hcju/isabelleps huggingface dataset:
term_variants_pred_refl: fixes P :: "'a \<Rightarrow> 'a list"   and t :: "('a, 'b) Term.term" shows "term_variants_pred P t t"
is_projection_on_zero: shows "is_projection_on (\<lambda>_. 0::'a) {0::'a}"
condensation_nodes_edges: fixes g :: "'g"   and P :: "'val set" shows "condensation_edges g P \<subseteq> condensation_nodes g P \<times> condensation_nodes g P"
finprod_cong': fixes A :: "'a set"   and B :: "'a set"   and g :: "'a \<Rightarrow> int"   and f :: "'a \<Rightarrow> int" assumes "A = B"   and "g \<in> B \<rightarrow> carrier R"   and "\<And>i. i \<in> B \<Longrightarrow> f i = g i" shows "finsum R f A = finsum R g B"
apex_def: shows "r.cod.apex \<equiv> C.dom r.cod.leg0"
infiter_annil: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and x :: "'b::refinement_lattice" assumes "infinite_iteration (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (infinite_iteration.infiter seq (c::'b::refinement_lattice)) (x::'b::refinement_lattice) = infinite_iteration.infiter seq c"
not_lindepD: fixes S :: "'a vec set"   and A :: "'a vec set"   and f :: "'a vec \<Rightarrow> 'a" assumes "N.lin_indpt S"   and "finite A"   and "A \<subseteq> S"   and "f \<in> A \<rightarrow> carrier class_ring"   and "N.lincomb f A = \<zero>\<^bsub>M n\<^esub>" shows "f \<in> A \<rightarrow> {\<zero>\<^bsub>class_ring\<^esub>}"
vd2: fixes x :: "real" assumes "x \<le> 1"   and "1 < x + a"   and "x \<notin> (\<lambda>x. x - a + 1) ` S" shows "vector_derivative (shiftpath a g) (at x) = vector_derivative g (at (x + a - 1))"
P_3: shows "path_conforms_with_strategy p P well_ordered_strategy"
assms_1: shows "crit_pair_cbelow_on d m F p q"
when1_def: fixes P :: "bool"   and m :: "'d \<Rightarrow> (unit \<times> 'd) \<times> bool" shows "when1 P m \<equiv> if P then m else return ()"
insert: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'a" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
ltlr_expand_Release: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> R\<^sub>r \<psi> = \<xi> \<Turnstile>\<^sub>r \<psi> and\<^sub>r \<phi> or\<^sub>r X\<^sub>r (\<phi> R\<^sub>r \<psi>)"
member_in_C: shows "G \<turnstile> m member_in C"
set_random_bst: fixes t :: "'a tree"   and A :: "'a set" assumes "t \<in> set_pmf (random_bst A)"   and "finite A" shows "set_tree t = A"
epred_enat_unfold: fixes stop :: "'c \<Rightarrow> bool"   and "next" :: "'c \<Rightarrow> 'c"   and a :: "'c" shows "epred (enat_unfold stop next a) = (if stop a then 0 else enat_unfold stop next (next a))"
inf_closed: fixes A :: "'a set" assumes "A \<subseteq> carrier (fpl L f)" shows "\<Sqinter>\<^bsub>fpl L f\<^esub>A \<in> carrier (fpl L f)"
coprime_factorize: fixes qs :: "rat poly list" shows "pairwise_coprime_list (fst (factorize_polys qs))"
Hermite_of_upt_row_preserves_below: fixes k :: "nat"   and a :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'c" assumes "k \<le> mod_type_class.to_nat a" shows "Hermite_of_upt_row_i A k ass res $ a $ b = A $ a $ b"
Card_lt_csucc_iff: fixes \<kappa>' :: "V"   and \<kappa> :: "V" assumes "Card \<kappa>'"   and "Card \<kappa>" shows "(\<kappa>' < csucc \<kappa>) = (\<kappa>' \<le> \<kappa>)"
submodule_finrank: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and N :: "'b set" assumes "R_scalar_mult.R_finrank UNIV smult M"   and "N \<subseteq> M" shows "R_scalar_mult.R_finrank UNIV smult N"
apartment_simplices: fixes Y :: "'a set set" assumes "Y \<in> \<A>" shows "order.greater_eq X Y"
levelsCounter_auxNotEmpty: fixes a :: "('b \<times> bool) list" shows "\<forall>l. l \<noteq> [] \<longrightarrow> levelsCounter_aux a l \<noteq> []"
atype_ground_term_ex: fixes a :: "'atom" shows "\<exists>t. fv t = {} \<and> \<Gamma> t = Var a \<and> wf\<^sub>t\<^sub>r\<^sub>m t"
frequently_def: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter" shows "frequently P F = (\<not> (\<forall>\<^sub>F x in F. \<not> P x))"
leg1_composite: shows "src_src.leg1 = src.leg1 \<cdot> src_src.prj\<^sub>1"
AssertionStatEqRefl: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>"
TYREL_DOMAIN_def: fixes i :: "'c itself" shows "TYREL_DOMAIN i \<equiv> True"
IH1: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> fst (simp r1) \<rightarrow> v" shows "s \<in> r1 \<rightarrow> snd (simp r1) v"
prefix_map_rightE: fixes xs :: "'a list"   and f :: "'b \<Rightarrow> 'a"   and ys :: "'b list" assumes "prefix xs (map f ys)" shows "\<exists>xs'. prefix xs' ys \<and> xs = map f xs'"
vals2: fixes l :: "nat"   and var :: "nat \<times> char list"   and p :: "lambda" assumes "R l var \<in> vals p" shows "var \<in> vars p"
simps_1: fixes f :: "'a vec"   and mus :: "'a list"   and g' :: "'a vec" shows "mus_adjuster f [] mus g' = (mus, g')"
rel: shows "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> TargetTerm TQ"
meet_lower1: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<sqsubseteq> x"
solve_subset_Minimal_Solutions: shows "set (solve a b) \<subseteq> local.Minimal_Solutions"
comp_ide_self: fixes a :: "'a \<times> 'a" assumes "VV.ide a" shows "VV.comp a a = a"
adapt_bexp_vars: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and s :: "'c \<Rightarrow> nat" shows "Bexp.vars (adapt_bexp e s) = (\<lambda>v. symvar v s) ` Bexp.vars e"
elims_2: fixes x :: "primfun"   and xa :: "primfun" assumes "x < xa" obtains xb :: "nat" and y :: "nat" where "x = LnChain xb" and "xa = LnChain y" and "xb < y"
power_zero_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and k :: "num" assumes "class.semiring_1 one times plus zero" shows "power.power one times zero (numeral k) = zero"
proj_P2R_raw: fixes \<phi> :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "proj \<phi> = (\<lambda>x. P2R \<phi> `` {x})"
m_ac_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G"   and "z \<in> carrier G" shows "x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
rec_triangle_def: shows "rec_triangle = CN rec_quo [CN rec_mult [recf.Id 1 0, S], constn 2]"
conf_xcp'_def: shows "conf_xcp' \<equiv> \<lambda>uu uua uub. rec_option (\<lambda>P h. True) (\<lambda>a P h. \<exists>D. typeof_addr h a = \<lfloor>Class_type D\<rfloor> \<and> P \<turnstile> D \<preceq>\<^sup>* Throwable) uub uu uua"
intros: fixes aexp_vars :: "'a \<Rightarrow> 'b set"   and bexp_vars :: "'c \<Rightarrow> 'b set"   and some_val :: "'d"   and some_val' :: "'d"   and ev\<^sub>A :: "('b \<Rightarrow> 'd) \<Rightarrow> 'a \<Rightarrow> 'd"   and ev\<^sub>B :: "('b \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> bool"   and c\<^sub>1 :: "('b, 'a, 'c) Stmt"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'd"   and c\<^sub>2 :: "('b, 'a, 'c) Stmt"   and mem\<^sub>2 :: "'b \<Rightarrow> 'd"   and dma :: "'b \<Rightarrow> Sec"   and \<Gamma>' :: "'b \<Rightarrow> Sec option"   and \<Gamma>\<^sub>1 :: "'b \<Rightarrow> Sec option"   and \<Gamma>\<^sub>2 :: "'b \<Rightarrow> Sec option" assumes "sifum_types aexp_vars bexp_vars some_val some_val' ev\<^sub>A ev\<^sub>B"   and "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_security.mm_equiv dma (sifum_lang.eval\<^sub>w ev\<^sub>A ev\<^sub>B)"   and "\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High"   and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>1 c\<^sub>1 \<Gamma>'"   and "sifum_types.has_type aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>\<^sub>2 c\<^sub>2 \<Gamma>'"   and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>1"   and "sifum_types.mds_consistent dma mds \<Gamma>\<^sub>2" shows "(((c\<^sub>1, mds), mem\<^sub>1), (c\<^sub>2, mds), mem\<^sub>2) \<in> sifum_types.\<R>\<^sub>2 aexp_vars bexp_vars dma ev\<^sub>A ev\<^sub>B \<Gamma>'"
crsp_step_dec_b_e_pre: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and lm :: "nat list"   and n :: "nat"   and la :: "cell list"   and ra :: "cell list"   and ires :: "cell list"   and e :: "nat" assumes "ly = layout_of ap"   and "inv_locate_b (as, lm) (n, la, ra) ires"   and "abc_lm_v lm n = 0"   and "abc_fetch as ap = Some (Dec n e)" shows "\<exists>stp lb rb. steps (Suc (start_of ly as) + 2 * n, la, ra) (ci ly (start_of ly as) (Dec n e), start_of ly as - Suc 0) stp = (start_of ly e, lb, rb) \<and> dec_inv_1 ly n e (as, lm) (start_of ly e, lb, rb) ires"
self_codomain_iff_ide: fixes a :: "'a \<times> 'a" shows "(a \<in> CC.codomains a) = CC.ide a"
tcn: shows "cnf F = set (map set (cnf_lists F))"
simps_1: fixes l :: "nat"   and i :: "nat" shows "Label l \<oplus> i = Label (l + i)"
simps_6: fixes t :: "'c local.term" shows "Dom \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Dom t"
not_empty_rootsE: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "domain R"   and "p \<in> carrier (poly_ring R)"   and "ring.roots R p \<noteq> {#}" obtains a :: "'c" where "a \<in> carrier R" and "a \<in># ring.roots R p" and "[\<one>\<^bsub>R\<^esub>, \<ominus>\<^bsub>R\<^esub> a] \<in> carrier (poly_ring R)" and "[\<one>\<^bsub>R\<^esub>, \<ominus>\<^bsub>R\<^esub> a] pdivides\<^bsub>R\<^esub> p"
simps: fixes a :: "nat"   and b :: "nat"   and c :: "nat"   and d :: "nat" shows "sum4 a b c d = a + b + c + d"
norm_cis: fixes a :: "real" shows "cmod (cis a) = 1"
span: shows "adjuster n w us + w \<notin> local.span (set (us @ ws))"
arctan_upper_14_def: shows "arctan_upper_14 \<equiv> \<lambda>x. pi / 2 - 3 * x / (1 + 3 * x\<^sup>2)"
ex_type_definition: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A"   and "bi_unique A" shows "\<exists>Rep Abs. type_definition Rep Abs (Collect (Domainp A)) \<and> (\<forall>b b'. A b b' = (b = Rep b'))"
foldr_def: shows "foldr \<equiv> \<lambda>uu uua. rec_list (\<lambda>f. id) (\<lambda>x xs xsa f. f x \<circ> xsa f) uua uu"
top_in_filter: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and top :: "'c"   and F :: "'c set" assumes "class.order_top less_eq less top"   and "ord.filter less_eq F" shows "top \<in> F"
Int_assoc: fixes A :: "'h set"   and B :: "'h set"   and C :: "'h set" shows "A \<inter> B \<inter> C = A \<inter> (B \<inter> C)"
Lex_unlex: fixes lxs :: "'b lexlist" shows "Lex (unlex lxs) = lxs"
tailR_def: fixes t :: "('c, 'd) FingerTree" shows "tailR t \<equiv> if t = local.empty then local.empty else Abs_FingerTree (FingerTreeStruc.tailR (Rep_FingerTree t))"
equality: fixes r :: "('b, 'c) monoid_scheme"   and r' :: "('b, 'c) monoid_scheme" assumes "carrier r = carrier r'"   and "(\<otimes>\<^bsub>r\<^esub>) = (\<otimes>\<^bsub>r'\<^esub>)"   and "\<one>\<^bsub>r\<^esub> = \<one>\<^bsub>r'\<^esub>"   and "monoid.more r = monoid.more r'" shows "r = r'"
CollectD: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" assumes "a \<in> {x. P x}" shows "P a"
leg0_composite_2: shows "HHH\<mu>\<nu>\<pi>\<rho>.dom.leg0 = \<rho>.leg0 \<cdot> \<mu>\<nu>_\<pi>_\<rho>.Prj\<^sub>0"
f5: fixes a :: "'a"   and aa :: "'a"   and ab :: "'a" shows "ad (a \<cdot> (aa + ab)) = ad (a \<cdot> (ab + aa))"
injectionsFromEmptyAreEmpty: fixes Z :: "'d set" shows "set (injections_alg [] Z) = {{}}"
validI: fixes M :: "msg"   and x :: "tag"   and A :: "agent"   and B :: "agent" assumes "M \<in> payload" shows "implem (Chan x A B M) \<in> valid"
fps_X_fps_compose_startby0: fixes a :: "'a fps" assumes "a $ 0 = (0::'a)" shows "fps_X oo a = a"
IP: fixes x :: "'d"   and it :: "'d set"   and \<sigma> :: "'e" assumes "x \<in> S"   and "x \<notin> it"   and "it \<subseteq> S"   and "I it \<sigma>" shows "f x \<le> SPEC (\<lambda>f'. c \<sigma> f' \<le> SPEC (I (insert x it)))"
lspasl_c_der: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a" assumes "class.heap_sep_algebra plus zero sep_disj"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0"   and "heap_sep_algebra.tern_rel plus sep_disj h1 h3 h0" shows "heap_sep_algebra.tern_rel plus sep_disj h1 h2 h0 \<and> h2 = h3"
preserves_ide: fixes a :: "'a" assumes "hom.ide a" shows "hom'.ide (trg f \<star> a \<star> e)"
update_by_pair_ko_def: fixes ko :: "'a key_order" shows "update_by_pair_ko ko \<equiv> comparator.update_by_pair (key_compare ko)"
card_of_set_type: shows "|UNIV| <o |UNIV|"
arc_simple_path_subpath: fixes g :: "real \<Rightarrow> 'b"   and u :: "real"   and v :: "real" assumes "simple_path g"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "g u \<noteq> g v" shows "arc (subpath u v g)"
f_1: fixes A :: "'a set"   and B :: "'a set" assumes "P A"   and "P B" shows "f (A + B) = f A + f B"
iffd_iffb_demorgan: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "fd\<^sup>-\<^sub>\<F> \<phi> = (fb\<^sup>-\<^sub>\<F> \<circ> \<partial>\<^sub>F) \<phi>"
lunit'_simps_2: fixes f :: "'a" assumes "S.ide f" shows "S.src (S.lunit' f) = S.src f"
segment_bound_1: fixes x :: "'a"   and a :: "'a"   and b :: "'a" assumes "x \<in> closed_segment a b" shows "norm (x - a) \<le> norm (b - a)"
simps_7: fixes x :: "octo"   and y :: "octo" shows "Im6 (x - y) = Im6 x - Im6 y"
converse_rtrancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P b"   and "\<And>y z. \<lbrakk>(y, z) \<in> r; (z, b) \<in> r\<^sup>*; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
assms: shows "\<forall>i. 0 < i \<and> i < Suc n \<longrightarrow> s (x + i) = []"
not\<^sub>n_semantics: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" shows "w \<Turnstile>\<^sub>n not\<^sub>n \<phi> = (\<not> w \<Turnstile>\<^sub>n \<phi>)"
r9': fixes s :: "trm"   and x :: "name" shows "s to x in [Var x] \<mapsto> s"
arcsin_less_mono: fixes x :: "real"   and y :: "real" assumes "\<bar>x\<bar> \<le> 1"   and "\<bar>y\<bar> \<le> 1" shows "(arcsin x < arcsin y) = (x < y)"
fst_snd_linear: shows "linear (\<lambda>(x, y). x + y)"
Subset_E: fixes H :: "fm set"   and t :: "tm"   and u :: "tm"   and a :: "tm"   and A :: "fm" assumes "H \<turnstile> t SUBS u"   and "H \<turnstile> a IN t"   and "insert (a IN u) H \<turnstile> A" shows "H \<turnstile> A"
eq_scons: fixes w :: "'a stream"   and a :: "'a"   and v :: "'a stream" shows "(w = a ## v) = (a = shd w \<and> v = stl w)"
bdd_above_Ico: fixes a :: "'a"   and b :: "'a" shows "bdd_above {a..<b}"
simps_1: fixes uu :: "variable"   and S :: "variable set" shows "variant_variables 0 uu S = ([], S)"
inv_character: shows "character (Characters G) (inv_character (principal_char (Characters G)))"
meet_monofun1: fixes y :: "'c" shows "monofun (\<lambda>x. x \<sqinter> y)"
counterexample_secure: shows "secure P\<^sub>c I\<^sub>c id"
Abs_seqauto_inject: fixes x :: "'a set \<times> 'a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) set \<times> ('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set"   and y :: "'a set \<times> 'a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) set \<times> ('a \<times> (('a, 'b) expr \<times> ('c data \<Rightarrow> bool) \<times> 'b set \<times> 'c pupdate) \<times> 'a) set" assumes "x \<in> seqauto"   and "y \<in> seqauto" shows "(Abs_seqauto x = Abs_seqauto y) = (x = y)"
wf\<^sub>s\<^sub>s\<^sub>tI': fixes S :: "('d, 'c) stateful_strand_step list"   and V :: "'c set" assumes "(\<Union>x\<in>set S. case x of receive\<langle>x\<rangle> \<Rightarrow> fv x | \<langle>assign: x \<doteq> t'\<rangle> \<Rightarrow> fv t' | \<langle>check: x \<doteq> t'\<rangle> \<Rightarrow> {} | insert\<langle>t,t'\<rangle> \<Rightarrow> fv t \<union> fv t' | _ \<Rightarrow> {}) \<subseteq> V" shows "wf'\<^sub>s\<^sub>s\<^sub>t V S"
rt_fresh_as_def: shows "rt_fresh_as \<equiv> \<lambda>dip rt1 rt2. rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2 \<and> rt2 \<sqsubseteq>\<^bsub>dip\<^esub> rt1"
component_of_SKIP: fixes F :: "'c program" shows "\<bottom> component_of F"
prod_one_def: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) Ring_scheme" shows "prod_one I A \<equiv> \<lambda>x\<in>I. 1\<^sub>r\<^bsub>A x\<^esub>"
runiq_wrt_ex1: fixes R :: "('a \<times> 'b) set" shows "runiq R = (\<forall>a\<in>Domain R. \<exists>!b. (a, b) \<in> R)"
assms_1: shows "R - 1\<^sub>\<pi> \<noteq> {}"
isSimplePath_def: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "Graph.isSimplePath c u p v \<equiv> Graph.isPath c u p v \<and> distinct (pathVertices u p)"
last_prog_hd_prog_same: fixes p' :: "'b"   and sb :: "'b memref list"   and p :: "'b" assumes "last_prog p' sb = p"   and "hd_prog p' sb = p'" shows "hd_prog p sb = p'"
pred_stateD: fixes P :: "'b \<Rightarrow> bool"   and m :: "('c, 'b) state"   and s :: "'c"   and a :: "'b"   and s' :: "'c" assumes "pred_state P m"   and "run_state m s = (a, s')" shows "P a"
rep_eq: fixes x :: "'c \<Rightarrow>\<^sub>L 'd" shows "norm x = onorm (blinfun_apply x)"
coboundedI1: fixes c :: "'e multiset"   and a :: "'e multiset"   and b :: "'e multiset" assumes "c \<subseteq># a" shows "c \<subseteq># a \<union># b"
filter_remove1: fixes Q :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "filter Q (remove1 x xs) = remove1 x (filter Q xs)"
T_LUB: fixes S :: "nat \<Rightarrow> 'b process" assumes "chain S" shows "T (lim_proc (range S)) = \<Inter> (T ` range S)"
R1_SOME_4: fixes f :: "nat \<Rightarrow> nat option"   and r :: "recf" assumes "f \<in> \<R>"   and "r = (SOME r'. recfn 1 r' \<and> Partial_Recursive.total r' \<and> f = (\<lambda>x. eval r' [x]))" shows "f = (\<lambda>x. eval r [x])"
i_def: shows "Max {\<bar>f i\<bar> |i. i \<in> UNIV} = \<parallel>diag_mat f *v \<e> i\<parallel>"
list_relD2: fixes x :: "'a list"   and y :: "'b list"   and a :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool" assumes "(x, y) \<in> \<langle>br a I\<rangle>list_rel" shows "y = map a x \<and> (\<forall>x\<in>set x. I x)"
preserve_coincide2: assumes "vid1 \<notin> ODE_vars I (OVar vid1)" shows "sterm_sem I (f1 fid1 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol t))) = sterm_sem I (f1 fid1 vid1) (sol 0)"
l_one: fixes x :: "'a" assumes "x \<in> carrier \<G>" shows "\<one> \<otimes> x = x"
assms: shows "ell_field_class.add a p\<^sub>1 p\<^sub>2 = Infinity"
repeat_mset_compower: fixes n :: "nat"   and A :: "'b multiset" shows "repeat_mset n A = ((+) A ^^ n) {#}"
DomainI: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f" assumes "r a b" shows "Domainp r a"
irred_idom_nz: 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"   and x :: "'b" assumes "class.idom times one plus zero minus uminus"   and "x \<noteq> zero" shows "Divisibility.irreducible \<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr> x = comm_semiring_1.irreducible times one zero x"
scalar_prod_minus_distrib: fixes v\<^sub>1 :: "'c vec"   and n :: "nat"   and v\<^sub>2 :: "'c vec"   and v\<^sub>3 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (v\<^sub>2 - v\<^sub>3) = v\<^sub>1 \<bullet> v\<^sub>2 - v\<^sub>1 \<bullet> v\<^sub>3"
gen_lossless_gpv_False: fixes co :: "bool"   and \<I> :: "('e, 'f) \<I>" assumes "co = False" shows "gen_lossless_gpv co \<I> \<equiv> lfp (\<lambda>gen_lossless_gpv gpv. \<exists>pa. gpv = GPV pa \<and> lossless_spmf pa \<and> (\<forall>out c input. IO out c \<in> set_spmf pa \<longrightarrow> input \<in> responses_\<I> \<I> out \<longrightarrow> gen_lossless_gpv (c input)))"
Seq_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (e;; e2, xs) (e';; e2, xs')"
ccw_sorted_implies_ccw'_sortedP: fixes Ps :: "(real \<times> real) list" assumes "\<And>y z. \<lbrakk>y \<in> set Ps; z \<in> set Ps; y \<noteq> z\<rbrakk> \<Longrightarrow> det3 0 y z \<noteq> 0"   and "ccw.sortedP 0 Ps"   and "distinct Ps" shows "ccw'.sortedP 0 Ps"
children_h: shows "h \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children_h"
prems_2: shows "bal_i n (height \<langle>l, y, r\<rangle> + d)"
finite_ReZ_segments_joinpaths: fixes g1 :: "real \<Rightarrow> complex"   and z :: "complex"   and g2 :: "real \<Rightarrow> complex" assumes "finite_ReZ_segments g1 z"   and "finite_ReZ_segments g2 z"   and "path g1"   and "path g2"   and "pathfinish g1 = pathstart g2" shows "finite_ReZ_segments (g1 +++ g2) z"
moebius_mu_code: fixes n :: "nat" shows "moebius_mu n = of_int (dirichlet_inverse (\<lambda>_. 1) 1 n)"
pivot_funD_4: fixes A :: "'b mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "f i < nc" shows "A $$ (i, f i) = (1::'b)"
objE: fixes a :: "'a" assumes "B.obj a" obtains "B.ide a" and "src a = a" and "trg a = a"
is_pure_assnE: fixes a :: "assn" assumes "is_pure_assn a" obtains P :: "bool" where "a = \<up> P"
bigthetaD2: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> \<Theta>[F](g)" shows "f \<in> \<Omega>[F](g)"
simps: fixes x :: "'m"   and f :: "'a \<Rightarrow> ('a, 'm) optionT" shows "local.bind_option (OptionT x) f = OptionT (bind x (\<lambda>x. case x of None \<Rightarrow> return None | Some y \<Rightarrow> run_option (f y)))"
transT_MtransT: fixes cf :: "('test, 'atom) com \<times> 'state"   and s' :: "'state" assumes "cf \<rightarrow>t s'" shows "cf \<rightarrow>*t s'"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "('i, 'f, 'g, 'j) SLin_state_scheme" shows "P7 (s1, s2) = (\<forall>c. status s1 c = Aborted \<and> \<not> initialized s2 \<longrightarrow> pending s2 c = pending s1 c \<and> status s2 c \<in> {Pending, Aborted})"
comp_ipresIGFreshAbs: fixes hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'ae, 'w, 'af) model_scheme"   and MOD' :: "('ag, 'ah, 'aa, 'ab, 'ai, 'ad, 'aj, 'x, 'ak) model_scheme"   and hA' :: "'x \<Rightarrow> 'al"   and MOD'' :: "('am, 'an, 'aa, 'ao, 'ap, 'ad, 'aq, 'al, 'ar) model_scheme" assumes "ipresIGWlsAbs hA MOD MOD'"   and "ipresIGFreshAbs hA MOD MOD'"   and "ipresIGFreshAbs hA' MOD' MOD''" shows "ipresIGFreshAbs (hA' \<circ> hA) MOD MOD''"
point_measure_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> ennreal" shows "point_measure A f = density (count_space A) f"
Abs_atoms_inverse: fixes y :: "'b set" assumes "y \<in> range atom_map" shows "Rep_atoms (Abs_atoms y) = y"
seqE: fixes g :: "'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data" assumes "V.seq g f" obtains "V.arr f" and "V.arr g" and "V.dom g = V.cod f"
affine_alt: fixes s :: "'c set" shows "affine s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
abs_eq: fixes x :: "16 word" shows "uint8_of_uint16 (Abs_uint16 x) = Abs_uint8 (ucast x)"