hcju/isabelleps huggingface dataset:
method_def2: fixes P :: "'b prog"   and C :: "char list"   and M :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and m :: "'b"   and D :: "char list" assumes "P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "method P C M = (D, b, Ts, T, m)"
antisym: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<preceq> y"   and "y \<preceq> x" shows "x = y"
ln_prod: fixes I :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> 0 < f i" shows "ln (prod f I) = (\<Sum>x\<in>I. ln (f x))"
filterlim_tendsto_add_at_top: fixes f :: "'c \<Rightarrow> real"   and c :: "real"   and F :: "'c filter"   and g :: "'c \<Rightarrow> real" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g at_top F" shows "LIM x F. f x + g x :> at_top"
atoms_imply_bounds_index_mono: fixes as :: "('a \<times> 'b atom) set"   and as' :: "('a \<times> 'b atom) set"   and bi :: "((nat \<Rightarrow> 'b option) \<times> (nat \<Rightarrow> 'b option)) \<times> (nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> 'a)" assumes "as \<subseteq> as'"   and "as \<Turnstile>\<^sub>i bi" shows "as' \<Turnstile>\<^sub>i bi"
has_integral_combine_division: fixes \<D> :: "'b set set"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "'b set \<Rightarrow> 'c" assumes "\<D> division_of S"   and "\<And>k. k \<in> \<D> \<Longrightarrow> (f has_integral i k) k" shows "(f has_integral sum i \<D>) S"
soconf_def: fixes P :: "'b prog"   and h :: "heap"   and C :: "char list"   and sfs :: "sfields" shows "P,h,C \<turnstile>\<^sub>s sfs \<surd> \<equiv> \<forall>F T. P \<turnstile> C has F,Static:T in C \<longrightarrow> (\<exists>v. sfs F = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
zero_neq_one_def: fixes one :: "'a"   and zero :: "'a" shows "class.zero_neq_one one zero \<equiv> zero \<noteq> one"
real_le_rsqrt: fixes x :: "real"   and y :: "real" assumes "x\<^sup>2 \<le> y" shows "x \<le> sqrt y"
thd3_simp: fixes a :: "'c"   and b :: "'d"   and c :: "'b" shows "thd3 (a, b, c) = c"
bisim1FAssNull: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and e2 :: "'addr expr1"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,e\<bullet>F{D} := e2,h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([v, Null], xs, length (compE2 e) + length (compE2 e2), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
hv_inj: fixes x :: "(real, 7) vec"   and y :: "(real, 7) vec" shows "(Hv x = Hv y) = (x = y)"
assms_1: shows "\<guillemotleft>\<phi> : r \<Rightarrow> r'\<guillemotright>"
independent_card_le: fixes S :: "'a set" assumes "independent S" shows "card S \<le> DIM('a)"
is_zero_row_def: fixes i :: "'d"   and A :: "(('e, 'f) vec, 'd) vec" shows "is_zero_row i A = is_zero_row_upt_k i (ncols A) A"
equality: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f"   and "ChamberComplexAutomorphism X g"   and "fun_eq_on f g (\<Union> X)" shows "f = g"
Pw: fixes w :: "W" shows "Preorder (Pw w) \<and> Total (Pw w)"
p2_6: shows "\<Union> {K. \<exists>x. (x, K) \<in> p2} = s2"
has_limits_if_has_products: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "category J"   and "has_products (Collect (partial_magma.ide J))"   and "has_products (Collect (partial_magma.arr J))" shows "has_limits_of_shape J"
orthonormal_system_l2norm: fixes S :: "'a set"   and w :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and i :: "'b" assumes "orthonormal_system S w" shows "l2norm S (w i) = 1"
map_bset_bsingleton: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e" shows "map_bset f (bsingleton x) = bsingleton (f x)"
IH_2: assumes "Oadmit \<sigma> ODE2 U" shows "osafe (Osubst ODE2 \<sigma>)"
simps_17: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x1 :: "('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option" shows "(case \<Up>x1 of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f1 x1"
antisym_conv2: fixes y :: "'d"   and x :: "'d" assumes "y \<le> x" shows "(\<not> y < x) = (x = y)"
Ext: fixes x :: "ZF"   and y :: "ZF" shows "(x = y) = (\<forall>z. z |\<in>| x = z |\<in>| y)"
arr_iff_in_hom: fixes f :: "'a1" shows "C1.arr f = \<guillemotleft>f : C1.dom f \<rightarrow>\<^sub>1 C1.cod f\<guillemotright>"
take_eq_Nil: fixes n :: "nat"   and xs :: "'b list" shows "(xs \<down> n = []) = (n = 0 \<or> xs = [])"
regular_weak_composition_axioms_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "regular_weak_composition_axioms V H \<equiv> (\<forall>a f. a \<in> weak_composition.sources V H f \<longrightarrow> partial_magma.ide V f \<longrightarrow> category.isomorphic V (H f a) f) \<and> (\<forall>b f. b \<in> weak_composition.targets V H f \<longrightarrow> partial_magma.ide V f \<longrightarrow> category.isomorphic V (H b f) f)"
class_of_def: fixes U :: "'a set"   and s :: "'b" shows "stk1.class_of U s \<equiv> e1.Class (U, s)"
in_hom: shows "\<guillemotleft>s : s \<Rightarrow> s\<guillemotright>"
reconstruct_poly_monic_defines_mset: fixes as :: "'a list"   and bs :: "'a list" assumes "(\<Prod>a\<leftarrow>as. [:- a, 1::'a:]) = (\<Prod>a\<leftarrow>bs. [:- a, 1::'a:])" shows "mset as = mset bs"
one_step_equiv: fixes N1 :: "('a literal multiset \<times> label) set"   and N2 :: "('a literal multiset \<times> label) set" assumes "N1 \<leadsto>GC N2" shows "N1 \<rhd>L N2"
mat_Ph_def: shows "mat_Ph = mat N N (\<lambda>(i, j). if i = j then if i = 0 then 1 else - 1 else 0)"
assoc_is_natural_1: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a>[\<mu>, \<nu>, \<tau>] = (\<mu> \<star> \<nu> \<star> \<tau>) \<cdot> \<a>[local.dom \<mu>, local.dom \<nu>, local.dom \<tau>]"
C: fixes a :: "'a" shows "a \<le> (1::'a) l\<rightarrow> a"
sub1: shows "P \<turnstile> C1 \<prec>\<^sup>1 D"
B44: shows "B4 = four_block_mat (mat m m (($$) B4)) (0\<^sub>m m 1) (mat 1 m (\<lambda>(i, j). B4 $$ (m, j))) (mat 1 1 (\<lambda>_. B4 $$ (m, m)))"
xeq: shows "x = (\<Sum>t\<in>insert a T. c t *\<^sub>R t)"
checker_r1_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "checker_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> \<equiv> let distance = s\<^sub>o - s\<^sub>e; precond = check_precond_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta>; vo_star = v\<^sub>o + a\<^sub>o * \<delta>; t_stop_o_star = - vo_star / a\<^sub>o; t_stop_e = - v\<^sub>e / a\<^sub>e; safe_dist0 = safe_distance_1r a\<^sub>e v\<^sub>e \<delta>; safe_dist1 = safe_distance_2r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta>; safe_dist2 = safe_distance_4r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta>; safe_dist3 = safe_distance_3r a\<^sub>e v\<^sub>e a\<^sub>o v\<^sub>o \<delta> in if \<not> precond then False else if safe_dist0 < distance \<or> safe_dist3 < distance then True else if a\<^sub>e < a\<^sub>o \<and> vo_star < v\<^sub>e \<and> t_stop_e < t_stop_o_star then safe_dist2 < distance else safe_dist1 < distance"
tl_append: fixes xs :: "'d list"   and ys :: "'d list" shows "tl (xs @ ys) = (case xs of [] \<Rightarrow> tl ys | z # zs \<Rightarrow> zs @ ys)"
produces_imp_in_interp: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and A :: "'b"   and C :: "'b literal multiset"   and N :: "'b literal multiset set"   and D :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "Neg A \<in># C"   and "ground_resolution_with_selection.production S N D = {A}" shows "A \<in> ground_resolution_with_selection.interp S N C"
simps_13: fixes v :: "\<I>"   and va :: "formula" shows "get_and_list (Next v va) = [Next v va]"
\<Z>_srngD_2: fixes \<alpha> :: "V"   and \<SS> :: "V" assumes "\<Z>_srng \<alpha> \<SS>" shows "vcard \<SS> = 3\<^sub>\<nat>"
hyps_3: assumes "F \<subseteq> \<Union> (range f)" shows "\<exists>n. F \<subseteq> f n"
mult_zero_uniq: fixes x :: "'a"   and a :: "real" assumes "x \<in> V"   and "x \<noteq> (0::'a)"   and "a \<cdot> x = (0::'a)" shows "a = 0"
CAST: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP" assumes "G \<rhd> c : twiddle" shows "G \<rhd> c : Sec (\<lambda>(x, y).  x \<approx> y )"
div_cancel_Suc: fixes c :: "nat"   and a :: "nat"   and b :: "nat" assumes "c dvd a"   and "1 < c" shows "Suc (a * b) div c = a div c * b"
pmdl_struct: fixes sel :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "'h \<times> nat \<times> 'g"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and aux :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data' :: "nat \<times> 'g" assumes "struct_spec sel ap ab compl"   and "compl_pmdl compl"   and "is_Groebner_basis (fst ` set gs)"   and "ps \<noteq> []"   and "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"   and "unique_idx (gs @ bs) (snd data)"   and "sps = sel gs bs ps (snd data)"   and "aux = compl gs bs (ps -- sps) sps (snd data)"   and "(hs, data') = add_indices aux (snd data)" shows "pmdl (fst ` set (gs @ ab gs bs hs data')) = pmdl (fst ` set (gs @ bs))"
times_enat_simps_1: fixes m :: "nat"   and n :: "nat" shows "enat m * enat n = enat (m * n)"
plain_floatarith_not_None: fixes N :: "nat"   and fa :: "floatarith"   and XS :: "((real \<times> real pdevs) \<times> real) list"   and p :: "nat" assumes "plain_floatarith N fa"   and "N \<le> length XS"   and "list_all (\<lambda>((a, b), c). is_float a \<and> is_float c) XS" shows "approx_floatarith p fa XS \<noteq> None"
eval_vmuaux_def: shows "eval_vmuaux = (\<lambda>args nt (t, auxlist). let (res, auxlist') = eval_until (args_ivl args) nt auxlist in (res, t, auxlist'))"
h_2: fixes x :: "nat" shows "(x \<in> B) = (\<exists>y. h x y = 0)"
simps: fixes to :: "(('a, 'b) pp \<times> nat) nat_term_order"   and bs :: "((('a, 'b) pp \<times> nat) \<times> (('a, 'b) pp \<Rightarrow>\<^sub>0 'c)) list"   and v :: "('a, 'b) pp \<times> nat"   and p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "sig_trd_spp_pprod to bs (v, p) = (v, sig_trd_spp_aux_pprod to bs v (p, 0))"
composite_simps_1: shows "src (\<mu> \<star> \<nu> \<star> \<pi>) = src \<pi>"
wf_add: fixes M :: "('c, 'd) insert_message set"   and m :: "('c, 'd) insert_message" assumes "wfP (depends_on M)"   and "\<And>n. n \<in> M \<union> {m} \<Longrightarrow> I m \<notin> deps (Insert n)"   and "m \<notin> M" shows "wfP (depends_on (M \<union> {m}))"
rel_with_source_impl_SRel_weakly_reflects_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and SRel :: "('procS \<times> 'procS) set" assumes "rel_weakly_reflects_barbs Rel (STCalWB SWB TWB)"   and "\<forall>S1 S2. (S1, S2) \<in> SRel \<longrightarrow> (SourceTerm S1, SourceTerm S2) \<in> Rel" shows "rel_weakly_reflects_barbs SRel SWB"
segment_sub: fixes a :: "'a" shows "segment D a \<subseteq> carrier D"
snd_quot_of_fract_Fract_whole: fixes y :: "'a"   and x :: "'a" assumes "y dvd x" shows "snd (quot_of_fract (Fraction_Field.Fract x y)) = (1::'a)"
le_less: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
chambersub_imp_sub: fixes Y :: "'a set set" assumes "ChamberSubcomplex Y" shows "order.greater_eq X Y \<and> SimplicialComplex Y"
SourcesS2_L2: shows "Sources level2 sS2 = {}"
span_subspace_induct: fixes x :: "'a"   and S :: "'a set"   and P :: "'a set" assumes "x \<in> More_Modules.ideal S"   and "ideal.subspace P"   and "\<And>x. x \<in> S \<Longrightarrow> x \<in> P" shows "x \<in> P"
dims_nths_le_n: fixes l :: "nat" assumes "l \<le> n" shows "nths dims {0..<l} = replicate l 2"
sq_mtx_vec_mult_def: shows "(*\<^sub>V) \<equiv> map_fun to_vec id (*v)"
ord0_less_le_trans: fixes x :: "ord0"   and y :: "ord0"   and z :: "ord0" assumes "x < y"   and "y \<le> z" shows "x < z"
IH_3: shows "tls1 [\<sim>] tls2"
UUinvU: shows "U o\<^sub>C\<^sub>L Uinv o\<^sub>C\<^sub>L U = U"
d: shows "d ` Ws \<subseteq> {x. (0::'a) \<le> x}"
intros_6: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<forall>s. P' s \<longrightarrow> P s"   and "\<turnstile> {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile> {P'} c {Q'}"
resrI: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y \<le> z" shows "y \<le> x \<rightarrow> z"
norm_eq_iff: fixes x :: "int"   and y :: "int" shows "(take_bit LENGTH('b) x = take_bit LENGTH('b) y) = (word_of_int x = word_of_int y)"
EH\<^sub>N_def: fixes n :: "nat" shows "EH\<^sub>N n = measure_pmf.expectation (H\<^sub>N n) real"
ee_length: fixes as :: "'a list"   and bs :: "'a list" assumes "essentially_equal G as bs" shows "length as = length bs"
eval_ternary_Not_UnknownD: fixes t :: "ternaryvalue" assumes "eval_ternary_Not t = TernaryUnknown" shows "t = TernaryUnknown"
not_subtype_Optional_Required: fixes \<tau> :: "'b basic_type"   and \<sigma> :: "'b OCL_Types.type"   and \<rho> :: "'b basic_type"   and P :: "bool" assumes "(\<sqsubset>)\<^sup>+\<^sup>+ \<tau>[?] \<sigma>"   and "\<sigma> = \<rho>[1]" shows "P"
bilinear_form_hermitean_commute: fixes H :: "complex \<times> complex \<times> complex \<times> complex"   and v1 :: "complex \<times> complex"   and v2 :: "complex \<times> complex" assumes "hermitean H" shows "bilinear_form v1 v2 H = cnj (bilinear_form v2 v1 H)"
distinct_12: fixes x3 :: "'b formula" shows "\<^bold>\<not> x3 \<noteq> \<bottom>"
leftderives_padback: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list" assumes "leftderives \<alpha> \<beta>"   and "is_sentence u" shows "leftderives (\<alpha> @ u) (\<beta> @ u)"
simps_1: fixes ys :: "'b list" shows "splice [] ys = ys"
A: shows "\<forall>\<^sub>F n in sequentially. (\<Sum>k<n. z / of_nat (Suc k) - Ln (1 + z / of_nat (Suc k))) - (ln_Gamma_series z n + complex_of_real (harm n - ln (real n)) * z + Ln z) = 0"
just_show_all_bigstep_semantics_equalities_with_start_Undecided: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs1 :: "'c rule list"   and t :: "state"   and rs2 :: "'c rule list"   and s :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs1, Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs2, Undecided\<rangle> \<Rightarrow> t" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs1, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs2, s\<rangle> \<Rightarrow> t"
induct: fixes x1 :: "'b"   and x2 :: "'b"   and r :: "('b \<times> 'b) set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "(x1, x2) \<in> r\<^sup>+"   and "\<And>a b. (a, b) \<in> r \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>(a, b) \<in> r\<^sup>+; P a b; (b, c) \<in> r\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
greek_less_unfold: fixes r :: "('a \<times> 'a) set" shows "greek_less r = nest r (greek_less r)"
eq1: shows "Suc (card X - Suc 0) = card X"
log_ln: fixes x :: "real" shows "real_ln x = log (exp 1) x"
normalize_mult_unit_right: fixes b :: "'a"   and a :: "'a" assumes "is_unit b" shows "normalize (a * b) = normalize a"
BijGroup_def: fixes S :: "'e set" shows "BijGroup S = \<lparr>carrier = Bij S, mult = \<lambda>g\<in>Bij S. restrict (compose S g) (Bij S), one = \<lambda>x\<in>S. x\<rparr>"
CoxeterSystem: shows "CoxeterSystem (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
prj_in_homs_5: shows "\<guillemotleft>\<p>\<^sub>1[HH\<mu>\<nu>\<pi>.leg0, \<rho>.leg1] : HHH\<mu>\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C HH\<mu>\<nu>\<pi>.chine\<guillemotright>"
other: fixes \<sigma> :: "'a"   and p :: "'b"   and \<sigma>' :: "'a" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "P \<sigma>"   and "U \<sigma> \<sigma>'" shows "P \<sigma>'"
fold_congs_6: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "Literal list"   and f :: "Literal list \<Rightarrow> Literal list"   and f' :: "Literal list \<Rightarrow> Literal list" assumes "r = r'"   and "getQ r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "getQ_update f r = getQ_update f' r'"
iter_narrow_def: fixes f :: "'a \<Rightarrow> 'a" shows "iter_narrow f = while_option (\<lambda>x. x \<triangle> f x < x) (\<lambda>x. x \<triangle> f x)"
it_dfs_def: fixes G :: "('a, 'c) graph_rec_scheme"   and param :: "('a, ('a, 'b) state_scheme, 'b) gen_parameterization" shows "param_DFS_defs.it_dfs G param \<equiv> param_DFS_defs.init param \<bind> WHILE (param_DFS_defs.cond G param) (param_DFS_defs.step G param)"
cmp_fun_image: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "cmp g f ` A = g ` f ` A"
bigE: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" obtains c :: "real" where "0 < c" and "\<forall>\<^sub>F x in F. R (norm (f x)) (c * norm (g x))"
trans: fixes a :: "('d, 'e) Term.term"   and b :: "('d, 'e) Term.term"   and c :: "('d, 'e) Term.term" assumes "a \<sqsubseteq> b"   and "b \<sqsubseteq> c" shows "a \<sqsubseteq> c"
eventually_ge_at_top: fixes c :: "'e" shows "eventually ((\<le>) c) at_top"
Top_in_Prof: fixes P :: "indi \<Rightarrow> (alt \<times> alt) set"   and S :: "alt set" assumes "P \<in> Prof" shows "Top S \<circ> P \<in> Prof"
cMax_finite_ex: fixes X :: "'a set" assumes "finite X"   and "X \<noteq> {}" shows "\<exists>x\<in>X. Max X = x"
Abs_fmap_inverse: fixes y :: "'a \<Rightarrow> 'b option" assumes "y \<in> {m. finite (dom m)}" shows "fmlookup (map_of_fmap y) = y"
22: shows "padic_mult p f g \<in> padic_set p"
simps_1: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" shows "scan f [] a = []"
phi_def: fixes g :: "'g"   and v :: "'val" shows "step.phi g v = p_g g (step.defNode g v, v)"
one_time_pad: fixes n :: "nat"   and xs :: "'h list" assumes "n \<le> length xs" shows "map_spmf (([\<oplus>]) xs) (spmf_of_set (nlists UNIV n)) = spmf_of_set (nlists UNIV n)"
simps_1: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and x :: "'a" shows "lmember\<cdot>eq\<cdot>x\<cdot>lnil = FF"
hoare_tripleD_1: fixes P :: "assn"   and c :: "'a Heap"   and Q :: "'a \<Rightarrow> assn"   and h :: "heap"   and as :: "nat set"   and \<sigma> :: "heap option"   and r :: "'a" assumes "<P> c <Q>"   and "(h, as) \<Turnstile> P"   and "run c (Some h) \<sigma> r" shows "\<not> is_exn \<sigma>"
list_all2_append: fixes xs :: "'b list"   and ys :: "'c list"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and us :: "'b list"   and vs :: "'c list" assumes "length xs = length ys" shows "list_all2 P (xs @ us) (ys @ vs) = (list_all2 P xs ys \<and> list_all2 P us vs)"
inject_12: fixes x121 :: "expr"   and x122 :: "expr"   and y121 :: "expr"   and y122 :: "expr" shows "(x121;; x122 = y121;; y122) = (x121 = y121 \<and> x122 = y122)"
assms_1: shows "m0 \<in> carrier (\<G> \<eta>)"
restrict0_scaleR: fixes A :: "'c set"   and c :: "real"   and f :: "'c \<Rightarrow> 'd" shows "restrict0 A (c *\<^sub>R f) = c *\<^sub>R restrict0 A f"
inv_in_hom: fixes f :: "'d"   and a :: "'d"   and b :: "'d" assumes "D.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>D b\<guillemotright>" shows "\<guillemotleft>D.inv f : b \<Rightarrow>\<^sub>D a\<guillemotright>"
hrealpow_two_le_add_order: fixes u :: "real star"   and v :: "real star" shows "0 \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"
ch_def: fixes X :: "'a set" shows "ch X \<equiv> \<exists>f. ch_by_ord f X"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case False of True \<Rightarrow> f1 | False \<Rightarrow> f2) = f2"
fbox_def: fixes x :: "'a"   and y :: "'a" shows "antidomain_semiringl.fbox a (\<cdot>) x y = a (x \<cdot> a y)"
continuous_cblinfun_matrix: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b \<Rightarrow>\<^sub>C\<^sub>L 'c"   and j :: "'b"   and i :: "'c" assumes "continuous F f" shows "continuous F (\<lambda>x. cblinfun_apply (f x) j \<bullet>\<^sub>C i)"
more_permissive_def: fixes p :: "'b \<Rightarrow> 'c decision option"   and q :: "'b \<Rightarrow> 'c decision option" shows "p \<sqsubseteq>\<^sub>A q = (\<forall>x. case q x of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> \<exists>z. p x = \<lfloor>allow z\<rfloor> | \<lfloor>deny y\<rfloor> \<Rightarrow> True)"
generalized: fixes t :: "Term.term"   and val :: "vals" assumes "prg Env\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (val, s1)"   and "Env\<turnstile>t\<Colon>Inl (PrimT Boolean)"   and "\<exists>expr. t = In1l expr" shows "assigns_if (the_Bool (the_In1 val)) (the_In1l t) \<subseteq> dom (locals (snd s1))"