hcju/isabelleps huggingface dataset:
proj_monomial: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c"   and v :: "'a"   and c :: "'d" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.proj_poly term_of_pair k (monomial c v) = (monomial c (term_powerprod.pp_of_term pair_of_term v) when term_powerprod.component_of_term pair_of_term v = k)"
remainder_recovery_imp: fixes Cn :: "'b set \<Rightarrow> 'b set"   and true_PL :: "'b"   and false_PL :: "'b"   and imp_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and not_PL :: "'b \<Rightarrow> 'b"   and conj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and equiv_PL :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and K :: "'b set"   and A :: "'b set"   and \<psi> :: "'b"   and \<phi> :: "'b"   and B :: "'b set" assumes "Supraclassical_logic Cn true_PL false_PL imp_PL not_PL conj_PL disj_PL equiv_PL"   and "K = Cn A"   and "\<psi> \<in> Cn K"   and "imp_PL \<psi> \<phi> \<in> Cn {}"   and "B \<in> Tarskian_logic.remainder Cn K \<phi>" shows "B \<in> Tarskian_logic.remainder Cn K \<psi>"
has_real_derivative_neg_dec_left: fixes f :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real"   and S :: "real set" assumes "(f has_real_derivative l) (at x within S)"   and "l < 0" shows "\<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
le_supI: fixes a :: "'c multiset"   and x :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># x"   and "b \<subseteq># x" shows "a \<union># b \<subseteq># x"
e_fro: shows "a + complex_of_real e \<in> frontier (inside (path_image p))"
pm: shows "bij_betw pm (sparsegrid dm lm) {..<card (sparsegrid dm lm)}"
simps: fixes factor_sq :: "nat"   and factor_pr :: "nat"   and limit :: "nat"   and n :: "nat"   and i :: "nat" shows "prime_product_factor_main factor_sq factor_pr limit n i = (if i \<le> limit \<and> 2 \<le> i then if i dvd n then let n' = n div i in if i dvd n' then let n'' = n' div i in prime_product_factor_main (factor_sq * i) factor_pr (nat (root_nat_floor 3 n'')) n'' i else case sqrt_nat n' of [] \<Rightarrow> prime_product_factor_main factor_sq (factor_pr * i) (nat (root_nat_floor 3 n')) n' (Suc i) | sn # x \<Rightarrow> (factor_sq * sn, factor_pr * i) else prime_product_factor_main factor_sq factor_pr limit n (Suc i) else (factor_sq, factor_pr * n))"
cinfinite_csum: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" assumes "cinfinite r1 \<or> cinfinite r2" shows "cinfinite (r1 +c r2)"
map_idI: fixes xs :: "'e list"   and f :: "'e \<Rightarrow> 'e" assumes "\<And>x. x \<in> set xs \<Longrightarrow> f x = x" shows "map f xs = xs"
scale_right_distrib: fixes a :: "'c"   and x :: "'b \<Rightarrow>\<^sub>0 'c"   and y :: "'b \<Rightarrow>\<^sub>0 'c" shows "a \<cdot> (x + y) = a \<cdot> x + a \<cdot> y"
seq1: fixes c1 :: "('a, 'b) MWLfCom"   and m :: "'b \<Rightarrow> 'c"   and m' :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and c2 :: "('a, 'b) MWLfCom" assumes "((c1, m), [], m') \<in> MWLf_semantics.MWLfSteps_det E BMap" shows "((c1;c2, m), [c2], m') \<in> MWLf_semantics.MWLfSteps_det E BMap"
C0: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<cdot> y = y \<cdot> z" shows "x\<^sup>\<star> \<cdot> y = y \<cdot> z\<^sup>\<star>"
div_0: fixes divide :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m" assumes "class.semidom_divide divide plus minus zero times one" shows "divide zero a = zero"
uminus: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "ring_ops ops R" shows "(R ===> R) (arith_ops_record.uminus ops) uminus_class.uminus"
assms_9: shows "R = (\<Union>(t, x)\<in>X. {merge xx t |xx. xx \<in> x})"
alphaAbs_qAbs_iff_ex_distinct_qFresh: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and xs :: "'e"   and x :: "'d"   and xs' :: "'e"   and x' :: "'d"   and X' :: "('f, 'g, 'e, 'd, 'h) qTerm" assumes "FixVars TYPE('d) TYPE('e)"   and "FixVars.qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
steps_reaches: fixes xs :: "'a list" assumes "steps xs" shows "hd xs \<rightarrow>* last xs"
RSpan_contains_RSpans_Cons_right: fixes m :: "'m"   and ms :: "'m list" assumes "m \<in> M" shows "RSpan ms \<subseteq> RSpan (m # ms)"
conf_Null: fixes G :: "prog"   and s :: "st"   and T :: "ty" shows "G,s\<turnstile>Null\<Colon>\<preceq>T = G\<turnstile>NT\<preceq>T"
low_proc_0\<^sub>1_correct: fixes p :: "prog" assumes "vars p = vars\<^sub>0" shows "low_proc_0\<^sub>1 p = low_proc_0 p"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set" assumes "\<Lambda>_PBD point_set block_collection 1 sizes" shows "PBD point_set block_collection sizes"
obj_src: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c" assumes "horizontal_homs V src trg"   and "partial_magma.arr V \<mu>" shows "horizontal_homs.obj V src (src \<mu>)"
r_cancel_one': fixes x :: "'a vec"   and a :: "'a vec" assumes "x \<in> carrier_vec n"   and "a \<in> carrier_vec n" shows "(x = a + x) = (a = 0\<^sub>v n)"
modif_clause: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call q (Modif \<sigma>),(ModifAbr \<sigma>)"
lowner_lub_prop: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "matrix_seq.lowner_is_lub f (matrix_seq.lowner_lub f)"
shiftr_zero_size: fixes x :: "'b word"   and n :: "nat" assumes "size x \<le> n" shows "x >> n = 0"
SafeDotCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and call :: "'a call_expr"   and b :: "'a expr" assumes "\<Gamma> \<turnstile> Call src SafeDotCall call \<Rrightarrow> b" obtains src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = expr.If (OperationCall src\<^sub>2__ DotCall (Inr (Inl (Inl NotEqualOp))) [Literal NullLiteral]) (Call (TypeOperationCall src\<^sub>2__ DotCall OclAsTypeOp (to_required_type \<tau>__)) DotCall call\<^sub>2__) (Literal NullLiteral)" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2__" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2__ : \<tau>__" and "OclVoid[?] \<le> \<tau>__" and "(\<Gamma>, to_required_type \<tau>__) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2__"   | src\<^sub>2 :: "'a expr" and \<tau> :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and call\<^sub>2 :: "'a call_expr" where "b = CollectIteratorCall (TypeOperationCall src\<^sub>2 ArrowCall SelectByKindOp (to_required_type \<sigma>)) ArrowCall [new_vname \<Gamma>] (Some (to_required_type \<sigma>)) (Call (Var (new_vname \<Gamma>)) DotCall call\<^sub>2)" and "\<Gamma> \<turnstile> src \<Rrightarrow> src\<^sub>2" and "\<Gamma> \<turnstile>\<^sub>E src\<^sub>2 : \<tau>" and "element_type \<tau> \<sigma>" and "OclVoid[?] \<le> \<sigma>" and "(\<Gamma>, to_required_type \<sigma>) \<turnstile>\<^sub>C call \<Rrightarrow> call\<^sub>2"
affine_rel_open: fixes S :: "'a set" assumes "affine S" shows "rel_open S"
wf\<^sub>s\<^sub>t\<^sub>s'_update\<^sub>s\<^sub>t_nil: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>" shows "wf\<^sub>s\<^sub>t\<^sub>s' (update\<^sub>s\<^sub>t \<S> []) \<A>"
oalist_inv_alt: fixes xs :: "('a \<times> 'd) list"   and ko :: "'o" shows "oalist_inv (xs, ko) = oalist_inv_raw ko xs"
IH: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "vc C Q" shows "\<turnstile>\<^sub>t {pre C Q} strip C {Q}"
mono_def: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" shows "prod.mono f = (\<forall>x y. less_eq_prod x y \<longrightarrow> f x \<le> f y)"
lens_override_id: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a" shows "S\<^sub>1 \<triangleleft>\<^bsub>\<Sigma>\<^esub> S\<^sub>2 = S\<^sub>2"
len_lexext_total: fixes B :: "'b set"   and A :: "'b set"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "\<forall>y\<in>B. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x"   and "ys \<in> lists B"   and "xs \<in> lists A" shows "len_lexext gt ys xs \<or> len_lexext gt xs ys \<or> ys = xs"
is_singleton_the_elem: fixes A :: "'b set" shows "is_singleton A = (A = {the_elem A})"
length_append: fixes xs :: "'c list"   and ys :: "'c list" shows "\<^bold>|xs \<cdot> ys\<^bold>| = \<^bold>|xs\<^bold>| + \<^bold>|ys\<^bold>|"
mod_mult_div_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "plus (modulo a b) (times b (divide a b)) = a"
wqo_letter_less: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "wqo_on (\<lambda>a b. (a, b) \<in> r\<^sup>=) UNIV" shows "wqo_on (\<lambda>a b. (a, b) \<in> (letter_less r)\<^sup>=) UNIV"
select_edge_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set" assumes "(s, p, D, pE) \<in> GS_rel"   and "p \<noteq> []" shows "select_edge_impl s \<le> \<Down> (Id \<times>\<^sub>r GS_rel) (select_edge (p, D, pE))"
less_asym': fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "bool" assumes "class.preorder less_eq less"   and "less a b"   and "less b a" shows "P"
preds_def: fixes rel :: "(nat \<times> nat) set"   and y :: "nat" shows "preds rel y = {x. (x, y) \<in> rel}"
hyps_4: shows "P \<turnstile> D \<preceq>\<^sup>* Object"
ltrm_monom: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "f = monom P a n" shows "monom P (f (deg R f)) (deg R f) = f"
aff_dim_le_card: fixes V :: "'a set" assumes "finite V" shows "aff_dim V \<le> int (card V) - 1"
Sbis_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis" shows "PL_Indis.Sbis tval aval indis \<equiv> bis (PL_Indis.Sretr tval aval indis)"
assms_2: shows "\<guillemotleft>\<theta> : (s\<^sub>0 \<star> p\<^sub>0) \<star> w \<Rightarrow> u\<guillemotright>"
sorted_snoc: fixes xs :: "'b list"   and y :: "'b" assumes "sorted_less (xs @ [y])" shows "sorted_less xs"
continuous_on_coordinatewise_then_product: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "\<And>i. continuous_on S (\<lambda>x. f x i)" shows "continuous_on S f"
iso_Inf_subdistl: fixes f :: "'a \<Rightarrow> 'c" assumes "mono f" shows "f \<circ> Inf \<le> Inf \<circ> (`) f"
comp_fun_idem_axioms: shows "comp_fun_idem (\<lambda>x. Mapping.update x p)"
RES_refine: fixes S :: "'c set"   and S' :: "'d set"   and R :: "('c \<times> 'd) set" assumes "\<And>s. s \<in> S \<Longrightarrow> \<exists>s'\<in>S'. (s, s') \<in> R" shows "RES S \<le> \<Down> R (RES S')"
q'_in: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" shows "q' g \<in> punit.dgrad_p_set d m'"
T_opt_def: fixes s :: "'a list"   and rs :: "'a list" shows "Move_to_Front.T_opt s rs = \<Sqinter> {Move_to_Front.T s rs as |as. length as = length rs}"
exhaust_sel: fixes strand_label :: "'a strand_label" obtains "strand_label = ln the_LabelN strand_label"   | "strand_label = \<star>"
P_def: fixes x :: "'a" shows "P x = (\<langle>t, x\<rangle> / \<langle>t, t\<rangle>) *\<^sub>C t"
vars2'_def: fixes vars1 :: "nat set"   and vars2 :: "nat set" assumes "partial_state2 vars1 vars2" shows "partial_state2.vars2' vars1 vars2 = ind_in_set (partial_state2.vars0 vars1 vars2) ` vars2"
C: shows "(xs, ys) \<in> rel_ipurge P I D u"
pred_rel: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a stream" shows "pred_stream P x = stream_all2 (eq_onp P) x x"
hyps_2: fixes i :: "nat" assumes "i < length T" shows "term_variants_pred P (T ! i) (S ! i)"
sel_3: fixes x21 :: "'g"   and x22 :: "'g list" shows "tl (x21 # x22) = x22"
splice_Nil2: fixes xs :: "'g list" shows "splice xs [] = xs"
res: shows "LLL_implementation.LLL_reconstruction p (p ^ find_exponent p (sqrt_int_ceiling (2 ^ (5 * (degree f - 1) * (degree f - 1)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f - 1))))) f (hensel_lifting p (find_exponent p (sqrt_int_ceiling (2 ^ (5 * (degree f - 1) * (degree f - 1)) * \<parallel>f\<parallel>\<^sup>2 ^ (2 * (degree f - 1))))) f gs) = gs"
IH1: shows "simple_fw (rmshadow' rs P) p = simple_fw rs p"
section_retraction_of_iso_1: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.iso f"   and "VV.ide (VV.comp g f)" shows "VV.inverse_arrows f g"
of_wstate_split: fixes St :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat" shows "(wN_of_wstate St, wP_of_wstate St, wQ_of_wstate St, n_of_wstate St) = St"
norm: shows "\<GG> ! i \<lhd> G\<lparr>carrier := \<GG> ! (i + 1)\<rparr>"
simps_3: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "successively P (x # y # xs) = (P x y \<and> successively P (y # xs))"
SubstAtomicP_Eq: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{SubstTermP v i x x', SubstTermP v i y y'} \<turnstile> SubstAtomicP v i (Q_Eq x y) (Q_Eq x' y')"
union_ring_def: fixes C :: "('c, 'd) ring_scheme set" shows "union_ring C = \<lparr>carrier = \<Union> (carrier ` C), monoid.mult = \<lambda>a b. a \<otimes>\<^bsub>SOME R. R \<in> C \<and> a \<in> carrier R \<and> b \<in> carrier R\<^esub> b, one = \<one>\<^bsub>SOME R. R \<in> C\<^esub>, zero = \<zero>\<^bsub>SOME R. R \<in> C\<^esub>, add = \<lambda>a b. a \<oplus>\<^bsub>SOME R. R \<in> C \<and> a \<in> carrier R \<and> b \<in> carrier R\<^esub> b\<rparr>"
eq_onp_to_Domainp: fixes P :: "'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and T :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "Domainp T = P"
post_before_pre_neighbors: fixes i :: "nat"   and j :: "nat"   and t :: "('a, 'b, 'c) event list" assumes "i < j"   and "j < length t"   and "\<forall>k. i < k \<and> k < j \<longrightarrow> \<not> regular_event (t ! k)"   and "postrecording_event t i"   and "prerecording_event t j"   and "trace init t final" shows "\<forall>ev\<in>set (take (j - (i + 1)) (drop (i + 1) t)). \<not> regular_event ev \<and> occurs_on ev \<noteq> occurs_on (t ! j)"
coeff_le_sq_norm: fixes f :: "'a poly"   and i :: "nat" shows "\<bar>coeff f i\<bar>\<^sup>2 \<le> \<parallel>f\<parallel>\<^sup>2"
abs_eq: fixes xa :: "'a ltl"   and x :: "'a set" shows "af_abs.f_abs (Abs xa) x = Abs (af_letter xa x)"
list_comb: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a"   and xs :: "'a list" assumes "simple_syntactic_or P" shows "P (list_comb f xs) = (P f \<or> list_ex P xs)"
I_minWeightBasisG: fixes T :: "'a set"   and E :: "'a set" assumes "I_minWeightBasis (T, E)" shows "no_smallest_element_skipped E T"
less_infI1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and x :: "'a"   and b :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less a x" shows "less (inf a b) x"
assms: shows "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"
fresh_smallstep_P: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term"   and x :: "Syntax.var" assumes "\<lless>\<pi>, e\<ggreater> P\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   and "atom x \<sharp> e" shows "atom x \<sharp> e'"
HDomain_Incl_subst: fixes r :: "tm"   and d :: "tm"   and i :: "name"   and t :: "tm" shows "(HDomain_Incl r d)(i::=t) = HDomain_Incl (subst i t r) (subst i t d)"
distinct_3: fixes x1 :: "name"   and x3 :: "term" shows "Const x1 \<noteq> \<Lambda> x3"
minus_cancel_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v - v = 0\<^sub>v n"
P_sup_False: fixes t :: "'s" shows "P_sup t (\<lambda>\<omega>. False) = 0"
cases: fixes x :: "'b list \<times> nat \<times> nat" obtains xs :: "'b list" and i :: "nat" and j :: "nat" where "x = (xs, i, j)"
disconnected_nodes_eq2_h3: fixes doc_ptr :: "(_) document_ptr" assumes "document_ptr \<noteq> doc_ptr" shows "|h3 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h' \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
has_bochner_integral_add: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and x :: "'d"   and g :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "has_bochner_integral M f x"   and "has_bochner_integral M g y" shows "has_bochner_integral M (\<lambda>x. f x + g x) (x + y)"
bheight_baliR: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "bheight l = bheight r" shows "bheight (baliR l a r) = Suc (bheight l)"
d1: shows "\<forall>p\<in>set Ps. \<exists>p'\<in>set ps. p = extend S p'"
assms_5: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"
real_cond_exp_intg_2: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M (\<lambda>x. f x * g x)"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "LINT x|M. f x * real_cond_exp M F g x = LINT x|M. f x * g x"
snd: fixes A :: "'b ptrm"   and B :: "'b ptrm" assumes "A \<approx> B" shows "PSnd A \<approx> PSnd B"
GB_imp_zero_reducibility: fixes G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_Groebner_basis G"   and "f \<in> pmdl G" shows "(red G)\<^sup>*\<^sup>* f 0"
inversions_def: shows "inversions \<equiv> \<lambda>xs. {(xa, x). inversionsp xs xa x}"
assms: shows "(s, a, s') \<in> automaton.trans (opnet onp \<langle>i; R\<rangle>)"
estep_Pr4: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Pr n f g, xs, ls) # fs, rv)"   and "ls = []" shows "estep_Pr (encode_config c) = encode_config (step c)"
R_equiv_entailment: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and lc\<^sub>1 :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and lc\<^sub>2 :: "(('a, 'c, 'd) Stmt \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)"   and \<Gamma>' :: "'a \<Rightarrow> 'd set option"   and \<S>' :: "'a set \<times> 'a set"   and P' :: "'d set"   and \<Gamma>'' :: "'a \<Rightarrow> 'd set option"   and P'' :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False"   and "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_types.\<R> ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma>' \<S>' P'"   and "sifum_types.context_equiv ev\<^sub>B \<Gamma>' P' \<Gamma>''"   and "sifum_types.pred_entailment ev\<^sub>B P' P''"   and "\<forall>mds. sifum_types.tyenv_wellformed bexp_vars \<C>_vars mds \<Gamma>' \<S>' P' \<longrightarrow> sifum_types.tyenv_wellformed bexp_vars \<C>_vars mds \<Gamma>'' \<S>' P''" shows "(lc\<^sub>1, lc\<^sub>2) \<in> sifum_types.\<R> ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type \<Gamma>'' \<S>' P''"
Suc_3: fixes i :: "nat"   and j :: "nat" assumes "i \<le> Suc n"   and "j \<le> Suc n"   and "i \<noteq> j" shows "I i <+> I j = carrier R"
B_N2_cases: fixes t1 :: "'a bro"   and a :: "'a"   and t2 :: "'a bro"   and n :: "nat" assumes "N2 t1 a t2 \<in> B (Suc n)" obtains "t1 \<in> B n" and "t2 \<in> B n"   | "t1 \<in> U n" and "t2 \<in> B n"   | "t1 \<in> B n" and "t2 \<in> U n"
supp_Arg: fixes v :: "Vars.var" shows "supp (Arg v) = supp v"
chine_assoc_def: shows "\<mu>_\<nu>\<pi>_\<rho>.chine_assoc \<equiv> \<langle>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1 \<lbrakk>\<mu>.leg0, \<nu>\<pi>.leg1 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>1\<rbrakk> \<langle>\<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<^sub>1 \<lbrakk>\<nu>\<pi>.leg0, \<rho>.leg1\<rbrakk> \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>0\<rangle>\<rangle>"
assms_5: shows "\<forall>k\<in>{l..<i}. a k < a i"
IH_2: shows "reduce_nested_mpoly (extract_var p2 v) = p2"
mset_heapify: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "mset_tree (fst (heapify n xs)) = mset (take n xs)"
range_of_var_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and x :: "nat" shows "ast_problem.range_of_var problem x \<equiv> {0..<ast_problem.numVals problem x}"
well_formed_product: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM" assumes "well_formed M1"   and "well_formed M2" shows "well_formed (product M2 M1)"
execute_raise: fixes s :: "String.literal" shows "execute (raise s) = Map.empty"
compact_eq_bounded_closed: fixes S :: "'b set" shows "compact S = (bounded S \<and> closed S)"
of_weak_ranking_Collect_ge_Cons_2: fixes y :: "'c"   and x :: "'c set"   and xs :: "'c set list" assumes "y \<notin> x" shows "of_weak_ranking_Collect_ge (x # xs) y = of_weak_ranking_Collect_ge xs y"
msum_aform_def: fixes n :: "nat"   and f :: "'a \<times> 'a pdevs"   and g :: "'a \<times> 'a pdevs" shows "msum_aform n f g = (fst f + fst g, msum_pdevs n (snd f) (snd g))"
\<iota>_in: shows "\<iota> \<in> Inf"
Union_atLeastAtMost_real_of_int: fixes a :: "int"   and b :: "int" assumes "a < b" shows "(\<Union>n\<in>{a..<b}. {real_of_int n..real_of_int (n + 1)}) = {real_of_int a..real_of_int b}"
S_transform_formula_Not: fixes x :: "('a, 'c, 'b) formula" shows "S_transform_formula (Formula.Not x) = Not\<^sub>\<alpha> (S_transform_formula x)"
language_eq_nextl: fixes M :: "'b dfa" assumes "dfa M" shows "dfa.language M = dfa.eq_nextl M `` \<Union> (dfa.left_lang M ` dfa.final M)"
Lcm_singleton: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize" shows "Lcm {a} = normalize a"