hcju/isabelleps huggingface dataset:
subtt_def: shows "subterms_tuples \<equiv> \<lambda>E. subterms\<^sub>s\<^sub>e\<^sub>t (fst ` set E) \<union> subterms\<^sub>s\<^sub>e\<^sub>t (snd ` set E)"
finite'_code_1: fixes xs :: "'a list" shows "finite' (set xs) = True"
append_Cons_nth_right: fixes xs :: "'c list"   and i :: "nat"   and u :: "'c"   and ys :: "'c list"   and z :: "'c" assumes "length xs < i" shows "(xs @ u # ys) ! i = (xs @ z # ys) ! i"
map_pmf_of_set_inj: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on f A"   and "A \<noteq> {}"   and "finite A" shows "map_pmf f (pmf_of_set A) = pmf_of_set (f ` A)"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b"   and TRel :: "('b \<times> 'b) set" shows "((a1, a2) \<in> encoding.indRelTEQ Enc TRel) = ((\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S)) \<or> (\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S) \<or> (\<exists>T1 T2. a1 = TargetTerm T1 \<and> a2 = TargetTerm T2 \<and> (T1, T2) \<in> TRel) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> (P, Q) \<in> encoding.indRelTEQ Enc TRel \<and> (Q, R) \<in> encoding.indRelTEQ Enc TRel))"
psimps_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and z :: "nat" assumes "rb_aux_dom ((bs, ss, []), z)" shows "rb_aux ((bs, ss, []), z) = ((bs, ss, []), z)"
subdivFace'_preserve_finals: fixes f :: "face"   and g :: "graph"   and f' :: "face"   and i :: "nat"   and n :: "nat"   and "is" :: "nat option list" assumes "f \<in> set (finals g)"   and "\<not> final f'" shows "f \<in> set (finals (subdivFace' g f' i n is))"
ang_circ_a_simp: fixes E :: "complex"   and \<mu>1 :: "complex"   and \<mu>2 :: "complex"   and p1 :: "bool"   and p2 :: "bool" assumes "E \<noteq> \<mu>1"   and "E \<noteq> \<mu>2" shows "ang_circ_a E \<mu>1 \<mu>2 p1 p2 = acute_ang \<bar>\<downharpoonright>arg (E - \<mu>2) - arg (E - \<mu>1) + sgn_bool p1 * pi / 2 - sgn_bool p2 * pi / 2\<downharpoonleft>\<bar>"
FINITE_1: shows "finite (\<Union> (A ` {1..m}))"
G: shows "map (inv p) (filter (\<lambda>x. x \<in> range p) xs') \<in> divergences P"
valid: shows "\<forall>M s. M, s \<Turnstile> p"
get_component_root_node_same: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and root_ptr :: "(_) object_ptr"   and x :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr"   and "x \<in> set c" shows "h \<turnstile> get_root_node x \<rightarrow>\<^sub>r root_ptr"
restriction_of_domain: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and A1 :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "A1 \<subseteq> A" shows "restrict f A1 \<in> A1 \<rightarrow> B"
mult_cancel_left': fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and F :: "'a filter"   and f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b"   and g2 :: "'a \<Rightarrow> 'b" assumes "landau_symbol L L' Lr"   and "eventually_nonzero F f1" shows "((\<lambda>x. f1 x * f2 x) \<in> L F (\<lambda>x. f1 x * g2 x)) = (f2 \<in> L F g2)"
diagonal_dvd_Gcd_minors: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n m"   and "Smith_normal_form_mat A" shows "(\<Prod>i = 0..<k. A $$ (i, i)) dvd Gcd (minors A k)"
dgrad_p_set_closed_tail: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "p \<in> dgrad_p_set d m" shows "tail p \<in> dgrad_p_set d m"
VectorSpaceHomI_fromaxioms: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'c"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "VectorSpace smult V"   and "\<And>g g'. \<lbrakk>g \<in> V; g' \<in> V\<rbrakk> \<Longrightarrow> T (g + g') = T g + T g'"   and "supp T \<subseteq> V"   and "\<And>r m. \<lbrakk>r \<in> UNIV; m \<in> V\<rbrakk> \<Longrightarrow> T (smult r m) = smult' r (T m)" shows "VectorSpaceHom smult V smult' T"
positional_strategy_exists: fixes v0 :: "'a" assumes "v0 \<in> V" shows "\<exists>p. v0 \<in> winning_region p"
sams_lea_lta456_suma2__lta123: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and D' :: "'a"   and E' :: "'a"   and F' :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a"   and G :: "'a"   and H :: "'a"   and I :: "'a"   and G' :: "'a"   and H' :: "'a"   and I' :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.LtA Bet Cong D' E' F' D E F"   and "Tarski_neutral_dimensionless.LeA Bet Cong G H I G' H' I'"   and "Tarski_neutral_dimensionless.SAMS Bet Cong A B C D E F"   and "Tarski_neutral_dimensionless.SumA Bet Cong A B C D E F G H I"   and "Tarski_neutral_dimensionless.SumA Bet Cong A' B' C' D' E' F' G' H' I'" shows "Tarski_neutral_dimensionless.LtA Bet Cong A B C A' B' C'"
IH: assumes "\<forall>a\<in>insert x (set xs). case a of (i, p) \<Rightarrow> \<forall>a\<in>insert x (set xs). case a of (i', p') \<Rightarrow> p = p' \<longrightarrow> i = i'" shows "map snd (List.insert x xs) = List.insert (snd x) (map snd xs)"
allDefs_disjoint': fixes n :: "'node"   and g :: "'g"   and m :: "'node"   and v :: "'val" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "v \<in> allDefs g n"   and "v \<in> allDefs g m" shows "n = m"
ceuclidean_representation: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and x :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleC (cinner b x) b) CBasis = x"
ACall_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> ('a, 'b) chan"   and p :: "PrIds"   and v :: "state \<Rightarrow> 'a" shows "Call ch p v \<equiv> \<not> $Calling ch p \<and> cbit<ch!p>$ \<noteq> $rbit<ch!p> \<and> arg<ch!p>$ = $v"
slimilar_bot_cases: fixes x :: "Value"   and y :: "CValue" assumes "x \<triangleleft>\<triangleright> y" obtains "x = \<bottom>" and "y = \<bottom>"   | b :: "bool" where "x = B\<cdot>(Discr b)" and "y = CB\<cdot>(Discr b)"   | f :: "Value \<rightarrow> Value" and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" where "x = Fn\<cdot>f" and "y = CFn\<cdot>g"
map_transfer: fixes R1b :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and S1d :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2b :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and S2d :: "'f \<Rightarrow> 'h \<Rightarrow> bool" shows "rel_fun (rel_fun R1b S1d) (rel_fun (rel_fun R2b S2d) (rel_fun (tllist_all2 R1b R2b) (tllist_all2 S1d S2d))) tmap tmap"
CondReds: fixes e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   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"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>if (e') e\<^sub>1 else e\<^sub>2,s'\<rangle>"
aligned_small_is_0: fixes x :: "'b word"   and n :: "nat" assumes "is_aligned x n"   and "x < 2 ^ n" shows "x = 0"
insert_before_list_node_in_set: fixes x :: "'m"   and ref :: "'m option"   and xs :: "'m list" shows "x \<in> set (insert_before_list x ref xs)"
fr_graph_impl_loc_axioms: shows "fr_graph_impl_loc mrel node_rel node_eq_impl node_hash_impl node_def_hash_size G_impl G"
diff_substs_yield_diff_trms: fixes x :: "'b"   and \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "Var x \<lhd> \<sigma> \<noteq> Var x \<lhd> \<eta>"   and "x \<in> vars_of t" shows "t \<lhd> \<sigma> \<noteq> t \<lhd> \<eta>"
d_Suc: fixes i :: "nat" shows "d (Suc i) = \<mu>' i i"
AxiomI: fixes A :: "(formula list \<times> formula list set) set"   and p :: "predicate"   and vs :: "vbl list" assumes "Axioms \<subseteq> A" shows "[FAtom Pos p vs, FAtom Neg p vs] \<in> deductions A"
preserves_arr: fixes f :: "'a arrow_of_spans_data" assumes "arr f" shows "arr (trg f)"
advantage_def: fixes "oracle" :: "'c \<Rightarrow> 'call \<Rightarrow> ('ret \<times> 'c) spmf"   and \<sigma> :: "'c"   and \<A> :: "('pubkey \<Rightarrow> (('plain \<times> 'plain) \<times> 'd, 'call, 'ret) gpv) \<times> ('cipher \<Rightarrow> 'd \<Rightarrow> (bool, 'call, 'ret) gpv)" shows "advantage oracle \<sigma> \<A> = \<bar>spmf (run_gpv oracle (ind_cpa \<A>) \<sigma>) True - 1 / 2\<bar>"
Units_r_inv: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
transpose_add: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "(A + B)\<^sup>T = A\<^sup>T + B\<^sup>T"
poincare_between_rev: fixes u :: "complex_homo"   and v :: "complex_homo"   and w :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "w \<in> unit_disc" shows "poincare_between u v w = poincare_between w v u"
path_to_pres_valid: fixes s :: "nat \<Rightarrow> nat option"   and \<pi>s :: "char list list"   and s' :: "nat \<Rightarrow> nat option" assumes "s \<in> valid_states"   and "path_to s \<pi>s s'" shows "s' \<in> valid_states"
newton_poly_impl_def: shows "local.newton_poly_impl = horner_composition (rev local.newton_coefficients) xs"
bind_resumption_eq_Done_iff: fixes f :: "('i, 'g, 'h) Resumption.resumption"   and g :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption"   and x :: "'f option" shows "(f \<bind> g = Resumption.resumption.Done x) = ((\<exists>y. f = DONE y \<and> g y = Resumption.resumption.Done x) \<or> f = ABORT \<and> x = None)"
succ_in_diff: fixes r :: "('c \<times> 'c) set"   and i :: "'c" assumes "wo_rel r"   and "aboveS r i \<noteq> {}" shows "(i, succ r i) \<in> r \<and> succ r i \<noteq> i"
strict_implies_order: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry" assumes "b \<prec> a" shows "b \<preceq> a"
distinct_3: fixes x3 :: "'b" shows "true\<^sub>n \<noteq> prop\<^sub>n(x3)"
card_Diff1_less_iff: fixes A :: "'a set"   and x :: "'a" shows "(card (A - {x}) < card A) = (finite A \<and> x \<in> A)"
closed_ereal_semiline: fixes a :: "ereal" shows "closed {y. a \<le> ereal y}"
units_of_pow: fixes x :: "'a"   and n :: "nat" assumes "x \<in> Units G" shows "x [^]\<^bsub>units_of G\<^esub> n = x [^] n"
cinsert_cDiff: fixes a :: "'f"   and A :: "'f cset" assumes "cin a A" shows "cinsert a (cDiff A (csingle a)) = A"
simps_16: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and f3 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b"   and f4 :: "PreSimplyTyped.type \<Rightarrow> PreSimplyTyped.type \<Rightarrow> 'b" shows "(case TUnit of TUnit \<Rightarrow> f1 | TVar x \<Rightarrow> f2 x | TArr x xa \<Rightarrow> f3 x xa | TPair x xa \<Rightarrow> f4 x xa) = f1"
prob_space_P: fixes J :: "'i set" assumes "finite J"   and "J \<subseteq> I" shows "prob_space (Pi\<^sub>M J M)"
iT_Minus_card_le: fixes I :: "nat set"   and k :: "nat" assumes "finite I" shows "card (k \<ominus> I) \<le> card I"
rel_coinduct: fixes P :: "('d, 'e, 'f) gpv \<Rightarrow> ('g, 'h, 'f) gpv \<Rightarrow> bool"   and x :: "('d, 'e, 'f) gpv"   and y :: "('g, 'h, 'f) gpv"   and R1 :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'h \<Rightarrow> bool" assumes "P x y"   and "\<And>gpv gpv'. P gpv gpv' \<Longrightarrow> rel_spmf (rel_generat R1 R2 (rel_fun (=) P)) (the_gpv gpv) (the_gpv gpv')" shows "rel_gpv R1 R2 x y"
subcocycle_avg_finite: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" shows "subcocycle_avg_ereal u < \<infinity>"
strip_3: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "\<And>x. x \<in> A \<Longrightarrow> P x" shows "\<forall>x\<in>A. P x"
CD_on: assumes "x \<in> CD_on ds (- Rf CH fp) - fp" shows "thesis"
m_inv_def: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" shows "inv\<^bsub>G\<^esub> x = (THE y. y \<in> carrier G \<and> x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub> \<and> y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>)"
ideal_distrib: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "ideal R A"   and "ideal R B"   and "ideal R C" shows "A \<diamondsuit>\<^sub>r (B \<minusplus> C) = A \<diamondsuit>\<^sub>r B \<minusplus> A \<diamondsuit>\<^sub>r C"
iMOD_add_neg: fixes r :: "nat"   and m :: "nat"   and k :: "nat" shows "[ r, mod m ] \<oplus>- k = (if k \<le> r then [ r - k, mod m ] else if 0 < m then [ (m + r mod m - k mod m) mod m, mod m ] else {})"
disc_1: fixes x1 :: "'a" shows "is_Prim \<^bold>\<langle>x1\<^bold>\<rangle>"
exists_ShiftP: fixes t :: "name"   and s :: "tm"   and k :: "tm"   and del :: "tm"   and H :: "fm set" assumes "atom t \<sharp> (s, k, del)" shows "H \<turnstile> SyntaxN.Ex t (ShiftP s k del (Var t))"
face_cycle_set_def: fixes a :: "'b \<times> 'b" shows "face_cycle_set a = orbit face_cycle_succ a"
divide_le_eq_1_neg: fixes a :: "'a"   and b :: "'a" assumes "a < (0::'a)" shows "(b / a \<le> (1::'a)) = (a \<le> b)"
r_sigma_def: shows "r_sigma \<equiv> Cn 2 r_phi [Cn 2 r_limr [recf.Id 2 0], recf.Id 2 1]"
c_def: shows "c = card {q \<in> set qs. m < card (snd q)}"
ioP_def: shows "ioP = take (length ioP) io"
t_notin_F: shows "t \<notin> Fault ` F"
strict_mono_imp_inj_on: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "class.linorder less_eq less"   and "order.strict_mono less f" shows "inj_on f A"
10: shows "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g) \<le> r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> * ((1::'a) \<squnion> e)"
lookup_RBT: fixes t :: "('a, 'b) RBT_Impl.rbt" assumes "is_rbt t" shows "lookup (RBT t) = rbt_lookup t"
lookup_impl_of: fixes t :: "('b, 'c) mapping_rbt" shows "rbt_comp_lookup ccomp (mapping_rbt.impl_of t) = RBT_Mapping2.lookup t"
simps_8: fixes P :: "pi"   and Q :: "pi" shows "hnf (P \<parallel> Q) = False"
assms_2: fixes x :: "'a" assumes "x \<in> M" shows "x \<le> m"
eval_det: fixes lc :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and lc' :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)"   and lc'' :: "((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)" assumes "eval_abv\<^sub>C lc lc'"   and "eval_abv\<^sub>C lc lc''" shows "lc' = lc''"
assms: shows "f : a \<mapsto>\<^bsub>\<AA>\<^esub> b"
distinct_43: fixes pi :: "pi"   and pi1' :: "pi"   and pi2' :: "pi" shows "\<tau>.pi \<noteq> pi1' \<parallel> pi2'"
simps_8: fixes f :: "'d \<Rightarrow> 'c" shows "map_tree f \<langle>\<rangle> = \<langle>\<rangle>"
ccpo: shows "class.ccpo tllist.lub_fun tllist.le_fun (mk_less tllist.le_fun)"
simps_1: fixes u :: "'a"   and v :: "'a" shows "cas u [] v = (u = v)"
p: shows "p = (\<Prod>a\<in>as. a ^ Suc (f a))"
fast_dice_roll_fixp_induct: fixes P :: "(nat \<Rightarrow> nat \<Rightarrow> nat spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>fast_dice_roll. P (curry fast_dice_roll))"   and "P (\<lambda>v c. return_pmf None)"   and "\<And>fdr. P fdr \<Longrightarrow> P (\<lambda>v c. if n \<le> v then if c < n then return_spmf c else fdr (v - n) (c - n) else coin_spmf \<bind> (\<lambda>b. fdr (2 * v) (2 * c + (if b then 1 else 0))))" shows "P local.fast_dice_roll"
vernichter_1: fixes r :: "'b wordinterval" shows "wordinterval_empty r = (wordinterval_to_set r = {})"
interval_lebesgue_integrable_mult_right: fixes c :: "'a"   and M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" assumes "c \<noteq> (0::'a) \<Longrightarrow> interval_lebesgue_integrable M a b f" shows "interval_lebesgue_integrable M a b (\<lambda>x. c * f x)"
fds_nth_newman: fixes n :: "nat" shows "fds_nth fds_newman n = complex_of_real (\<MM> (real n))"
dsr1: fixes x :: "'a" shows "x + x \<cdot> ar ar x = x \<cdot> ar ar x"
seql_onl_swap: fixes i :: "'a"   and \<Gamma> :: "'e \<Rightarrow> ('c, 'd, 'e, 'f) seqp"   and P :: "'b \<times> 'f \<Rightarrow> bool" shows "seql i (onl \<Gamma> P) = onl \<Gamma> (seql i P)"
connected_intermediate_closure: fixes s :: "'a set"   and t :: "'a set" assumes "connected s"   and "s \<subseteq> t"   and "t \<subseteq> closure s" shows "connected t"
elims_3: fixes x :: "'b extB"   and xa :: "'b extB" assumes "\<not> x \<le> xa" obtains uv_ :: "'b" where "xa__ = extB.Val uv_" and "xa = extB.Bot"   | x :: "'b" and y :: "'b" where "xa__ = extB.Val x" and "xa = extB.Val y" and "\<not> x \<le> y"
fresh_fact2: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "a \<sharp> P[a::=b]"
Sadmit: shows "Sadmit \<sigma> (\<Gamma>, \<Delta>)"
Act_eq_iff_perm_renaming: fixes \<alpha>1 :: "'d"   and x1 :: "('b, 'c, 'd) formula"   and \<alpha>2 :: "'d"   and x2 :: "('b, 'c, 'd) formula" shows "(Act \<alpha>1 x1 = Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2 \<and> supp p \<subseteq> bn \<alpha>1 \<union> p \<bullet> bn \<alpha>1)"
aff_dim_affine_independent: fixes B :: "'b set" assumes "\<not> affine_dependent B" shows "int (card B) = aff_dim B + 1"
rep_eq: fixes x :: "'a"   and xa :: "'b"   and xb :: "('a, 'b) fmap" shows "fmlookup (fmupd x xa xb) = map_upd x xa (fmlookup xb)"
arrI: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>1 b\<guillemotright>" shows "C1.arr f"
arev_arev_raw: fixes a :: "'b" assumes "a \<in> arcs G" shows "edge_rev GM (edge_rev GM a) = a"
cnj_sum: fixes f :: "'b \<Rightarrow> complex"   and s :: "'b set" shows "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))"
fail_state_def: fixes fail :: "'e" shows "fail_state fail = StateT (\<lambda>s. fail)"
CFG_intra_edge_transfer_uses_only_Use: fixes a :: "'edge"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" assumes "valid_edge a"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V"   and "intra_kind (kind a)"   and "pred (kind a) s"   and "pred (kind a) s'" shows "\<forall>V\<in>Def (sourcenode a). state_val (transfer (kind a) s) V = state_val (transfer (kind a) s') V"
induct: fixes P :: "'c list \<Rightarrow> bool"   and a0 :: "'c list" assumes "P []"   and "\<And>x xs. \<lbrakk>P (filter (\<lambda>y. y < x) xs); P (filter ((<) x) xs)\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
assms_2: shows "h \<turnstile> adopt_node owner_document node_ptr \<rightarrow>\<^sub>h h'"
S_def: shows "S = {lt f |f. f \<in> pmdl F \<and> f \<in> dgrad_p_set d m \<and> f \<noteq> 0}"
absorb_iff2: fixes b :: "'a::type"   and a :: "'a::type" shows "((b::'a::type) \<sqsubseteq> (a::'a::type)) = (a \<sqinter> b = b)"
h3: shows "dim_row (take_rows A [0..<length (pivot_positions A)]) + (dim_row A - length (pivot_positions A)) = dim_row A"
dghm_proj_ObjMap_vdomain: fixes I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" shows "\<D>\<^sub>\<circ> (\<pi>\<^sub>D\<^sub>G I \<AA> i\<lparr>ObjMap\<rparr>) = dg_prod I \<AA>\<lparr>Obj\<rparr>"
assms: fixes w :: "('a, 'b) obs_event action" assumes "w \<in> set obs"   and "is_write_action w"   and "adal \<in> action_loc_aux P w" shows "case vs adal of None \<Rightarrow> False | \<lfloor>(v, b)\<rfloor> \<Rightarrow> b \<and> is_new_action w"
poly2_poly_lift: fixes p :: "'a poly"   and x :: "'a"   and y :: "'a" shows "poly2 (poly_lift p) x y = poly p y"
effect_freezeI: fixes h' :: "heap"   and h :: "heap"   and r :: "'a list"   and a :: "'a array" assumes "h' = h"   and "r = Array.get h a" shows "effect (Array.freeze a) h h' r"
rev_existence_ivl_eq0: fixes y :: "'a" shows "rev.existence_ivl0 y = uminus ` existence_ivl0 y"
det_transpose: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A\<^sup>T = det A"
lookup_zero: fixes k :: "'c" shows "lookup 0 k = (0::'b)"
bisim1CondThen: fixes P :: "'a prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e :: "'addr expr1"   and e2 :: "'addr expr1" assumes "P,e1,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,if (e) e1 else e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (length (compE2 e) + pc), xcp)"
1: shows "degree p1 \<le> length zs_ws - 1 \<and> (\<forall>x y. (x, y) \<in> set zs_ws \<longrightarrow> poly p1 x = y)"
mul_ext_mono_1: fixes xs :: "'b list"   and ys :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and P' :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys; fst (P x y)\<rbrakk> \<Longrightarrow> fst (P' x y)"   and "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set ys; snd (P x y)\<rbrakk> \<Longrightarrow> snd (P' x y)"   and "fst (mul_ext P xs ys)" shows "fst (mul_ext P' xs ys)"
rep_props_3: fixes r :: "'a" assumes "B.ide r" shows "r \<cong>\<^sub>B B.VVV.iso_class_rep \<lbrakk>r\<rbrakk>\<^sub>B"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
assms: shows "low_equal s1 s2 \<and> t1 = snd (fst (sub_instr instr s1)) \<and> t2 = snd (fst (sub_instr instr s2))"
power_set_func_nat_trans: fixes f :: "'a \<Rightarrow> 'b" shows "\<eta> \<circ> id f = \<P> f \<circ> \<eta>"
axioms_1: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c" assumes "ChamberComplexIsomorphism X Y f" shows "ChamberComplexMorphism X Y f"