cancel_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "nat"   and b :: "nat"   and w :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "less_eq (semiring_1.of_nat one plus zero x) (power.power one times (semiring_1.of_nat one plus zero b) w) = (x \<le> b ^ w)"
linf_norm_vec_of_list: fixes xs :: "'b list" shows "\<parallel>vec_of_list xs\<parallel>\<^sub>\<infinity> = Missing_Lemmas.max_list (map abs xs @ [0::'b])"
concatenates_Cons: fixes x :: "brick"   and xs :: "brick list"   and ys :: "brick list" shows "(x # xs) \<otimes> ys = x # xs \<otimes> ys"
unl_after: fixes v :: "state \<Rightarrow> 'a"   and s :: "state"   and t :: "state" shows "(v$) (s, t) \<equiv> v t"
measure_down: fixes \<Omega> :: "'b set"   and N :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal"   and M :: "'b set set" assumes "measure_space \<Omega> N \<mu>"   and "sigma_algebra \<Omega> M"   and "M \<subseteq> N" shows "measure_space \<Omega> M \<mu>"
assms: shows "((\<sigma>, s), a, \<sigma>', s') \<in> onode_sos S"
mayDiverge_If_L: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'a"   and s :: "'b"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com" assumes "tval tst s"   and "PL.mayDiverge tval aval c1 s" shows "PL.mayDiverge tval aval (if tst then c1 else c2) s"
cod_eqI: fixes b :: "'a"   and f :: "'a" assumes "L.ide b"   and "L.seq b f" shows "L.cod f = b"
mult_ac_3: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b * (a * c) = a * (b * c)"
EH\<^sub>N_sum_1: shows "EH\<^sub>N n = (\<Sum>\<^sub>ai\<in>{1..}. measure_pmf.prob (H\<^sub>N n) {i..})"
chfin: shows "\<forall>M\<in>Chains leq. finite M"
prime_square_sum_nat_decomp_correct_1: fixes p :: "nat" assumes "prime p"   and "p = 2 \<or> [p = 1] (mod 4)" shows "(fst (prime_square_sum_nat_decomp p))\<^sup>2 + (snd (prime_square_sum_nat_decomp p))\<^sup>2 = p"
ex_ex1I: fixes P :: "'a \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
RDirProd_list_carrier_def': fixes Rs :: "('c, 'd) ring_scheme list" shows "carrier (RDirProd_list Rs) = carrier (DirProd_list Rs)"
neqx: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "bl \<noteq> {x}"
distinct_map_mirror_point_eq: fixes x :: "'a"   and xs :: "'a list" shows "distinct (map (mirror_point x) xs) = distinct xs"
psubstEnv_vsubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[y1 // y]_ys &[rho'] = rho &[rho' [y \<leftarrow> Var ys y1 #[rho']]_ys]"
inner_add_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "inner x (y + z) = inner x y + inner x z"
build_rel_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool" shows "br \<alpha> I \<equiv> {(c, a). a = \<alpha> c \<and> I c}"
InterpExprWellDefined: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and A :: "'a"   and e :: "('a, 'b) Expression"   and B :: "'a"   and i :: "('c, 'd) IType" assumes "Interpretation I"   and "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> i" shows "Sig iS\<^bsub>I\<^esub> \<triangleright> Vx : A \<turnstile> e : B"
Aheap_simp1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" assumes "\<not> nonrec \<Gamma>" shows "Aheap \<Gamma> e\<cdot>a = Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>) f|` domA \<Gamma>"
mult_poly_raw_mult_poly_p: fixes p :: "(char list list \<times> int) list"   and p' :: "(char list multiset \<times> int) multiset"   and q :: "(char list list \<times> int) list"   and q' :: "(char list multiset \<times> int) multiset" assumes "(p, p') \<in> sorted_poly_rel"   and "(q, q') \<in> sorted_poly_rel" shows "\<exists>r. (PAC_Polynomials_Operations.mult_poly_raw p q, r) \<in> unsorted_poly_rel \<and> (mult_poly_p q')\<^sup>*\<^sup>* (p', {#}) ({#}, r)"
distinct_51: fixes x6 :: "(char list \<times> trm) list"   and x71 :: "trm"   and x72 :: "char list" shows "Rcd x6 \<noteq> x71..x72"
A: shows "(\<Union>i. {inverse (real (Suc i))..}) = {0<..}"
dim_vec_eq: fixes X :: "(real, 'b) vec set" shows "vec.dim X = dim X"
assms_2: shows "vsv (r \<restriction>\<^sup>l\<^sub>\<circ> \<R>\<^sub>\<circ> s)"
order_iff: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># b) = (a = a \<inter># b)"
mult: fixes f :: "nat \<Rightarrow> 'a"   and a :: "nat"   and b :: "nat" assumes "completely_multiplicative_function f" shows "f (a * b) = f a * f b"
AAss1Red1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and a' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and i :: "'addr expr1"   and e :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>a,s\<rangle> -ta\<rightarrow> \<langle>a',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>a\<lfloor>i\<rceil> := e,s\<rangle> -ta\<rightarrow> \<langle>a'\<lfloor>i\<rceil> := e,s'\<rangle>"
concat_morph: fixes xs :: "'b list list"   and ys :: "'b list list" shows "concat xs \<cdot> concat ys = concat (xs \<cdot> ys)"
ide_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.ide (free_monoidal_category.comp C) f = (free_monoidal_category.can C f \<and> partial_magma.dom (free_monoidal_category.comp C) f = partial_magma.cod (free_monoidal_category.comp C) f)"
bernoulli_pmf_half_conv_pmf_of_set: shows "bernoulli_pmf (1 / 2) = coin_pmf"
simps_2: fixes x :: "'ubx1" shows "box_operand (OpUbx1 x) = OpDyn (box_ubx1 x)"
lt_strict_2: fixes compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a" assumes "class.Ord_strict compare" shows "Ord.lt compare\<cdot>x\<cdot>\<bottom> = \<bottom>"
simps_5: fixes \<sigma> :: "nat \<Rightarrow> exp option"   and C :: "nat"   and e :: "exp" shows "substs \<sigma> (Cast C e) = Cast C (substs \<sigma> e)"
rev_morph_idemp: fixes f :: "'b list \<Rightarrow> 'c list" shows "rev_morph (rev_morph f) = f"
nn_integral_monotone_convergence_SUP: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "incseq f"   and "\<And>i. f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (SUP i. f i x) \<partial>M = (SUP i. integral\<^sup>N M (f i))"
simps_2: fixes tp :: "nat"   and tp' :: "nat" shows "max_tstp (Inr tp) (Inr tp') = Inr (max tp tp')"
Irr_def: fixes K :: "'a set"   and x :: "'a" shows "Irr K x = (THE p. p \<in> carrier (K [X]) \<and> pirreducible K p \<and> local.eval p x = \<zero> \<and> lead_coeff p = \<one>)"