hcju/isabelleps huggingface dataset:
map_def: shows "Poly_Mapping.map \<equiv> map_fun id (map_fun lookup Abs_poly_mapping) (\<lambda>g f k. g (f k) when f k \<noteq> (0::'b))"
min_elt_mem: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "minimal_element P A"   and "B \<subseteq> A"   and "B \<noteq> {}" shows "minimal_element.min_elt P A B \<in> B"
GrpE: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "BNF_Def.Grp A f x y" obtains "y = f x" and "x \<in> A"
emb_step_subst: fixes t :: "('c, 'd) tm"   and s :: "('c, 'd) tm"   and \<rho> :: "'d \<Rightarrow> ('c, 'd) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "subst \<rho> t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b subst \<rho> s"
word_nat_cases: fixes x :: "'a word" obtains n :: "nat" where "x = word_of_nat n" and "n < 2 ^ LENGTH('a)"
f_def: shows "f = Float (denormal_mantissa x) (denormal_exponent TYPE(('e, 'f) IEEE.float))"
VectorSpaceHomI: fixes T :: "'v \<Rightarrow> 'b"   and smult' :: "'f \<Rightarrow> 'b \<Rightarrow> 'b" assumes "GroupHom V T"   and "\<And>r m. \<lbrakk>r \<in> UNIV; m \<in> V\<rbrakk> \<Longrightarrow> T (r \<cdot> m) = smult' r (T m)" shows "VectorSpaceHom (\<cdot>) V smult' T"
pow_subgroup_indicator: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := H'\<rparr>)"   and "a \<in> carrier (G\<lparr>carrier := H'\<rparr>)" shows "a [^]\<^bsub>G\<lparr>carrier := H'\<rparr>\<^esub> subgroup_indicator (G\<lparr>carrier := H'\<rparr>) H a \<in> H"
has_derivative_continuous: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "(f has_derivative f') (at x within s)" shows "continuous (at x within s) f"
normalize_const_poly: fixes c :: "'b" shows "normalize [:c:] = [:normalize c:]"
vec_comp_allI: fixes v :: "'a vec"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w :: "'a vec" assumes "\<And>i. i < dim_vec v \<Longrightarrow> r (v $ i) (w $ i)" shows "vec_comp_all r v w"
elims_1: fixes x :: "value"   and y :: "bool" assumes "is_Num x = y" obtains uu_ :: "int" where "x = Num uu_" and "y"   | uv_ :: "String.literal" where "x = Str uv_" and "\<not> y"
cindex_pathE_circlepath_upper_1: fixes z0 :: "complex"   and z :: "complex"   and r :: "real" assumes "cmod (z0 - z) < r" shows "cindex_pathE (part_circlepath z r 0 pi) z0 = - 1"
primeness_condition: fixes G :: "('c, 'd) monoid_scheme" assumes "gcd_condition_monoid G" shows "primeness_condition_monoid G"
map_matrix_def: fixes f :: "'d \<Rightarrow> 'a"   and A :: "(('d, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "HMA_Connect.map_matrix f A \<equiv> \<chi>i. map_vector f (A $h i)"
NS_no_Notes: fixes evs :: "event list"   and A :: "agent"   and X :: "msg" assumes "evs \<in> ns_public" shows "Notes A X \<notin> set evs"
subcocycle_ineq: fixes u :: "nat \<Rightarrow> (int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> real"   and n :: "nat"   and m :: "nat"   and x :: "int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real" assumes "I.subcocycle u" shows "u (n + m) x \<le> u n x + u m ((TI ^^ n) x)"
sub_bot_def: shows "bot = - (THE x. \<forall>y. x = - y \<squnion> - - y)"
weaken_post: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P} c {Q}"   and "\<forall>s. Q s \<longrightarrow> Q' s" shows "\<turnstile>\<^sub>t {P} c {Q'}"
fo: shows "(\<lambda>z. f (a + complex_of_real r * z)) = f \<circ> (\<lambda>z. a + complex_of_real r * z)"
f_g: fixes i :: "nat" assumes "i < length ss1" shows "f (g i) = i"
inversions_between_def: fixes xs :: "'a list"   and ys :: "'a list" shows "inversions_between xs ys = {(i, j). (i, j) \<in> {..<length xs} \<times> {..<length ys} \<and> ys ! j < xs ! i}"
comp_simp: fixes g :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data"   and f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VV.seq g f" shows "VV.comp g f = VxV.comp g f"
vec_of_basis_enum_scaleR: fixes r :: "real"   and b :: "'a" shows "vec_of_basis_enum (r *\<^sub>R b) = complex_of_real r \<cdot>\<^sub>v vec_of_basis_enum b"
image_Collect_subsetI: fixes P :: "'b::type \<Rightarrow> bool"   and f :: "'b::type \<Rightarrow> 'c::type"   and B :: "'c::type set" assumes "\<And>x::'b::type. (P::'b::type \<Rightarrow> bool) x \<Longrightarrow> (f::'b::type \<Rightarrow> 'c::type) x \<in> (B::'c::type set)" shows "(f::'b::type \<Rightarrow> 'c::type) ` {x::'b::type. (P::'b::type \<Rightarrow> bool) x} \<subseteq> (B::'c::type set)"
map_index: fixes f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list" shows "map_index f xs = map2 f [0..<length xs] xs"
nn_integral_uniform_measure: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure"   and S :: "'b set" assumes "f \<in> borel_measurable M"   and "S \<in> sets M" shows "integral\<^sup>N (uniform_measure M S) f = set_nn_integral M S f / emeasure M S"
in_chain_finite: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "finite A"   and "A \<noteq> {}" shows "Sup A \<in> A"
inverse_eq_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and 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.field divide inverse times one plus zero minus uminus" shows "(inverse x = one) = (x = one)"
components_are_iso: fixes a :: "'d" assumes "D.ide a" shows "D.iso (D.R a)"
implies_e1: fixes \<Gamma> :: "'b constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'b TESL_atomic list"   and \<Phi> :: "'b TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>"
UP_r_one: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "p \<otimes>\<^bsub>P\<^esub> \<one>\<^bsub>P\<^esub> = p"
sorted_wrt2: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and zs :: "'g list" assumes "transp P" shows "sorted_wrt P (x # y # zs) = (P x y \<and> sorted_wrt P (y # zs))"
simps_1: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "bst_wrt P \<langle>\<rangle> = True"
induct: fixes P :: "(int \<times> int) \<times> int \<times> int \<Rightarrow> (int \<times> int) list \<Rightarrow> bool"   and a0 :: "(int \<times> int) \<times> int \<times> int"   and a1 :: "(int \<times> int) list" assumes "\<And>c\<^sub>0 c\<^sub>1. P (c\<^sub>0, c\<^sub>1) []"   and "\<And>c\<^sub>0 c\<^sub>1 uu_. P (c\<^sub>0, c\<^sub>1) [uu_]"   and "\<And>c\<^sub>0 c\<^sub>1 p\<^sub>0 v va. \<lbrakk>\<And>x. \<lbrakk>x = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va); dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (v # va); \<And>x. \<lbrakk>x = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (v # va); \<not> dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) \<le> dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))\<rbrakk> \<Longrightarrow> P (p\<^sub>0, x) (v # va)\<rbrakk> \<Longrightarrow> P (c\<^sub>0, c\<^sub>1) (p\<^sub>0 # v # va)" shows "P a0 a1"
nat_to_ce_set_into_ce: fixes n :: "nat" shows "nat_to_ce_set n \<in> ce_sets"
matrix_change_of_basis_def: fixes X :: "(('b, 'c) vec, 'c) vec"   and Y :: "(('b, 'c) vec, 'c) vec" shows "matrix_change_of_basis X Y = (\<chi>i j. coord Y (X $ j) $ i)"
span_zero: fixes S :: "'c set" shows "(0::'c) \<in> cspan S"
Beta_def: shows "{\<rightarrow>\<^sub>\<beta>} \<equiv> {(xa, x). Betap xa x}"
vquot_fm_insert: fixes i :: "name"   and A :: "fm"   and V :: "name set" assumes "atom i \<notin> supp A" shows "\<lfloor>A\<rfloor>(insert i V) = \<lfloor>A\<rfloor>V"
inject_6: fixes x61 :: "'c"   and x62 :: "('b, 'c) fm"   and y61 :: "'c"   and y62 :: "('b, 'c) fm" shows "((\<^bold>@ x61 x62) = (\<^bold>@ y61 y62)) = (x61 = y61 \<and> x62 = y62)"
CSP1_Sup: fixes A :: "('c, 'd) action set" assumes "A \<noteq> {}" shows "\<lambda>A. A \<in> \<Inter> {{p. P p} |P. P \<in> action.relation_of ` A} is CSP1 healthy"
is_valid_problem_sas_plus_then_strips_transformation_too_iv: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>x. (((\<phi> \<Psi> )\<^sub>I) x \<noteq> None) = ListMem x ((\<phi> \<Psi> )\<^sub>\<V>)"
length_dropWhile_le: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "length (dropWhile P xs) \<le> length xs"
is_lasso_prpl_conv: fixes m :: "'a"   and prpl :: "'Q list \<times> 'Q list" shows "gb_graph.is_lasso_prpl (to_gbg_ext m) prpl = (snd prpl \<noteq> [] \<and> gb_graph.is_lasso (to_gbg_ext m) (lasso_of_prpl prpl))"
neg_eq: fixes x :: "tr"   and y :: "tr" shows "(neg\<cdot>x = neg\<cdot>y) = (x = y)"
x_dom_1: shows "x \<notin> subst_domain \<sigma>"
a_closed: fixes H :: "'a set"   and G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "additive_subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<oplus>\<^bsub>G\<^esub> y \<in> H"
obtain_const_trm: fixes e :: "(name, nat) finfun"   and x :: "nat" obtains t :: "trm" where "eval_trm e t = x" and "t \<in> num"
SuppA_def: 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"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.SuppA Bet Cong A B C D E F \<equiv> A \<noteq> B \<and> (\<exists>A'. Bet A B A' \<and> Tarski_neutral_dimensionless.CongA Bet Cong D E F C B A')"
sort_Cons_append_swap: fixes x :: "'a"   and xs :: "'a list" shows "sort (x # xs) = sort (xs @ [x])"
set_of_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "set_of P = {(x, y). P x y}"
case_2: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "'a" shows "(case \<infinity> of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f2"
\<A>2_def: shows "\<A>2 = (SIGMA A:{A \<in> Pow X. card A = k}. A)"
weak_group_morphism_is_iso: fixes f :: "'a \<Rightarrow> 'd"   and H :: "'a set" assumes "weak_group_morphism f H G" shows "(\<lambda>x. the_elem (f ` x)) \<in> Group.iso (G Mod H) (image_group f G)"
Abs_inverse: fixes y :: "bool list" assumes "y \<in> {bl. length bl = LENGTH('b)}" shows "to_bl (of_bl y) = y"
inject_4: fixes x4 :: "'a"   and y4 :: "'a" shows "(Snapshot x4 = Snapshot y4) = (x4 = y4)"
subs_def: shows "subs = (\<lambda>pseq. if SATAxiom (sequent pseq) then {} else let (atms, nforms) = pseq in case nforms of [] \<Rightarrow> {} | nA # nAs \<Rightarrow> let (n, A) = nA in case A of FAtom z P vs \<Rightarrow> subsFAtom atms nAs z P vs | FConj z A0 A1 \<Rightarrow> subsFConj atms nAs z A0 A1 | FAll z A \<Rightarrow> subsFAll atms nAs n z A (freeVarsFL (sequent pseq)))"
t: fixes i :: "nat" shows "simple_bochner_integrable M (t i)"
cong: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = g x"   and "P g" shows "P f"
bounded_measure: fixes p :: "'a spmf"   and A :: "'a set" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> weight_spmf p"
count_node_def: shows "count_node \<equiv> rec_Node (\<lambda>_ a. 1) (\<lambda>_ a b. (+)) (\<lambda>_ a b c aa ba. (+) (aa + ba))"
chainTransitionAppend_1: fixes P :: "pi"   and P'' :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>\<tau> P''"   and "P'' \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
leg1_composite: shows "SPN_fgh.\<nu>\<pi>.leg1 = SPN_fgh.\<nu>.leg1 \<odot> SPN_fgh.\<nu>\<pi>.prj\<^sub>1"
eventually_mp: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and F :: "'c filter" assumes "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x"   and "eventually P F" shows "eventually Q F"
sourcesD_3: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "a \<in> weak_composition.sources V H \<mu>" shows "H \<mu> a \<noteq> partial_magma.null V"
SystemCEFml2_def: shows "SystemCEFml2 = Geq ($' vid1) (Const 0)"
fls_mult_left_inverse_base_factor: fixes x :: "'a"   and f :: "'a fls" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)" shows "fls_left_inverse (fls_base_factor f) x * f = fls_X_intpow (fls_subdegree f)"
induct: fixes P :: "'b tree \<Rightarrow> bool"   and a0 :: "'b tree" assumes "\<And>hsx x hsy y hs. P hs \<Longrightarrow> P \<langle>hsx, x, \<langle>hsy, y, hs\<rangle>\<rangle>"   and "P \<langle>\<rangle>"   and "\<And>v va. P \<langle>v, va, \<langle>\<rangle>\<rangle>" shows "P a0"
basis_augment: fixes B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and x :: "'a" assumes "\<E>.basis B\<^sub>1"   and "\<E>.basis B\<^sub>2"   and "x \<in> B\<^sub>1 - B\<^sub>2" shows "\<exists>y\<in>B\<^sub>2 - B\<^sub>1. \<E>.basis (insert y (B\<^sub>1 - {x}))"
measurable_emeasure_Pair1: fixes Q :: "(('i \<Rightarrow> 'a) \<times> ('i \<Rightarrow> 'a)) set" assumes "Q \<in> sets (Pi\<^sub>M I M \<Otimes>\<^sub>M Pi\<^sub>M J M)" shows "(\<lambda>x. emeasure (Pi\<^sub>M J M) (Pair x -` Q)) \<in> borel_measurable (Pi\<^sub>M I M)"
pres_comp: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "pres y z"   and "pres x z" shows "pres (x \<cdot> y) z"
is_fmapI: fixes M :: "('d \<times> 'e) fset" assumes "\<And>a b\<^sub>1 b\<^sub>2. \<lbrakk>(a, b\<^sub>1) |\<in>| M; (a, b\<^sub>2) |\<in>| M\<rbrakk> \<Longrightarrow> b\<^sub>1 = b\<^sub>2" shows "is_fmap M"
inv_solve_right: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "c \<in> carrier (K [X])" shows "(a = b \<oplus>\<^bsub>K [X]\<^esub> \<ominus>\<^bsub>K [X]\<^esub> c) = (b = a \<oplus>\<^bsub>K [X]\<^esub> c)"
ffilter_empty: fixes f :: "'a \<Rightarrow> bool" shows "ffilter f {||} = {||}"
le_def: fixes compare :: "'b \<rightarrow> 'b \<rightarrow> Ordering" shows "Ord.le compare = (\<Lambda> x y. case compare\<cdot>x\<cdot>y of LT \<Rightarrow> TT | EQ \<Rightarrow> TT | GT \<Rightarrow> FF)"
fds_truncate_def: fixes m :: "nat"   and f :: "'a fds" shows "fds_truncate m f = fds (\<lambda>n. if n \<le> m then fds_nth f n else (0::'a))"
eq: fixes J :: "'a set"   and A :: "'a \<Rightarrow> 'b set" assumes "finite J"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> A i \<in> sets (M i)" shows "emeasure P (prod_emb I M J (Pi\<^sub>E J A)) = emeasure Q (prod_emb I M J (Pi\<^sub>E J A))"
If_def: fixes P :: "bool"   and x :: "'c"   and y :: "'c" shows "if P then x else y \<equiv> THE z. (P = True \<longrightarrow> z = x) \<and> (P = False \<longrightarrow> z = y)"
gets_the_return: fixes x :: "'c" shows "gets_the (oreturn x) = return x"
dest_segment: fixes x :: "real"   and b :: "real"   and x0 :: "real"   and y0 :: "real"   and x1 :: "real"   and y1 :: "real" assumes "(x, b) \<in> closed_segment (x0, y0) (x1, y1)"   and "x0 \<noteq> x1" shows "b = (y1 - y0) * (x - x0) / (x1 - x0) + y0"
quasi_inversesE: fixes f :: "'d"   and g :: "'d" assumes "D.quasi_inverses f g" obtains "D.ide f" and "D.ide g" and "src\<^sub>D f \<cong>\<^sub>D g \<star>\<^sub>D f" and "f \<star>\<^sub>D g \<cong>\<^sub>D trg\<^sub>D f"
norm_F_x: shows "\<exists>K. \<forall>x. norm (F x) \<le> norm x * K"
from_nat_0: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Rep :: "'c \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'c" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.from_nat Abs 0 = zero"
tuple_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "HF'.tuple f g \<equiv> if span f g then THE l. local.comp (pr1 (cod f) (cod g)) l = f \<and> local.comp (pr0 (cod f) (cod g)) l = g else null"
dom_delete_aux: fixes ps :: "('a \<times> 'b) list"   and k :: "'a" assumes "distinct (map fst ps)" shows "fst ` list.set (AList.delete_aux k ps) = fst ` list.set ps - {k}"
list_of_vec_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "list_of_vec (Matrix.vec n f) = map f [0..<n]"
step: fixes r :: "Term_Class.term \<times> Term_Class.term"   and rs :: "(Term_Class.term \<times> Term_Class.term) fset"   and t :: "Term_Class.term"   and u :: "Term_Class.term" assumes "r |\<in>| rs"   and "r \<turnstile> t \<rightarrow> u" shows "rs \<turnstile> t \<longrightarrow> u"
lift_pref_profile_permute_agents: fixes \<pi> :: "'a \<Rightarrow> 'a"   and agents :: "'a set"   and agents' :: "'a set"   and alts :: "'b set"   and alts' :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<pi> permutes agents"   and "agents \<subseteq> agents'" shows "lift_pref_profile agents alts agents' alts' (R \<circ> \<pi>) = lift_pref_profile agents alts agents' alts' R \<circ> \<pi>"
hb_consistent_technical: fixes cs :: "('msgid \<times> 'op) event list"   and i :: "nat" assumes "\<And>m n. \<lbrakk>m < length cs; n < m\<rbrakk> \<Longrightarrow> cs ! n \<sqsubset>\<^sup>i cs ! m" shows "hb.hb_consistent (node_deliver_messages cs)"
preserves_arr: fixes f :: "'a" assumes "L.arr f" shows "L.arr (L.map f)"
lm043: fixes PP :: "'c set set set" assumes "is_non_overlapping PP"   and "is_non_overlapping (\<Union> PP)" shows "is_non_overlapping (\<Union> ` PP)"
is_extensional: fixes f :: "'a" assumes "\<not> Left_a.arr f" shows "Left_a.L f = Left_a.null"
sqUn_lem: fixes A :: "char list set option"   and A' :: "char list set option"   and B :: "char list set option" assumes "A \<sqsubseteq> A'" shows "A \<squnion> B \<sqsubseteq> A' \<squnion> B"
dghm_const_is_tiny_dghm: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<DD> :: "V"   and a :: "V"   and f :: "V" assumes "tiny_digraph \<alpha> \<CC>"   and "tiny_digraph \<alpha> \<DD>"   and "f : a \<mapsto>\<^bsub>\<DD>\<^esub> a" shows "dghm_const \<CC> \<DD> a f : \<CC> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<DD>"
cases: fixes x :: "int poly list" obtains f :: "int poly" and fs :: "int poly list" where "x = f # fs"   | "x = []"
rel_setD1: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'f set"   and B :: "'g set"   and x :: "'f" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
state_rank_sink: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q" shows "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = None"
in_restricted_store_in_unrestricted_store_2: fixes l' :: "'d"   and \<sigma> :: "'d \<Rightarrow> ('e, 'd, 'f) val option"   and l :: "'d" assumes "l' \<in> LID\<^sub>S (\<sigma>(l := None))" shows "l' \<in> LID\<^sub>S \<sigma>"
hyps: fixes t' :: "('sequent \<times> 'rule) tree" assumes "t' |\<in>| cont t" shows "tfinite t'"
bij_is_surj: fixes f :: "'q \<Rightarrow> 'r" assumes "bij f" shows "surj f"
arr_cod: fixes f :: "'a" assumes "R.arr f" shows "R.arr (R.cod f)"
apply_empty_None: fixes x2 :: "'b" shows "<> $ x2 = None"
cr_spmf_gpv_def: fixes p :: "'d spmf"   and gpv :: "('d, 'e, 'f) gpv" shows "spmf_to_gpv.cr_spmf_gpv p gpv = (gpv = lift_spmf p)"
p: shows "extreme_bound A (\<sqsubseteq>) (Collect derivable) p"
induct: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x1 x2"   and "\<And>a b. r a b \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>r\<^sup>+\<^sup>+ a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
liftT_liftT: fixes i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat"   and T :: "POPLmark.type" assumes "i \<le> j"   and "j \<le> i + m" shows "\<up>\<^sub>\<tau> n j (\<up>\<^sub>\<tau> m i T) = \<up>\<^sub>\<tau> (m + n) i T"
dprodI: fixes M :: "('a, 'b) node set"   and M' :: "('a, 'b) node set"   and r :: "(('a, 'b) node set \<times> ('a, 'b) node set) set"   and N :: "('a, 'b) node set"   and N' :: "('a, 'b) node set"   and s :: "(('a, 'b) node set \<times> ('a, 'b) node set) set" assumes "(M, M') \<in> r"   and "(N, N') \<in> s" shows "(Scons M N, Scons M' N') \<in> dprod r s"
simps_6: fixes r :: "'b mrexp2"   and uy :: "bool" shows "strip2 (Star2 r uy) = Star (strip2 r)"
option_rule: fixes v :: "'c option"   and S1 :: "'d nres"   and \<Phi> :: "'d \<Rightarrow> bool"   and f2 :: "'c \<Rightarrow> 'd nres" assumes "v = None \<Longrightarrow> S1 \<le> SPEC \<Phi>"   and "\<And>x. v = Some x \<Longrightarrow> f2 x \<le> SPEC \<Phi>" shows "(case v of None \<Rightarrow> S1 | Some x \<Rightarrow> f2 x) \<le> SPEC \<Phi>"
obj_simps_2: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.obj a" shows "S.src a = a"
intro: fixes G :: "('a, 'b, 'c) igba_rec_scheme"   and S :: "('d, 'b, 'e) sa_rec_scheme" assumes "igba G"   and "sa S" shows "igba_sys_prod_precond G S"
valofd_def: fixes x :: "('a, 'b) IEEE.float" shows "valofd x = 2 / 2 ^ bias TYPE(('a, 'b) IEEE.float) * (real (fraction x) / 2 ^ LENGTH('b))"
vrrestriction_vconst_on: fixes c :: "V"   and B :: "V"   and A :: "V" assumes "c \<in>\<^sub>\<circ> B" shows "vconst_on A c \<restriction>\<^sup>r\<^sub>\<circ> B = vconst_on A c"
inject_2: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c"   and y21 :: "nat"   and y22 :: "'a"   and y23 :: "'a"   and y24 :: "'b"   and y25 :: "'b"   and y26 :: "'c" shows "(Send x21 x22 x23 x24 x25 x26 = Send y21 y22 y23 y24 y25 y26) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25 \<and> x26 = y26)"
toStates_inv_1: fixes steps :: "step list"   and ss :: "(integer \<times> edge list) iarray"   and start :: "edgeIndex"   and lbls :: "(String.literal, nat) assoc_list" assumes "toStates steps = (ss, start, lbls)" shows "\<exists>s. start = Index s \<and> s < IArray.length ss"
simps_3: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a"   and xs :: "'a list" shows "sorted_by cmp (x1 # x2 # xs) = (cmp x1 x2 \<and> sorted_by cmp (x2 # xs))"
comm_monoid_set_and: shows "comm_monoid_set (\<and>) True"
box_eq_empty_2: fixes a :: "'b"   and b :: "'b" shows "(cbox a b = {}) = (\<exists>i\<in>Basis. b \<bullet> i < a \<bullet> i)"
prems: shows "\<forall>x\<in>free_vars (IF b THEN e1 ELSE e2). val_type (\<sigma> x) = \<Gamma> x"