hcju/isabelleps huggingface dataset:
Inf_empty: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "(Inf::'b::type set \<Rightarrow> 'b::type) {} = (top::'b::type)"
x: shows "\<forall>S\<in>I. x \<in> rel_interior S"
if_then_else_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y fi = d p \<cdot> x + ad p \<cdot> y"
strict_linear_irreflexive: fixes x :: "'a" assumes "strict_linear x" shows "irreflexive x"
long_in: shows "\<alpha> * Up + (1 - \<alpha>) * Uq \<in> {0..1}"
r_phi'': fixes i :: "nat"   and x :: "nat" shows "eval r_phi [i, x] = eval r_univ [i, singleton_encode x]"
sees_field_i_i_i_o_o_iE: fixes x :: "'a prog"   and xa :: "String.literal"   and xb :: "String.literal"   and xe :: "String.literal"   and xc :: "ty"   and xd :: "fmod" assumes "pred.eval (sees_field_i_i_i_o_o_i x xa xb xe) (xc, xd)" obtains "x \<turnstile> xa sees xb:xc (xd) in xe"
add_node_correct: fixes v :: "'b"   and G :: "'b list_graph" shows "FiniteGraph.add_node v (list_graph_to_graph G) = list_graph_to_graph (FiniteListGraph.add_node v G)"
c1_4: shows "lst_conv (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) = fst \<delta>3"
LeastI_ex: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and P :: "'f \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "\<exists>x. P x" shows "P (ord.Least less_eq P)"
Union_disjoint: fixes C :: "'c set set"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "sum g (\<Union> C) = (sum \<circ> sum) g C"
swaps_filtered: fixes xs :: "'b list"   and ys :: "nat list" shows "swaps (filter (\<lambda>x. Suc x < length xs) ys) xs = swaps ys xs"
eq_\<I>_converter_reflI: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter" assumes "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>" shows "\<I>,\<I>' \<turnstile>\<^sub>C conv \<sim> conv"
group_isomorphisms_sym: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme"   and f :: "'d \<Rightarrow> 'f"   and g :: "'f \<Rightarrow> 'd" assumes "group_isomorphisms G H f g" shows "group_isomorphisms H G g f"
assms: shows "unifier\<^sub>l\<^sub>s \<epsilon> L"
distinctPermCancel: fixes p :: "(name \<times> name) list"   and T :: "'d" assumes "distinctPerm p" shows "p \<bullet> p \<bullet> T = T"
cospan_\<mu>0_H\<nu>H\<pi>\<rho>1: shows "C.cospan \<mu>.leg0 H\<nu>H\<pi>\<rho>.leg1"
finite_g_in: fixes A :: "'v set" shows "\<integral>\<^sup>+y\<in>A. g (y, x)\<partial>count_space UNIV \<noteq> \<top>"
tan_pos_pi2_le: fixes x :: "real" assumes "0 \<le> x"   and "x < pi / 2" shows "0 \<le> tan x"
coeff_dvd_poly: fixes a :: "'a poly" shows "[:lead_coeff a:] dvd a"
winding_number_subpath_trivial: fixes z :: "complex"   and g :: "real \<Rightarrow> complex"   and x :: "real" assumes "z \<noteq> g x" shows "winding_number (subpath x x g) z = 0"
set_iterator_genord_product_correct: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_a :: "'e set"   and R_a :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and it_b :: "'e \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S_b :: "'e \<Rightarrow> 'f set"   and R_b :: "'e \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_genord it_a S_a R_a"   and "\<And>a. a \<in> S_a \<Longrightarrow> set_iterator_genord (it_b a) (S_b a) (R_b a)" shows "set_iterator_genord (set_iterator_product it_a it_b) (Sigma S_a S_b) (set_iterator_product_order R_a R_b)"
pmf_pred_map: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and p :: "'d pmf" shows "pred_pmf P (map_pmf f p) = pred_pmf (P \<circ> f) p"
eventually_at_right_field: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b" shows "eventually P (at_right x) = (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
fls_const_nonzero: fixes c :: "'a" assumes "c \<noteq> (0::'a)" shows "fls_const c \<noteq> 0"
topology_eq_iff: fixes t :: "'b topology"   and s :: "'b topology" shows "(t = s) = (topspace t = topspace s \<and> (\<forall>x\<subseteq>topspace t. openin t x = openin s x))"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C G"
run_reachable: fixes r :: "nat \<Rightarrow> 'q \<times> 's" assumes "prod.is_run r" shows "range r \<subseteq> prod.reachable"
runit_in_hom: fixes a :: "'a" assumes "ide a" shows "\<guillemotleft>\<r>[a] : a \<otimes> \<one> \<rightarrow> a\<guillemotright>"
unfoldr_simps_2: fixes g :: "('a, 'b) generator"   and s :: "'b" assumes "\<not> list.has_next g s" shows "list.unfoldr g s = []"
retraction_is_epi: fixes g :: "'a \<times> 'a" assumes "VoV.retraction g" shows "VoV.epi g"
cases: fixes a1 :: "pat"   and a2 :: "trm"   and a3 :: "(name \<times> trm) list" assumes "\<turnstile> a1 \<rhd> a2 \<Rightarrow> a3" obtains x :: "name" and T :: "ty" and t :: "trm" where "a1 = PVar x T" and "a2 = tb__" and "a3 = [(x, tb__)]"   | p :: "pat" and t :: "trm" and \<theta> :: "(name \<times> trm) list" and q :: "pat" and u :: "trm" and \<theta>' :: "(name \<times> trm) list" where "a1 = \<langle>\<langle>p, q\<rangle>\<rangle>" and "a2 = \<langle>t, u\<rangle>" and "a3 = \<theta> @ \<theta>'" and "\<turnstile> p \<rhd> t \<Rightarrow> \<theta>" and "\<turnstile> q \<rhd> u \<Rightarrow> \<theta>'"
ltree_del_min: fixes t :: "('b \<times> nat) tree" assumes "ltree t" shows "ltree (del_min t)"
sconjunctive_def: fixes S :: "'d \<Rightarrow> 'e" shows "sconjunctive S = (\<forall>Q. (\<exists>x. x \<in> Q) \<longrightarrow> S (Inf Q) = Inf (S ` Q))"
work: shows "work = fix\<cdot>(unwrap oo body oo wrap)"
path_decomp_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "path (xs @ ys)" shows "path xs"
Cignorabimus_equiv: fixes \<alpha> :: "game"   and V :: "variable set" shows "Cignorabimus \<alpha> V = {M. \<forall>I \<omega> \<omega>' X. Vagree \<omega> \<omega>' (- M) \<longrightarrow> (\<omega> \<in> game_sem I \<alpha> (restrictto X V)) = (\<omega>' \<in> game_sem I \<alpha> (restrictto X V))}"
rsp: shows "((\<approx>) ===> (\<approx>) ===> (\<approx>)) plus_int_raw plus_int_raw"
b: fixes x :: "'b"   and y :: "'b" shows "bob x y \<longrightarrow> bob' x y"
valid_dbm_V': fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "valid_dbm M" shows "[M]\<^bsub>v,n\<^esub> \<in> V'"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" shows "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> a2 = ((\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>S1 S2. S1 \<in>S a1 \<and> S2 \<in>S a2 \<and> (S1, S2) \<in> SRel) \<or> (\<exists>T1 T2. T1 \<in>T a1 \<and> T2 \<in>T a2 \<and> (T1, T2) \<in> TRel))"
real_cond_exp_F_meas: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f"   and "f \<in> borel_measurable F" shows "AE x in M. real_cond_exp M F f x = f x"
PAC_checker_l_step_PAC_checker_step: fixes Ast :: "char list code_status \<times> char list set \<times> (nat, (char list list \<times> int) list) fmap"   and Bst :: "status \<times> nat set \<times> (nat, int mpoly) fmap"   and st :: "((char list list \<times> int) list, char list, nat) pac_step"   and st' :: "(int mpoly, nat, nat) pac_step"   and spec :: "(char list list \<times> int) list"   and spec' :: "int mpoly" assumes "(Ast, Bst) \<in> code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel"   and "(st, st') \<in> p2rel (\<langle>nat_rel, fully_unsorted_poly_rel O mset_poly_rel, var_rel\<rangle>pac_step_rel_raw)"   and "(spec, spec') \<in> sorted_poly_rel O mset_poly_rel" shows "PAC_checker_l_step spec Ast st \<le> \<Down> (code_status_status_rel \<times>\<^sub>r \<langle>var_rel\<rangle>set_rel \<times>\<^sub>r \<langle>nat_rel, sorted_poly_rel O mset_poly_rel\<rangle>fmap_rel) (PAC_checker_step spec' Bst st')"
RG_aezdeltafun_closed: fixes g :: "'g"   and r :: "'a" assumes "g \<in> G" shows "r \<delta>\<delta> g \<in> ActingGroup.group_ring"
simps_2: fixes P :: "pi" shows "valid (\<tau>.P) = valid P"
ring_distribs_1: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a * (b + c) = a * b + a * c"
assms_2: shows "simplex (mat_leqb_eqc A b c) = Inr X"
Done_eq_DONE: fixes b :: "'f option"   and a :: "'f" shows "(Resumption.resumption.Done b = DONE a) = (b = Some a)"
substEnv_updEnv: fixes ys :: "'varSort"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs &[Y / y]_ys = rho &[Y / y]_ys [x \<leftarrow> X #[Y / y]_ys]_xs"
bernoulli_real_cond_exp_AE: fixes N :: "bool stream measure"   and q :: "real"   and X :: "bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "integrable N X" shows "real_cond_exp N (fct_gen_subalgebra N (stream_space borel) (proj_stoch_proc geom_proc n)) X w = expl_cond_expect N (proj_stoch_proc geom_proc n) X w"
eq: shows "snd (SG ! i) ! j = (p \<leftrightarrow> q)"
Chernoff_on_consistency_on_path_independent_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "(Chernoff_on A f \<and> consistency_on A f) = path_independent_on A f"
changes_pCons: fixes f :: "'a poly" shows "changes (coeffs (pCons (0::'a) f)) = changes (coeffs f)"
bijective_reverse: fixes p :: "'a"   and q :: "'a"   and r :: "'a" assumes "bijective p"   and "bijective q" shows "(p \<le> r * q) = (q \<le> r[[p]])"
in_set_cyc_permutations: fixes xs :: "'b list"   and ys :: "'b list" assumes "distinct xs" shows "(ys \<in> set (cyc_permutations xs)) = (distinct ys \<and> set xs = set ys \<and> hd ys = hd xs)"
bindE_def: fixes f :: "'d \<Rightarrow> (('e + 'g) \<times> 'd) \<times> bool"   and g :: "'g \<Rightarrow> 'd \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool" shows "f >>=E g \<equiv> f >>= lift g"
eval_hom: fixes K :: "'a list set"   and a :: "'a list"   and p :: "'a list list" assumes "subring K (K [X])"   and "a \<in> carrier (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "rupture_surj K p (UP.eval p a) = Hom.eval (map (rupture_surj K p) p) (rupture_surj K p a)"
dim_col_of_cjn_prod: fixes M :: "complex mat"   and N :: "complex mat" shows "dim_col (M\<^sup>\<star> * N\<^sup>\<star>) = dim_col N"
less_eq_acom_def: fixes C1 :: "'a acom"   and C2 :: "'a acom" shows "(C1 \<le> C2) = (strip C1 = strip C2 \<and> (\<forall>p<length (annos C1). anno C1 p \<le> anno C2 p))"
distinct_77: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and pi' :: "pi" shows "[name1\<noteq>name2]pi \<noteq> !pi'"
lQuots_simps_1: fixes A :: "'b list set" shows "lQuots [] A = A"
Match: fixes P :: "pi"   and V :: "residual"   and b :: "name" assumes "P \<longmapsto> V" shows "[b\<frown>b]P \<longmapsto> V"
ord_embed_def: fixes f :: "'b \<Rightarrow> 'c" shows "ord_embed f = (\<forall>x y. (f x \<le> f y) = (x \<le> y))"
SUP_mult_left_ennreal: fixes c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal"   and I :: "'a set" shows "c * Sup (f ` I) = (SUP i\<in>I. c * f i)"
is_greatest_postfixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_greatest_postfixpoint f x \<equiv> x \<le> f x \<and> (\<forall>y. y \<le> f y \<longrightarrow> y \<le> x)"
set_take_drop: fixes xs :: "'a list"   and n :: "nat" shows "set xs = set (take n xs) \<union> set (drop n xs)"
product_def: fixes A :: "'b fset set"   and B :: "'b fset set" shows "A \<otimes> B = {a |\<union>| b |a b. a \<in> A \<and> b \<in> B}"
reduced_word_for_arg_min: fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a" shows "reduced_word_for A a (arg_min length (word_for A a))"
filterlim_def: fixes f :: "'d \<Rightarrow> 'e"   and F2 :: "'e filter"   and F1 :: "'d filter" shows "filterlim f F2 F1 = (filtermap f F1 \<le> F2)"
open_segment_linear_image: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "linear f"   and "inj f" shows "open_segment (f a) (f b) = f ` open_segment a b"
cong_lt_per2__lt: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Per A B C"   and "Per A' B' C'"   and "Cong A B A' B'"   and "B C Lt B' C'" shows "A C Lt A' C'"
fmset_perm_cong: fixes as :: "'a list"   and bs :: "'a list" assumes "as <~~> bs" shows "fmset G as = fmset G bs"
simps_2: fixes n :: "nat"   and c :: "int"   and b :: "MIR.num" shows "split_int (CN n c b) = (let (bv, y) = split_int b in (CN n c bv, y))"
simps: fixes as :: "'c atom set"   and lb :: "nat \<Rightarrow> 'c option"   and ub :: "nat \<Rightarrow> 'c option" shows "as \<doteq> (lb, ub) = (\<forall>v. v \<Turnstile>\<^sub>a\<^sub>s as = v \<Turnstile>\<^sub>b (lb, ub))"
simps_2: fixes x :: "'a"   and y :: "'a"   and p :: "'b" shows "destNet (AllowPortFromTo x y p) = y"
proportionality_def: shows "proportionality \<equiv> {(x, y). x \<noteq> (0::'a) \<and> y \<noteq> (0::'a) \<and> (\<exists>k. x = k *\<^sub>R y)}"
OrdE: fixes v :: "i"   and x :: "\<kappa>" assumes "[\<lparr>O!,x\<rparr> in v]" shows "\<exists>o\<^sub>1 y. Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 = \<omega>\<nu> y"
invalidate: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b" shows "R x y \<Longrightarrow>\<^sub>A invalid_assn R x y"
path_via: shows "P \<turnstile> Path last Cs to C via Cs' "
QQ4_a_def: shows "QQ4_a \<equiv> [: p, t, left, right, atom, mrk \<leadsto> p', t', left', right', atom', mrk' . p \<noteq> nil \<and> gg mrk atom right p \<and> p' = right p \<and> t' = p \<and> mrk' = mrk \<union> {right p} \<and> left' = left \<and> right' = right(p := t) \<and> atom' = atom(p := True) :]"
Notes_imp_used: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Notes A X \<in> set evs" shows "X \<in> used evs"
a_meet_b_eq_d: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "a_aux a b c \<sqinter> b_aux a b c = d_aux a b c"
scaleR_cong_right: fixes x :: "'c"   and r :: "real"   and p :: "real" assumes "x \<noteq> (0::'c) \<Longrightarrow> r = p" shows "r *\<^sub>R x = p *\<^sub>R x"
cong_sym_eq: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "[b = c] (mod a) = [c = b] (mod a)"
uses_in_\<alpha>n: fixes v :: "'a"   and g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b" assumes "v \<in> gen_wf_uses g n" shows "n \<in> set (gen_wf_\<alpha>n g)"
subcls1D: fixes P :: "'a prog"   and C :: "String.literal"   and D :: "String.literal" assumes "P \<turnstile> C \<prec>\<^sup>1 D" shows "C \<noteq> Object \<and> (\<exists>fs ms. class P C = \<lfloor>(D, fs, ms)\<rfloor>)"
Lcm_eqI: fixes a :: "'b"   and A :: "'b set" assumes "normalize a = a"   and "\<And>b. b \<in> A \<Longrightarrow> b dvd a"   and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> b dvd c) \<Longrightarrow> a dvd c" shows "normalization_euclidean_semiring_class.Lcm A = a"
1: fixes u :: "V" assumes "u \<in> elts y" shows "elts (lift (x * u) x) \<approx> elts x"
compls_subls: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" shows "L\<^sup>C \<cdot>\<^sub>l\<^sub>s \<sigma> = (L \<cdot>\<^sub>l\<^sub>s \<sigma>)\<^sup>C"
sdnets2: fixes x :: "('b, 'c) Combinators" assumes "\<exists>a b. sdnets x = {(a, b), (b, a)}"   and "\<not> member DenyAll x" shows "sdnets x = {(first_srcNet x, first_destNet x), (first_destNet x, first_srcNet x)}"
X2_X_polynomial_bool_mult_in: fixes x1 :: "nat"   and p :: "int mpoly" shows "Var x1 * (Var x1 * p) - Var x1 * p \<in> More_Modules.ideal polynomial_bool"
sublist_appendI: fixes xs :: "'b list"   and ps :: "'b list"   and ss :: "'b list" shows "sublist xs (ps @ xs @ ss)"
tendsto_imp_filterlim_at_left: fixes f :: "'a \<Rightarrow> 'b"   and L :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> L) F"   and "\<forall>\<^sub>F x in F. f x < L" shows "filterlim f (at_left L) F"
equivalence_functor_axioms: shows "equivalence_functor (\<bullet>) (\<bullet>) R"
l8_14_2_1b: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and Y :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"   and "Tarski_neutral_dimensionless.Col Bet Y A B"   and "Tarski_neutral_dimensionless.Col Bet Y C D" shows "X = Y"
B2: fixes x :: "'a" assumes "x \<in> B" shows "f (T x) \<le> f x"
quasinorm_subsetI: fixes N1 :: "'c quasinorm"   and N2 :: "'c quasinorm"   and C :: "real" assumes "\<And>f. f \<in> space\<^sub>N N1 \<Longrightarrow> eNorm N2 f \<le> ennreal C * eNorm N1 f" shows "N1 \<subseteq>\<^sub>N N2"
Notes_imp_parts_subset_used: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Notes A X \<in> set evs" shows "parts {X} \<subseteq> used evs"
iso_\<psi>: shows "D.iso ((\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<r>\<^sub>D[g]) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D g \<star>\<^sub>D \<sigma>.counit a) \<cdot>\<^sub>D (\<sigma>.map\<^sub>0 a' \<star>\<^sub>D \<a>\<^sub>D[g, \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a]) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>.map\<^sub>0 a', g \<star>\<^sub>D \<tau>\<^sub>0 a, \<sigma>.map\<^sub>0 a])"
alternativelistconc2: fixes a :: "'b"   and x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators" assumes "a \<in> set (net_list_aux [x])" shows "a \<in> set (net_list_aux [y, x])"
25: shows "prim_invariant t v g r \<and> v * - v\<^sup>T \<sqinter> g = bot"
list_updates_nth_notmem: fixes xs :: "nat list"   and ys :: "'a list"   and i :: "nat"   and vs :: "'a list" assumes "length xs = length ys"   and "i \<notin> set xs" shows "list_updates xs ys vs ! i = vs ! i"
IH_2: assumes "is_cupcake_all_env (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env)"   and "is_cupcake_exp e'" shows "if_rval is_cupcake_value bv"
eq: shows "f ` S \<inter> g -` T = f ` (S \<inter> T)"
terminal_TNil: fixes x1 :: "'f" shows "terminal (TNil x1) = x1"
rational_preference_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" shows "rational_preference carrier relation \<equiv> preference carrier relation \<and> rational_preference_axioms carrier relation"
ltl_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "ltl (xs $ ys) = (if lnull xs then ltl ys else ltl xs $ ys)"
P3: shows "\<forall>x. Abstract x \<longrightarrow> Dependent x"
prime_power_eq_imp_eq: fixes p :: "'a"   and q :: "'a"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "prime q"   and "0 < m"   and "p ^ m = q ^ n" shows "p = q"
poincare_line_circline_set_2: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z \<noteq> w" shows "w \<in> circline_set (poincare_line z w)"
nonempty_rule_def: shows "nonempty_rule = (LG {} {}, LG {} {0})"
plus_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "x + y = (fst x + fst y, snd x + snd y)"
lead_coeff_comp: fixes q :: "'a poly"   and p :: "'a poly" assumes "0 < degree q" shows "lead_coeff (p \<circ>\<^sub>p q) = lead_coeff p * lead_coeff q ^ degree p"
leq_linear: fixes x :: "'b"   and y :: "'b" shows "x \<sqsubseteq> y \<or> y \<sqsubseteq> x"
assms_2: shows "ipresIGSwapAll h hA MOD MOD'"
Units_l_inv_ex: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "\<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
2: fixes g :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "g \<in> G" shows "\<exists>F' q. finite F' \<and> F' \<subseteq> F \<and> g = (\<Sum>f\<in>F'. q f * f) \<and> (\<forall>f. indets (q f) \<subseteq> \<Union> (indets ` F) \<and> poly_deg (q f * f) \<le> b \<and> (f \<notin> F' \<longrightarrow> q f = 0))"
Q_zip: fixes z :: "('a \<times> 'c) H" assumes "Q (map_H fst z)"   and "Q (map_H snd z)" shows "Q z"
Block: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>0 :: "char list exp"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>0 :: "char list \<Rightarrow> val option"   and V :: "char list"   and e\<^sub>1 :: "char list exp"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l\<^sub>1 :: "char list \<Rightarrow> val option"   and T :: "ty" assumes "P \<turnstile> \<langle>e\<^sub>0,(h\<^sub>0, l\<^sub>0(V := None))\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1)\<rangle>" shows "P \<turnstile> \<langle>{V:T; e\<^sub>0},(h\<^sub>0, l\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, l\<^sub>1(V := l\<^sub>0 V))\<rangle>"
existence_ivl_initial_time_iff: fixes t0 :: "real"   and x0 :: "'a" shows "(t0 - t0 \<in> existence_ivl0 x0) = (t0 \<in> UNIV \<and> x0 \<in> X)"
CH_def: fixes X :: "'a set" shows "Bossy''.CH X = (\<Union>h. BCh' h X)"
checked_idx_def: fixes x :: "'k" shows "checked_idx x \<equiv> if in_bound index_class.less_eq index_class.less bound x then idx bound x else local.size"
Catch_1: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate" assumes "(a1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile>a1 \<down> s" shows "\<Gamma>\<turnstile>c \<down> s"
elims_2: fixes x :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "qGoodAbs x" obtains xs :: "'varSort" and x :: "'var" and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm" where "xa__ = qAbs xs x X" and "qGood X"
generalizes_lit_def: fixes L :: "'a literal"   and M :: "'a literal" shows "generalizes_lit L M = (\<exists>\<sigma>. L \<cdot>l \<sigma> = M)"
no_msgs_change_if_no_channel: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "channel i = None" shows "msgs c i = msgs c' i"
insert_subset: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" shows "(Set.insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
inv\<pi>: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A"   and "(\<pi> x, \<pi> y) \<in> s" shows "(x, y) \<in> r"
prems_1: shows "insert \<psi> G \<subseteq> fmla"
subst_floatarith_Var_FDERIV_floatarith: fixes xs :: "nat list"   and ds :: "floatarith list"   and s :: "nat \<Rightarrow> nat"   and fa :: "floatarith" assumes "length xs = DIM('a)"   and "length ds = DIM('a)"   and "\<And>x y. x \<in> set xs \<Longrightarrow> (y = x) = (s y = x)" shows "subst_floatarith (\<lambda>x. Var (s x)) (FDERIV_floatarith fa xs ds) = FDERIV_floatarith (subst_floatarith (\<lambda>x. Var (s x)) fa) xs (map (subst_floatarith (\<lambda>x. Var (s x))) ds)"