hcju/isabelleps huggingface dataset:
reach: shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow>\<^sup>* (cs1, css1, s1)"
CFG_edge_transfer_uses_only_Use: fixes a :: "'a"   and s :: "'c"   and s' :: "'c" assumes "valid_edge a"   and "\<forall>V\<in>Use (sourcenode a). state_val s V = state_val s' V"   and "pred (kind a) s"   and "pred (kind a) s'" shows "\<forall>V\<in>Def (sourcenode a). state_val (transfer (kind a) s) V = state_val (transfer (kind a) s') V"
has_vderiv_on_union: fixes f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and s :: "real set"   and t :: "real set" assumes "(f has_vderiv_on g) (s \<union> closure s \<inter> closure t)"   and "(f has_vderiv_on g) (t \<union> closure s \<inter> closure t)" shows "(f has_vderiv_on g) (s \<union> t)"
distr_restrict: fixes N :: "'b measure"   and L :: "'b measure"   and K :: "'c measure"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" assumes "sets N = sets L"   and "sets K \<subseteq> sets M"   and "\<And>X. X \<in> sets K \<Longrightarrow> emeasure M X = emeasure K X"   and "\<And>X. \<lbrakk>X \<in> sets M; X \<subseteq> space M - space K\<rbrakk> \<Longrightarrow> emeasure M X = 0"   and "f \<in> M \<rightarrow>\<^sub>M N"   and "f \<in> K \<rightarrow>\<^sub>M L" shows "distr M N f = distr K L f"
sel_1: fixes x1 :: "'c extended"   and x2 :: "'c"   and x3 :: "'c extended"   and x4 :: "'d" shows "P (InsertMessage x1 x2 x3 x4) = x1"
in_mset_sum_list_iff: fixes a :: "'b"   and \<A> :: "'b multiset list" shows "(a \<in># sum_list \<A>) = (\<exists>A\<in>set \<A>. a \<in># A)"
subst_bv_def: fixes u :: "Core.term"   and t :: "Core.term" shows "subst_bv u t \<equiv> subst_bv1 t 0 u"
bnd: fixes s :: "'a set"   and f :: "'a set" assumes "s \<in> simplices"   and "face f s"   and "bnd f" shows "card {s \<in> simplices. face f s} = 1"
affine_hull_Int_subset: fixes S :: "'a set"   and T :: "'a set" assumes "\<not> affine_dependent (S \<union> T)" shows "affine hull S \<inter> affine hull T \<subseteq> affine hull (S \<inter> T)"
sup_bot_right: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> \<bool>"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> \<bool>"   and bot :: "'a"   and x :: "'a" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
fprodrg_def: fixes n :: "nat"   and B :: "nat \<Rightarrow> ('d, 'e) Ring_scheme" shows "fprodrg n B = \<lparr>carrier = ac_fProd_Rg n B, pop = prod_pOp {i. i \<le> n} B, mop = prod_mOp {i. i \<le> n} B, zero = prod_zero {i. i \<le> n} B, tp = prod_tOp {i. i \<le> n} B, un = prod_one {i. i \<le> n} B\<rparr>"
functor_axioms: shows "functor D.CC.comp (\<cdot>\<^sub>D) T\<^sub>D"
connectedin_connected_components_of: fixes C :: "'b set"   and X :: "'b topology" assumes "C \<in> connected_components_of X" shows "connectedin X C"
DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P_DF_refine_FD: fixes A :: "'b set" shows "DF\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F\<^sub>D DF A"
H'_def: fixes t :: "'s"   and n :: "nat" shows "H' t n = {\<omega> \<in> space (stream_space (count_space UNIV)). enat n \<le> scount (HLD {t}) \<omega>}"
plus_matrix_def': fixes A :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c"   and B :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c" shows "Matrix A + Matrix B = Matrix (\<lambda>i j. A i j + B i j)"
hom_decompI: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "\<And>h U. (h, U) \<in> set ps \<Longrightarrow> homogeneous h" shows "hom_decomp ps"
ideD_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "category C"   and "partial_magma.ide C a" shows "partial_magma.dom C a = a"
Domt: fixes S :: "('a, 'b) Signature"   and f :: "'b" assumes "Signature S"   and "f \<in> Fn\<^bsub>S\<^esub>" shows "sDom\<^bsub>S\<^esub> f \<in> Ty\<^bsub>S\<^esub>"
pp_total: fixes x :: "'a" shows "times_top_class.total (- - x) = dense (x * top)"
map_prod_surj_on: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and A' :: "'a set"   and g :: "'d \<Rightarrow> 'c"   and B :: "'d set"   and B' :: "'c set" assumes "f ` A = A'"   and "g ` B = B'" shows "map_prod f g ` (A \<times> B) = A' \<times> B'"
L2_Controller_One: fixes y :: "nat \<Rightarrow> nat list"   and l :: "nat \<Rightarrow> bit"   and z :: "nat \<Rightarrow> bit list"   and t :: "nat" assumes "Controller_L y (fin_inf_append [Zero] l) l z"   and "l t = One"   and "hd (y (Suc t)) < 700" shows "z (Suc t) = []"
acyclic_5e_down_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y"   and "acyclic_5e y" shows "acyclic_5e x"
proper_itI: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and l :: "'b list"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "it = foldli l \<and> it' = foldli l" shows "proper_it it it'"
subdg_dom_simp: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = \<CC>\<lparr>Dom\<rparr>\<lparr>f\<rparr>"
indep_vars_def: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> (nat \<Rightarrow> real) \<Rightarrow> 'c"   and I :: "'b set" shows "M'.indep_vars M' X I = ((\<forall>i\<in>I. M'.random_variable (M' i) (X i)) \<and> M'.indep_sets (\<lambda>i. sigma_sets (space M'__) {X i -` A \<inter> space M'__ |A. A \<in> sets (M' i)}) I)"
left_tree_in_V: fixes s :: "'a"   and t :: "'a" shows "left_tree s t \<subseteq> V"
indexed_pmult_def: fixes P :: "'c multiset \<Rightarrow> ('a list \<times> nat) multiset \<Rightarrow> 'a"   and i :: "'c" shows "M.indexed_pmult P i = (\<lambda>m. if i \<in># m then P (m - {#i#}) else \<zero>\<^bsub>M\<^esub>)"
data_refinement_hoare: fixes S :: "'a \<Rightarrow> 'b set"   and D :: "'c \<Rightarrow> 'a"   and p :: "'d set"   and Q :: "'d \<Rightarrow> 'c"   and R :: "'b \<Rightarrow> 'd set" assumes "mono S"   and "mono D" shows "DataRefinement ({. p .} \<circ> [: Q :]) {: R :} D S = (\<forall>s. \<Turnstile> {s'. s \<in> R s' \<and> s \<in> p}{| S |}D (Q s))"
ND0aux4: fixes x :: "'b"   and y :: "'b"   and p :: "'c"   and b :: "('b, 'c) Combinators list" assumes "AllowPortFromTo x y p \<in> set b" shows "y \<in> set (net_list_aux b)"
ppos: fixes n :: "nat" shows "(0::'a) \<le> p n"
llength_llist_of_tllist: fixes xs :: "('c, 'd) tllist" shows "llength (llist_of_tllist xs) = tlength xs"
pnr: shows "d \<in> D \<longrightarrow> q \<noteq> r"
flip_altsublist_adjacent_length: fixes S :: "'b set"   and ss :: "'b list"   and ts :: "'b list" assumes "PreCoxeterSystem S"   and "PreCoxeterSystem.flip_altsublist_adjacent ss ts" shows "length ts = length ss"
no_possible_steps_rejects: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and t :: "(String.literal \<times> value list) list" assumes "possible_steps e s r l i = {||}" shows "\<not> recognises_execution e s r ((l, i) # t)"
has_lock_may_lock_t_eq: fixes l :: "('a \<times> nat) option"   and t :: "'a"   and t' :: "'a" assumes "has_lock l t"   and "may_lock l t'" shows "t = t'"
scalar_prod_left_sum_distrib: fixes vvs :: "'a vec list"   and w :: "'a vec" assumes "\<And>v. v \<in> set vvs \<Longrightarrow> v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "M.sumlist vvs \<bullet> w = (\<Sum>v\<leftarrow>vvs. v \<bullet> w)"
red_single_lookup: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single p q f t" shows "lookup q (local.punit.splus t (punit.lt f)) = (0::'c)"
mydiv_cancel: fixes k :: "nat"   and i :: "nat" assumes "0 < k" shows "mydiv (k * i) k = i"
left_minus_one_mult_self: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and n :: "nat"   and a :: "'c" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "times (power.power one times (uminus one) n) (times (power.power one times (uminus one) n) a) = a"
x_swap: shows "term_brackets vs (unlift_vars n x) = Some (unlift_vars n x')"
matrix_to_iarray_snd_Gauss_Jordan_in_ij_iarrays_PA: fixes i :: "'c"   and j :: "'d"   and A :: "(('b, 'd) vec, 'c) vec"   and P :: "(('b, 'c) vec, 'c) vec" assumes "\<not> vector_all_zero_from_index (mod_type_class.to_nat i, vec_to_iarray (column j A))" shows "matrix_to_iarray (snd (Gauss_Jordan_in_ij_PA (P, A) i j)) = snd (Gauss_Jordan_in_ij_iarrays_PA (matrix_to_iarray P, matrix_to_iarray A) (mod_type_class.to_nat i) (mod_type_class.to_nat j))"
lexp_prepend: fixes order :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and zs :: "'a list"   and xs :: "'a list" assumes "lexp order ys zs" shows "lexp order (xs @ ys) (xs @ zs)"
insert: fixes A :: "'d set"   and x :: "'d"   and g :: "'d \<Rightarrow> 'e" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
remove_comp_values: fixes x :: "'c"   and y :: "'c"   and pf1 :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" assumes "x \<noteq> y" shows "\<forall>n w. pf1 x n w = qty_rem_comp pf1 y x n w"
simps: fixes y :: "nat"   and x :: "nat" shows "c_assoc_value y x = (if y = 0 then 0 else if c_fst (c_hd y) = x then c_snd (c_hd y) else c_assoc_value (c_tl y) x)"
has_integral_periodic_offset: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and i :: "real"   and c :: "real" assumes "has_bochner_integral (lebesgue_on {a..b}) f i"   and "\<And>x. f (x + (b - a)) = f x" shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f (x + c)) i"
accepts_plus: fixes A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and w :: "'b list" shows "accepts (RegExp2NA.plus A) w = (\<exists>us. us \<noteq> [] \<and> w = concat us \<and> (\<forall>u\<in>set us. accepts A u))"
tail_invar_cons_up_invar: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.tail_invar (t # bq)"   and "SkewBinomialHeapStruc.rank t' \<le> SkewBinomialHeapStruc.rank t"   and "SkewBinomialHeapStruc.tree_invar t'" shows "SkewBinomialHeapStruc.invar (t' # t # bq)"
inf_compl_bot: fixes x :: "'a upred" shows "(x \<and> \<not> x) = false"
simps_2: fixes a :: "'d"   and x :: "'d list" shows "mset (a # x) = add_mset a (mset x)"
sup_aci_2: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd"   and z :: "'c \<times> 'c \<Rightarrow> 'd" shows "x \<oplus> y \<oplus> z = x \<oplus> (y \<oplus> z)"
elims_2: fixes x :: "'a Point"   and xa :: "'a Line" assumes "inLine x xa" obtains p :: "'a Point" and l :: "'a Line" where "x = p" and "xa = l" and "collinear p (basepoint l) (basepoint l \<leadsto> direction l)"
reachable_def: fixes SES :: "('a, 'b) SES_rec"   and s :: "'a" shows "reachable SES s \<equiv> \<exists>t. s0\<^bsub>SES\<^esub> t\<Longrightarrow>\<^bsub>SES\<^esub> s"
nth_equalityI: fixes xs :: "'g list"   and ys :: "'g list" assumes "length xs = length ys"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! i" shows "xs = ys"
natlist_trivial_1: fixes n :: "nat" shows "natpermute n 1 = {[n]}"
vec_index_conjugate: fixes i :: "nat"   and v :: "'c vec" assumes "i < dim_vec v" shows "conjugate v $ i = conjugate (v $ i)"
fact_binomial_limit: fixes k :: "nat" shows "(\<lambda>n. of_nat (k + n choose n) / of_nat (n ^ k)) \<longlonglongrightarrow> (1::'a) / fact k"
module_vec_simps_2: fixes ty :: "'a itself"   and n :: "nat" shows "\<zero>\<^bsub>module_vec ty n\<^esub> = 0\<^sub>v n"
when_when: fixes zero :: "'b"   and a :: "'b"   and P :: "bool"   and Q :: "bool" shows "zero.when zero (zero.when zero a P) Q = zero.when zero a (P \<and> Q)"
set_eqD1: fixes R :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and A' :: "'d set"   and a :: "'d" assumes "A {.=}\<^bsub>R\<^esub> A'"   and "a \<in> A" shows "\<exists>a'\<in>A'. a .=\<^bsub>R\<^esub> a'"
sb_step: shows "storebuffer_step\<^sup>*\<^sup>* (m, sb, \<O>, \<R>, \<S>) (m', sb', \<O>', \<R>', \<S>')"
gen_llength_code_2: fixes n :: "nat"   and x :: "'a"   and xs :: "'a llist" shows "gen_llength n (LCons x xs) = gen_llength (n + 1) xs"
c1: shows "((of_int_poly (pCons (coeff (pCons b c2) (degree q) div lead_coeff q) c1), of_int_poly (pCons b c2 - smult (coeff (pCons b c2) (degree q) div lead_coeff q) q)) = div_rat_poly_step (of_int_poly q) (rat_of_int b) (of_int_poly c1, of_int_poly c2)) = (pCons (rat_of_int (coeff (pCons b c2) (degree q) div lead_coeff q)) (of_int_poly c1) = pCons (coeff (pCons (rat_of_int b) (of_int_poly c2)) (degree q) / coeff (of_int_poly q) (degree q)) (of_int_poly c1) \<and> of_int_poly (pCons b c2 - smult (coeff (pCons b c2) (degree q) div lead_coeff q) q) = pCons (rat_of_int b) (of_int_poly c2) - smult (coeff (pCons (rat_of_int b) (of_int_poly c2)) (degree q) / coeff (of_int_poly q) (degree q)) (of_int_poly q))"
openin_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set" shows "openin (product_topology X I) = arbitrary union_of (finite intersection_of (\<lambda>F. \<exists>i U. F = {f. f i \<in> U} \<and> i \<in> I \<and> openin (X i) U) relative_to topspace (product_topology X I))"
Local: fixes G :: "'b triple set"   and P :: "'b assn"   and c :: "com"   and Q :: "'b assn"   and X :: "loc"   and s' :: "state"   and a :: "state \<Rightarrow> nat" assumes "G|-{P}. c .{\<lambda>Z s. Q Z (s[Loc X::=s'<X>])}" shows "G|-{\<lambda>Z s. s' = s \<and> P Z (s[Loc X::=a s])}. LOCAL X:=a IN c .{Q}"
least_mod_type: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and n :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "less_eq zero n"
foldr_cong: fixes a :: "'a"   and b :: "'a"   and l :: "'b list"   and k :: "'b list"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a" assumes "a = b"   and "l = k"   and "\<And>a x. x \<in> set l \<Longrightarrow> f x a = g x a" shows "foldr f l a = foldr g k b"
supp_set_mset: fixes M :: "'a multiset" shows "supp (set_mset M) \<subseteq> supp M"
primepow_power_iff: fixes p :: "'a"   and n :: "nat" assumes "unit_factor p = (1::'a)" shows "primepow (p ^ n) = (primepow p \<and> 0 < n)"
a_1: shows "list_all stateful_strand_step.is_Receive (unlabel (transaction_receive T))"
as0_14: shows "D0 \<cdot> \<eta>0 = D"
moebius_mu_power: fixes a :: "nat"   and n :: "nat" assumes "1 < a"   and "1 < n" shows "moebius_mu (a ^ n) = (0::'b)"
simps_1: fixes f :: "'a \<Rightarrow> 'a"   and as :: "'a list" shows "mapAt [] f as = as"
generate_empty: shows "generate (Z 1) {} = {\<one>\<^bsub>Z 1\<^esub>}"
chine_hcomp_def: fixes \<nu> :: "(('a, 'a set) Maps.arr, 'b) arrow_of_spans_data_scheme"   and \<mu> :: "(('a, 'a set) Maps.arr, 'c) arrow_of_spans_data_scheme" shows "Span.chine_hcomp \<nu> \<mu> \<equiv> Maps.tuple (Chn \<nu> \<odot> Maps.PRJ\<^sub>1 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>))) (Leg0 (Cod \<nu>)) (Leg1 (Cod \<mu>)) (Chn \<mu> \<odot> Maps.PRJ\<^sub>0 (Leg0 (Dom \<nu>)) (Leg1 (Dom \<mu>)))"
inf_img_fin_dom': fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "finite (f ` A)"   and "infinite A" shows "\<exists>y\<in>f ` A. infinite (f -` {y} \<inter> A)"
compare_Eq_is_eq: fixes compare :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b" assumes "class.compare compare" shows "(compare x y = Eq) = (x = y)"
add_scalar_prod_distrib: fixes v\<^sub>1 :: "'a vec"   and n :: "nat"   and v\<^sub>2 :: "'a vec"   and v\<^sub>3 :: "'a vec" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n"   and "v\<^sub>3 \<in> carrier_vec n" shows "(v\<^sub>1 + v\<^sub>2) \<bullet> v\<^sub>3 = v\<^sub>1 \<bullet> v\<^sub>3 + v\<^sub>2 \<bullet> v\<^sub>3"
open_UN: fixes "open" :: "'b set \<Rightarrow> bool"   and A :: "'c set"   and B :: "'c \<Rightarrow> 'b set" assumes "class.topological_space open"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Union> (B ` A))"
targets_def: fixes \<mu> :: "'a" shows "targets \<mu> \<equiv> {b. weak_unit b \<and> b \<star> \<mu> \<noteq> null}"
hyps_4: shows "\<forall>f\<in>funs_term (Fun f S). \<not> is_Abs f"
max_step_r: fixes j :: "nat" assumes "t j \<le> t1" shows "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (t1 - t 0) + 1) - 1))"
id: shows "{\<bar>fs_init ! i $ j\<bar> |i j. i < m \<and> j < n} \<union> {0} = (\<lambda>(i, j). \<bar>fs_init ! i $ j\<bar>) ` ({0..<m} \<times> {0..<n}) \<union> {0}"
measure_Pi_pmf_PiE_dflt: fixes A :: "'a set"   and dflt :: "'b"   and p :: "'a \<Rightarrow> 'b pmf"   and B :: "'a \<Rightarrow> 'b set" assumes "finite A" shows "measure_pmf.prob (Pi_pmf A dflt p) (PiE_dflt A dflt B) = (\<Prod>x\<in>A. measure_pmf.prob (p x) (B x))"
edom_join: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "edom (f \<squnion> g) = edom f \<union> edom g"
hash_view_zipper_eqI: fixes h :: "'b \<Rightarrow> 'a"   and p :: "('b \<times> ('b, 'a) rose_tree\<^sub>m list \<times> ('b, 'a) rose_tree\<^sub>m list) list"   and h' :: "'c \<Rightarrow> 'a"   and p' :: "('c \<times> ('c, 'a) rose_tree\<^sub>m list \<times> ('c, 'a) rose_tree\<^sub>m list) list"   and v :: "('b, 'a) rose_tree\<^sub>m"   and v' :: "('c, 'a) rose_tree\<^sub>m" assumes "map (hash_path_elem h) p = map (hash_path_elem h') p'"   and "hash_tree h v = hash_tree h' v'" shows "hash_tree h (tree_of_zipper\<^sub>m (p, v)) = hash_tree h' (tree_of_zipper\<^sub>m (p', v'))"
is_RealVal_def: fixes val :: "val" shows "is_RealVal val = (\<exists>x4. val = RealVal x4)"
UN_iff: fixes b :: "'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
ltlc_eq_on: fixes \<phi> :: "'b ltlc"   and w :: "nat \<Rightarrow> 'b set"   and w' :: "nat \<Rightarrow> 'b set" assumes "pw_eq_on (atoms_ltlc \<phi>) w w'" shows "w \<Turnstile>\<^sub>c \<phi> = w' \<Turnstile>\<^sub>c \<phi>"
simps_12: fixes f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and f3 :: "('d \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'c"   and x3 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" shows "(case Apply x3 of Const x \<Rightarrow> f1 x | Load x \<Rightarrow> f2 x | Apply x \<Rightarrow> f3 x) = f3 x3"
inf_chain_subset: fixes p :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and q :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "inf_chain p f"   and "p \<le> q" shows "inf_chain q f"
cases: fixes x :: "((int \<times> int) set set, int) Combinators" obtains "x = DenyAll"   | xa :: "(int \<times> int) set set" and y :: "(int \<times> int) set set" where "x = DenyAllFromTo xa__ y__"   | xa :: "(int \<times> int) set set" and y :: "(int \<times> int) set set" and p :: "int" where "x = AllowPortFromTo xaa__ ya__ p"   | xa :: "((int \<times> int) set set, int) Combinators" and y :: "((int \<times> int) set set, int) Combinators" where "x = xa \<oplus> y"
wfE_min': fixes R :: "('c \<times> 'c) set"   and Q :: "'c set" assumes "wf R"   and "Q \<noteq> {}" obtains z :: "'c" where "z \<in> Q" and "\<And>y. (y, z) \<in> R \<Longrightarrow> y \<notin> Q"
indep_events_def: fixes A :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "indep_events A I = (A ` I \<subseteq> events \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j))))"
fun_rel_id_simp: shows "Id \<rightarrow> Id = Id"
order_iff: fixes a :: "'g"   and b :: "'g" shows "(a \<le> b) = (a = a \<sqinter> b)"
Done_bind: fixes a :: "'i option"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "Resumption.resumption.Done a \<bind> f = (case a of None \<Rightarrow> Resumption.resumption.Done None | Some a \<Rightarrow> f a)"
from_hma\<^sub>v_diff: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec"   and w :: "('c, 'd) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v - from_hma\<^sub>v w = from_hma\<^sub>v (v - w)"
rel_gpv''_map_gpv1: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and f :: "'o \<Rightarrow> 'i"   and g :: "'p \<Rightarrow> 'k"   and gpv :: "('o, 'p, 'm) gpv"   and gpv' :: "('j, 'l, 'n) gpv" shows "rel_gpv'' A C R (map_gpv f g gpv) gpv' = rel_gpv'' (\<lambda>a. A (f a)) (\<lambda>c. C (g c)) R gpv gpv'"
util_imp_trans: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" assumes "ordinal_utility carrier relation u" shows "trans relation"
Diff_iff: fixes c :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(c \<in> A - B) = (c \<in> A \<and> c \<notin> B)"
unitor_coincidence_1: shows "\<l>\<^sub>C[\<I>\<^sub>C] = \<iota>\<^sub>C"
UP_pre_univ_propI: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "cring R"   and "cring S"   and "h \<in> ring_hom R S" shows "UP_pre_univ_prop R S h"
pref_profile_from_tableI: fixes agents :: "'a set"   and alts :: "'b set"   and xss :: "('a \<times> 'b set list) list" assumes "prefs_from_table_wf agents alts xss" shows "pref_profile_wf agents alts (prefs_from_table xss)"
inverse_arrowsE: fixes f :: "'a"   and g :: "'a" assumes "inverse_arrows f g" obtains "ide (g \<cdot> f)" and "ide (f \<cdot> g)"
p_prod_iff: fixes ps :: "'a list"   and X :: "'a list set"   and Y :: "'a list set" shows "(ps \<in> p_prod X Y) = (\<exists>qs rs. ps = p_fusion qs rs \<and> qs \<in> X \<and> rs \<in> Y \<and> p_filter qs rs)"
invertible_mult: fixes A :: "(('d, 'e) vec, 'f) vec"   and B :: "(('d, 'g) vec, 'e) vec" assumes "invertible A"   and "invertible B" shows "invertible (A ** B)"
simps_13: fixes x51 :: "'b formula"   and x52 :: "'b formula"   and x1 :: "'b" shows "x51 \<^bold>\<or> x52 \<noteq> Atom x1"
lambda_predicates_2_1: fixes \<phi> :: "\<kappa> \<Rightarrow> \<o>"   and x :: "\<nu>" assumes "IsProperInX \<phi>" shows "[[\<lparr>\<^bold>\<lambda>x. \<phi> (x\<^sup>P),x\<^sup>P\<rparr> \<^bold>\<equiv> \<phi> (x\<^sup>P)]]"
ty_init: shows "li'.wf_gglobal (init_gc prog)"
axioms_1: 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 \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "category A"
eq_refl: fixes x :: "'a option \<Rightarrow>\<^sub>0 nat"   and y :: "'a option \<Rightarrow>\<^sub>0 nat" assumes "x = y" shows "extended_ord x y"
eij_neq_0: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and j :: "nat" assumes "hlde a b"   and "i < length a"   and "j < length b" shows "Linear_Diophantine_Equations.eij a b i j \<noteq> 0"
simps_3: fixes prg :: "prog"   and cls :: "qtname"   and lcl :: "lenv"   and more :: "'a" shows "lcl \<lparr>prg = prg, cls = cls, lcl = lcl, \<dots> = more\<rparr> = lcl"