hcju/isabelleps huggingface dataset:
enc_eq_welldef: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "enc (w, I) = enc (w', I')"   and "wf_formula (length I) \<phi>"   and "wf_interp_for_formula (w, I) \<phi>"   and "wf_interp_for_formula (w', I') \<phi>" shows "((w, I) \<Turnstile> \<phi>) = ((w', I') \<Turnstile> \<phi>)"
compute_mult_scalar_poly_mapping: fixes xs :: "('a \<times> 'b) list"   and q :: "'a \<Rightarrow>\<^sub>0 'b" shows "punit.mult_scalar (Pm_fmap (fmap_of_list xs)) q = (case xs of [] \<Rightarrow> Pm_fmap fmempty | (t, c) # ys \<Rightarrow> punit.monom_mult c t q + punit.mult_scalar (except (Pm_fmap (fmap_of_list ys)) {t}) q)"
pid_def: fixes x :: "'d \<times> 'e" shows "pm_prod.rel_partial_monoid.pid x = (if x \<in> pm_prod_E then 1::'c else \<bottom>)"
elims: fixes x :: "('a, 'b) fmap"   and xa :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and y :: "('a, 'b) fmap list" assumes "state_list x xa = y" obtains s :: "('a, 'b) fmap" where "x = s__" and "xa = []" and "y = [s__]"   | s :: "('a, 'b) fmap" and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" and as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" where "x = s" and "xa = a # as" and "y = s # state_list (state_succ s a) as"
cases: fixes AP' :: "'aprop set"   and a :: "('state \<times> 'aprop set) stream" assumes "wfp AP' a" obtains s :: "'state" and A :: "'aprop set" and \<pi> :: "('state \<times> 'aprop set) stream" where "a = (s, A) ## \<pi>" and "s \<in> S" and "A \<subseteq> AP'" and "A \<inter> AP = L s" and "stateOf \<pi> \<in> \<delta> s" and "wfp AP' \<pi>"
continuous_on_Gen_Shleg: fixes A :: "real set"   and k :: "nat" shows "continuous_on A (local.Gen_Shleg k)"
sample_cong: fixes p :: "'p pmf"   and q :: "'p pmf"   and f :: "'p \<Rightarrow> 'm"   and g :: "'p \<Rightarrow> 'm" assumes "p = q"   and "\<And>x. x \<in> set_pmf p \<Longrightarrow> f x = g x" shows "sample p f = sample q g"
tight_subseq_weak_converge: fixes \<mu> :: "nat \<Rightarrow> real measure"   and M :: "real measure" assumes "\<And>n. real_distribution (\<mu> n)"   and "real_distribution M"   and "tight \<mu>"   and "\<And>s \<nu>. \<lbrakk>strict_mono s; real_distribution \<nu>; weak_conv_m (\<mu> \<circ> s) \<nu>\<rbrakk> \<Longrightarrow> weak_conv_m (\<mu> \<circ> s) M" shows "weak_conv_m \<mu> M"
le_iff_sup: fixes x :: "'a multiset"   and y :: "'a multiset" shows "(x \<subseteq># y) = (x \<union># y = y)"
left_idem: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<oplus> (a \<oplus> b) = a \<oplus> b"
assms: shows "(u \<longlongrightarrow> 1) F"
closest_supchamber_def: fixes F :: "'a set"   and D :: "'a set" shows "domain.closest_supchamber F D = (ARG_MIN (\<lambda>C. domain.chamber_distance C D) C. domain.chamber C \<and> order.greater_eq C F)"
dvd_associated1: fixes a :: "'b"   and b :: "'b" assumes "\<exists>u. u dvd (1::'b) \<and> a = u * b" shows "a dvd b \<and> b dvd a"
simps_6: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>]\<rbrace> = \<ll>' \<lbrace>t\<rbrace>"
bangPres: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P \<leadsto>\<^sup>^<Rel> Q"   and "\<And>P Q R T. \<lbrakk>(P, Q) \<in> Rel; (R, T) \<in> Rel'\<rbrakk> \<Longrightarrow> (P \<parallel> R, Q \<parallel> T) \<in> Rel'"   and "\<And>P Q x. (P, Q) \<in> Rel' \<Longrightarrow> (<\<nu>x>P, <\<nu>x>Q) \<in> Rel'"   and "\<And>P Q. (P \<parallel> !P, Q) \<in> Rel' \<Longrightarrow> (!P, Q) \<in> Rel'"   and "bangRel Rel \<subseteq> Rel'"   and "eqvt Rel'" shows "!P \<leadsto>\<^sup>^<Rel'> !Q"
induct: fixes x1 :: "binding list"   and x2 :: "ty"   and x3 :: "ty"   and P :: "binding list \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool" assumes "x1\<turnstile>x2<:x3"   and "\<And>\<Gamma> S. \<lbrakk>\<turnstile> \<Gamma> ok; S closed_in \<Gamma>\<rbrakk> \<Longrightarrow> P \<Gamma> S Top"   and "\<And>\<Gamma> X. \<lbrakk>\<turnstile> \<Gamma> ok; X \<in> ty_dom \<Gamma>\<rbrakk> \<Longrightarrow> P \<Gamma> (Tvar X) (Tvar X)"   and "\<And>X S \<Gamma> T. \<lbrakk>TVarB X S \<in> set \<Gamma>; \<Gamma>\<turnstile>S<:T; P \<Gamma> S T\<rbrakk> \<Longrightarrow> P \<Gamma> (Tvar X) T"   and "\<And>\<Gamma> T\<^sub>1 S\<^sub>1 S\<^sub>2 T\<^sub>2. \<lbrakk>\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1; P \<Gamma> T\<^sub>1 S\<^sub>1; \<Gamma>\<turnstile>S\<^sub>2<:T\<^sub>2; P \<Gamma> S\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P \<Gamma> (S\<^sub>1 \<rightarrow> S\<^sub>2) (T\<^sub>1 \<rightarrow> T\<^sub>2)"   and "\<And>\<Gamma> T\<^sub>1 S\<^sub>1 X S\<^sub>2 T\<^sub>2. \<lbrakk>\<Gamma>\<turnstile>T\<^sub>1<:S\<^sub>1; P \<Gamma> T\<^sub>1 S\<^sub>1; (TVarB X T\<^sub>1 # \<Gamma>)\<turnstile>S\<^sub>2<:T\<^sub>2; P (TVarB X T\<^sub>1 # \<Gamma>) S\<^sub>2 T\<^sub>2\<rbrakk> \<Longrightarrow> P \<Gamma> (\<forall>X<:S\<^sub>1. S\<^sub>2) (\<forall>X<:T\<^sub>1. T\<^sub>2)" shows "P x1 x2 x3"
sq_norm_zero_vec: fixes n :: "nat" shows "\<parallel>0\<^sub>v n\<parallel>\<^sup>2 = (0::'b)"
Seq: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c\<^sub>1 :: "('b, 'a, 'c) com"   and s :: "'b"   and c\<^sub>2 :: "('b, 'a, 'c) com" assumes "\<Gamma>\<turnstile>c\<^sub>1 \<down> Normal s"   and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c\<^sub>2 \<down> s'" shows "\<Gamma>\<turnstile>Seq c\<^sub>1 c\<^sub>2 \<down> Normal s"
elims_2: fixes x :: "('a, 'b) Combinators list" assumes "noDenyAll1 x" obtains xs :: "('a, 'b) Combinators list" where "x = DenyAll # xs" and "noDenyAll xs"   | "x = []" and "noDenyAll []"   | vb :: "'a" and vc :: "'a" and va :: "('a, 'b) Combinators list" where "x = DenyAllFromTo vb__ vc__ # va__" and "noDenyAll (DenyAllFromTo vb__ vc__ # va__)"   | vb :: "'a" and vc :: "'a" and vd :: "'b" and va :: "('a, 'b) Combinators list" where "x = AllowPortFromTo vba__ vca__ vd # vaa__" and "noDenyAll (AllowPortFromTo vba__ vca__ vd # vaa__)"   | vb :: "('a, 'b) Combinators" and vc :: "('a, 'b) Combinators" and va :: "('a, 'b) Combinators list" where "x = vb \<oplus> vc # va" and "noDenyAll (vb \<oplus> vc # va)"
simps_9: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and \<nu> :: "'b set" shows "af_letter (\<phi> R\<^sub>n \<psi>) \<nu> = af_letter \<psi> \<nu> and\<^sub>n af_letter \<phi> \<nu> or\<^sub>n (\<phi> R\<^sub>n \<psi>)"
dupe_monic_dynamic_def: fixes p :: "int" shows "dupe_monic_dynamic p = (if p \<le> 65535 then dupe_monic_i_int (finite_field_ops32 (uint32_of_int p)) else if p \<le> 4294967295 then dupe_monic_i_int (finite_field_ops64 (uint64_of_int p)) else dupe_monic_i_int (finite_field_ops_integer (integer_of_int p)))"
exhaust: fixes y :: "preal" obtains x1 :: "real" where "y = preal x1"   | "y = preal.PInfty"
mat_of_cols_carrier_1: fixes n :: "nat"   and vs :: "'b vec list" shows "mat_of_cols n vs \<in> carrier_mat n (length vs)"
Ug_def: fixes G :: "('c, 'd) Group_scheme" shows "Ug G = \<natural>\<^bsub>G\<^esub>{\<one>\<^bsub>G\<^esub>}"
unbounded_all: fixes R :: "('c \<Rightarrow> real) set"   and u :: "'c \<Rightarrow> real"   and x :: "'c" assumes "R \<in> \<R>"   and "u \<in> R"   and "unbounded x R"   and "x \<in> \<X>" shows "real (k x) < u x"
has_domain_iff_arr: fixes f :: "'c" shows "(C.domains f \<noteq> {}) = C.arr f"
simps_56: fixes x5 :: "char list" shows "Return \<noteq> Call x5"
create_name_gba_aimpl_def: fixes \<phi> :: "'a ltlr" shows "create_name_gba_aimpl \<phi> \<equiv> create_graph_aimpl \<phi> \<bind> (\<lambda>nds. ASSERT (nds_invars nds) \<bind> (\<lambda>_. cr_rename_gba nds \<phi>))"
v11_vconverse: shows "v11 ((\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>)"
finite_deg_le_sect: fixes X :: "'b set"   and d :: "nat" assumes "finite X" shows "finite (deg_le_sect X d)"
color_of_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)"   and "0 < n" shows "color_of (fst (rbtreeify_g n kvs)) = B"
map_fst_zip: fixes xs :: "'g list"   and ys :: "'h list" assumes "length xs = length ys" shows "map fst (zip xs ys) = xs"
DBM_reset_neg_cycle_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and k :: "nat"   and d :: "'c"   and M' :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and i :: "nat"   and xs :: "nat list" assumes "DBM_reset M n k d M'"   and "len M i i xs < Le (0::'c)"   and "set (k # i # xs) \<subseteq> {0..n}" shows "\<exists>j ys. set (j # ys) \<subseteq> {0..n} \<and> len M' j j ys < Le (0::'c)"
uminus_of_uminus_id: fixes x :: "'b literal" shows "- (- x) = x"
nonzero_imp_inverse_nonzero: fixes a :: "'a" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>" shows "inv a \<noteq> \<zero>"
lF_char_2: fixes f :: "'c" assumes "C.ide f" shows "F (trg\<^sub>C f) \<star>\<^sub>D lF f = (F \<i>\<^sub>C[trg\<^sub>C f] \<cdot>\<^sub>D \<Phi> (trg\<^sub>C f, trg\<^sub>C f) \<star>\<^sub>D F f) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[F (trg\<^sub>C f), F (trg\<^sub>C f), F f]"
ginitial_impl_def: fixes v0 :: "'q"   and s0 :: "('q set \<times> 'q set) option \<times> ('q \<Rightarrow> node_state option)" shows "ginitial_impl v0 s0 \<equiv> ([acc v0], initial_impl v0 (snd s0))"
wordinterval_to_set_ipcidr_tuple_to_wordinterval_simple_match_ip_set: fixes ip :: "'a word \<times> nat" shows "wordinterval_to_set (ipcidr_tuple_to_wordinterval ip) = {d. simple_match_ip ip d}"
is_locally_frac_subset: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and V :: "'a set set" assumes "is_locally_frac s U"   and "V \<subseteq> U" shows "is_locally_frac s V"
cSup_eq_maximum: fixes z :: "'a multiset"   and X :: "'a multiset set" assumes "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<subseteq># z" shows "Sup X = z"
mask_eq_take_bit_minus_one: fixes n :: "nat" shows "mask n = take_bit n (- (1::'a))"
inner_eq_zero_iff: fixes x :: "real" shows "(x * x = 0) = (x = 0)"
Rxa_def: fixes R :: "('e, 'f) Ring_scheme"   and a :: "'e" shows "R \<diamondsuit>\<^sub>p a = {x. \<exists>r\<in>carrier R. x = r \<cdot>\<^sub>r\<^bsub>R\<^esub> a}"
fBallE: fixes A :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"
iter_dom_invariant_complete: fixes ss' :: "nat list list"   and w' :: "nat list"   and i :: "nat"   and res :: "nat list"   and k :: "nat" shows "iter f step start (unstables r step start) = (ss', w') \<longrightarrow> i < n \<and> ss' ! i = res \<and> k \<notin> set res \<longrightarrow> non_strict_dominate k i"
I_notcarr: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme" assumes "primeideal I R" shows "carrier R \<noteq> I"
blaschke_ounit_circle_fix': fixes a :: "complex" assumes "cmod a < 1" shows "moebius_ocircline (blaschke a) ounit_circle = ounit_circle"
comparator_compare: fixes compare :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "class.compare compare" shows "comparator compare"
is_ChurchRosser_def: shows "is_ChurchRosser \<equiv> \<forall>a b. a \<leftrightarrow>\<^sup>* b \<longrightarrow> a \<down>\<^sup>* b"
A: shows "c_fst (c_fst key) mod 7 = 0"
mbind_unit: fixes f :: "'f \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'd) option" shows "mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e [] f = unit\<^sub>S\<^sub>E []"
attach_shadow_root_impl: shows "attach_shadow_root = l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_attach_shadow_root set_shadow_root set_mode get_tag_name get_shadow_root"
hull_redundant_eq: fixes a :: "'b"   and S :: "'b set \<Rightarrow> bool"   and s :: "'b set" shows "(a \<in> S hull s) = (S hull insert a s = S hull s)"
psubstInp_empInp: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" shows "empInp %[rho] = empInp"
DirProds_comm_group_iff: fixes G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme"   and I :: "'d set" shows "comm_group (DirProds G I) = (\<forall>i\<in>I. comm_group (G i))"
ocompleteTEs_2: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and i' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), disconnect(i, i'), \<sigma>', s') \<in> ocnet_sos S" obtains "((\<sigma>, s), disconnect(i, i'), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
minimal_unsat_core_real_def: fixes I :: "'b set"   and ics :: "('b \<times> constraint) list" shows "minimal_unsat_core_real I ics = (I \<subseteq> fst ` set ics \<and> (\<nexists>v. (I, v) \<Turnstile>\<^sub>r\<^sub>i\<^sub>c\<^sub>s set ics) \<and> (distinct_indices ics \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set ics)))"
OUTfromVCorrect1_data13: shows "OUTfromVCorrect1 data13"
derangements_no_fixpoint: fixes d :: "nat \<Rightarrow> nat"   and A :: "nat set"   and x :: "nat" assumes "d \<in> derangements A"   and "x \<in> A" shows "d x \<noteq> x"
Suc: shows "\<K> ^ n = (\<Sum>w | length w = n \<and> set w \<subseteq> L. 1 / b ^ cw_len_concat w)"
distinct_adj_Cons: fixes x :: "'g"   and xs :: "'g list" shows "distinct_adj (x # xs) = (xs = [] \<or> x \<noteq> hd xs \<and> distinct_adj xs)"
ms: shows "matrix_seq d (matrix_sum d (\<lambda>n. denote_while_n (x1 0) (x1 1) (denote x2) n \<rho>))"
crel_vs_success: fixes P :: "heap \<Rightarrow> bool"   and Q :: "heap \<Rightarrow> bool"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and dp :: "'c \<Rightarrow> 'd"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'e"   and b :: "'f Heap"   and heap :: "heap" assumes "dp_heap P Q lookup lookup_st update update_st"   and "dp_heap.crel_vs P Q dp lookup_st R a b"   and "P heap"   and "Q heap"   and "dp_consistency.cmem lookup_st dp heap" shows "success b heap"
assms: shows "\<forall>n. set (perm2 l n) = set l"
lemma_poly_length_mult: fixes k :: "'a"   and p :: "'a list"   and h :: "'a"   and a :: "'a" shows "length (k %* p +++ (h # a %* p)) = Suc (length p)"
finite_entropy_integrable_transform: fixes S :: "'d measure"   and X :: "'a \<Rightarrow> 'd"   and Px :: "'d \<Rightarrow> real"   and T :: "'e measure"   and Y :: "'a \<Rightarrow> 'e"   and Py :: "'e \<Rightarrow> real"   and f :: "'e \<Rightarrow> 'd" assumes "finite_entropy S X Px"   and "distributed M T Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space T \<Longrightarrow> 0 \<le> Py x"   and "X = (\<lambda>x. f (Y x))"   and "f \<in> T \<rightarrow>\<^sub>M S" shows "integrable T (\<lambda>x. Py x * log b (Px (f x)))"
multiset_diff_union_assoc: fixes C :: "'b multiset"   and B :: "'b multiset"   and A :: "'b multiset" assumes "C \<subseteq># B" shows "A + B - C = A + (B - C)"
odd_succ_div_two: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment"   and "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus a one) (numeral.numeral one plus (num.Bit0 num.One)) = plus (divide a (numeral.numeral one plus (num.Bit0 num.One))) one"
map_nth_eq_self: fixes fas :: "'d list"   and l :: "nat" assumes "length fas = l" shows "map ((!) fas) [0..<l] = fas"
coprime_prime_exists: fixes n :: "nat" assumes "0 < n" obtains p :: "nat" where "prime p" and "coprime p n"
prime_factors_one: shows "prime_factors (Suc 0) = {}"
fresh_floatariths_fresh_floatarithI: fixes fas :: "floatarith list"   and xs :: "nat list"   and fa :: "floatarith" assumes "freshs_floatariths fas xs"   and "fa \<in> set fas" shows "freshs_floatarith fa xs"
lt_ex: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.no_bot less_eq less" shows "\<exists>y. less y x"
ipresAbs_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresAbs h hA MOD \<equiv> \<forall>xs x s X. isInBar (xs, s) \<and> wls s X \<longrightarrow> hA (Abs xs x X) = igAbs MOD xs x (h X)"
scale_left_distrib_on: fixes x :: "'b"   and a :: "'a"   and b :: "'a" assumes "x \<in> U\<^sub>V\<^sub>S" shows "(a + b) *s x = a *s x + b *s x"
qstar_unfoldl: fixes x :: "'b" shows "(1::'b) \<squnion> x \<cdot> qstar x \<le> qstar x"
card_disjoint_shuffles: fixes xs :: "'b list"   and ys :: "'b list" assumes "set xs \<inter> set ys = {}" shows "card (shuffles xs ys) = length xs + length ys choose length xs"
splitFace_faces_1: fixes f12 :: "face"   and f21 :: "face"   and newGraph :: "graph"   and g :: "graph"   and ram1 :: "nat"   and ram2 :: "nat"   and oldF :: "face"   and newVertexList :: "nat list" assumes "(f12, f21, newGraph) = splitFace g ram1 ram2 oldF newVertexList"   and "oldF \<in> \<F> g" shows "\<F> newGraph \<union> {oldF} = {f12, f21} \<union> \<F> g"
Abs1_eq_iff_fresh_2: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and x :: "'b"   and y :: "'b" assumes "atom c \<sharp> (a, b, x, y)" shows "([{atom a}]res. x = [{atom b}]res. y) = ((a \<leftrightarrow> c) \<bullet> x = (b \<leftrightarrow> c) \<bullet> y)"
gcd_mono: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c"   and "dvd.dvd times b d" shows "dvd.dvd times (gcd a b) (gcd c d)"
imp_triv_I: fixes H :: "fmla set"   and B :: "fmla"   and A :: "fmla" assumes "H \<turnstile> B" shows "H \<turnstile> A IMP B"
Min_insert2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and a :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> less_eq a b" shows "linorder.Min less_eq (Set.insert a A) = a"
\<PP>_wellformed: fixes p :: "(symbol \<times> character list) list" assumes "p \<in> \<PP>" shows "wellformed_tokens p"
inverse_arrows_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "local.iso \<mu>"   and "local.iso \<nu>"   and "src \<nu> = trg \<mu>" shows "inverse_arrows (\<nu> \<star> \<mu>) (local.inv \<nu> \<star> local.inv \<mu>)"
R: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "VarB S \<Colon> \<Gamma> \<turnstile> s : S'"   and "\<Gamma> \<turnstile> \<down>\<^sub>\<tau> 1 0 S' <: U'" shows "P"
fold_infinite: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" assumes "infinite A" shows "Finite_Set.fold f z A = z"
simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.numeral plus" shows "numeral.numeral one plus (num.Bit0 n) = plus (numeral.numeral one plus n) (numeral.numeral one plus n)"
execute_parallel_operators_strips_none_if: fixes ops :: "'a strips_operator list"   and v :: "'a"   and s :: "'a \<Rightarrow> bool option" assumes "\<forall>op\<in>set ops. v \<notin> set (add_effects_of op) \<and> v \<notin> set (delete_effects_of op)"   and "s v = None" shows "execute_parallel_operator s ops v = None"
g_comp_is_pr: shows "g_comp \<in> PrimRec2"
OclOr_false2: fixes Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(Y or false) = Y"
and_mask_eq_iff_le_mask: fixes w :: "'b word"   and n :: "nat" shows "(w AND mask n = w) = (w \<le> mask n)"
OclOr_idem: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X or X) = X"
map_of_SomeI: fixes kxs :: "('a \<times> 'b) list"   and k :: "'a"   and x :: "'b" assumes "distinct_fst kxs"   and "(k, x) \<in> set kxs" shows "map_of kxs k = \<lfloor>x\<rfloor>"
amtx_assn_def: fixes N :: "nat"   and M :: "nat"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "amtx_assn N M A \<equiv> hr_comp (IICF_Array_Matrix.is_amtx N M) (\<langle>the_pure A\<rangle>mtx_rel)"
assms_2: shows "\<pi> (real p) = real n + 1"
has_integral_sin_nx: fixes n :: "int" shows "((\<lambda>x. sin (real_of_int n * x)) has_integral 0) {- pi..pi}"
eval_simps_1: fixes f :: "'c local.term" assumes "Arr f" shows "arr \<lbrace>f\<rbrace>"
arith_simps_1: shows "num.One + num.One = num.Bit0 num.One"
envObs_def: fixes a :: "'a"   and s :: "('a, 'es, 'as) BEState" shows "envObs a s \<equiv> (envObsC (es s), ODList.lookup (ps s) a)"
type_aexpr_type_wellformed: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'AExp"   and t :: "'BExp set" assumes "types_wellformed \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>a e \<in> t" shows "type_wellformed t"
pc1: shows "Some (prop_config c1 p) = propagate_all (prop_config c0 p)"
real_cond_exp_add: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M F (\<lambda>x. f x + g x) x = real_cond_exp M F f x + real_cond_exp M F g x"
Compl_disjoint2: fixes A :: "'b set" shows "- A \<inter> A = {}"
rev_drop: fixes i :: "nat"   and xs :: "'b list" shows "rev (drop i xs) = take (length xs - i) (rev xs)"
prod_mset_prime_factorization_weak: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero" shows "normalize (comm_monoid_mult.prod_mset times one (factorial_semiring.prime_factorization zero times one normalize x)) = normalize x"
measurable_emeasure_Pair: fixes Q :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "Q \<in> sets (N \<Otimes>\<^sub>M M2)" shows "(\<lambda>x. emeasure M2 (Pair x -` Q)) \<in> borel_measurable N"
A_vertex: shows "A \<Gamma> \<subseteq> \<^bold>V"
I_l_closed: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and a :: "'c"   and x :: "'c" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> a \<in> I"
sinpoly_upper_nonneg: fixes x :: "real"   and n :: "nat" assumes "0 \<le> x" shows "sin x \<le> sinpoly (Suc (Suc (4 * n))) x"
l1_inv2_init: shows "init l1 \<subseteq> l1_inv2"
winning_path_drop: fixes p :: "Player"   and P :: "'a llist"   and n :: "nat" assumes "winning_path p P"   and "enat n < llength P" shows "winning_path p (ldropn n P)"