hcju/isabelleps huggingface dataset:
sbintrunc_uint_ucast: fixes n :: "nat" assumes "Suc n = LENGTH('c)" shows "signed_take_bit n (uint (UCAST('a \<rightarrow> 'c) w)) = signed_take_bit n (uint w)"
stable_on_CD_on: fixes ds :: "'d set"   and X :: "'x set" assumes "stable_on ds X" shows "CD_on ds X = X"
divisor_count_naive: fixes n :: "nat" shows "divisor_count n = (if n = 0 then 0 else fold_atLeastAtMost_nat (\<lambda>d acc. if d dvd n then Suc acc else acc) 1 n 0)"
monoid_axioms: shows "Group_Theory.monoid stk1.carrier_stalk stk1.mult_stalk (stk1.one_stalk U)"
continuous_at_open: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b" shows "isCont f x = (\<forall>t. open t \<and> f x \<in> t \<longrightarrow> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x'\<in>s. f x' \<in> t)))"
plus_or_minus_sqrt: fixes y :: "real"   and x :: "real" assumes "0 \<le> y" shows "(x\<^sup>2 = y) = (x = sqrt y \<or> x = - sqrt y)"
higher_deriv_lnGamma_stirling: fixes m :: "nat"   and j :: "nat" assumes "0 < m" shows "(\<lambda>x. (deriv ^^ j) ln_Gamma x - Polygamma_approx j m x) \<in> O(\<lambda>x. 1 / x ^ (m + j))"
D\<sigma>_subset_C: shows "D \<cdot> \<sigma> \<subset># C"
rel_sum_simps_1: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c"   and b1 :: "'d" shows "rel_sum R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
Polys_closed_map_scale: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and X :: "'c set"   and c :: "'d" assumes "p \<in> P[X]" shows "c \<cdot> p \<in> P[X]"
is_zero_iarray_def: fixes A :: "'b iarray" shows "is_zero_iarray A = IArray_Addenda_QR.all (\<lambda>i. A !! i = (0::'b)) (IArray [0..<IArray.length A])"
RedSCallNone: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool" assumes "\<nexists>b Ts T m D. P \<turnstile> C sees M, b :  Ts\<rightarrow>T = m in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),s,b\<rangle> \<rightarrow> \<langle>THROW NoSuchMethodError,s,False\<rangle>"
assms_2: shows "l' \<in> labels \<Gamma> q"
brw_cond_def: fixes Qi :: "'d set" shows "brw_cond Qi \<equiv> {(Q, W, rcm). W \<noteq> {} \<and> Qi \<inter> dom Q = {}}"
int_simps_16: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> (P \<and> #True) = P"
red_Nml: fixes a :: "'c local.term" assumes "Nml a" shows "a\<^bold>\<down> = a"
field_def2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F:T in D" shows "field P C F = (D, T)"
distinct_3: fixes x1 :: "'b"   and x31 :: "'b trm"   and x32 :: "'b trm" shows "Var x1 \<noteq> x31 \<cdot> x32"
intTI_def: shows "intTI \<equiv> MonotProblem.intTI TE_wtFsym wtPsym TE_arOf TE_resOf parOf TE_\<Phi>"
set_sel_2: fixes a :: "'b tree" assumes "a \<noteq> \<langle>\<rangle>" shows "value a \<in> set_tree a"
PositiveImplies_def: fixes v :: "'b zmultiset"   and w :: "'b zmultiset" shows "PositiveImplies v w = (\<forall>t. 0 < zcount v t \<longrightarrow> 0 < zcount w t)"
Limit_def: fixes i :: "V" shows "Limit i \<equiv> Ord i \<and> []\<^sub>\<circ> \<in>\<^sub>\<circ> i \<and> (\<forall>y. y \<in>\<^sub>\<circ> i \<longrightarrow> ZFC_in_HOL.succ y \<in>\<^sub>\<circ> i)"
abs_eq: fixes xa :: "'a ltl set"   and x :: "'a ltl" shows "xa \<up>\<Turnstile>\<^sub>P Abs x = xa \<Turnstile>\<^sub>P x"
r1: shows "remove1 (i - 1) [3..<i] = [3..<i - 1]"
vsubstAbs_preserves_good: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and xs :: "'varSort"   and x1 :: "'var"   and x :: "'var" assumes "goodAbs A" shows "goodAbs (A $[x1 // x]_xs)"
valid_nodes: shows "nodeOf ` fset vertices \<subseteq> i.R nodes"
rb_aux_term2_def: shows "rb_aux_term2 \<equiv> {(x, y). (fst x, fst y) \<in> rb_aux_term1 \<or> fst x = fst y \<and> length (snd (snd x)) < length (snd (snd y))}"
2: fixes E :: "(nat \<times> nat) set" shows "one_final_but g E"
mu_roll: fixes g :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "isotone g"   and "has_least_fixpoint (f \<circ> g)"   and "has_least_fixpoint (g \<circ> f)" shows "\<mu> (g \<circ> f) = g (\<mu> (f \<circ> g))"
sst_vars_proj_subset_1: fixes n :: "'c"   and A :: "('c strand_label \<times> ('b, 'a) stateful_strand_step) list" shows "fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
2: shows "{i. n cd\<^bsup>\<pi>\<^esup>\<rightarrow> i \<and> i < k} = {}"
select_firstD1: fixes x :: "'a"   and p :: "'a \<Rightarrow> real"   and y :: "'a" assumes "select_first x p y" shows "y \<in> I x"
eq_abs_iff': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "(a = abs b) = (less_eq zero a \<and> (b = a \<or> b = uminus a))"
apply_operations_Snoc: fixes xs :: "'a list"   and x :: "'a" shows "apply_operations (xs @ [x]) = apply_operations xs \<rhd> \<langle>x\<rangle>"
concat: shows "concat (map (\<lambda>(x, i). replicate (scf i) x) (zip bia [0..<length bia] @ [(a, length bia)])) = concat (map2 (\<lambda>x i. replicate (scf i) x) bia [0..<length bia]) @ replicate (scf (length bia)) a"
SLC_fork: fixes s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and \<sigma> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<tau> :: "'l \<Rightarrow> ('r, 'l, 'v) val option"   and \<E> :: "('r, 'l, 'v) cntxt"   and e :: "('r, 'l, 'v) expr"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and left_forkee :: "'r"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"   and "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>, \<E> [VE (Rid left_forkee)]), left_forkee \<mapsto> (\<sigma>;;\<tau>, \<epsilon>, e))"   and "left_forkee \<notin> RID\<^sub>G s\<^sub>1"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "r \<noteq> r'"   and "reachable s\<^sub>1"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
rem_implicit_pres_ops_enabled: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op" shows "ast_problem.enabled (rem_implicit_pres_ops prob) name s = ast_problem.enabled prob name s"
ground_fun: fixes s :: "('d, 'c) tm" assumes "ground s" shows "ground (fun s)"
subgame_def: fixes V' :: "'a set" shows "G'.subgame V' \<equiv> G'\<lparr>verts := V\<^bsub>G'\<^esub> \<inter> V', arcs := Restr E\<^bsub>G'\<^esub> V', player0 := V0\<^bsub>G'\<^esub> \<inter> V'\<rparr>"
well_founded_iff_ex_extremal: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "well_founded A less = (\<forall>X\<subseteq>A. X \<noteq> {} \<longrightarrow> (\<exists>x\<in>X. \<forall>z\<in>X. \<not> less z x))"
equiv_imp_dvd_card: fixes A :: "'a set"   and r :: "('a \<times> 'a) set"   and k :: "nat" assumes "finite A"   and "equiv A r"   and "\<And>X. X \<in> A // r \<Longrightarrow> k dvd card X" shows "k dvd card A"
DBM_zone_repr_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'b \<Rightarrow> nat"   and n :: "nat" shows "[M]\<^bsub>v,n\<^esub> = {u. u \<turnstile>\<^bsub>v,n\<^esub> M}"
in_second_summand: fixes z :: "i"   and x :: "i"   and y :: "i" assumes "P z (x \<oplus> y) \<and> \<not> O z x" shows "P z y"
evaluation_functor_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>D :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>D :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>D :: "'d"   and V :: "'b \<Rightarrow> 'd" shows "FreeMonoidalCategory.evaluation_functor C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V \<equiv> (category C \<and> monoidal_category D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D) \<and> evaluation_map C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V \<and> free_monoidal_category C"
equiv_relation_of_partition_of: fixes A :: "'a set"   and R :: "('a \<times> 'a) set" assumes "equiv A R" shows "{(x, y). \<exists>X\<in>A // R. x \<in> X \<and> y \<in> X} = R"
when_commute: fixes zero :: "'a"   and a :: "'a"   and Q :: "bool"   and P :: "bool" shows "zero.when zero (zero.when zero a Q) P = zero.when zero (zero.when zero a P) Q"
ide_char: fixes a :: "'b" shows "D.ide a = (D.arr a \<and> D.dom a = a \<and> D.cod a = a)"
add_raw_def: shows "add_raw \<equiv> \<lambda>(x, y) (u, v). (x + u, y + v)"
exists_superset_carrier: fixes C :: "'c ring set"   and S :: "'c set" assumes "\<And>R. R \<in> C \<Longrightarrow> field R"   and "\<And>R S. \<lbrakk>R \<in> C; S \<in> C\<rbrakk> \<Longrightarrow> R \<lesssim> S \<or> S \<lesssim> R"   and "finite S"   and "S \<noteq> {}"   and "S \<subseteq> carrier (union_ring C)" shows "\<exists>R\<in>C. S \<subseteq> carrier R"
choose_2: fixes S :: "'a set" assumes "card S = 2" shows "\<exists>x y. S = {x, y}"
rep_eq: fixes x :: "('e, 'f) uexpr"   and xa :: "('e, 'f) uexpr" shows "(x \<le> xa) = (\<forall>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<le> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
llexord_conv: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'c llist" shows "llexord r xs ys = (xs = ys \<or> (\<exists>zs xs' y ys'. lfinite zs \<and> xs = lappend zs xs' \<and> ys = lappend zs (LCons y ys') \<and> (xs' = LNil \<or> r (lhd xs') y)))"
is_Write\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Write\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
atLeast_Suc_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "prod g {Suc m..Suc n} = prod (g \<circ> Suc) {m..n}"
rw_add_is_rewrite_ord: shows "is_rewrite_ord rw_add"
Invoke1: fixes p :: "'b"   and c :: "'c"   and i :: "nat" assumes "Inv p c s s' \<and> t = t'"   and "i < id1"   and "a = Invoke i p c" shows "thesis"
root_int_floor_pos_pos: fixes x :: "int"   and p :: "nat" assumes "0 \<le> x" shows "0 \<le> root_int_floor_pos p x"
dgrad_p_set_le_subset: fixes F :: "('t \<Rightarrow>\<^sub>0 'e) set"   and G :: "('t \<Rightarrow>\<^sub>0 'e) set"   and d :: "'a \<Rightarrow> nat" assumes "F \<subseteq> G" shows "dgrad_p_set_le d F G"
assms_1: shows "B.arr (fst \<mu>\<nu>)"
c_failures_def: shows "c_failures \<equiv> \<lambda>step out s\<^sub>0. {(xa, x). c_failuresp step out s\<^sub>0 xa x}"
sin_differentiable_at: fixes f :: "'b \<Rightarrow> real"   and x :: "'b"   and X :: "'b set" assumes "f differentiable at x within X" shows "(\<lambda>x. sin (f x)) differentiable at x within X"
fps_const_inverse: fixes a :: "'b" shows "inverse (fps_const a) = fps_const (inverse a)"
unit_imp_dvd: fixes b :: "'a"   and a :: "'a" assumes "b dvd (1::'a)" shows "b dvd a"
additiveD: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> 'c"   and x :: "'b set"   and y :: "'b set" assumes "Measure_Space.additive M f"   and "x \<inter> y = {}"   and "x \<in> M"   and "y \<in> M" shows "f (x \<union> y) = f x + f y"
that_7: fixes b\<^sub>1 :: "bool"   and b\<^sub>2 :: "bool" assumes "x = B\<cdot>(Discr b\<^sub>1)"   and "y = CB\<cdot>(Discr b\<^sub>2)" shows "thesis"
simps: fixes r :: "rat"   and cc :: "constraint" shows "r *R cc = (if r = 0 then LEQ 0 0 else case cc of LT p c \<Rightarrow> if r < 0 then GT (r *R p) (r *R c) else LT (r *R p) (r *R c) | GT p c \<Rightarrow> if 0 < r then GT (r *R p) (r *R c) else LT (r *R p) (r *R c) | LEQ p c \<Rightarrow> if r < 0 then GEQ (r *R p) (r *R c) else LEQ (r *R p) (r *R c) | GEQ p c \<Rightarrow> if 0 < r then GEQ (r *R p) (r *R c) else LEQ (r *R p) (r *R c) | EQ p c \<Rightarrow> LEQ (r *R p) (r *R c) | LTPP p q \<Rightarrow> if r < 0 then GT (r *R (p - q)) 0 else LT (r *R (p - q)) 0 | GTPP p q \<Rightarrow> if 0 < r then GT (r *R (p - q)) 0 else LT (r *R (p - q)) 0 | LEQPP p q \<Rightarrow> if r < 0 then GEQ (r *R (p - q)) 0 else LEQ (r *R (p - q)) 0 | GEQPP p q \<Rightarrow> if 0 < r then GEQ (r *R (p - q)) 0 else LEQ (r *R (p - q)) 0 | EQPP p q \<Rightarrow> LEQ (r *R (p - q)) 0)"
ChamberComplexMorphism_axioms: shows "ChamberComplexMorphism X X f"
P_def: shows "P \<equiv> \<lambda>l u x. f x \<in> {l1 x..u1 x} \<longrightarrow> g x \<in> {l2 x..u2 x} \<longrightarrow> f x powr g x \<in> {l..u}"
pprogress_eq: fixes \<pi> :: "(String.literal \<times> event_data list) prefix"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and \<phi> :: "Formula.formula" assumes "prefix_of \<pi> \<sigma>" shows "pprogress \<phi> \<pi> = Monitor.progress \<sigma> Map.empty \<phi> (plen \<pi>)"
VectorSpaceHomI: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'c"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "VectorSpace smult V"   and "GroupHom V T"   and "\<And>r m. \<lbrakk>r \<in> UNIV; m \<in> V\<rbrakk> \<Longrightarrow> T (smult r m) = smult' r (T m)" shows "VectorSpaceHom smult V smult' T"
convex_hull_insert_alt: fixes a :: "'b"   and S :: "'b set" shows "convex hull insert a S = (if S = {} then {a} else {(1 - u) *\<^sub>R a + u *\<^sub>R x |x u. 0 \<le> u \<and> u \<le> 1 \<and> x \<in> convex hull S})"
expected_value_mix_pmf: fixes p :: "'a pmf"   and q :: "'a pmf"   and a :: "real"   and f :: "'a \<Rightarrow> real" assumes "finite (set_pmf p)"   and "finite (set_pmf q)"   and "a \<in> {0..1}" shows "measure_pmf.expectation (mix_pmf a p q) f = a * measure_pmf.expectation p f + (1 - a) * measure_pmf.expectation q f"
a2_eq: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> d y) = ad (x \<cdot> y)"
simps_2: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and p :: "'a process" shows "CH2 sys (PROC p) = PROC (ch2Of p)"
PRelQ: fixes \<phi> :: "'c"   and Q :: "('a, 'b, 'c) psi" assumes "(\<phi>, Q) mem CsQ" shows "\<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> (\<Psi>, P, Q) \<in> Rel"
insert_order: fixes k :: "nat"   and t :: "'a btree"   and x :: "'a" assumes "0 < k"   and "root_order k t" shows "root_order k (local.insert k x t)"
normal_invE_1: fixes G :: "('c, 'd) monoid_scheme"   and N :: "'c set" assumes "Group.group G"   and "N \<lhd> G" shows "subgroup N G"
FAcc: fixes A :: "triple set"   and P :: "assn"   and e :: "expr"   and Q :: "vassn"   and f :: "fname" assumes "A \<turnstile>\<^sub>e {P} e {\<lambda>v s. \<forall>a. v = Addr a \<longrightarrow> Q (get_field s a f) s}" shows "A \<turnstile>\<^sub>e {P} e..f {Q}"
set_primes_upto: fixes n :: "nat" shows "set (primes_upto n) = {m. m \<le> n \<and> prime m}"
moebius_pt_moebius_similarity': fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> 0" shows "moebius_pt (moebius_similarity a b) = (\<lambda>z. of_complex a *\<^sub>h z +\<^sub>h of_complex b)"
AA: shows "Esigma 69615 = 13 * 8 * 14 * 6 * 18"
add_right_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.ordered_ab_semigroup_add plus less_eq less"   and "less_eq a b" shows "less_eq (plus a c) (plus b c)"
ide_mkIde: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f0 :: "'b"   and f1 :: "'b" assumes "span_vertical_category C"   and "partial_magma.arr C f0 \<and> partial_magma.arr C f1 \<and> partial_magma.dom C f0 = partial_magma.dom C f1" shows "partial_magma.ide (span_vertical_category.vcomp C) \<lparr>Chn = partial_magma.dom C f0, Dom = \<lparr>Leg0 = f0, Leg1 = f1\<rparr>, Cod = \<lparr>Leg0 = f0, Leg1 = f1\<rparr>\<rparr>"
simps_1: fixes l :: "formula list" shows "get_and_list (Ands l) = l"
preorder_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "preorder_on A r \<equiv> refl_on A r \<and> trans r"
intro: fixes R :: "('b, 'c) ring_scheme" assumes "ring R"   and "Group.comm_monoid R" shows "cring R"
simps: fixes a :: "trm" shows "valuep a = ((\<exists>T t. a = (\<lambda>:T. t)) \<or> (\<exists>T t. a = (\<lambda><:T. t)))"
lappend_snocL1_conv_LCons2: fixes xs :: "'b llist"   and y :: "'b"   and ys :: "'b llist" shows "(xs $ <y>) $ ys = xs $ y % ys"
accepting_pair\<^sub>G\<^sub>R_abstract: fixes \<Sigma> :: "'e set"   and \<delta> :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and \<delta>' :: "'f \<Rightarrow> 'e \<Rightarrow> 'f"   and q\<^sub>0' :: "'f"   and w :: "nat \<Rightarrow> 'e"   and f :: "'f \<times> 'e \<times> 'f \<Rightarrow> 'd \<times> 'e \<times> 'd"   and F :: "('d \<times> 'e \<times> 'd) set"   and F' :: "('f \<times> 'e \<times> 'f) set"   and \<I> :: "'g set"   and I :: "'g \<Rightarrow> ('d \<times> 'e \<times> 'd) set"   and I' :: "'g \<Rightarrow> ('f \<times> 'e \<times> 'f) set" assumes "finite (reach\<^sub>t \<Sigma> \<delta> q\<^sub>0)"   and "finite (reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0')"   and "range w \<subseteq> \<Sigma>"   and "run\<^sub>t \<delta> q\<^sub>0 w = f \<circ> run\<^sub>t \<delta>' q\<^sub>0' w"   and "\<And>t. t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0' \<Longrightarrow> (f t \<in> F) = (t \<in> F')"   and "\<And>t i. \<lbrakk>i \<in> \<I>; t \<in> reach\<^sub>t \<Sigma> \<delta>' q\<^sub>0'\<rbrakk> \<Longrightarrow> (f t \<in> I i) = (t \<in> I' i)" shows "accepting_pair\<^sub>G\<^sub>R \<delta> q\<^sub>0 (F, {I i |i. i \<in> \<I>}) w = accepting_pair\<^sub>G\<^sub>R \<delta>' q\<^sub>0' (F', {I' i |i. i \<in> \<I>}) w"
v_S2: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set"   and cfg :: "'a cfg" assumes "Reachability_Problem K S S1 S2"   and "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"   and "state cfg \<in> S2" shows "Reachability_Problem.v K S S1 S2 cfg = 1"
p_1: shows "g \<turnstile> defNode g p-ns\<rightarrow>n"
has_derivative_in_compose: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x within s)"   and "(g has_derivative g') (at (f x) within f ` s)" shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
dynclass_SuperM: fixes s :: "st"   and a' :: "val"   and statT :: "ref_ty" shows "invocation_class SuperM s a' statT = the_Class (RefT statT)"
lrev3_2_syntactic: shows "lrev_body3 oo unwrapH oo wrapH = lrev_body2"
ode_d_expr_def: fixes N :: "nat"   and n :: "nat" shows "ode_d_expr N n = FDERIV_floatariths (FDERIV_n_floatariths ode_e [0..<N] (map floatarith.Var [N..<2 * N]) n) [0..<N] (map floatarith.Var [2 * N..<3 * N])"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "subseqs (x # xs) = (let xss = subseqs xs in map ((#) x) xss @ xss)"
other_otherD: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'e"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd" assumes "other P {i} \<sigma> \<sigma>'" shows "\<forall>j. j \<noteq> i \<longrightarrow> P (\<sigma> j) (\<sigma>' j)"
path2_tl_in_\<alpha>n: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "m \<in> set (\<alpha>n g)"
StaticCastThrow: fixes E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and r :: "nat \<times> char list list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   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" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>Throw r,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
P: fixes a :: "'a"   and T :: "real" assumes "a \<in> S"   and "0 \<le> T"   and "T \<le> 1" shows "P (\<lambda>x. (1 - T) *\<^sub>R x + T *\<^sub>R a)"
intros_11: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile>While b c \<down> Normal s"
sconcat_scons: fixes x :: "[:'a:]"   and xs :: "[:[:'a:]:]" shows "sconcat\<cdot>(x :# xs) = x :@ sconcat\<cdot>xs"
set_join: fixes l :: "('b \<times> color) tree"   and x :: "'b"   and r :: "('b \<times> color) tree" shows "Tree2.set_tree (Set2_Join_RBT.join l x r) = Tree2.set_tree l \<union> {x} \<union> Tree2.set_tree r"
insert_absorb: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "{a} \<union> A = A"
subst_fm_RestrictedP: fixes f :: "tm"   and k :: "tm"   and g :: "tm"   and i :: "name"   and u :: "tm" shows "(RestrictedP f k g)(i::=u) = RestrictedP (subst i u f) (subst i u k) (subst i u g)"
Nml_HcompD_3: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg (t \<^bold>\<star> u)" shows "bicategorical_language.Nml V src trg t"
pcompose_0: fixes q :: "'a poly" shows "0 \<circ>\<^sub>p q = 0"
mem_rename_set_iff: fixes h :: "'c \<Rightarrow> 'd"   and z :: "'d"   and A :: "'c set" assumes "bij h" shows "(z \<in> h ` A) = (inv h z \<in> A)"
HBL1: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "prv (subst P \<langle>\<phi>\<rangle> xx)"
isRight_def: fixes x :: "'b + 'c" shows "isRight x \<equiv> \<exists>y. x = Inr y"
map_on_empty_is_regular: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set" shows "is_regular s {}"
normal_form_plus_normal_form: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "normal_form F (normal_form F p + normal_form F q) = normal_form F p + normal_form F q"
ptensor_mat_def: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "st2.ptensor_mat m1 m2 = tensor_mat st2.dims0 st2.vars1' m1 m2"
argmin_red_ivl: fixes i :: "int"   and i' :: "int"   and f :: "int \<Rightarrow> 'a"   and j :: "int"   and j' :: "int" assumes "i \<le> i'"   and "argmin f [i..j] \<in> {i'..j'}"   and "j' \<le> j" shows "argmin f [i'..j'] = argmin f [i..j]"
linord_of_list_range: fixes xs :: "'a list" shows "linord_of_list xs \<subseteq> set xs \<times> set xs"
lfilter_conv_lnths: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "lfilter P xs = lnths xs {n. enat n < llength xs \<and> P (xs $ n)}"
naturality: fixes f :: "'a" assumes "arr f" shows "trg (cod f) \<cdot> trg f = trg f \<cdot> trg (local.dom f)"
abs_summable_on_comparison_test: fixes g :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'f" assumes "g abs_summable_on A"   and "\<And>x. x \<in> A \<Longrightarrow> norm (f x) \<le> norm (g x)" shows "f abs_summable_on A"
exhaust: fixes y :: "'g LDCFG_node" obtains x1 :: "'g" where "y = Node x1"   | "y = NewEntry"   | "y = NewExit"