hcju/isabelleps huggingface dataset:
lim_ereal: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and net :: "'a filter" shows "((\<lambda>n. ereal (f n)) \<longlongrightarrow> ereal x) net = (f \<longlongrightarrow> x) net"
interval_lebesgue_integrable_cong_AE: fixes f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and a :: "ereal"   and b :: "ereal" assumes "f \<in> borel_measurable lborel"   and "g \<in> borel_measurable lborel"   and "AE x\<in>einterval (min a b) (max a b) in lborel. f x = g x" shows "interval_lebesgue_integrable lborel a b f = interval_lebesgue_integrable lborel a b g"
lexn_length: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set"   and n :: "nat" assumes "(xs, ys) \<in> lexn r n" shows "length xs = n \<and> length ys = n"
globalsimp: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "(nat \<Rightarrow> 'd) \<times> 'e" shows "global P s = P (fst s)"
saturated_def: fixes N :: "'a literal multiset set" shows "saturated N = (concls_of (inferences_from N) \<subseteq> N)"
IH: fixes V :: "nat set" assumes "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. cexpr_sem \<sigma> e) \<in> state_measure V (\<lambda>a. case a of 0 \<Rightarrow> t | Suc x \<Rightarrow> \<Gamma> x) \<rightarrow>\<^sub>M stock_measure REAL"
1: fixes type :: "String.literal"   and y :: "String.literal \<Rightarrow> String.literal set list option" assumes "exe_osig_conds sa"   and "trans (fst (translate_osig sa))"   and "snd (translate_osig sa) type = Some y" shows "\<exists>y. (type, y) \<in> set tao"
add_vert_def: fixes G :: "('c, 'd) pre_digraph"   and v :: "'c" shows "pre_digraph.add_vert G v = \<lparr>verts = insert v (verts G), arcs = arcs G, tail = tail G, head = head G\<rparr>"
not_prime_0: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "\<not> normalization_semidom.prime zero times one normalize zero"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set"   and index :: "int" assumes "K_block_design point_set block_collection sizes"   and "GDD point_set block_collection groups index" shows "K_\<Lambda>_GDD point_set block_collection sizes groups index"
uniformity_riemann_sphere_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
inj_enumerate: fixes S :: "'a set" assumes "infinite S" shows "inj (enum S)"
\<Phi>_simp: fixes yx :: "'d \<times> 'c" assumes "\<Phi>\<Psi>.DopxC.ide yx" shows "\<phi>\<psi>.\<Phi> yx = \<Phi>\<Psi>.S.mkArr (\<Phi>\<Psi>.HomC.set (F (fst yx), snd yx)) (\<Phi>\<Psi>.HomD.set (fst yx, G (snd yx))) (\<phi>\<psi>.inD \<circ> \<phi> (fst yx) \<circ> \<Phi>\<Psi>.\<psi>C (F (fst yx), snd yx))"
emeasure_PiM: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (count_space (B i))" shows "emeasure (Pi\<^sub>M I (\<lambda>i. count_space (B i))) (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (count_space (B i)) (A i))"
healthy_scalingD: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "scaling t"
cong_inner_transitivity: shows "\<forall>a b p q r s. Cong a b p q \<and> Cong a b r s \<longrightarrow> Cong p q r s"
ge_prv_neg_eql_pls_Num_zer: fixes t :: "'trm"   and k :: "nat"   and m :: "nat" assumes "t \<in> atrm"   and "k < m" shows "prv (neg (eql (pls t (Num m)) (Num k)))"
in_keys_iff: fixes x :: "'a"   and s :: "'a \<Rightarrow>\<^sub>0 'b" shows "(x \<in> Poly_Mapping.keys s) = (poly_mapping.lookup s x \<noteq> (0::'b))"
top_rule_def: fixes t :: "'b" shows "top_rule t = (LG {} {0, 1}, LG {(t, 0, 1)} {0, 1})"
fUnionM_funion: fixes A :: "'m fset"   and B :: "'m fset" shows "fUnionM (A |\<union>| B) = funionM (fUnionM A) (fUnionM B)"
axioms_2: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme" assumes "vectorspace K V" shows "field K"
fmap_inf: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b formula"   and y :: "'b formula" shows "fmap f (x \<sqinter> y) = fmap f x \<sqinter> fmap f y"
dense: fixes l :: "real"   and y :: "real" assumes "\<forall>y\<in>{l<..<x}. y \<notin> L"   and "l < x"   and "l < y"   and "y \<le> x" shows "P y"
lepoll_trans: fixes A :: "'b set"   and B :: "'c set"   and C :: "'d set" assumes "A \<lesssim> B"   and "B \<lesssim> C" shows "A \<lesssim> C"
dirichlet_prod_commutes: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "f \<star> g = g \<star> f"
connected_Icc: fixes a :: "'a"   and b :: "'a" shows "connected {a..b}"
f: assumes "\<not> x < y in xs" shows "y < x in Lxy xs {x, y}"
d_OUT_def: fixes g :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" shows "d_OUT g x = (\<Sum>\<^sup>+ y. g (x, y))"
lc_punit_def: fixes cmp_term :: "'a nat_term_order" shows "lc_punit cmp_term \<equiv> ordered_term.lc fst (le_of_nat_term_order cmp_term)"
right_neutral: fixes a :: "'a hrel" shows "a ;; II = a"
simps_2: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a"   and e :: "'b"   and t :: "'b list" shows "path SES s1 (e # t) = (if \<exists>s2. s1 e\<longrightarrow>\<^bsub>SES\<^esub> s2 then path SES (the (T\<^bsub>SES\<^esub> s1 e)) t else None)"
assms: shows "c1 @ qs # c2 = c1' @ qs' # c2'"
cr_real_eq: shows "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
A2: shows "A2 \<in> carrier_mat (dim_row A - 1) (dim_col A)"
igWlsAbsIsInBar_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gWlsAbsIsInBar MOD" shows "igWlsAbsIsInBar (fromMOD MOD)"
theFTOne_length_finished: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and n :: "nat"   and cl :: "('c, 'd, 'e) cmd list"   and dl :: "('f, 'g, 'h) cmd list" assumes "PL_Indis cval indis"   and "n \<in> theFTOne cl dl" shows "n < length cl \<and> finished (cl ! n) \<or> n < length dl \<and> finished (dl ! n)"
saturated_def: fixes N :: "'f set" shows "no_labels.empty_ord.saturated N = (no_labels.Inf_from N \<subseteq> no_labels.Red_I_\<G> N)"
a_def: shows "a = 8295936325956147794769600190539918304 / 2626685325478320010006427764892578125"
\<psi>_ide_simp: fixes f' :: "'a" assumes "\<guillemotleft>f' : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>"   and "ide f'" shows "\<psi> f' = \<r>[f'] \<cdot> \<l>[f' \<star> trg e\<^sub>0] \<cdot> (\<epsilon>\<^sub>1 \<star> f' \<star> \<epsilon>\<^sub>0) \<cdot> ((e\<^sub>1 \<star> d\<^sub>1) \<star> \<a>[f', e\<^sub>0, d\<^sub>0]) \<cdot> \<a>\<^sup>-\<^sup>1[e\<^sub>1, d\<^sub>1, (f' \<star> e\<^sub>0) \<star> d\<^sub>0] \<cdot> (e\<^sub>1 \<star> \<a>[d\<^sub>1, f' \<star> e\<^sub>0, d\<^sub>0])"
distinct_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "[] \<noteq> Cons x21 x22"
Cons_1: assumes "sorted_wrt r xs" shows "sorted_wrt r (insort_wrt r x xs)"
finsum_vec: fixes n :: "nat" shows "finsum_vec TYPE('c) n = finsum (module_vec TYPE('c) n)"
Var: fixes env :: "nat \<Rightarrow> LambdaType.type"   and x :: "nat"   and T :: "LambdaType.type" assumes "env x = T" shows "env \<turnstile> Var x : T"
nprt_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.nprt zero inf x = inf x zero"
Gromov_completion_rel_to_const: fixes u :: "nat \<Rightarrow> 'b"   and x :: "'b"   and n :: "nat" assumes "Gromov_completion_rel u (\<lambda>n. x)" shows "u n = x"
sp_ends_in_tgt_imp_mem: fixes e :: "'c edge"   and g :: "('c, 'd) rgraph_scheme"   and v :: "'c"   and es :: "'c edge list" assumes "tgt e \<notin> Graph.vertices g"   and "v \<noteq> tgt e"   and "Graph.subpath (add_edge g e) v es (tgt e)" shows "e \<in> set es"
vunionCI: fixes x :: "V"   and B :: "V"   and A :: "V" assumes "x \<notin>\<^sub>\<circ> B \<Longrightarrow> x \<in>\<^sub>\<circ> A" shows "x \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B"
sum_norm_le: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (comm_monoid_add.sum plus zero f S) \<le> sum g S"
vec_of_ell2_timesScalarVec: fixes a :: "complex"   and \<psi> :: "'a ell2" shows "vec_of_ell2 (a *\<^sub>C \<psi>) = a \<cdot>\<^sub>v vec_of_ell2 \<psi>"
dghm_ObjMap_vsv: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "vsv (\<FF>\<lparr>ObjMap\<rparr>)"
ccpo_Sup_upper: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and x :: "'c" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "x \<in> A" shows "less_eq x (Sup A)"
flatten_matrix_def: fixes A :: "'b mat"   and k :: "nat" shows "flatten_matrix A k = A $$ (k div dim_col A, k mod dim_col A)"
AxiomP_fresh_iff: fixes a :: "atom"   and x :: "tm" shows "a \<sharp> AxiomP x = a \<sharp> x"
CombineStrip_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and n :: "nat" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P strip_guards (- F) c UNIV,UNIV" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/{}\<^esub> P c Q,A"
compile_load: fixes p1 :: "('fenv_std, 'b, 'fun) prog"   and p2 :: "('fenv_inca, 'b, 'fun) prog"   and s2 :: "('fenv_inca, 'b, ('fun, 'c) frame) state" assumes "compile p1 = Some p2"   and "Sinca.load p2 s2" shows "\<exists>s1. Sstd.load p1 s1 \<and> s1 \<sim> s2"
intros_1: fixes ars :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "seqp ars a []"
exe_trap_st_pc_def: fixes uu :: "unit" shows "exe_trap_st_pc uu \<equiv> do annul \<leftarrow> gets annul_val; pc_val \<leftarrow> gets (cpu_reg_val PC); npc_val \<leftarrow> gets (cpu_reg_val nPC); curr_win \<leftarrow> get_curr_win (); if annul = False then do write_reg pc_val curr_win (word_of_int 17); write_reg npc_val curr_win (word_of_int 18); return () od else do write_reg npc_val curr_win (word_of_int 17); write_reg (npc_val + 4) curr_win (word_of_int 18); set_annul False; return () od od"
subst_atoms_simp: fixes k :: "'b"   and G :: "'b formula"   and F :: "'b formula" shows "atoms (F[G/k]) = atoms F - {k} \<union> (if k \<in> atoms F then atoms G else {})"
square_eq_iff: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<otimes> x = y \<otimes> y) = (x = y \<or> x = \<ominus> y)"
linear: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> \<F> \<and> T \<in> \<F>" shows "S \<subseteq> T \<or> T \<subseteq> S"
cong_bigtheta: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> \<Theta>[F](g)" shows "\<Theta>[F](f) = \<Theta>[F](g)"
pivot_funI: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat" assumes "dim_row A = nr"   and "\<And>i. i < nr \<Longrightarrow> f i \<le> nc"   and "\<And>i j. \<lbrakk>i < nr; j < f i\<rbrakk> \<Longrightarrow> A $$ (i, j) = (0::'c)"   and "\<And>i. \<lbrakk>i < nr; Suc i < nr\<rbrakk> \<Longrightarrow> f i < f (Suc i) \<or> f (Suc i) = nc"   and "\<And>i. \<lbrakk>i < nr; f i < nc\<rbrakk> \<Longrightarrow> A $$ (i, f i) = (1::'c)"   and "\<And>i i'. \<lbrakk>i < nr; f i < nc; i' < nr; i' \<noteq> i\<rbrakk> \<Longrightarrow> A $$ (i', f i) = (0::'c)" shows "pivot_fun A f nc"
distinct_33: fixes x4 :: "nat"   and x61 :: "msg"   and x62 :: "msg" shows "Key x4 \<noteq> \<lbrace>x61, x62\<rbrace>"
f_Exec_Stream_length: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and c :: "'a" shows "length (f_Exec_Comp_Stream trans_fun xs c) = length xs"
CallThrowObj: fixes e :: "char list exp"   and M :: "char list"   and es :: "char list exp list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P \<turnstile> \<langle>throw e\<bullet>M(es),s\<rangle> \<rightarrow> \<langle>throw e,s\<rangle>"
U'_eq_suminf: fixes x :: "'s"   and y :: "'s" assumes "recurrent x"   and "(x, y) \<in> acc" shows "U' x y = (\<Sum>i. ennreal (u x y i * real (Suc i)))"
le_divide_eq_1_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less_eq one (divide b a) = less_eq b a"
rep_eq: fixes x :: "('a, 'b) lifted_pair"   and xa :: "('a, 'b) lifted_pair" shows "Rep_lifted_pair (x \<sqinter> xa) = (\<lambda>f. triple.pairs_inf (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
set_less_eq_aux''_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<sqsubseteq>''' B"   and "B \<sqsubseteq>''' C" shows "A \<sqsubseteq>''' C"
mult_right_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times a c) (times b c)"
isSucc_succ: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}" shows "isSucc (succ i)"
avg_def: fixes s :: "'s" shows "avg s \<equiv> case avg_aux s of (c, s) \<Rightarrow> s div c"
nhds_LNil: shows "nhds LNil = {A\<^sup>\<infinity>}"
intros_5: fixes b :: "bexp"   and A :: "char list set"   and c :: "com"   and A' :: "char list set" assumes "vars b \<subseteq> A"   and "D A c A'" shows "D A (WHILE b DO c) A"
GodNecExists_v2: shows "universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x)))"
uint_of_uint8_def: shows "uint_of_uint8 \<equiv> map_fun Rep_uint8 Abs_uint ucast"
dghm_proj_ObjMap_app: fixes x :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "x \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>" shows "\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ObjMap\<rparr>\<lparr>x\<rparr> = x\<lparr>i\<rparr>"
wp_xs: shows "wellformed_policy2 (removeShadowRules2 xs)"
Q_initQ_def: shows "Q_initQ \<equiv> Q_update s 0 Q_empty"
d: shows "p mod [:- c, 1::'a:] = [:d:]"
f_expand_drop_mod: fixes n :: "nat"   and k :: "nat"   and xs :: "'d message_af list" assumes "n mod k = 0" shows "xs \<odot> k \<up> n = xs \<up> (n div k) \<odot> k"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>ys. P [] ys"   and "\<And>v va. P (v # va) []"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
inside_no_overlap: fixes S :: "'b set" shows "inside S \<inter> S = {}"
SUP_cong: fixes A :: "'b::type set"   and B :: "'b::type set"   and C :: "'b::type \<Rightarrow> 'c::type"   and D :: "'b::type \<Rightarrow> 'c::type"   and Sup :: "'c::type set \<Rightarrow> 'c::type" assumes "(A::'b::type set) = (B::'b::type set)"   and "\<And>x::'b::type. x \<in> (B::'b::type set) \<Longrightarrow> (C::'b::type \<Rightarrow> 'c::type) x = (D::'b::type \<Rightarrow> 'c::type) x" shows "(Sup::'c::type set \<Rightarrow> 'c::type) ((C::'b::type \<Rightarrow> 'c::type) ` (A::'b::type set)) = Sup ((D::'b::type \<Rightarrow> 'c::type) ` (B::'b::type set))"
empty: shows "ordered_powerprod_lin.sorted_list_of_set {} = []"
MGFn_Init: fixes n :: "nat"   and G :: "prog"   and A :: "state triple set"   and C :: "qtname" assumes "\<forall>m. Suc m \<le> n \<longrightarrow> (\<forall>t. G,A\<turnstile>{=:m} t\<succ> {G\<rightarrow>})" shows "G,A\<turnstile>{=:n} In1r (Init C)\<succ> {G\<rightarrow>}"
divide_floatarith_def: fixes f :: "floatarith"   and g :: "floatarith" shows "f / g = f * Inverse g"
tabs_2: shows "table (args_n args) (args_R args) rel2"
cinner_minus_left: fixes x :: "'a"   and y :: "'a" shows "cinner (- x) y = - cinner x y"
has_subprob_density_distr_Snd: fixes M :: "val measure"   and t1 :: "pdf_type"   and t2 :: "pdf_type"   and f :: "val \<Rightarrow> ennreal" assumes "has_subprob_density M (stock_measure (PRODUCT t1 t2)) f" shows "has_subprob_density (distr M (stock_measure t2) (op_sem Snd)) (stock_measure t2) (\<lambda>y. \<integral>\<^sup>+ x. f <|x, y|> \<partial>stock_measure t1)"
i: fixes vl :: "'value list"   and vl1 :: "'value list" assumes "B vl vl1" shows "\<Delta>0 istate vl istate vl1"
assms_2: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
subring_def: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "subring S R addition multiplication zero unit \<equiv> subgroup S R addition zero \<and> submonoid S R multiplication unit"
comp_simp: fixes g :: "'b1 \<times> 'b2"   and f :: "'b1 \<times> 'b2" assumes "B1.seq (fst g) (fst f)"   and "B2.seq (snd g) (snd f)" shows "g \<cdot>\<^sub>B\<^sub>1\<^sub>x\<^sub>B\<^sub>2 f = (fst g \<cdot>\<^sub>B\<^sub>1 fst f, snd g \<cdot>\<^sub>B\<^sub>2 snd f)"
simps_2: fixes dm :: "nat"   and lm :: "nat"   and d :: "nat"   and \<alpha> :: "(nat \<times> int) list \<Rightarrow> real" shows "updown' dm lm (Suc d) \<alpha> = sum_vector (updown' dm lm d (up dm lm d \<alpha>)) (down dm lm d (updown' dm lm d \<alpha>))"
emb_step_under_args_emb: fixes p :: "dir list"   and t :: "('c, 'd) tm"   and d :: "dir" assumes "\<not> list_all (\<lambda>x. x = dir.Left) p"   and "position_of t (p @ [d])" shows "\<forall>i<num_args t. args t ! i \<unrhd>\<^sub>e\<^sub>m\<^sub>b args (emb_step_at p d t) ! i"
sdrop_add: fixes n :: "nat"   and m :: "nat"   and s :: "'a stream" shows "sdrop n (sdrop m s) = sdrop (m + n) s"
if_dens_det_def: fixes \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and b :: "bool" shows "if_dens_det \<delta> e b \<equiv> \<lambda>\<sigma>. \<delta> \<sigma> * (if expr_sem_rf \<sigma> e = BoolVal b then 1 else 0)"
uf: shows "pl.unique_factorization_m f' (lead_coeff f', mset gs')"
borel_measurable_Pair: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and g :: "'a \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. (f x, g x)) \<in> borel_measurable M"
s2r_union_hom: fixes A :: "'b set"   and B :: "'b set" shows "s2r (A \<union> B) = s2r A \<union> s2r B"
freshTauTransition: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
var_comp_2: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term" shows "Var \<circ>\<^sub>s \<sigma> = \<sigma>"
b: shows "homotopic_paths S (f \<circ> r) (linepath b b)"
int_pow_mult: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and i :: "int"   and j :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> (i + j) = x [^]\<^bsub>G\<^esub> i \<otimes>\<^bsub>G\<^esub> x [^]\<^bsub>G\<^esub> j"
simps_3: fixes a :: "'b"   and b :: "'b"   and as :: "'b list" shows "cycle (a # b # as) = cycle (a # as) * swap a b"
bezout_ring_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" shows "class.bezout_ring times one plus zero minus uminus \<equiv> class.comm_ring_1 times one plus zero minus uminus \<and> class.bezout_ring_axioms times plus"
L\<alpha>\<omega>: shows "Limit (\<alpha> + \<omega>)"
filterR_def: shows "filterR \<equiv> \<Lambda> v r. R\<cdot>(resultR\<cdot>r)\<cdot>(SetInsert\<cdot>v\<cdot>(exceptR\<cdot>r))"
nonEmpty_contain: fixes \<Gamma> :: "'a multiset" assumes "\<Gamma> \<noteq> \<Empt>" shows "\<exists>a. a \<in># \<Gamma>"
conjugate_preserve_poincare_distance: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc" shows "poincare_distance (conjugate u) (conjugate v) = poincare_distance u v"
cross_once_notin: fixes Q :: "'a set"   and R :: "'a set"   and a :: "'a"   and b :: "'a" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "a \<in> Q"   and "b \<in> Q"   and "b \<in> R"   and "a \<noteq> b"   and "Q \<noteq> R" shows "a \<notin> R"
norm_commute: fixes x :: "'d"   and y :: "'e" shows "norm (x, y) = norm (y, x)"
is_unit_power: fixes x :: "'a"   and n :: "nat" assumes "is_unit x" shows "is_unit (x ^ n)"
comp_identity_functor: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" assumes "functor A B F" shows "identity_functor.map B \<circ> F = F"
igConsIPresIGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igConsIPresIGWls MOD \<equiv> igVarIPresIGWls MOD \<and> igAbsIPresIGWls MOD \<and> igOpIPresIGWls MOD"
hyp: fixes L :: "'a language"   and K :: "'a language" assumes "R L K" shows "\<oo> L = \<oo> K \<and> rel_fun (=) (Plus_cong R) (\<dd> L) (\<dd> K)"
bres_top_top: shows "prod.bres top_prod top_prod = top_prod"
tpOf_lOfFax: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and f :: "'a" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "Signature.tpOf (ProblemIkTpart.TE_resOf resOf) (ProblemIkTpart.lOfFax arOf resOf f) = resOf f"
rec_by_index_main: fixes n1 :: "nat"   and n2 :: "nat" shows "nat_to_pr (rec_by_index n1 n2) = UnaryRecOp (nat_to_pr n1) (nat_to_pr n2)"
pick_asAbs: fixes qA :: "('e, 'f, 'varSort, 'var, 'g) qAbs" assumes "qGoodAbs qA" shows "pick (asAbs qA) $= qA"