hcju/isabelleps huggingface dataset:
IH_5: fixes as :: "accent list"   and bs :: "accent list" assumes "d # ds = as @ ([Macron] @ [Grave]) @ bs" shows "thesis"
uGraph_from_list_invar_conv: fixes L :: "(nat \<times> int \<times> nat) list" shows "uGraph_from_list_invar L = fromlist L"
mult_scaleR_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and a :: "real"   and y :: "'a" assumes "class.real_algebra plus zero minus uminus scaleR times" shows "times x (scaleR a y) = scaleR a (times x y)"
upd_left_comm: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "comp_fun_commute (\<lambda>aa. If (a = aa) (f aa))"
104: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> - y = x"
g: shows "g \<in> M \<rightarrow>\<^sub>M L"
wt_sub: fixes \<Gamma> :: "(nat \<times> ty) list"   and e :: "exp"   and A :: "ty"   and B :: "ty" assumes "\<Gamma> \<turnstile> e : A"   and "A <: B" shows "\<Gamma> \<turnstile> e : B"
map_indets_injI: fixes f :: "'c \<Rightarrow> 'd" assumes "inj f" shows "inj (map_indets f)"
mat_hom_pow: fixes A :: "real mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "of_real_hom.mat_hom (A ^\<^sub>m k) = of_real_hom.mat_hom A ^\<^sub>m k"
t_def: shows "t \<equiv> \<lambda>x. if s x \<up> then None else if s x \<down>= 0 then Some 0 else Some (Suc (the (c (the (s x) - 1))))"
phiArg_def: fixes \<alpha>n :: "'b \<Rightarrow> 'c list"   and defs :: "'b \<Rightarrow> 'c \<Rightarrow> 'd set"   and phis :: "'b \<Rightarrow> 'c \<times> 'd \<Rightarrow> 'd list option"   and g :: "'b"   and v :: "'d"   and v' :: "'d" shows "CFG_SSA_wf_base.phiArg \<alpha>n defs phis g v v' \<equiv> \<exists>vs. CFG_SSA_wf_base.phi \<alpha>n defs phis g v = Some vs \<and> v' \<in> set vs"
length_it_def: fixes xs :: "'b list" shows "length_it xs = length_it' 0 xs"
vconn_implies_hpar_4: fixes \<mu> :: "'c"   and f :: "'c"   and g :: "'c" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C g = trg\<^sub>C f"
intro: fixes V :: "'a set"   and R :: "('a, 'b) ring_scheme"   and K :: "'a set" assumes "subgroup V (add_monoid R)"   and "subalgebra_axioms K V R" shows "subalgebra K V R"
count_single: fixes b :: "'d"   and a :: "'d" shows "count {#b#} a = (if b = a then 1 else 0)"
typing_inversion_9: fixes \<Gamma> :: "(name \<times> ty) list"   and n :: "nat"   and T :: "ty" assumes "\<Gamma> \<turnstile> NUM n : T" obtains "T = tINT"
WtFT_le_1: fixes cl :: "('d, 'e, 'f) cmd list" shows "WtFT cl \<le> 1"
A_invariant: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Bisimulation_Invariant A B sim PA PB"   and "PA a"   and "A a b" shows "PA b"
infsetsum_nonneg: fixes A :: "'d set"   and f :: "'d \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> 0 \<le> f x" shows "0 \<le> infsetsum f A"
fun_eq_on_restrict1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'b" shows "fun_eq_on (restrict1 f A) f A"
R_lin_independent_ConsI: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and ms :: "'b list"   and m :: "'b" assumes "R_scalar_mult R"   and "R_scalar_mult.R_lin_independent R smult ms"   and "\<And>r rs. \<lbrakk>set (r # rs) \<subseteq> R; scalar_mult.lincomb smult (r # rs) (m # ms) = (0::'b)\<rbrakk> \<Longrightarrow> r = (0::'a)" shows "R_scalar_mult.R_lin_independent R smult (m # ms)"
fmsubset_eq: fixes s1 :: "('d, 'e) fmap"   and s2 :: "('d, 'e) fmap" assumes "s1 \<subseteq>\<^sub>f s2" shows "\<forall>a. a |\<in>| fmdom s1 \<longrightarrow> fmlookup s1 a = fmlookup s2 a"
relcomppE: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'a"   and c :: "'d" assumes "(r OO s) a c" obtains b :: "'b" where "r a b" and "s b c"
surjective: fixes r :: "('b, 'c) arrow_of_spans_data_scheme" shows "r = \<lparr>Chn = Chn r, Dom = Dom r, Cod = Cod r, \<dots> = arrow_of_spans_data.more r\<rparr>"
heq_1: shows "fs ! (i - 1) = fs' ! i"
pmf_bernoulli_False: fixes p :: "real" assumes "0 \<le> p"   and "p \<le> 1" shows "pmf (bernoulli_pmf p) False = 1 - p"
ThinChamberComplexFolding_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" shows "ThinChamberComplexFolding X f \<equiv> ThinChamberComplex X \<and> ChamberComplexFolding X f"
sum_character: shows "sum (principal_char C) (carrier C) = (if principal_char C = principal_char C then of_nat (order C) else 0)"
is_unit_monom_trivial: fixes p :: "'a poly" assumes "is_unit p" shows "monom (lead_coeff p) 0 = p"
real_cond_exp_int_1: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "integrable M (real_cond_exp M F f)"
tllist_all2_mono: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist"   and P' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "tllist_all2 P Q xs ys"   and "\<And>x y. P x y \<Longrightarrow> P' x y"   and "\<And>x y. Q x y \<Longrightarrow> Q' x y" shows "tllist_all2 P' Q' xs ys"
ncols_eq_card_columns: fixes A :: "(('b, 'c) vec, 'd) vec" shows "ncols_iarray (matrix_to_iarray A) = CARD('c)"
irreducibleE: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a" assumes "Divisibility.irreducible G a" obtains "a \<notin> Units G" and "\<forall>b. b \<in> carrier G \<and> Divisibility.properfactor G b a \<longrightarrow> b \<in> Units G"
all_nat_less_eq: fixes n :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<forall>m<n. P m) = (\<forall>m\<in>{0..<n}. P m)"
type_impl_card_n_enum: fixes n :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<n}" obtains enuma :: "'b list" and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool" and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" where "class.enum enuma enum_all enum_ex \<and> n = CARD('b)"
typeof_Class_Subo: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val"   and C :: "char list" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Class C\<rfloor>" shows "\<exists>a Cs D S. v = Ref (a, Cs) \<and> h a = \<lfloor>(D, S)\<rfloor> \<and> Subobjs P D Cs \<and> last Cs = C"
induct: fixes P :: "'b heap list \<Rightarrow> bool"   and a0 :: "'b heap list" assumes "P []"   and "\<And>hs. P hs \<Longrightarrow> P (heap.Empty # hs)"   and "\<And>x hsl hsr. \<lbrakk>P hsl; P hsr\<rbrakk> \<Longrightarrow> P (Hp x hsl # hsr)" shows "P a0"
fresh_2: fixes a :: "name"   and x2 :: "ty"   and x1 :: "ty" shows "a \<sharp> x2\<rightarrow>x1 = (a \<sharp> x2 \<and> a \<sharp> x1)"
ipv4addr_of_nat_nat_of_ipv4addr: fixes addr :: "32 word" shows "ipv4addr_of_nat (nat_of_ipv4addr addr) = addr"
v11_vimage_vpsubset_neq: fixes A :: "V"   and B :: "V" assumes "A \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> f"   and "B \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> f"   and "A \<noteq> B" shows "f `\<^sub>\<circ> A \<noteq> f `\<^sub>\<circ> B"
intro: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" assumes "PreFunctor F"   and "FunctorM_axioms F" shows "FunctorM F"
simps: fixes f :: "'c \<Rightarrow> 'b"   and a :: "'c"   and b :: "'c" shows "pairself f (a, b) = (f a, f b)"
int_ops_5: fixes a :: "nat"   and b :: "nat" shows "int (a + b) = int a + int b"
path_emptyI: fixes s :: "'c" shows "lts.path s [] s"
Ex_rev_conv: fixes P :: "'b list \<Rightarrow> bool" shows "(\<exists>x. P (rev x)) = (\<exists>y. P y)"
wnf_lemma_1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(- x \<squnion> - y) \<sqinter> (- - x \<squnion> - z) \<squnion> - x = - x \<squnion> - y"
endpoints: shows "minimum (i \<squnion> j) = minimum k \<and> maximum (i \<squnion> j) = maximum k"
simps_1: fixes I :: "('b, 'c, 'a) interp"   and p :: "'a" shows "prog_sem I ($\<alpha> p) = Programs I p"
consistent_maps_def: fixes D :: "'e \<Rightarrow> 'f"   and E :: "'g \<Rightarrow> 'f"   and p :: "'e \<Rightarrow> 'h"   and q :: "'g \<Rightarrow> 'h" shows "consistent_maps D E p q \<equiv> \<forall>x\<in>range p \<inter> range q. D (inv p x) = E (inv q x)"
mult_less_iff1: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "(0::'c) < z" shows "(x * z < y * z) = (x < y)"
Red_F_Bot: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set"   and B :: "'a"   and N :: "'a set" assumes "calculus Bot Inf entails Red_I Red_F"   and "B \<in> Bot"   and "entails N {B}" shows "entails (N - Red_F N) {B}"
permutations_of_setI: fixes xs :: "'b list"   and A :: "'b set" assumes "set xs = A"   and "distinct xs" shows "xs \<in> permutations_of_set A"
assms_2: shows "M *\<^sub>m\<^sub>m A = M *\<^sub>m\<^sub>m B"
circ_mult_sub: fixes x :: "'a"   and y :: "'a" shows "Rf (x * (y * x)\<^sup>\<star>) y \<le> (x * y)\<^sup>\<star>"
t4: shows "typ_of (subst_bv (Fv x \<tau>') s) = typ_of (Abs \<tau>' t $ Fv x \<tau>')"
simps_2: shows "cospan_shape.comp cospan_shape.AT cospan_shape.AA = cospan_shape.AT"
noprishr_parts_knowledge_equiv: fixes X :: "msg"   and A :: "agent"   and evs :: "event list"   and B :: "agent" assumes "X \<notin> {Key (priEK A), Key (priSK A), Key (shrK A)}"   and "X \<in> parts (knows A evs)" shows "X \<in> parts (knows B evs) \<union> parts {X. Notes A X \<in> set evs}"
wf_mdecl_def: fixes wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, Ts, T, mb). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> T \<noteq> NT \<and> wf_md P C (M, Ts, T, mb)"
Union_orbits: fixes f :: "'b perm" shows "\<Union> (orbits f) = affected f"
atomic_Nonce: fixes X :: "nonce_t" shows "atomic (Nonce X)"
v1': shows "wfactors G (as' ! i # take i as' @ drop (Suc i) as') (as' ! i \<otimes> (aa_1 \<otimes> aa_2))"
cases: fixes a1 :: "'a literal multiset"   and a2 :: "'a literal multiset"   and a3 :: "'a literal multiset" assumes "unord_resolve a1 a2 a3" obtains C :: "'a literal multiset" and n :: "nat" and A :: "'a" and D :: "'a literal multiset" where "a1 = C + replicate_mset (Suc n) (Pos A)" and "a2 = add_mset (Neg A) D" and "a3 = C + D"
finite_varsA: fixes at :: "('c, 'd) atm" shows "finite (varsA at)"
RDirProd_list_zero: fixes Rs :: "('a, 'b) ring_scheme list" shows "\<zero>\<^bsub>RDirProd_list Rs\<^esub> = foldr (\<lambda>R. (#) \<zero>\<^bsub>R\<^esub>) Rs []"
is_path_memb: fixes v :: "'v"   and p :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_path v p v'" shows "v \<in> V \<and> v' \<in> V"
pointer_def: fixes nil :: "'a" shows "pointer nil \<equiv> nil = nil"
sums_summable: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "f sums l" shows "summable f"
hoareQ_complete: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<Turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}"
Posix_Plus1: fixes s :: "'b list"   and r1 :: "'b rexp"   and v :: "'b val"   and r2 :: "'b rexp" assumes "s \<in> r1 \<rightarrow> v" shows "s \<in> Plus r1 r2 \<rightarrow> val.Left v"
rep_eq: shows "(\<le>) = (\<lambda>X Y. Rep_downset X \<subseteq> Rep_downset Y)"
sel_4: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "args (Fun x21 x22) = x22"
eval_map_all: fixes f :: "'a \<Rightarrow> atomUni fmUni"   and L :: "'a list"   and x :: "real" shows "(\<forall>l\<in>set (map f L). evalUni l x) = (\<forall>l\<in>set L. evalUni (f l) x)"
vec_comp_allE: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a vec"   and w :: "'a vec"   and i :: "nat" assumes "vec_comp_all r v w"   and "i < dim_vec v" shows "r (v $ i) (w $ i)"
sign_r_pos_at_left: fixes p :: "real poly"   and c :: "real" assumes "p \<noteq> 0" shows "if even (order c p) = sign_r_pos p c then \<forall>\<^sub>F x in at_left c. 0 < poly p x else \<forall>\<^sub>F x in at_left c. poly p x < 0"
simps: fixes a :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node"   and get_return_edges :: "'l \<Rightarrow> 'l set"   and valid_edge :: "'l \<Rightarrow> bool"   and src :: "'l \<Rightarrow> 'g"   and trg :: "'l \<Rightarrow> 'g"   and knd :: "'l \<Rightarrow> ('h, 'i, 'j, 'k) edge_kind"   and e :: "'g LDCFG_node \<times> ('h, 'i, 'j, 'k) edge_kind \<times> 'g LDCFG_node" shows "(a \<in> lift_get_return_edges get_return_edges valid_edge src trg knd e) = (\<exists>aa a' e'. a = e' \<and> e = (Node (src aa), knd aa, Node (trg aa)) \<and> valid_edge aa \<and> a' \<in> get_return_edges aa \<and> e' = (Node (src a'), knd a', Node (trg a')))"
append_eq_conv_conj: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
Min_def: shows "Min \<equiv> semilattice_set.F min"
111: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- x \<sqinter> y \<squnion> (- - x \<sqinter> y \<squnion> z) = y \<squnion> z"
prv_eql_subst_trm2: fixes x :: "'var"   and \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "prv (eql t1 t2)" shows "prv (imp (subst \<phi> t1 x) (subst \<phi> t2 x))"
step: fixes y :: "'a set" assumes "y \<subseteq> S" shows "f\<cdot>y \<subseteq> S"
subcategory_axioms: shows "subcategory (\<cdot>) (left b)"
1: shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"
ln_one: fixes dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and ln :: "'e \<Rightarrow> 'e" assumes "class.ln dist uniformity open minus norm plus zero uminus scaleR sgn times one ln" shows "ln one = zero"
last_seg_code_def: shows "last_seg_code \<equiv> \<lambda>(x, xa, xb, xc). WHILE\<^sub>T (\<lambda>(xe, xf). xe < (if as_length xa - 1 + 1 = as_length xa then as_length x else as_get xa (as_length xa - 1 + 1))) (\<lambda>(xe, xf). let y' = as_get x xe in RETURN (Suc xe, y' # xf)) (as_get xa (as_length xa - 1), []) \<bind> (\<lambda>(xd, y). RETURN y)"
hyps_2: fixes y :: "'a" assumes "y \<in> set xs" shows "compare cmp x y \<noteq> Greater"
bounded_countable_network_axioms: shows "bounded_countable_network \<Delta>"
arctan_lower_bound: fixes x :: "real" assumes "0 \<le> x" shows "x / (1 + x\<^sup>2) \<le> arctan x"
of_int_eq_numeral_power_cancel_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and y :: "int"   and x :: "num"   and n :: "nat" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(ring_1.of_int one plus zero minus y = power.power one times (numeral.numeral one plus x) n) = (y = numeral x ^ n)"
simps_1: fixes T :: "Unary_PCF.type"   and i :: "nat" shows "\<pi> T i 0 = (if i < ar T then nth_drop i (dest_fun T) \<rightarrow>\<rightarrow> \<B> else \<B>)"
hasread: shows "\<forall>d. hasRead s' q d qq = hasRead s q d qq"
intros_17: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>2 :: "('b, 'a, 'c) com"   and s :: "'b" shows "\<Gamma>\<turnstile> (Catch Throw c\<^sub>2, Normal s) \<rightarrow> (c\<^sub>2, Normal s)"
neg_le_minus_divideC_eq: fixes c :: "complex"   and a :: "'b"   and b :: "'b" assumes "c < 0" shows "(a \<le> - (b /\<^sub>C c)) = (- b \<le> c *\<^sub>C a)"
MP_setI: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and x :: "('d, 'e) strand_step"   and S :: "('d, 'e) strand_step list" assumes "typed_model arity public Ana \<Gamma>"   and "x \<in> set S" shows "trms\<^sub>s\<^sub>t\<^sub>p x \<subseteq> trms\<^sub>s\<^sub>t S"
closed_orbit_recurrence_times_bdd_below: fixes x :: "'a" shows "bdd_below {T \<in> existence_ivl0 x. 0 < T \<and> flow0 x T = x}"
parts_crypt_key_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and K :: "key"   and K' :: "key" assumes "s \<turnstile> s'"   and "Crypt K (Key K') \<in> parts (used s')"   and "Crypt K (Key K') \<notin> parts (used s)"   and "K' \<notin> range PriK \<union> range PubK" shows "{Key K', Key K} \<subseteq> spied s'"
headR_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "headR t = (case viewR t of Some (a, x) \<Rightarrow> a)"
less: fixes j :: "int" assumes "nat \<bar>j\<bar> < nat \<bar>ja__\<bar>" shows "P j"
raw_match'_tvsT_subset_dom_res: fixes T :: "typ"   and U :: "typ"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "raw_match' T U subs = Some subs'" shows "tvsT T \<subseteq> dom subs'"
power_le_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a n) one"
cr_fract_def: shows "cr_fract \<equiv> \<lambda>x y. fractrel x x \<and> abs_fract x = y"
has_ticket_unique: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and t :: "nat"   and k :: "nat"   and t' :: "nat" assumes "A.reachable (c, n, ts)"   and "t < N"   and "has_ticket (ts t) k"   and "t' < N"   and "has_ticket (ts t') k" shows "t' = t"
borel_measurable_restrict_space_iff: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "\<Omega> \<inter> space M \<in> sets M" shows "(f \<in> borel_measurable (restrict_space M \<Omega>)) = ((\<lambda>x. indicat_real \<Omega> x *\<^sub>R f x) \<in> borel_measurable M)"
s_def: fixes n :: "nat"   and i :: "nat" shows "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))"
inv_matching_def: fixes E :: "'b set set"   and k :: "nat"   and C :: "'c set"   and F :: "'b set set"   and M :: "'b set set" assumes "Bounded_Hypergraph E k" shows "Bounded_Hypergraph.inv_matching E k C F M = (matching M \<and> M \<subseteq> E \<and> card C \<le> k * card M \<and> (\<forall>e\<in>M. \<forall>f\<in>F. e \<inter> f = {}))"
pmc_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi> = Cn (K \<div>\<^bsub>\<gamma>\<^sub>T\<^sub>R\<^esub> \<phi>)"
co_in2c: shows "connected (inside (path_image c2 \<union> path_image c))"
adapt: shows "P \<subseteq> {s. init s \<in> P' s}"
mult_closed: fixes x :: "'a"   and a :: "real" assumes "x \<in> E" shows "a \<cdot> x \<in> E"
rep_eq: fixes x :: "'a st_mat"   and xa :: "'a" shows "st_vec (transition_vec_of_st_mat x xa) = column xa (st_mat x)"
Le_le_LeI: fixes a :: "'c"   and b :: "'c" assumes "a \<le> b" shows "Le a \<le> Le b"
saturated_upto_def: fixes N :: "'a literal multiset set" shows "sr_ext.saturated_upto N = (sr_ext.inferences_from (N - sr.Rf N) \<subseteq> sr_ext_Ri N)"
finite_reachable: fixes as :: "'b list"   and r :: "'b rexp" shows "finite (reachable_a as r)"
simple_valid_cfg: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and ct :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" assumes "Finite_Markov_Decision_Process K S"   and "ct \<in> Pi S K"   and "s \<in> S" shows "Finite_Markov_Decision_Process.simple K S ct s \<in> Finite_Markov_Decision_Process.valid_cfg K S"
1: shows "{y \<in> elts \<alpha>. (y, x) \<in> VWF} = elts x"
units_of_mult: fixes G :: "('b, 'c) monoid_scheme" shows "(\<otimes>\<^bsub>units_of G\<^esub>) = (\<otimes>\<^bsub>G\<^esub>)"
rbt_lookup_Empty: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "ord.rbt_lookup less rbt.Empty = Map.empty"
CBlinfun_inject: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "x \<in> {f. bounded_clinear f}"   and "y \<in> {f. bounded_clinear f}" shows "(CBlinfun x = CBlinfun y) = (x = y)"
prism_lens_inverse: fixes X :: "'c \<Longrightarrow>\<^sub>\<triangle> 'd" assumes "wb_prism X" shows "lens_prism (prism_lens X) = X"
invfun_surj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "surj_to f A B" shows "surj_to (f\<inverse>\<^bsub>B,A\<^esub>) B A"
cond_spmf_fst_inverse: fixes p :: "('a \<times> 'b) spmf" shows "map_spmf fst p \<bind> (\<lambda>x. map_spmf (Pair x) (cond_spmf_fst p x)) = p"
fB_C_1: shows "f ` B \<subseteq> vs2.span C"
sel_1: fixes x1 :: "('a, 'b) hd" shows "head (Hd x1) = x1"
matrix_to_iarray_Gram_Schmidt_column_k_efficient: fixes k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "k < ncols A" shows "matrix_to_iarray (Gram_Schmidt_column_k_efficient A k) = Gram_Schmidt_column_k_iarrays_efficient (matrix_to_iarray A) k"
p_converg: assumes "n \<noteq> 0" shows "eval p [f \<triangleright> n] \<down>"
iteratei_map_op_list_it_rm_ops_def: shows "iteratei_map_op_list_it_rm_ops \<equiv> rm.iteratei"
phull_closed_mult_scalar: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'c" assumes "p \<in> phull B" shows "monomial c (0::'a) \<odot> p \<in> phull B"
simps_2: fixes x :: "brick"   and y :: "brick list" shows "domain_block (x # y) = Preliminaries.domain x + domain_block y"
config_config_set: fixes A :: "('c list \<Rightarrow> 'd) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> (nat \<times> nat list) \<times> 'd)"   and init :: "'c list"   and qs :: "'c list" shows "set (fst (config A init qs)) = set init"
prj_simps_6: shows "C.dom trg_trg_f.Prj\<^sub>0 = Chn ((trg f \<star> trg f) \<star> f)"
count_mset_set_3: fixes x :: "'a"   and A :: "'a set" assumes "x \<notin> A" shows "count (mset_set A) x = 0"
assms: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) < b" shows "P (Fract a b)"
univ_preserves_predicate: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool" assumes "equiv A r"   and "P respects r" shows "{x \<in> A. P x} // r = {X \<in> A // r. univ P X}"
intro: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>y x. f y \<circ> f x = f x \<circ> f y" shows "comp_fun_commute f"
impl_bool_def: fixes a :: "bool"   and b :: "bool" shows "a l\<rightarrow> b = (a \<longrightarrow> b)"
pathfinish_reversepath: fixes g :: "real \<Rightarrow> 'c" shows "pathfinish (reversepath g) = pathstart g"