hcju/isabelleps huggingface dataset:
weakCongRes: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs"   and x :: "name" assumes "P \<Longrightarrow>\<alpha> \<prec> P'"   and "x \<sharp> \<alpha>" shows "\<lparr>\<nu>x\<rparr>P \<Longrightarrow>\<alpha> \<prec> \<lparr>\<nu>x\<rparr>P'"
invert_opposite_sides_of_square: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and h :: "'a \<times> 'a"   and k :: "'a \<times> 'a" assumes "VV.seq f g"   and "VV.comp f g = VV.comp h k"   and "VV.iso f"   and "VV.iso k" shows "VV.seq g (VV.inv k) \<and> VV.seq (VV.inv f) h \<and> VV.comp g (VV.inv k) = VV.comp (VV.inv f) h"
countable_image_inj_on: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "countable (f ` A)"   and "inj_on f A" shows "countable A"
return_node_call_of_return_node: fixes n :: "'node" assumes "return_node n" shows "\<exists>!n'. call_of_return_node n n'"
gi_is_fi_minus_sum_mu_gso: fixes i :: "nat" assumes "i < length fs" shows "gso i = fs ! i - M.sumlist (map (\<lambda>j. \<mu> i j \<cdot>\<^sub>v gso j) [0..<i])"
many_strongly_connected_iff_5_eq: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>\<star> ; x\<^sup>T = x\<^sup>+)"
convert: fixes t1 :: "t"   and t2 :: "t"   and sx :: "sx option"   and \<C> :: "t_context" assumes "t1 \<noteq> t2"   and "(sx = None) = (is_float_t t1 \<and> is_float_t t2 \<or> is_int_t t1 \<and> is_int_t t2 \<and> t_length t1 < t_length t2)" shows "\<C> \<turnstile> [Cvtop t1 Convert t2 sx] : [t2] _> [t1]"
sum_lessThan_dcharacter: fixes m :: "nat" assumes "principal_dchar n \<noteq> principal_dchar n" shows "sum (principal_dchar n) {..<m} = sum (principal_dchar n) {..<m mod n}"
coinduct: fixes X :: "'a stream \<Rightarrow> bool"   and x :: "'a stream"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>xs. x = xs \<and> \<phi> xs \<and> (X (stl xs) \<or> alw \<phi> (stl xs))" shows "alw \<phi> x"
is_path_nonempty: fixes n :: "nat"   and S :: "(nat \<times> nat) set"   and p :: "nat list" assumes "is_path n S p" shows "p \<noteq> []"
conjugate_smult_vec: fixes k :: "'c"   and v :: "'c vec" shows "conjugate (k \<cdot>\<^sub>v v) = conjugate k \<cdot>\<^sub>v conjugate v"
axioms_1: fixes step1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step2 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and final1 :: "'b \<Rightarrow> bool"   and final2 :: "'c \<Rightarrow> bool"   and order :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and match :: "'d \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "backward_simulation step1 step2 final1 final2 order match" shows "semantics step1 final1"
simps_2: 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and es :: "expr list"   and T :: "ty"   and Ts :: "ty list" shows "(P,E,h \<turnstile> e # es [:]\<^bsub>NT\<^esub> T # Ts) = (P,E,h \<turnstile> e :\<^bsub>NT\<^esub> T \<and> P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> Ts)"
simplical_face_def: fixes k :: "nat"   and x :: "nat \<Rightarrow> 'c" shows "simplical_face k x \<equiv> \<lambda>i. if i < k then x i else if i = k then 0::'c else x (i - 1)"
setEqualityAsDifference: fixes x :: "'c"   and y :: "'c" shows "({x} - {y} = {}) = (x = y)"
oalist_inv_list_of_oalist: fixes x :: "'y" shows "oalist_inv (list_of_oalisty x)"
AE_T_enabled: fixes s :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "almost_everywhere (T s) (enabled s)"
f4_red_aux_dgrad_p_set_le: fixes d :: "'a \<Rightarrow> nat"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" assumes "dickson_grading d" shows "dgrad_p_set_le d (set (f4_red_aux bs ps)) (args_to_set ([], bs, ps))"
nonzero_mult_divide_mult_cancel_left2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and 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 c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times c a) (times b c) = divide a b"
shift_prefix_cases: fixes xl :: "'a list"   and xs :: "'a stream"   and yl :: "'a list"   and ys :: "'a stream" assumes "xl @- xs = yl @- ys" shows "prefix xl yl \<or> prefix yl xl"
add_closed_inverse_right: fixes x :: "'g"   and g :: "'g" assumes "x \<notin> G"   and "g \<in> G" shows "g + x \<notin> G"
Quotient3_abs_rep: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and a :: "'b" assumes "Quotient3 R Abs Rep" shows "Abs (Rep a) = a"
cfg_onD_state: fixes cfg :: "'s cfg"   and s :: "'s" assumes "cfg \<in> cfg_on s" shows "state cfg = s"
q0_def: shows "q0 = (\<lambda>h. SOME qf. P h qf)"
lfilter_lmap: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "lfilter P (lmap f xs) = lmap f (lfilter (P \<circ> f) xs)"
set_bind_spmf: fixes M :: "'d spmf"   and f :: "'d \<Rightarrow> 'c spmf" shows "set_spmf (M \<bind> f) = set_spmf M \<bind> set_spmf \<circ> f"
representation_eqI: fixes basis :: "'b set"   and v :: "'b"   and f :: "'b \<Rightarrow> complex" assumes "cindependent basis"   and "v \<in> cspan basis"   and "\<And>b. f b \<noteq> 0 \<Longrightarrow> b \<in> basis"   and "finite {b. f b \<noteq> 0}"   and "(\<Sum>b | f b \<noteq> 0. f b *\<^sub>C b) = v" shows "crepresentation basis v = f"
cases: fixes x :: "nat \<times> bdd" obtains bdd :: "bdd" where "x = (0, bdd__)"   | bdd :: "bdd" where "x = (Suc 0, bdda__)"   | n :: "nat" and bdd :: "bdd" where "x = (Suc (Suc n), bdd)"
wp_equiv: fixes c :: "com"   and c' :: "com"   and \<pi> :: "char list \<Rightarrow> com option" assumes "c \<sim> c'" shows "wp \<pi> c = wp \<pi> c'"
fps_divide_unit_factor_numer: fixes f :: "'a fps"   and g :: "'a fps" shows "unit_factor f / g = fps_shift (subdegree f) (f / g)"
1: fixes x :: "('fun, 'atom) Term.term \<times> nat" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set M" shows "\<exists>y. Var y \<in> set M \<and> \<Gamma> (Var y) = \<Gamma> (Var x)"
hcomp_simps_1: fixes \<nu> :: "('a S.E.term, 'a) S.arr"   and \<mu> :: "('a S.E.term, 'a) S.arr" assumes "S.hseq \<nu> \<mu>" shows "S.src (\<nu> \<star>\<^sub>S \<mu>) = S.src \<mu>"
r_def: shows "r = (\<Sum>y\<in>{v. p v \<noteq> 0}. p y *R v y) + p x *R (c - v x)"
simps_24: fixes vc :: "'b rexp"   and v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Star vc) (Plus v va) = True"
OPT: shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = 1"
lt: shows "ltake (enat n) (lalternate (ltl (ltl xs))) = lalternate (ltake (enat (2 * n)) (ltl (ltl xs)))"
space_Sup_measure': fixes M :: "'b measure set"   and A :: "'b measure" assumes "\<And>m. m \<in> M \<Longrightarrow> sets m = sets A"   and "M \<noteq> {}" shows "space (Sup_measure' M) = space A"
index_add_vec_2: fixes v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" shows "dim_vec (v\<^sub>1 + v\<^sub>2) = dim_vec v\<^sub>2"
of_int_mult: fixes one :: "'q"   and times :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and uminus :: "'q \<Rightarrow> 'q"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w * z) = times (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
simps_8: fixes P :: "pol"   and x :: "nat"   and R :: "pol"   and y :: "nat"   and Q :: "pol" shows "PX P x R \<langle>*\<rangle> Pinj y Q = (if y = 0 then PX P x R \<langle>*\<rangle> Q else if y = 1 then mkPX (Pinj y Q \<langle>*\<rangle> P) x (R \<langle>*\<rangle> Q) else mkPX (Pinj y Q \<langle>*\<rangle> P) x (R \<langle>*\<rangle> Pinj (y - 1) Q))"
left_divisibility: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (\<exists>c. a = b * c)"
4: shows "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
weakenBisim_def: shows "weakenBisim \<equiv> {(xb, xa, x). weakenBisimp xb xa x}"
dropped_seq_is_exact_seq: fixes G :: "'b monoid list"   and F :: "('b \<Rightarrow> 'b) list"   and i :: "nat" assumes "exact_seq (G, F)"   and "i < length F" shows "exact_seq (drop i G, drop i F)"
inject_2: fixes x21 :: "nat"   and x22 :: "exp"   and x23 :: "(nat \<times> bval) list"   and y21 :: "nat"   and y22 :: "exp"   and y23 :: "(nat \<times> bval) list" shows "(BClos x21 x22 x23 = BClos y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
assms_1: shows "decompose_literal (subst_lit L \<sigma>) t s polarity"
Int_emptyI: fixes A :: "'e set"   and B :: "'e set" assumes "\<And>x. \<lbrakk>x \<in> A; x \<in> B\<rbrakk> \<Longrightarrow> False" shows "A \<inter> B = {}"
ordinal_times_not_0: fixes x :: "ordinal"   and y :: "ordinal" shows "(0 < x * y) = (0 < x \<and> 0 < y)"
tc_eq: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.left_kleene_algebra plus times less_eq less one star"   and "times x x = x" shows "times (star x) x = x"
ring_distribs_3: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a - b) * c = a * c - b * c"
drop_bit_of_0: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit n zero = zero"
tr12: shows "trace (\<rho>\<^sub>1 + \<rho>\<^sub>2) \<le> 1"
similar_mat_witD2_5: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "A \<in> carrier_mat n m"   and "similar_mat_wit A B P Q" shows "B \<in> carrier_mat n n"
impl_of_inverse: fixes x :: "('b, 'c) assoc_list" shows "Assoc_List (assoc_list.impl_of x) = x"
simps_2: fixes x :: "'c"   and v :: "'c"   and va :: "'c list" shows "Missing_List.min_list (x # v # va) = min x (Missing_List.min_list (v # va))"
simps_5: fixes f :: "'c \<Rightarrow> 'a"   and p :: "('c, 'b) form"   and q :: "('c, 'b) form" shows "psubst f (Or p q) = Or (psubst f p) (psubst f q)"
sub: shows "subgroup (H \<times> N) (G \<times>\<times> K)"
homeomorphic_subspace_Euclidean_space: fixes S :: "'a set"   and n :: "nat" assumes "subspace S" shows "(top_of_set S homeomorphic_space Euclidean_space n) = (dim S = n)"
E: shows "(r + (Suc (Suc (foldl (+) 0 ns + n)) - bn_count (Suc (Suc n) # ns)) * p) mod q = 0"
rref_k: shows "reduced_row_echelon_form_upt_k (Gauss_Jordan A) k"
map_of_map: fixes f :: "'g \<Rightarrow> 'f"   and xs :: "('e \<times> 'g) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
tarski_s_euclid_implies_euclid_5: assumes "TarskiSParallelPostulate" shows "Euclid5"
order_trans: fixes x :: "('c, 'd) Term.term"   and y :: "('c, 'd) Term.term"   and z :: "('c, 'd) Term.term" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z" shows "x \<sqsubseteq> z"
vec_nth_cases: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> UNIV" obtains x :: "('e, 'd) vec" where "y = ($) x"
ltrm_rep_X_pow: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "monom P (p (deg R p)) (deg R p) = p (deg R p) \<odot>\<^bsub>P\<^esub> X_poly R [^]\<^bsub>P\<^esub> deg R p"
aux0_1: fixes xs :: "('a, 'b) Combinators list"   and x :: "('a, 'b) Combinators" assumes "singleCombinators xs"   and "singleCombinators [x]" shows "singleCombinators (x # xs)"
elims_1: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal"   and y :: "bool" assumes "validate_ground_lit x xa = y" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and E :: "'b equation" where "x = I__" and "xa = Pos E__" and "y = validate_ground_eq I__ E__"   | I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and E :: "'b equation" where "x = I" and "xa = Neg E" and "y = (\<not> validate_ground_eq I E)"
le_left_mono: fixes x :: "'h"   and y :: "'h"   and a :: "'h" assumes "x \<le> y" shows "y \<le> a \<longrightarrow> x \<le> a"
wls_Abs_vsubst_all_aux: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<forall>y. (y = x \<or> fresh xs y X) \<and> (y = x' \<or> fresh xs y X') \<longrightarrow> X #[y // x]_xs = X' #[y // x']_xs)"
Rm_1: shows "integrable M (\<lambda>x. fm x * real_transfer_operator g x)"
hyps_2: shows "L (substL (lang e2) \<sigma>) = lang (subst e2 \<sigma>)"
nat_ivl_Suc': fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'c" assumes "m \<le> Suc n" shows "prod g {m..Suc n} = g (Suc n) * prod g {m..n}"
isInAlternativeListb: fixes aa :: "'b"   and p :: "('b, 'c) Combinators list"   and a :: "('b, 'c) Combinators" assumes "aa \<in> set (net_list_aux p)" shows "aa \<in> set (net_list_aux (a # p))"
smultfunsetI: fixes G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<forall>a. \<forall>g\<in>G. \<forall>x\<in>FH. f (a \<delta>\<delta> g * x) = smult (a \<delta>\<delta> g) (f x)" shows "f \<in> aezfun_scalar_mult.smultfunset smult G FH"
bisimOutputPushRes: fixes x :: "name"   and M :: "'a"   and N :: "'a"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> M"   and "x \<sharp> N" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<langle>N\<rangle>.P) \<sim> M\<langle>N\<rangle>.\<lparr>\<nu>x\<rparr>P"
polyfun_linear_factor_root: fixes c :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "(\<Sum>i\<le>n. c i * a ^ i) = (0::'a)" obtains b :: "nat \<Rightarrow> 'a" where "\<And>z. (\<Sum>i\<le>n. c i * z ^ i) = (z - a) * (\<Sum>i<n. b i * z ^ i)"
induct: fixes x1 :: "recf"   and x2 :: "nat"   and P :: "recf \<Rightarrow> nat \<Rightarrow> bool" assumes "primerec x1 x2"   and "P z (Suc 0)"   and "P s (Suc 0)"   and "\<And>n m. n < m \<Longrightarrow> P (recf.id m n) m"   and "\<And>f k gs m n. \<lbrakk>primerec f k; P f k; length gs = k; \<forall>i<length gs. primerec (gs ! i) m \<and> P (gs ! i) m; m = n\<rbrakk> \<Longrightarrow> P (Cn n f gs) m"   and "\<And>f n g m. \<lbrakk>primerec f n; P f n; primerec g (Suc (Suc n)); P g (Suc (Suc n)); m = Suc n\<rbrakk> \<Longrightarrow> P (Pr n f g) m" shows "P x1 x2"
simps: fixes x :: "real"   and y :: "real" shows "ordinate (Point x y) = y"
signed_word_eqI: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and v :: "'b Word.word"   and w :: "'b Word.word" assumes "class.ring_char_0 one times plus zero minus uminus"   and "ring_1.signed one plus zero minus v = ring_1.signed one plus zero minus w" shows "v = w"
r_into_rtrancl: fixes p :: "'g \<times> 'g"   and r :: "('g \<times> 'g) set" assumes "p \<in> r" shows "p \<in> r\<^sup>*"
transform_simps_4: fixes a :: "Arity"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "\<T>\<^bsub>a\<^esub> (Terms.Let \<Gamma> e) = Terms.Let (map_transform Aeta_expand (Aheap \<Gamma> e\<cdot>a) (map_transform transform_syn (Aheap \<Gamma> e\<cdot>a) \<Gamma>)) (\<T>\<^bsub>a\<^esub> e)"
cases: fixes a1 :: "'b set"   and a2 :: "'b set"   and a3 :: "'b set"   and a4 :: "'b set"   and R :: "('b \<times> 'b) set" assumes "((a1, a2), a3, a4) \<in> sse_step R" obtains \<sigma> :: "'b" and W :: "'b set" and \<Sigma>' :: "'b set" and \<Sigma> :: "'b set" and W' :: "'b set" where "a1 = \<Sigma>" and "a2 = W" and "a3 = \<Sigma>'" and "a4 = W'" and "\<sigma> \<in> W" and "\<Sigma>' = \<Sigma> \<union> R `` {\<sigma>}" and "W' = W - {\<sigma>} \<union> (R `` {\<sigma>} - \<Sigma>)"
distr_bij_count_space: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "distr (count_space A) (count_space B) f = count_space B"
intTF: fixes \<sigma> :: "'b"   and a :: "'c" shows "intTF \<sigma> a"
weight_cong: fixes g\<^sub>1 :: "'a ugraph"   and g\<^sub>2 :: "'a ugraph"   and w :: "'a set \<Rightarrow> nat" assumes "edges g\<^sub>1 = edges g\<^sub>2" shows "weight w g\<^sub>1 = weight w g\<^sub>2"
set_eq_fold: fixes xs :: "'a list" shows "gcd_list xs = fold gcd xs (0::'a)"
integrable_sinc': fixes T :: "real"   and \<theta> :: "real" shows "interval_lebesgue_integrable lborel (ereal 0) (ereal T) (\<lambda>t. sin (t * \<theta>) / t)"
prems_4: fixes k :: "nat" assumes "k < Suc n" shows "f k = g k + h k"
repv_self: fixes \<omega> :: "variable \<Rightarrow> real"   and x :: "variable" shows "repv \<omega> x (\<omega> x) = \<omega>"
trans_S_point: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "(x, y) \<in> S"   and "(y, z) \<in> S" shows "(x, z) \<in> S"
assms_3: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> C P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> C Q"
union_filter_mset_complement: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and M :: "'b multiset" assumes "\<forall>x. P x = (\<not> Q x)" shows "filter_mset P M + filter_mset Q M = M"
f_def: fixes i :: "nat" shows "f i = isolate_variable_sparse p x (i + 1) * Var x ^ i * Const (real i + 1)"
id: shows "of_int = of_rat \<circ> rat_of_int"
relB: shows "HMA_M (from_hma\<^sub>m B) B"
prems: shows "\<Gamma> \<turnstile> App e\<^sub>1 e\<^sub>2 : T"
mult_column_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and n :: "'e"   and q :: "'d" shows "mult_column A n q = (\<chi>i j. if j = n then A $ i $ j * q else A $ i $ j)"
s: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "s (f \<triangleright> n) \<down>= Suc (the (f 0))"
mono_step'_top: fixes C :: "(char list \<Rightarrow> 'av) option acom"   and C' :: "(char list \<Rightarrow> 'av) option acom" assumes "C \<le> C'" shows "step' \<top> C \<le> step' \<top> C'"
simple_bochner_integral_norm_bound: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" shows "norm (simple_bochner_integral M f) \<le> simple_bochner_integral M (\<lambda>x. norm (f x))"
rep_eq: fixes x :: "('c, 'd) mapping" shows "Mapping.lookup x = mapping.rep x"
noWhile_MtransT: fixes c :: "('test, 'atom) com"   and s :: "'state" assumes "noWhile c" shows "\<exists>s'. (c, s) \<rightarrow>*t s'"
vectorize_uminus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "local.punit.vectorize_poly (- p) = - local.punit.vectorize_poly p"
lookup_rep_nat_pp_pp: fixes t :: "('a, 'b) pp" shows "lookup_pp (rep_nat_pp t) = (\<lambda>n. rep_nat (lookup_pp t (abs_nat n)))"
summandEquiv: fixes P :: "pi"   and Q :: "pi" assumes "uhnf P"   and "uhnf Q"   and "\<forall>P'\<in>summands P. \<exists>Q'\<in>summands Q. P' \<equiv>\<^sub>e Q'"   and "\<forall>Q'\<in>summands Q. \<exists>P'\<in>summands P. Q' \<equiv>\<^sub>e P'" shows "P \<equiv>\<^sub>e Q"
snd3_simp: fixes a :: "'e"   and b :: "'d"   and c :: "'f" shows "snd3 (a, b, c) = b"
simps_1: fixes uu :: "'a list" shows "p_fusion [] uu = []"
coeff_lowerPoly: fixes b :: "nat"   and i :: "nat"   and mp :: "'a mpoly"   and x :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (lowerPoly b i mp) x = MPoly_Type.coeff mp (lowerPowers b i x)"
continuous_on_empty: fixes f :: "'a \<Rightarrow> 'b" shows "continuous_on {} f"
m_inst_isl: fixes xs :: "'a option list"   and tstp :: "nat + nat" assumes "Mapping.lookup m xs = Some tstp" shows "isl tstp = (0 < left I)"
INF_const: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'d set"   and f :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<noteq> {}" shows "Inf ((\<lambda>i. f) ` A) = f"
monad_writer_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and tell :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" shows "monad_writer return bind tell \<equiv> monad return bind \<and> monad_writer_axioms bind tell"
inner_Basis_0_1: fixes a :: "'a" shows "local.inner_Basis (0::'a) a = 0"
lm037: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" assumes "a \<in> injections (Domain a) (Range a)"   and "Range a \<in> all_partitions G"   and "Domain a \<subseteq> N" shows "a \<in> allAllocations N G"
invertible_eq_full_rank: fixes A :: "(('b, 'c) vec, 'c) vec" shows "invertible A = (rank A = nrows A)"
app6: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and err :: "v error_result"   and op0 :: "op0" assumes "evaluate_list ck env s1 (rev es) (s2, Rerr err)" shows "evaluate ck env s1 (App op0 es) (s2, Rerr err)"
mpath_weight'_correct: fixes wl :: "'V \<Rightarrow> 'W infty option"   and res :: "'V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option"   and v :: "'V" assumes "dinvarm (wl, res)" shows "mpath_weight' (res v) = path_weight' (\<alpha>r res v)"
Deduction_theorem: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "AX0 \<union> (F \<triangleright> \<Gamma>) \<turnstile>\<^sub>H G" shows "AX0 \<union> \<Gamma> \<turnstile>\<^sub>H F \<^bold>\<rightarrow> G"
id_is_homeomorphism: shows "homeomorphism Spec is_zariski_open Spec is_zariski_open (identity Spec)"
elims: fixes x :: "interpolation_algorithm"   and xa :: "(int \<times> int) list"   and y :: "int poly option" assumes "interpolation_poly_int x xa = y" obtains xs_ys :: "(int \<times> int) list" where "x = Newton" and "xa = xs_ysa__" and "y = newton_interpolation_poly_int xs_ysa__"   | xs_ys :: "(int \<times> int) list" where "x = Lagrange" and "xa = xs_ysb__" and "y = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ysb__; rp = interpolation_poly Lagrange rxs_ys in if Ball (set (coeffs rp)) is_int_rat then Some (map_poly int_of_rat rp) else None)"   | xs_ys :: "(int \<times> int) list" where "x = Neville_Aitken" and "xa = xs_ys" and "y = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys; rp = interpolation_poly Neville_Aitken rxs_ys in if Ball (set (coeffs rp)) is_int_rat then Some (map_poly int_of_rat rp) else None)"
components_maximal: fixes c :: "'b set"   and s :: "'b set"   and t :: "'b set" assumes "c \<in> components s"   and "connected t"   and "t \<subseteq> s"   and "c \<inter> t \<noteq> {}" shows "t \<subseteq> c"