hcju/isabelleps huggingface dataset:
transpose_infmatrix_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and j :: "nat"   and i :: "nat" shows "transpose_infmatrix A j i \<equiv> A i j"
de_morgan_1: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot"   and "\<exists>ya. sup (inf x y) ya = top \<and> inf (inf x y) ya = bot \<and> inf ya (inf x y) = bot" shows "il_semiring.neg inf sup bot top (inf x y) = sup (il_semiring.neg inf sup bot top x) (il_semiring.neg inf sup bot top y)"
mult_map_poly_imp_map_poly: fixes q :: "real poly"   and r :: "complex poly"   and p :: "real poly" assumes "map_poly complex_of_real q = r * map_poly complex_of_real p"   and "p \<noteq> 0" shows "r = map_poly complex_of_real (map_poly Re r)"
that: fixes p :: "('c \<times> 'c) list" assumes "path PM (io || p) (initial PM) \<and> length p = length io \<and> target (io || p) (initial PM) = FAIL" shows "thesis"
nat_power_mult_distrib: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and nat_power :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.nat_power one times plus zero nat_power"   and "0 < m"   and "0 < n" shows "nat_power (m * n) a = times (nat_power m a) (nat_power n a)"
val_le_fun_le_trans: fixes n :: "nat" shows "\<forall>v2 t2. n = val_size v2 + fun_size t2 \<longrightarrow> (\<forall>v1 v3. v1 \<sqsubseteq> v2 \<longrightarrow> v2 \<sqsubseteq> v3 \<longrightarrow> v1 \<sqsubseteq> v3) \<and> (\<forall>t1 t3. t1 \<lesssim> t2 \<longrightarrow> t2 \<lesssim> t3 \<longrightarrow> t1 \<lesssim> t3)"
ZWhile: fixes I :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and b :: "bexp"   and c :: "com" assumes "\<turnstile>\<^sub>Z {\<lambda>s. I s + \<up> (bval b s)} c {\<lambda>t. I t + 1}" shows "\<turnstile>\<^sub>Z {\<lambda>s. I s + 1} WHILE b DO c {\<lambda>s. I s + \<up> (\<not> bval b s)}"
rel_spmf_spmf_of_pmf: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" shows "rel_spmf R (spmf_of_pmf p) (spmf_of_pmf q) = rel_pmf R p q"
list_distinct_filter: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "equal_base.list_distinct equal xs" shows "equal_base.list_distinct equal (List.filter P xs)"
open_segment_subsegment: fixes x1 :: "'b"   and x0 :: "'b"   and x3 :: "'b"   and x2 :: "'b" assumes "x1 \<in> {x0<--<x3}"   and "x2 \<in> {x1<--<x3}" shows "x1 \<in> {x0<--<x2}"
rank_of_def: fixes carrier' :: "'a set" shows "\<E>.rank_of carrier' \<equiv> Min {card B |B. \<E>.basis_in carrier' B}"
norm_sgn: fixes x :: "'c" shows "norm (sgn x) = (if x = (0::'c) then 0 else 1)"
UnionE: fixes A :: "'a"   and C :: "'a set set" assumes "A \<in> \<Union> C" obtains X :: "'a set" where "A \<in> X" and "X \<in> C"
DiaN': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and i :: "'c"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (\<^bold>\<diamond> p)) at a in' set ((ps, a) # branch)"   and "(\<^bold>\<diamond> Nom i) at a in' set ((ps, a) # branch)"   and "A, n \<turnstile> ((\<^bold>\<not> (\<^bold>@ i p)) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
COMP: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c1 :: "IMP"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and c2 :: "IMP"   and \<Psi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "G \<rhd> c1 : Sec \<Phi>"   and "G \<rhd> c2 : Sec \<Psi>" shows "G \<rhd> Comp c1 c2 : Sec (\<lambda>(s, t). \<exists>r. \<Phi> (r, t) \<and> (\<forall>w.  r \<approx> w  \<longrightarrow> \<Psi> (s, w)))"
z_def: shows "z = Re (zeta (complex_of_real s))"
pair_by_index_is_pr: shows "pair_by_index \<in> PrimRec2"
add_pos_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "less zero a"   and "less zero b" shows "less zero (plus a b)"
ImageE: fixes b :: "'d"   and r :: "('e \<times> 'd) set"   and A :: "'e set" assumes "b \<in> r `` A" obtains x :: "'e" where "(x, b) \<in> r" and "x \<in> A"
measure_ge_1_iff: fixes A :: "'a set" shows "(1 \<le> prob A) = (prob A = 1)"
prv_neg_zer_suc_var: shows "prv (neg (eql zer (suc (Var xx))))"
make_cert_def: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and \<tau>s :: "'c list"   and B :: "'c" shows "make_cert step \<tau>s B = map (\<lambda>pc. if is_target step \<tau>s pc then \<tau>s ! pc else B) [0..<length \<tau>s] @ [B]"
cmp_simps_5: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi> (f, g)) = F (f \<star>\<^sub>C g)"
lc_eq_zero_iff: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "(punit.lc p = (0::'c)) = (p = 0)"
inducts: fixes x :: "'b formula"   and P :: "'b formula \<Rightarrow> bool" assumes "AX0p x"   and "\<And>F G. P (F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> F)"   and "\<And>F G H. P ((F \<^bold>\<rightarrow> G \<^bold>\<rightarrow> H) \<^bold>\<rightarrow> (F \<^bold>\<rightarrow> G) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> H)" shows "P x"
invertible_def: fixes u :: "'a" assumes "u \<in> I" shows "maxI.additive.sub.invertible u = (\<exists>v\<in>I. addA u v = zeroA \<and> addA v u = zeroA)"
ffold_finsert_fremove: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'a"   and A :: "'a fset" assumes "comp_fun_commute f" shows "ffold f z (finsert x A) = f x (ffold f z (A |-| {|x|}))"
subsetI: fixes A :: "'f set"   and B :: "'f set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "A \<subseteq> B"
Linear: fixes b :: "'b"   and r :: "('a, 'b, 'c, 'd) resumption" assumes "x = Linear b r" shows "thesis"
min_0_1_4: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.min less_eq (numeral.numeral one plus x) zero = zero"
elims: fixes x :: "nat list"   and xa :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and xb :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec"   and y :: "(('a, 'b) vec, 'b) vec \<times> (('a, 'c) vec, 'b) vec \<times> (('a, 'c) vec, 'c) vec" assumes "diagonal_to_Smith_aux_PQ x xa xb = y" obtains bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" and P :: "(('a, 'b) vec, 'b) vec" and A :: "(('a, 'c) vec, 'b) vec" and Q :: "(('a, 'c) vec, 'c) vec" where "x = []" and "xa = bezout__" and "xb = (P__, A__, Q__)" and "y = (P__, A__, Q__)"   | i :: "nat" and xs :: "nat list" and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" and P :: "(('a, 'b) vec, 'b) vec" and A :: "(('a, 'c) vec, 'b) vec" and Q :: "(('a, 'c) vec, 'c) vec" where "x = i # xs" and "xa = bezout" and "xb = (P, A, Q)" and "y = diagonal_to_Smith_aux_PQ xs bezout (Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q))"
subdegree_div_unit': fixes q :: "'a fps"   and p :: "'a fps" assumes "fps_nth q 0 \<noteq> (0::'a)"   and "fps_nth p (subdegree p) * inverse (fps_nth q 0) \<noteq> (0::'a)" shows "subdegree (p / q) = subdegree p"
GreatestB_le: fixes x :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "x \<le> n"   and "P x" shows "x \<le> (GREATEST k. k \<le> n \<and> P k)"
linear: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_linear f" shows "linear f"
assms: shows "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) s'"
od_fm_fun_def: fixes R :: "('j, 'k) Ring_scheme"   and M :: "('h, 'j, 'l) Module_scheme"   and N :: "('i, 'j, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'i"   and H :: "'h set" shows "od_fm_fun R M N f H = \<lparr>carrier = fsps R M N f H, rel = {Y \<in> fsps R M N f H \<times> fsps R M N f H. fst (fst Y) \<subseteq> fst (snd Y)}\<rparr>"
assms_3: shows "\<not> set (take (length Us) Ks) \<subseteq> {\<zero>}"
nn_integral_ge_point: fixes x :: "'k"   and A :: "'k set"   and p :: "'k \<Rightarrow> ennreal" assumes "x \<in> A" shows "p x \<le> integral\<^sup>N (count_space A) p"
all_def: fixes P :: "'a \<Rightarrow> (nat \<Rightarrow> cnf) \<Rightarrow> nat \<Rightarrow> bool" shows "all P \<equiv> \<lambda>t n. \<forall>y. P y t n"
determOn_def: fixes phi :: "'a \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "determOn phi r \<equiv> \<forall>a b b'. phi a \<and> r a b \<and> r a b' \<longrightarrow> b = b'"
linearizationD: fixes p :: "(nat \<times> int) list" assumes "p \<in> sparsegrid dm lm" shows "pm p < card (sparsegrid dm lm)"
bind_singleton_conv_image: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd" shows "Set.bind A (\<lambda>x. {f x}) = f ` A"
continuous_componentwise: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "continuous F f = (\<forall>i\<in>Basis. continuous F (\<lambda>x. f x \<bullet> i))"
conj_bot_right: fixes c :: "'b::refinement_lattice" shows "(c::'b::refinement_lattice) \<iinter> \<bottom> = \<bottom>"
bis_imp_3: fixes indis :: "('a \<times> 'a) set"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis"   and "(c1, c2) \<in> PL_Indis.ZObisT tval aval indis" shows "(c1, c2) \<in> PL_Indis.WbisT tval aval indis"
residue_UnitsI: fixes n :: "nat"   and k :: "int" assumes "1 \<le> n"   and "0 < k"   and "k < p ^ n"   and "coprime k p" shows "k \<in> Units (residue_ring (p ^ n))"
combine_nest_1: fixes s :: "char list \<Rightarrow> int \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "<s|<s'|t>> = <s|t>"
assms: assumes "\<Phi>" shows "S \<le> \<Down> R (ASSERT \<Phi> \<bind> (\<lambda>_. S'))"
kjsub: shows "k ` j ` (S - {c}) \<subseteq> S"
irrefl: fixes a :: "'c DBMEntry" shows "\<not> a \<prec> a"
zer: fixes b :: "complex"   and c :: "complex" assumes "b \<in> S"   and "c \<in> S" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
gu2: fixes x :: "'b" assumes "x \<in> vertices G" shows "f x \<in> vertices G"
sums: fixes z :: "'a" assumes "norm z < K" shows "(\<lambda>n. c n * z ^ n) sums f z"
log_eq_one: fixes a :: "real" assumes "0 < a"   and "a \<noteq> 1" shows "Transcendental.log a a = 1"
neq_le_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.order less_eq less"   and "a \<noteq> b"   and "less_eq a b" shows "less a b"
uminus_freeword_Abs_eq: fixes xs :: "('a \<times> bool) list" assumes "proper_signed_list xs" shows "- Abs_freeword xs = Abs_freeword (rev (map flip_signed xs))"
hcomplexpow_Suc: fixes z :: "complex star"   and n :: "nat" shows "z ^ Suc n = z * z ^ n"
odd_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b)) = ((\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) \<noteq> (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b))"
simps_1: fixes n :: "nat" shows "local.wf n Zero = True"
scaleC_left: fixes r :: "complex"   and a :: "'a"   and b :: "'b" shows "r *\<^sub>C a ** b = cnj r *\<^sub>C (a ** b)"
s_1: fixes i :: "'a" assumes "i \<in> Basis" shows "x \<bullet> i = s i \<bullet> i"
StepWithoutP: shows "cfg \<turnstile> m' \<mapsto> cfg2"
acquire_all_iff: fixes ls :: "('d, 'c) locks"   and t :: "'c"   and ln :: "'d \<Rightarrow>f nat"   and l :: "'d" shows "acquire_all ls t ln $ l = acquire_locks (ls $ l) t (ln $ l)"
starfun_Re: fixes f :: "'b \<Rightarrow> complex" shows "*f* (\<lambda>x. Re (f x)) = (\<lambda>x. hRe ((*f* f) x))"
s2_2: shows "g2 = s2 \<otimes> r2"
elims: fixes x :: "('b, 'c) arr"   and y :: "'b" assumes "Cod x = y" obtains uu_ :: "'b" and B :: "'b" and uv_ :: "'c" where "x = MkArr uu_ B uv_" and "y = B"   | "x = Null" and "y = undefined"
const: fixes c :: "'a"   and F :: "'b filter" assumes "c \<noteq> (0::'a)" shows "O[F](\<lambda>_. c) = O[F](\<lambda>_. 1::'a)"
dim_row_conjugate: fixes m :: "'a mat" shows "dim_row (conjugate m) = dim_row m"
simps_1: fixes s :: "'d stream" shows "stake 0 s = []"
map_sel: fixes f :: "'d \<Rightarrow> 'c"   and a :: "('b, 'd) state" shows "run_state (map_state f a) = map_prod f id \<circ> run_state a"
cCons_not_0_eq: fixes x :: "'a"   and xs :: "'a list" assumes "x \<noteq> (0::'a)" shows "cCons x xs = x # xs"
intro: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "\<And>f. F (oLimit f) = oLimit (\<lambda>n. F (f n))" shows "continuous F"
case_5: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x51 :: "'b ltln"   and x52 :: "'b ltln" shows "(case x51 and\<^sub>n x52 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f5 x51 x52"
that_2: assumes "x \<noteq> y"   and "{x--y} \<subseteq> X"   and "\<And>z. z \<in> {x--y} \<Longrightarrow> 0 < f z \<bullet> rot (y - x)" shows "thesis"
diff_add_assoc: fixes k :: "nat"   and j :: "nat"   and i :: "nat" assumes "k \<le> j" shows "i + j - k = i + (j - k)"
subspace_def: fixes K :: "('e, 'f) ring_scheme"   and W :: "'g set"   and V :: "('e, 'g, 'h) module_scheme" shows "subspace K W V \<equiv> vectorspace K V \<and> LinearCombinations.submodule K W V"
Compl_iff: fixes c :: "'c"   and A :: "'c set" shows "s2p (- A) c = (c \<notin> A)"
s_def: shows "s \<equiv> {(x, y). nss x y False}"
seq_mono_right: fixes seq :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and nil :: "'b::refinement_lattice"   and c\<^sub>0 :: "'b::refinement_lattice"   and c\<^sub>1 :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "seq_distrib_left (seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (nil::'b::refinement_lattice)"   and "(c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'b::refinement_lattice)" shows "(seq::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (d::'b::refinement_lattice) (c\<^sub>0::'b::refinement_lattice) \<sqsubseteq> seq d (c\<^sub>1::'b::refinement_lattice)"
dim_ty_sem_Current_def: fixes uu :: "I itself" shows "dim_ty_sem uu = \<^bold>I"
rev_filter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "rev (filter P xs) = filter P (rev xs)"
Act\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c) Tree\<^sub>\<alpha>"   and \<alpha> :: "'c" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "hereditarily_fs (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>)"
On_CTRL_States_def: shows "On_CTRL_States = {''TunerMode'', ''CDMode''}"
exCI: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<forall>x. \<not> P x \<Longrightarrow> P a" shows "\<exists>x. P x"
trivialideals_fieldI: assumes "carrier R \<noteq> {\<zero>}"   and "{I. ideal I R} = {{\<zero>}, carrier R}" shows "field R"
rational_root_test_def: fixes p :: "rat poly" shows "rational_root_test p = rational_root_test_main divisors_int divisors_int_pos p"
simps_1: fixes q :: "('c, 'd) pq" shows "bts_augment [] q = q"
simps_52: fixes x8 :: "('a, 'b, 'c) expr"   and x4 :: "('a, 'b, 'c) expr" shows "Rjoin x8 \<noteq> Ref x4"
paths_plus_one_disjoint: fixes xs :: "'a list"   and ys :: "'a list"   and v :: "'a" assumes "xs \<in> paths_with_new"   and "ys \<in> paths_with_new"   and "xs \<noteq> ys"   and "v \<in> set xs"   and "v \<in> set ys" shows "v = v0 \<or> v = v1 \<or> v = new_last"
x: shows "a11 * fst x + a12 * snd x = b1"
has_laplace_derivative_time_domain: fixes f :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex"   and f' :: "real \<Rightarrow> complex"   and f0 :: "complex"   and M :: "real"   and c :: "real" assumes "(f has_laplace L) s"   and "\<And>t. 0 < t \<Longrightarrow> (f has_vector_derivative f' t) (at t)"   and "(f \<longlongrightarrow> f0) (at_right 0)"   and "exponential_order M c f"   and "c < Re s" shows "(f' has_laplace s * L - f0) s"
abs_summable_on_cong: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and B :: "'c set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "(f abs_summable_on A) = (g abs_summable_on B)"
lm03: fixes a :: "(integer \<times> 'b set) set"   and N :: "integer set"   and \<Omega> :: "'b set" shows "(a \<in> soldAllocations'' N \<Omega>) = (a \<in> allocationsUniverse \<and> a \<in> {aa. Domain aa \<subseteq> N - {seller} \<and> \<Union> (Range aa) \<subseteq> \<Omega>})"
Max_ge_iff: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le> Max A) = (\<exists>a\<in>A. x \<le> a)"
two_set: fixes ls :: "'c list"   and a :: "'c"   and b :: "'c" assumes "distinct ls"   and "set ls = set [a, b]"   and "a \<noteq> b" shows "ls = [a, b] \<or> ls = [b, a]"
ax_2: shows "p = \<lbrakk>\<guillemotleft>A IMP B\<guillemotright>\<rbrakk>e"
monoid_cancelI: assumes "\<And>a b c. \<lbrakk>c \<otimes> a = c \<otimes> b; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b"   and "\<And>a b c. \<lbrakk>a \<otimes> c = b \<otimes> c; a \<in> carrier G; b \<in> carrier G; c \<in> carrier G\<rbrakk> \<Longrightarrow> a = b" shows "monoid_cancel G"
class_changed_def: fixes P1 :: "'b prog"   and P2 :: "'b prog"   and cn :: "char list" shows "class_changed P1 P2 cn = (class P1 cn \<noteq> class P2 cn)"
eval_trm_fv_cong: fixes t :: "trm"   and v :: "event_data list"   and v' :: "event_data list" assumes "\<forall>x\<in>fv_trm t. v ! x = v' ! x" shows "eval_trm v t = eval_trm v' t"
prime: fixes p :: "nat" assumes "prime p" shows "f p = f_prime p"
prod_mem_prod_ideals: fixes I :: "'a set"   and J :: "'a set"   and i :: "'a"   and j :: "'a" assumes "ideal R I"   and "ideal R J"   and "i \<in> I"   and "j \<in> J" shows "i \<cdot>\<^sub>r j \<in> I \<diamondsuit>\<^sub>r J"
A: shows "out (foldl step (foldl step s\<^sub>0 (xs @ [y])) ys) x = out (foldl step (foldl step s\<^sub>0 (xs @ [y])) (c_ipurge I D (D x) ys)) x"
strict_trans2: fixes a :: "'a Word.word"   and b :: "'a Word.word"   and c :: "'a Word.word" assumes "a <s b"   and "b \<le>s c" shows "a <s c"
DomainpE: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d" assumes "Domainp r a" obtains b :: "'e" where "r a b"
intros_3: fixes n :: "nat"   and l :: "'b tree234"   and m :: "'b tree234"   and r :: "'b tree234"   and p :: "'b"   and q :: "'b" assumes "full n l"   and "full n m"   and "full n r" shows "full (Suc n) \<langle>l, p, m, q, r\<rangle>"
proper_implies_overlap: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "O x y"
r_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier R"   and "x \<in> carrier R" shows "H #>\<^bsub>add_monoid R\<^esub> x \<subseteq> carrier R"
sum_rel_simp_4: fixes a :: "'b"   and a' :: "'c"   and Rl :: "('a \<times> 'c) set"   and Rr :: "('b \<times> 'd) set" shows "(Inr a, Inl a') \<notin> \<langle>Rl, Rr\<rangle>sum_rel"
containtwo: fixes X :: "'b set set"   and \<A> :: "'b set set set"   and C :: "'b set"   and D :: "'b set" assumes "ChamberComplexWithApartmentSystem X \<A>"   and "SimplicialComplex.maxsimp X C"   and "SimplicialComplex.maxsimp X D" shows "\<exists>A\<in>\<A>. C \<in> A \<and> D \<in> A"
conc_assoc: fixes xs :: "'b seq"   and ys :: "'b seq"   and zs :: "'b seq" shows "conc (conc xs ys) zs = conc xs (conc ys zs)"
assms_4: shows "h \<turnstile> insert_before ptr' child ref \<rightarrow>\<^sub>h h'"
c_def_rec: fixes i :: "int"   and j :: "int" assumes "i < j" shows "c i j = (MIN k\<in>{i + 1..j}. c i (k - 1) + c k j + w i j)"
minus_equation_iff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (uminus b = a)"
is_shortest_pathD1: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and p :: "nat list" assumes "is_shortest_path G m n p" shows "p \<in> path_set G m n"
word_ao_absorbs_4: fixes y :: "'b word"   and x :: "'b word" shows "y AND x OR x = x"
gam1: shows "S = ( \<Gamma> \<Rightarrow>* \<Delta>)"
a_host_shadow_root_rel_shadow_root: fixes h :: "(_) heap"   and host :: "(_) element_ptr"   and shadow_root :: "(_) shadow_root_ptr" assumes "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> local.a_host_shadow_root_rel h"
None_notin_image_Some: fixes A :: "'e set" shows "None \<notin> Some ` A"
sym: fixes x :: "'a"   and y :: "'a" shows "invert_order (comparator_of x y) = comparator_of y x"
true_imp: fixes p :: "'b upred" shows "(true \<Rightarrow> p) = p"
flists_def: fixes A :: "'b fset" shows "flists A = {xs. distinct xs \<and> finset xs = A}"
assms_3: shows "nsqn (rt \<xi>) i = nsqn (rt \<xi>') i"
if_SE_D1': fixes P :: "'a \<Rightarrow> bool"   and \<sigma> :: "'a"   and B\<^sub>1 :: "'a \<Rightarrow> ('b \<times> 'a) option"   and B\<^sub>2 :: "'a \<Rightarrow> ('b \<times> 'a) option"   and M :: "'a \<Rightarrow> (bool \<times> 'a) option" assumes "P \<sigma>" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = (\<sigma> \<Turnstile> B\<^sub>1 ;- M)"
linear_frechet_derivative: fixes f :: "'a \<Rightarrow> 'b"   and net :: "'a filter" assumes "f differentiable net" shows "linear (frechet_derivative f net)"
M_index: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j < length fs" shows "M (length fs) $$ (i, j) = \<mu> i j"