hcju/isabelleps huggingface dataset:
denote_while_n_sum_lowner_le: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat"   and n :: "nat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M0 * M0 + adjoint M1 * M1 = 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>"   and "\<rho> \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) n \<le>\<^sub>L matrix_sum d (\<lambda>n. denote_while_n M0 M1 DS n \<rho>) (Suc n)"
induct: fixes P :: "'a spmf \<Rightarrow> ('b \<times> 'c \<times> 'd list + 'e \<times> 'f \<times> 'd list) list \<Rightarrow> bool"   and rest :: "('a, 'd, 'b, 'e, 'c, 'f, 'g) rest_scheme"   and a0 :: "'a spmf"   and a1 :: "('b \<times> 'c \<times> 'd list + 'e \<times> 'f \<times> 'd list) list" assumes "\<And>p. P p []"   and "\<And>p ia oaes tr. P (cond_spmf_fst (p \<bind> (\<lambda>s. rfunc_adv rest s ia)) oaes) tr \<Longrightarrow> P p (Inl (ia, oaes) # tr)"   and "\<And>p iu oues tr. P (cond_spmf_fst (p \<bind> (\<lambda>s. rfunc_usr rest s iu)) oues) tr \<Longrightarrow> P p (Inr (iu, oues) # tr)" shows "P a0 a1"
inf_sup_aci_1: fixes inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g"   and y :: "'g" assumes "class.lattice inf less_eq less sup" shows "inf x y = inf y x"
intros_2: fixes out :: "'e"   and rpv :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and input :: "'f"   and \<I> :: "('e, 'f) \<I>"   and x :: "'e" assumes "IO out rpv \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<I> out"   and "x \<in> outs_gpv \<I> (rpv input)" shows "x \<in> outs_gpv \<I> gpv"
leq_def: shows "leq \<equiv> \<lambda>A i. \<forall>k<i. A ! k \<le> pivot"
ge_zero: fixes x :: "'a" assumes "x \<in> E" shows "0 \<le> norm x"
nat_filtration_subalgebra: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat" assumes "infinite_coin_toss_space p M" shows "subalgebra M (infinite_coin_toss_space.nat_filtration M n)"
simps_3: fixes a :: "'b"   and l :: "('a, 'b) Digit"   and m :: "('a, 'b) FingerTreeStruc"   and r :: "('a, 'b) Digit" shows "FingerTreeStruc.is_measured_ftree (Deep a l m r) = ((FingerTreeStruc.is_measured_digit l \<and> FingerTreeStruc.is_measured_ftree m \<and> FingerTreeStruc.is_measured_digit r) \<and> a = FingerTreeStruc.gmd l + FingerTreeStruc.gmft m + FingerTreeStruc.gmd r)"
is_ground_cls_is_ground_atms_atms_of: fixes C :: "'a literal multiset" assumes "is_ground_cls C" shows "is_ground_atms (atms_of C)"
enum_Suc: fixes i :: "nat" assumes "i < n" shows "t.enum (Suc i) = (t.enum i)(u_t i := Suc (t.enum i (u_t i)))"
lc_Obj: fixes L :: "char list set"   and f :: "Label \<Rightarrow> sterm option"   and T :: "Sigma.type" assumes "finite L"   and "\<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> lc (the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup>)" shows "lc (Obj f T)"
int: shows "((\<lambda>x. \<Sum>b\<in>basis. F (g x) \<bullet> b * (vector_derivative g (at x within {0..1}) \<bullet> b)) has_integral c) {0..1}"
ccSup_upper: fixes A :: "'a set"   and x :: "'a" assumes "countable A"   and "x \<in> A" shows "x \<le> Sup A"
atLeast_Suc_lessThan: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'b" assumes "m < n" shows "prod g {m..<n} = g m * prod g {Suc m..<n}"
compliant_stateful_ACS: shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"
insert_first_part: fixes ref :: "'b option"   and r :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and oid :: "'b" assumes "ref = None \<or> ref = Some r \<and> r \<in> set xs" shows "insert_spec (xs @ ys) (oid, ref) = insert_spec xs (oid, ref) @ ys"
get_root_node_si_same_no_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child" shows "h \<turnstile> get_parent child \<rightarrow>\<^sub>r None"
case2: assumes "x \<le> 0"   and "0 \<le> y" shows "x * y \<le> 4611686018427387904"
preserves_comp: fixes g :: "'c"   and f :: "'c" assumes "C.seq g f" shows "C.map (g \<cdot>\<^sub>C f) = C.map g \<cdot>\<^sub>C C.map f"
has_derivative_iff_Ex: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative f') (at x) = (bounded_linear f' \<and> (\<exists>e. (\<forall>h. f (x + h) = f x + f' h + e h) \<and> (\<lambda>h. norm (e h) / norm h) \<midarrow>0::'a\<rightarrow> 0))"
ncoplanar_perm_13: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet C A D B"
inv_chart_restrict_chart: fixes S :: "'d set"   and c :: "('d, 'c) chart" shows "inv_chart (restrict_chart S c) = inv_chart c"
preserves_dom: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and f :: "'c" assumes "functor A B F"   and "partial_magma.arr A f" shows "partial_magma.dom B (F f) = F (partial_magma.dom A f)"
ntsem_def: 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 eintT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and eintF :: "('a, 'c) efsym \<Rightarrow> univ list \<Rightarrow> univ"   and eintP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c" assumes "ProblemIkTpart_TEModel wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw eintT eintF eintP" shows "ProblemIkTpart_TEModel.ntsem prot protFw eintT eintF \<sigma> \<equiv> if ProblemIkTpart.unprot prot protFw \<sigma> \<or> protFw \<sigma> then id else iimg {b. eintT \<sigma> b} (eintF (Tag \<sigma>) \<circ> singl)"
invrules_2: assumes "V \<inter> L \<noteq> {}" shows "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> L \<noteq> {}"
wt_instr_def: fixes P :: "'a prog"   and T :: "ty"   and mxs :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and i :: "instr"   and pc :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option list" shows "P,T,mxs,mpc,xt \<turnstile> i,pc :: \<tau>s \<equiv> app i P mxs T pc mpc xt (\<tau>s ! pc) \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt (\<tau>s ! pc)). P \<turnstile> \<tau>' \<le>' \<tau>s ! pc')"
rtip: shows "rt ip = Some (dsn, dsk, f, hops, nhop)"
graph_Suc: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and k :: "nat" shows "graph A w (Suc k) = (if even k then clean A w else prune A w) (graph A w k)"
axioms_1: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
le_diff_conv2: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b" shows "(c \<le> b - a) = (c + a \<le> b)"
assms_5: shows "j icd\<^bsup>\<pi>\<^esup>\<rightarrow> 0"
implInsec_synth_analz: fixes H :: "msg set"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "H \<subseteq> payload \<union> valid \<union> range LtK \<union> Tags"   and "implInsec A B M \<in> synth (analz H)" shows "implInsec A B M \<in> H \<or> M \<in> synth (analz H)"
subint1_irrefl_lemma2: fixes G :: "prog" assumes "ws_prog G" shows "\<forall>x. (x, x) \<notin> (subint1 G)\<^sup>+"
mult_cancel_le: fixes b :: "nat"   and a :: "nat"   and c :: "nat" assumes "b \<noteq> 0"   and "a * b \<le> c * b" shows "a \<le> c"
intro: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b"   and Y :: "'b set" assumes "CLF cl f"   and "Tarski_axioms cl f Y" shows "Tarski cl f Y"
radical_eq_0_iff: 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"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(factorial_semiring.radical zero times one normalize x = zero) = (x = zero)"
const_on_subset: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and y :: "'c"   and S' :: "'b set" assumes "const_on f S y"   and "S' \<subseteq> S" shows "const_on f S' y"
sorted2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and zs :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # y # zs) = (less_eq x y \<and> linorder.sorted less_eq (y # zs))"
normalize_in_carrier: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R"   and "set p \<subseteq> carrier R" shows "set (ring.normalize R p) \<subseteq> carrier R"
simps: fixes a :: "(('form fset \<times> 'form) \<times> ('rule \<times> 'form) NatRule) tree" shows "local.wf a = (\<exists>t. a = t \<and> snd (tree.root t) \<in> R \<and> effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t) \<and> (\<forall>x. x |\<in>| cont t \<longrightarrow> local.wf x))"
before_in_index: fixes l :: "'b list"   and x :: "'b"   and y :: "'b" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "x < y in l = (index l x = 0)"
subgroupE_1: fixes H :: "'a set set" assumes "subgroup H (G Mod N)" shows "H \<subseteq> carrier (G Mod N)"
abs_eq: fixes xa :: "'c poly"   and x :: "'c poly" shows "xa - x = Abs_poly (\<lambda>n. poly.coeff xa n - poly.coeff x n)"
Bleast_def: fixes S :: "'g set"   and P :: "'g \<Rightarrow> bool" shows "Bleast S P = (LEAST x. x \<in> S \<and> P x)"
arr_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.arr (\<F>C.cod f)"
8: shows "coreflexive (e * prim_P w v e\<^sup>T\<^sup>T)"
\<gamma>_st_def: fixes \<gamma> :: "'b \<Rightarrow> 'a set"   and F :: "'b st" shows "\<gamma>_st \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (fun F x)}"
AKcryptSKI: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and tt :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Says Tgs A \<lbrace>Crypt authK \<lbrace>Key servK, Agent B, tt\<rbrace>, X\<rbrace> \<in> set evs"   and "evs \<in> kerbV" shows "AKcryptSK authK servK evs"
snds_of_triples_form_comlist: fixes i :: "nat"   and \<Pi> :: "('a, 'b) chain" assumes "i < chainlen \<Pi>" shows "snd3 (nthtriple \<Pi> i) = comlist \<Pi> ! i"
NAKTr2: fixes j :: "nat"   and k :: "nat" assumes "Lcg R M j"   and "k < (LEAST j. Lcg R M j)" shows "\<not> Lcg R M k"
lspasl_c: fixes Gamma :: "bool"   and h1 :: "'a"   and h2 :: "'a"   and h0 :: "'a"   and h3 :: "'a"   and Delta :: "bool" assumes "Gamma \<and> (h1,h2\<triangleright>h0) \<and> h2 = h3 \<longrightarrow> Delta" shows "Gamma \<and> (h1,h2\<triangleright>h0) \<and> (h1,h3\<triangleright>h0) \<longrightarrow> Delta"
neg_inf_h: assumes "sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} \<noteq> []"   and "x1 < sorted_list_of_set {x. \<exists>q\<in>set qs. rpoly q x = 0} ! 0" shows "sga = sign_vec qs (real_of_int (- crb (prod_list (cast_rat_list qs))))"
phl_while: fixes p :: "'a"   and s :: "'a"   and w :: "'a"   and x :: "'a" assumes "\<lparr>p\<rparr>s\<lparr>s \<cdot> p \<cdot> s\<rparr>"   and "\<lparr>p\<rparr>w\<lparr>w \<cdot> p \<cdot> w\<rparr>"   and "\<lparr>p \<cdot> s\<rparr>x\<lparr>x \<cdot> p\<rparr>" shows "\<lparr>p\<rparr>(s \<cdot> x)\<^sup>\<star> \<cdot> w\<lparr>(s \<cdot> x)\<^sup>\<star> \<cdot> w \<cdot> (p \<cdot> w)\<rparr>"
derived_subgroup_is_normal: fixes H :: "'a set" assumes "subgroup H G" shows "derived G H \<lhd> G\<lparr>carrier := H\<rparr>"
evaln_exprs_eq: fixes G :: "prog"   and s :: "state"   and t :: "expr list"   and n :: "nat"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>s \<midarrow>In3 t\<succ>\<midarrow>n\<rightarrow> (w, s') = (\<exists>vs. w = In3 vs \<and> G\<turnstile>s \<midarrow>t\<doteq>\<succ>vs\<midarrow>n\<rightarrow> s')"
HEndPhase2_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
List_upd_same_lemma: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" assumes "p \<noteq> Null" shows "\<not> List p (h(p := p)) ps"
map: fixes y :: "nat" shows "map_fm_binders (\<lambda>a x. liftAtom (i + x) amount a) F (y + x1) = map_fm_binders (\<lambda>a x. liftAtom (i + x1 + x) amount a) F y"
arr_char: fixes f :: "'a" shows "Right_a.arr f = right a f"
map_sel_1: fixes a :: "'g list"   and f :: "'g \<Rightarrow> 'h" assumes "a \<noteq> []" shows "hd (map f a) = f (hd a)"
size_of_rhs_R: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "(nat list \<times> nat list) list" shows "dim_vec (v_vec_R p qs subsets) = length subsets"
image_ideal_subset: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c set" assumes "\<And>x y. h (x + y) = h x + h y"   and "\<And>x y. h (x * y) = h x * h y" shows "h ` ideal F \<subseteq> ideal (h ` F)"
equivalence_refines_to_adjoint_equivalence: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a" assumes "equivalence_map f"   and "\<guillemotleft>g : trg f \<rightarrow> src f\<guillemotright>"   and "ide g"   and "\<guillemotleft>\<eta> : src f \<Rightarrow> g \<star> f\<guillemotright>"   and "local.iso \<eta>" shows "\<exists>!\<epsilon>. adjoint_equivalence_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
that: fixes T :: "'a set" assumes "T \<subseteq> S"   and "closed T"   and "T \<noteq> {}"   and "\<not> connected_component (- T) a b"   and "\<And>U. U \<subset> T \<Longrightarrow> connected_component (- U) a b" shows "thesis"
preserves_reflects_arr: fixes f :: "'c" shows "D.arr (src\<^sub>D f) = D.arr f"
assms_2: shows "b \<in>\<^sub>\<circ> dg_prod (I -\<^sub>\<circ> J) \<AA>\<lparr>Obj\<rparr>"
tendsto_iff: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and l :: "'b"   and F :: "'c filter" assumes "class.metric_space dist uniformity open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>e>0. \<forall>\<^sub>F x in F. dist (f x) l < e)"
radical_mono: fixes F :: "'c set"   and G :: "'c set" assumes "F \<subseteq> G" shows "\<surd>F \<subseteq> \<surd>G"
transformation_by_components_def: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and t :: "'d \<Rightarrow> 'e" shows "transformation_by_components A B F G t \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> transformation_by_components_axioms A B F G t"
order_root: fixes p :: "'a poly"   and a :: "'a" shows "(poly p a = (0::'a)) = (p = 0 \<or> order a p \<noteq> 0)"
exec_move_InstanceOfI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and T :: "ty" assumes "exec_move ci P t e h s ta h' s'" shows "exec_move ci P t (e instanceof T) h s ta h' s'"
Upper_elemI: fixes A :: "'a set"   and x :: "'a" assumes "\<And>y. y \<in> A \<Longrightarrow> y \<sqsubseteq> x"   and "x \<in> carrier L" shows "x .\<in> Upper L A"
calc_ideal_points_cvec_different: fixes A :: "complex"   and B :: "complex" assumes "(cmod A)\<^sup>2 < (cmod B)\<^sup>2"   and "is_real A" shows "\<not> calc_ideal_point1_cvec A B \<approx>\<^sub>v calc_ideal_point2_cvec A B"
dlist_eq_def: shows "dlist_eq = BNF_Def.vimage2p remdups remdups (=)"
rev_alternating_list: fixes n :: "nat"   and s :: "'a"   and t :: "'a" shows "rev (alternating_list n s t) = (if gcd_nat.greater_eq n 2 then alternating_list n t s else alternating_list n s t)"
set_OAlist_tc_sorted_domain: fixes xs :: "('b, 'c) oalist_tc" shows "set (OAlist_tc_sorted_domain xs) = fst ` set (list_of_oalist_tc xs)"
\<section>: shows "homotopic_with_canon Q t U (f \<circ> h \<circ> k) ((\<lambda>x. c) \<circ> k)"
ipd_2: shows "\<pi>' l' = ipd (\<pi>' k')"
add_div_trivial: fixes c :: "int"   and b :: "int"   and a :: "int" assumes "0 \<le> c"   and "c < b" shows "(a * b + c) div b = a"
boundary_of_region_is_subset_of_partition_boundaries_1: shows "one_chain_typeI \<subseteq> two_chain_boundary two_chain_typeI"
assms: fixes x :: "'v" shows "\<not> edge \<Delta> x x"
H: fixes xy :: "'v \<times> 'v" shows "(\<lambda>n. h (k n) xy) \<longlonglongrightarrow> H xy"
st_mult: fixes x :: "real star"   and y :: "real star" assumes "x \<in> HFinite"   and "y \<in> HFinite" shows "st (x * y) = st x * st y"
induct: fixes P :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> 'c \<times> 'd \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a1 :: "'c \<times> 'd"   and a2 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>uu_ uv_ t1 a1 t2 a2 t3 a3 t4. P (R uu_ (R uv_ t1 a1 t2) a2 t3) a3 t4"   and "\<And>a1 uw_ t2 a2 ux_ t3 a3 t4. P (R uw_ \<langle>\<rangle> a1 (R ux_ t2 a2 t3)) a3 t4"   and "\<And>v vc vf vb a1 uw_ t2 a2 ux_ t3 a3 t4. P (R uw_ (B vf v vc vb) a1 (R ux_ t2 a2 t3)) a3 t4"   and "\<And>a t2. P \<langle>\<rangle> a t2"   and "\<And>v ve vb a t2. P (B ve \<langle>\<rangle> v vb) a t2"   and "\<And>va a t2. P \<langle>\<langle>\<rangle>, va, \<langle>\<rangle>\<rangle> a t2"   and "\<And>va v ve vh vd a t2. P \<langle>\<langle>\<rangle>, va, B vh v ve vd\<rangle> a t2"   and "\<And>vc vf vi ve v vh vb a t2. P (B vh (B vi vc vf ve) v vb) a t2"   and "\<And>vc vf vi ve va a t2. P \<langle>B vi vc vf ve, va, \<langle>\<rangle>\<rangle> a t2"   and "\<And>vc vf vi ve va v vh vl vg a t2. P \<langle>B vi vc vf ve, va, B vl v vh vg\<rangle> a t2"   and "\<And>v vc vf vb a t2. P (B vf v vc vb) a t2" shows "P a0 a1 a2"
gcd_self: fixes a :: "'a" shows "gcd a a = normalize a"
powr_ge_pzero: fixes x :: "real"   and y :: "real" shows "0 \<le> x powr y"
a1: shows "Sample_L req dt a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st buffer"
linorder_wlog: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less"   and "\<And>a b. less_eq a b \<Longrightarrow> P a b"   and "\<And>a b. P b a \<Longrightarrow> P a b" shows "P a b"
qAFreshAll_qSwapAll_compose: fixes zs :: "'a"   and y :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and z :: "'b"   and x :: "'b"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" shows "(qAFresh zs y X \<and> qAFresh zs z X \<longrightarrow> X #[[y \<and> x]]_zs #[[z \<and> y]]_zs = X #[[z \<and> x]]_zs) \<and> (qAFreshAbs zs y A \<and> qAFreshAbs zs z A \<longrightarrow> A $[[y \<and> x]]_zs $[[z \<and> y]]_zs = A $[[z \<and> x]]_zs)"
qtimes_left_unit: fixes x :: "'e['f, 'g]" shows "\<one> \<^bold>\<cdot> x \<cong>\<^sub>Q x"
pref_pref_lcp': fixes w :: "'b list"   and r :: "'b list"   and w' :: "'b list"   and s :: "'b list" assumes "w \<le>p r"   and "w' \<le>p s" shows "w \<and>\<^sub>p w' \<le>p r \<and>\<^sub>p s"
paracompact: fixes S :: "'a set"   and \<C> :: "'a set set" assumes "S \<subseteq> \<Union> \<C>"   and "\<And>T. T \<in> \<C> \<Longrightarrow> open T" obtains \<C>' :: "'a set set" where "S \<subseteq> \<Union> \<C>'" and "\<And>U. U \<in> \<C>' \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<C> \<and> U \<subseteq> T)" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>'. U \<inter> V \<noteq> {}}"
scale: fixes A :: "(('b, 'd) vec, 'c) vec"   and r :: "'b"   and b :: "('b, 'd) vec" shows "A *v (r *s b) = r *s (A *v b)"
exists_fresh': fixes x :: "'a" assumes "finite (supp x)" shows "\<exists>c. c \<sharp> x"
carrier_product_group: fixes I :: "'f set"   and G :: "'f \<Rightarrow> ('g, 'h) monoid_scheme" shows "carrier (product_group I G) = (\<Pi>\<^sub>E i\<in>I. carrier (G i))"
eSuc_max: fixes x :: "enat"   and y :: "enat" shows "eSuc (max x y) = max (eSuc x) (eSuc y)"
is_cnf_encode_all_operator_preconditions: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "is_cnf (encode_all_operator_preconditions \<Pi> (\<Pi>\<^sub>\<O>) t)"
pmf_positive_iff: fixes p :: "'a pmf"   and x :: "'a" shows "(0 < pmf p x) = (x \<in> set_pmf p)"
1: shows "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<^bold>\<not>\<phi> \<alpha>) in v]"
simps_2: fixes x :: "('a, 'b) Node"   and xs :: "('a, 'b) Node list"   and t :: "('a, 'b) FingerTreeStruc" shows "FingerTreeStruc.lconsNlist (x # xs) t = FingerTreeStruc.nlcons x (FingerTreeStruc.lconsNlist xs t)"
assms: shows "map (\<lambda>L. L \<cdot>l \<eta>) (map Neg As') = map Neg As"
extends_def: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list" shows "\<Delta> extends \<Gamma> \<equiv> \<forall>X Q. TVarB X Q \<in> set \<Gamma> \<longrightarrow> TVarB X Q \<in> set \<Delta>"
circle_polar_smooth: shows "circle_polar C1_differentiable_on {0..1}"
div_field_poly_impl: shows "(div) = div_field_poly_impl"
coboundedI1: fixes a :: "'a word"   and c :: "'a word"   and b :: "'a word" assumes "a \<le>s c" shows "signed.min a b \<le>s c"
step_7_correct: fixes p :: "prog" assumes "spec\<^sub>7 p" shows "spec\<^sub>6 p"
iMODb_cut_less: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and t :: "nat" shows "[ r, mod m, c ] \<down>< t = (if t \<le> r then {} else if r + m * c < t then [ r, mod m, c ] else [ r, mod m, (t - Suc r) div m ])"
length_negGorbit_list_sublist: fixes fs :: "'w list"   and gs :: "'g list"   and T :: "'a \<Rightarrow> 'w"   and as :: "'a list" assumes "fs \<in> set (HRep.negGorbit_list gs T as)" shows "length fs = length as"
simps_1: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.ins t [] = [t]"
no_disequation_in_cl_def: fixes t :: "'a trm"   and C :: "'a literal set" shows "no_disequation_in_cl t C = (\<forall>u v. Neg (Eq u v) \<in> C \<longrightarrow> u \<noteq> t \<and> v \<noteq> t)"
qGoodEnv_iff: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" shows "qGoodEnv rho = ((\<forall>xs. liftAll qGood (rho xs)) \<and> (\<forall>ys. |{y. rho ys y \<noteq> None}| <o |UNIV|) \<and> |{X. \<exists>ys y. rho ys y = Some X}| <o |UNIV|)"
unit_vecs_carrier: fixes n :: "nat" shows "set (unit_vecs n) \<subseteq> carrier_vec n"