hcju/isabelleps huggingface dataset:
poly_mapping_eq_iff: fixes a :: "'c \<Rightarrow>\<^sub>0 'd"   and b :: "'c \<Rightarrow>\<^sub>0 'd" shows "(a = b) = (lookup a = lookup b)"
simps_11: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> U\<^sub>c \<psi> = (\<exists>i. suffix i \<xi> \<Turnstile>\<^sub>c \<psi> \<and> (\<forall>j<i. suffix j \<xi> \<Turnstile>\<^sub>c \<phi>))"
i_leq: fixes i' :: "nat" assumes "i' \<in> set (i # il)" shows "i \<le> i'"
goodAbs_imp_qGoodAbs_pick: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set" assumes "goodAbs A" shows "qGoodAbs (pick A)"
mod_add_right_eq: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a + b mod c) mod c = (a + b) mod c"
induct: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and x1 :: "ty list"   and x2 :: "val list"   and x3 :: "val list"   and Pa :: "ty list \<Rightarrow> val list \<Rightarrow> val list \<Rightarrow> bool" assumes "P \<turnstile> x1 Casts x2 to x3 "   and "Pa [] [] []"   and "\<And>T v v' Ts vs vs'. \<lbrakk>P \<turnstile> T casts v to v' ; P \<turnstile> Ts Casts vs to vs' ; Pa Ts vs vs'\<rbrakk> \<Longrightarrow> Pa (T # Ts) (v # vs) (v' # vs')" shows "Pa x1 x2 x3"
tanh_real_at_top: shows "(tanh \<longlongrightarrow> 1) at_top"
arith_special_1: fixes n :: "num" shows "numeral n + (1::'a) = numeral (n + num.One)"
dvdm_Mp: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.dvdm m f (poly_mod.Mp m g) = poly_mod.dvdm m f g"
assms_2: shows "desargues_config A B C A' B' C' M N P R"
abs_eq: fixes x :: "64 word" shows "uint16_of_uint64 (Abs_uint64 x) = Abs_uint16 (ucast x)"
diff_0_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.group_add minus plus zero uminus" shows "minus a zero = a"
BoxSFI: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> SF(A)_v \<longrightarrow> \<box>SF(A)_v"
ldrop_eSuc_conv_ltl: fixes n :: "enat"   and xs :: "'c llist" shows "ldrop (eSuc n) xs = ltl (ldrop n xs)"
distinct_17: fixes x2 :: "nat"   and x61 :: "exp"   and x62 :: "exp"   and x63 :: "exp" shows "ENat x2 \<noteq> EIf x61 x62 x63"
simps_1: shows "not\<^sub>n true\<^sub>n = false\<^sub>n"
Abs_induct: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and P :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "type_definition Rep Abs A"   and "\<And>y. y \<in> A \<Longrightarrow> P (Abs y)" shows "P x"
zero_not_in_prime_factors: 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" shows "zero \<notin># factorial_semiring.prime_factorization zero times one normalize x"
comp_parallel_wiring: fixes C1 :: "('y, 'ac, 'ag, 'ak) converter"   and C2 :: "('z, 'ad, 'ai, 'am) converter"   and C3 :: "('aa, 'ae, 'ah, 'al) converter"   and C4 :: "('ab, 'af, 'aj, 'an) converter" shows "((C1 |\<^sub>= C2) |\<^sub>= C3 |\<^sub>= C4) \<odot> parallel_wiring = parallel_wiring \<odot> ((C1 |\<^sub>= C3) |\<^sub>= C2 |\<^sub>= C4)"
eq_name_eq_pres': fixes s :: "nat \<Rightarrow> nat option"   and name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer :: "nat"   and name' :: "char list"   and pres' :: "(nat \<times> nat) list"   and effs' :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and layer' :: "nat" assumes "s \<in> valid_states"   and "is_standard_operator' (name, pres, effs, layer)"   and "(name, pres, effs, layer) \<in> set ast\<delta>"   and "lookup_operator ((fst \<circ> the \<circ> lookup_action \<circ> abs_ast_operator) (name, pres, effs, layer)) = Some (name', pres', effs', layer')" shows "pres = pres'"
set_plus_mono2: fixes C :: "'a set"   and D :: "'a set"   and E :: "'a set"   and F :: "'a set" assumes "C \<subseteq> D"   and "E \<subseteq> F" shows "C + E \<subseteq> D + F"
select_convs_2: fixes es :: "'d"   and ps :: "('b \<times> 'c) odlist"   and more :: "'e" shows "ps \<lparr>es = es, ps = ps, \<dots> = more\<rparr> = ps"
comp_def: fixes g :: "'a"   and f :: "'a" shows "Right_a.comp g f = (if right a f \<and> right a g \<and> cod f = local.dom g then g \<cdot> f else null)"
tendsto_componentwise_iff: fixes f :: "'d \<Rightarrow> 'e"   and l :: "'e"   and F :: "'d filter" shows "(f \<longlongrightarrow> l) F = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) \<longlongrightarrow> l \<bullet> i) F)"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "suffix a b"   and "suffix b c" shows "suffix a c"
distinct_append: fixes xs :: "'b list"   and ys :: "'b list" shows "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
ccw_self_2: fixes b :: "real \<times> real"   and a :: "real \<times> real" shows "ccw b a a"
monic_smult_add_small: fixes f :: "'a poly"   and g :: "'a poly"   and q :: "'a" assumes "f = 0 \<or> degree f < degree g"   and "monic g" shows "monic (g + Polynomial.smult q f)"
expand: fixes item :: "item"   and item' :: "item" assumes "item_rule item = item_rule item' \<and> item_dot item = item_dot item' \<and> item_origin item = item_origin item' \<and> item_end item = item_end item'" shows "item = item'"
pfun_canc: fixes z :: "'b \<Rightarrow> 'b option"   and x :: "'b \<Rightarrow> 'b option"   and y :: "'b \<Rightarrow> 'b option" assumes "ortho z x"   and "ortho z y"   and "z ++ x = z ++ y" shows "x = y"
COD_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "\<F>C.COD (\<F>C.mkarr t) = Cod t"
sum_power2_gt_zero_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (plus (power.power one times x 2) (power.power one times y 2)) = (x \<noteq> zero \<or> y \<noteq> zero)"
ast\<delta>_def: shows "ast\<delta> \<equiv> case problem of (D, I, G, \<delta>) \<Rightarrow> \<delta>"
prime_eq: fixes p :: "nat" shows "prime p = (1 < p \<and> (\<forall>m. m dvd p \<longrightarrow> 1 < m \<longrightarrow> m = p))"
rep_eq: fixes x :: "'a" shows "blinfun_apply (blinfun.prod_left x) = (\<lambda>a. blinfun_apply a x)"
bMkk: fixes k :: "nat" shows "b k (k, k - 1) \<lless> M (Suc k)"
is_zero_MP_oalist: fixes xs :: "('b, 'c) oalist_ntm" shows "is_zero (MP_oalist xs) = List.null (fst (list_of_oalist_ntm xs))"
forest_maximally_connected_incl_max1: fixes G :: "('a, 'b) graph"   and H :: "('a, 'b) graph" assumes "valid_graph G"   and "forest H"   and "subgraph H G"   and "\<forall>(a, w, b)\<in>edges G - edges H. \<not> forest (add_edge a w b H)" shows "maximally_connected H G"
L_D_def: fixes x :: "'a"   and y :: "'b" shows "L_D x y = seq [Atom x, Atom x]"
add_nonpos_eq_0_iff: fixes x :: "'a multiset"   and y :: "'a multiset" assumes "x \<subseteq># {#}"   and "y \<subseteq># {#}" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
infinite_fst_vimage: fixes a :: "'a" shows "infinite (fst -` {a})"
gcd_dvd2: fixes a :: "'a"   and b :: "'a" shows "gcd a b dvd b"
widen_trans: fixes P :: "'a prog"   and S :: "ty"   and U :: "ty"   and T :: "ty" assumes "P \<turnstile> S \<le> U"   and "P \<turnstile> U \<le> T" shows "P \<turnstile> S \<le> T"
Min_insert2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and a :: "'a" 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 (insert a A) = a"
generate_def: fixes A :: "nat"   and B :: "nat"   and m :: "nat"   and n :: "nat" shows "generate A B m n = tl (gen2 A B m n)"
lcm_least_iff: 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 b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (lcm a b) c = (dvd.dvd times a c \<and> dvd.dvd times b c)"
prj_simps_1: shows "C.arr \<nu>\<pi>_\<rho>.prj\<^sub>1"
last_ps_sgn_const: fixes x :: "real"   and y :: "real" shows "sgn (poly (last (sturm_squarefree p)) x) = sgn (poly (last (sturm_squarefree p)) y)"
f: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra B"
tagged_division_split_right_inj: fixes d :: "('a \<times> 'a set) set"   and i :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set"   and c :: "real"   and k :: "'a" assumes "d tagged_division_of i"   and "(x1, K1) \<in> d"   and "(x2, K2) \<in> d"   and "K1 \<noteq> K2"   and "K1 \<inter> {x. c \<le> x \<bullet> k} = K2 \<inter> {x. c \<le> x \<bullet> k}" shows "interior (K1 \<inter> {x. c \<le> x \<bullet> k}) = {}"
a_ht: shows "steps0 (1, tp) A n = (0, tp')"
limp_ppath_def: fixes Z :: "'a ppath set"   and Y :: "'a ppath set" shows "limp_ppath Z Y = \<Union> {X. pp_prod X Y \<subseteq> Z}"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> arr f" shows "\<ll>' f = null"
measurable_embed_measure1: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and M :: "'a measure"   and N :: "'b measure" assumes "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M N" shows "g \<in> embed_measure M f \<rightarrow>\<^sub>M N"
map_upds_Nil1: fixes m :: "'c \<Rightarrow> 'd option"   and bs :: "'d list" shows "m([] [\<mapsto>] bs) = m"
in_own_span: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "S \<subseteq> span S"
is_arrow_of_tabulations_in_maps: shows "arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (t\<^sub>0\<^sup>* \<star> u\<^sub>1) t\<^sub>0u\<^sub>1.tab t\<^sub>0u\<^sub>1.p\<^sub>0 t\<^sub>0u\<^sub>1.p\<^sub>1 (t\<^sub>0\<^sup>* \<star> u\<^sub>1) t\<^sub>0u\<^sub>1.tab t\<^sub>0u\<^sub>1.p\<^sub>0 t\<^sub>0u\<^sub>1.p\<^sub>1 (t\<^sub>0\<^sup>* \<star> u\<^sub>1)"
adjhalfchsys_image_gf: fixes C :: "'a set"   and D :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>"   and "D \<in> f \<turnstile> folding_g.\<C>"   and "C \<sim> D" shows "g ` D = C"
prod_set_defs_1: shows "Basic_BNFs.fsts = (\<lambda>p. {fst p})"
Int_ac_3: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> B = B \<inter> A"
simps_4: fixes n :: "nat"   and \<phi> :: "'a formula" shows "pre_wf_formula n (FNot \<phi>) = pre_wf_formula n \<phi>"
ring_simprules_9: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "ring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
last_drop: fixes n :: "nat"   and xs :: "'g list" assumes "n < length xs" shows "last (drop n xs) = last xs"
target_iso_trg: fixes \<mu> :: "'a"   and b :: "'a" assumes "arr \<mu>"   and "b \<in> targets \<mu>" shows "b \<cong> trg \<mu>"
nat_diff_split: fixes P :: "nat \<Rightarrow> bool"   and a :: "nat"   and b :: "nat" shows "P (a - b) = ((a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d))"
subset_Pow_Union: fixes A :: "'c set set" shows "A \<subseteq> Pow (\<Union> A)"
outer_invar_def: shows "outer_invar \<equiv> outer_invar_loc G"
unique_NB: fixes B :: "agent"   and NA :: "msg"   and NB :: "msg"   and A :: "agent"   and evs :: "event list"   and NC :: "msg"   and C :: "agent" assumes "Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>NC, NB, Agent C, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> otway"   and "B \<notin> bad" shows "NC = NA \<and> C = A"
Pi'_I': fixes f :: "'d \<Rightarrow>\<^sub>F 'e"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'e set" assumes "domain f = A"   and "\<And>x. x \<in> A \<longrightarrow> (f)\<^sub>F x \<in> B x" shows "f \<in> Pi' A B"
simulation1_rtrancl: fixes s1 :: "'s1"   and tls1 :: "'tl1 list"   and s1' :: "'s1"   and s2 :: "'s2" assumes "s1 -1-tls1\<rightarrow>* s1'"   and "s1 \<approx> s2" shows "\<exists>s2' tls2. s2 -2-tls2\<rightarrow>* s2' \<and> s1' \<approx> s2' \<and> tls1 [\<sim>] tls2"
denote_while_n_positive: fixes M0 :: "complex mat"   and M1 :: "complex mat"   and DS :: "complex mat \<Rightarrow> complex mat"   and \<rho> :: "complex mat"   and n :: "nat" assumes "M0 \<in> carrier_mat d d"   and "M1 \<in> carrier_mat d d"   and "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m d"   and "\<And>\<rho>. \<lbrakk>\<rho> \<in> carrier_mat d d; partial_density_operator \<rho>\<rbrakk> \<Longrightarrow> positive (DS \<rho>) \<and> trace (DS \<rho>) \<le> trace \<rho> \<and> DS \<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho> \<and> \<rho> \<in> carrier_mat d d" shows "positive (denote_while_n M0 M1 DS n \<rho>)"
arctan_monotone': fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "arctan x \<le> arctan y"
axioms_1: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "semicategory \<alpha> \<CC>" shows "\<Z> \<alpha>"
eq: shows "1\<^sub>m K - proj_fst_k k = M k"
maddux_142: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "(x\<^sup>T ; z \<cdot> y = 0) = (z ; y\<^sup>T \<cdot> x = 0)"
lookup_single_not_eq: fixes k :: "'b"   and k' :: "'b"   and v :: "'c" assumes "k \<noteq> k'" shows "lookup (Poly_Mapping.single k v) k' = (0::'c)"
induct: fixes x1 :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and x2 :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and x3 :: "nat"   and x4 :: "nat"   and x5 :: "lrule"   and P :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list \<Rightarrow> ('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> lrule \<Rightarrow> bool" assumes "lrule_ok x1 x2 x3 x4 x5"   and "\<And>SG i j C p q. fst (SG ! i) ! j = (p && q) \<Longrightarrow> P SG C i j AndL"   and "\<And>SG i j C p q. fst (SG ! i) ! j = (p \<rightarrow> q) \<Longrightarrow> P SG C i j ImplyL"   and "\<And>SG i j C p q. fst (SG ! i) ! j = (p \<leftrightarrow> q) \<Longrightarrow> P SG C i j EquivForwardL"   and "\<And>SG i j C p q. fst (SG ! i) ! j = (p \<leftrightarrow> q) \<Longrightarrow> P SG C i j EquivBackwardL" shows "P x1 x2 x3 x4 x5"
is_dghmD_10: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
upto_enum_set_conv2: fixes a :: "'b word"   and b :: "'b word" shows "set [a .e. b] = {a..b}"
fset_both_sides: fixes s :: "'a set"   and f :: "'a fset" shows "(Abs_fset s = f) = (fset (Abs_fset s) = fset f)"
rec_newrgt0_def: shows "rec_newrgt0 = Cn 2 rec_minus [recf.id 2 1, Cn 2 rec_scan [recf.id 2 1]]"
is_Delete_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "is_Delete stateful_strand_step = (\<exists>x51 x52. stateful_strand_step = delete\<langle>x51,x52\<rangle>)"
infinite_solutions_homogeneous_imp_dim_solution_set_not_zero: fixes A :: "(('b, 'c) vec, 'd) vec" assumes "infinite (solution_set A 0)" shows "0 < vec.dim (solution_set A 0)"
rel_with_target_impl_TRel_weakly_preserves_barbs: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "rel_weakly_preserves_barbs Rel (STCalWB SWB TWB)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel" shows "rel_weakly_preserves_barbs TRel TWB"
Root_unRoot: fixes x :: "complex" assumes "x \<in> Roots" shows "Root (unRoot x) = x"
exec_instr_CmpEq: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr CmpEq P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (Bool (hd (tl stk) = hd stk) # tl (tl stk), loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
alt_effective_matrix_tensor_elements: fixes i :: "nat"   and M2 :: "'a list list"   and M3 :: "'a list list"   and j :: "nat" assumes "(i < row_length M2 * row_length M3 \<and> j < length M2 * length M3) \<and> mat (row_length M2) (length M2) M2 \<and> mat (row_length M3) (length M3) M3" shows "(M2 \<otimes> M3) ! j ! i = M2 ! (j div length M3) ! (i div row_length M3) * M3 ! (j mod length M3) ! (i mod row_length M3)"
l_zero: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<zero> \<oplus> x = x"
IH: fixes s :: "('a, 'b) fmap"   and as' :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "sat_precond_as s (as @ as')" shows "sat_precond_as s as"
Schroeder_Bernstein: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "inj_on g B"   and "g ` B \<subseteq> A" shows "\<exists>h. bij_betw h A B"
wls_psubst_idEnv: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X" shows "X #[idEnv] = X"
combine_common_factor: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and e :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
fast_ppoint_mult_0: fixes m :: "int"   and a :: "int"   and p :: "int \<times> int \<times> int" shows "fast_ppoint_mult m a 0 p = (0, 0, 0)"
select_convs_1: fixes store :: "'c \<Rightarrow> nat"   and pred :: "(('c \<times> nat \<Rightarrow> 'd) \<Rightarrow> bool) set"   and more :: "'e" shows "store \<lparr>store = store, pred = pred, \<dots> = more\<rparr> = store"
rtranclp_into_tranclp2: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
rbt_less_prop: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_less less k t = (\<forall>x\<in>set (RBT_Impl.keys t). less x k)"
simps_2: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr" shows "conf_xcp' P h \<lfloor>a\<rfloor> = (\<exists>D. typeof_addr h a = \<lfloor>Class_type D\<rfloor> \<and> P \<turnstile> D \<preceq>\<^sup>* Throwable)"
sys_block_sizes_obtain_bl: fixes x :: "int" assumes "x \<in> sys2.sys_block_sizes" shows "\<exists>bl\<in>#\<B>'. int (card bl) = x"
callee_invariant_on_fuse: fixes \<I>_adv_core :: "('iadv_core, 'oadv_core) \<I>"   and \<I>_usr_core :: "('iusr_core, 'ousr_core) \<I>"   and I_core :: "'s_core \<Rightarrow> bool"   and \<I>_adv_rest :: "('f, 'g) \<I>"   and \<I>_usr_rest :: "('h, 'i) \<I>"   and I_rest :: "'j \<Rightarrow> bool"   and rest :: "('j, 'event, 'f, 'h, 'g, 'i, 'j) rest_scheme" assumes "WT_core \<I>_adv_core \<I>_usr_core I_core core"   and "WT_rest \<I>_adv_rest \<I>_usr_rest I_rest rest" shows "callee_invariant_on (fuse rest) (pred_prod I_core I_rest) ((\<I>_adv_core \<oplus>\<^sub>\<I> \<I>_adv_rest) \<oplus>\<^sub>\<I> (\<I>_usr_core \<oplus>\<^sub>\<I> \<I>_usr_rest))"
EM_remainder_converges_cnj: fixes a :: "int"   and f :: "real \<Rightarrow> complex"   and n :: "nat" assumes "continuous_on {real_of_int a..} f"   and "EM_remainder_converges n f a" shows "EM_remainder_converges n (\<lambda>x. cnj (f x)) a"
span_zero: fixes S :: "'b set" shows "(0::'b) \<in> m1.span S"
intv1: fixes i :: "'b" assumes "\<I> i" shows "i \<in> BEGIN i"
ordLeq_cexp1: fixes r :: "('d \<times> 'd) set"   and q :: "('e \<times> 'e) set" assumes "(r, czero) \<notin> ordIso \<and> Card_order r"   and "Card_order q" shows "q \<le>o q ^c r"
bet_le_eq: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Tarski_neutral_dimensionless.Le Bet Cong A C B C" shows "A = B"
aff_dim_affine_Int_hyperplane: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "affine S" shows "aff_dim (S \<inter> {x. a \<bullet> x = b}) = (if S \<inter> {v. a \<bullet> v = b} = {} then - 1 else if S \<subseteq> {v. a \<bullet> v = b} then aff_dim S else aff_dim S - 1)"
nec_russell_axiom: fixes \<psi> :: "\<kappa> \<Rightarrow> \<o>"   and v :: "i"   and \<phi> :: "\<nu> \<Rightarrow> \<o>" assumes "SimpleExOrEnc \<psi>" shows "[\<psi> (\<^bold>\<iota>x. \<phi> x) \<^bold>\<equiv> (\<^bold>\<exists>x. \<^bold>\<A>\<phi> x \<^bold>& (\<^bold>\<forall>z. \<^bold>\<A>\<phi> z \<^bold>\<rightarrow> z \<^bold>= x) \<^bold>& \<psi> (x\<^sup>P)) in v]"
eval_if_inv: fixes \<rho> :: "(nat \<times> bval) list"   and e1 :: "exp"   and e2 :: "exp"   and e3 :: "exp"   and v :: "bval" assumes "\<rho> \<turnstile> EIf e1 e2 e3 \<Down> v" obtains "\<rho> \<turnstile> e1 \<Down> BNat 0" and "\<rho> \<turnstile> e3 \<Down> v"   | n :: "nat" where "\<rho> \<turnstile> e1 \<Down> BNat n" and "0 < n" and "\<rho> \<turnstile> e2 \<Down> v"
yun_rel_def: fixes F :: "int poly"   and c :: "rat"   and f :: "rat poly" shows "yun_rel F c f = (yun_wrel F c f \<and> content F = 1 \<and> 0 < lead_coeff F \<and> monic f)"
insert_code_2: fixes x :: "'a"   and xs :: "'a list" shows "insert x (List.coset xs) = List.coset (removeAll x xs)"
Class_widen_Class: fixes P :: "'a prog"   and C :: "char list"   and D :: "char list" shows "subtype P (Class C) (Class D) = P \<turnstile> C \<preceq>\<^sup>* D"
set_pmf_not_empty: fixes M :: "'b pmf" shows "set_pmf M \<noteq> {}"
rell_F_Grp: fixes A1 :: "'n set"   and f1 :: "'n \<Rightarrow> 'x"   and A2 :: "'o set"   and f2 :: "'o \<Rightarrow> 'y"   and A3 :: "'p set"   and f3 :: "'p \<Rightarrow> 'z" shows "rell_F (Grp A1 f1) (Grp A2 f2) (Grp A3 f3) = Grp {x. set1_F x \<subseteq> A1 \<and> set2_F x \<subseteq> A2 \<and> set3_F x \<subseteq> A3} (mapl_F f1 f2 f3)"
subdivision_pair_simps: fixes G :: "'b pair_pre_digraph"   and a :: "'b pair_pre_digraph" shows "subdivision_pair G a = (a = G \<and> pair_bidirected_digraph G \<or> (\<exists>e w H. a = subdivide H e w \<and> e \<in> parcs H \<and> w \<notin> pverts H \<and> subdivision_pair G H))"
IsProperInXYZ_def: shows "IsProperInXYZ \<equiv> map_fun (map_fun id (map_fun id (map_fun id eval\<o>))) id (\<lambda>\<phi>. \<forall>x y z v. (\<exists>a b c. \<nu>\<upsilon> a = \<nu>\<upsilon> x \<and> \<nu>\<upsilon> b = \<nu>\<upsilon> y \<and> \<nu>\<upsilon> c = \<nu>\<upsilon> z \<and> \<phi> (a\<^sup>P) (b\<^sup>P) (c\<^sup>P) dj v) = \<phi> (x\<^sup>P) (y\<^sup>P) (z\<^sup>P) dj v)"
pseudominimal_inI: fixes x :: "'b"   and P :: "'b set" assumes "x \<in> P"   and "x \<noteq> order.bottom P"   and "\<And>z. \<lbrakk>z \<in> P; order.greater x z\<rbrakk> \<Longrightarrow> z = order.bottom P" shows "order.pseudominimal_in P x"
x: shows "x = \<lbrakk>quot_dbfm B\<rbrakk>e"
map_replicate_trivial: fixes x :: "'b"   and i :: "nat" shows "list.map (\<lambda>i. x) [0..<i] = replicate i x"