hcju/isabelleps huggingface dataset:
2138: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - (x \<squnion> y) = - (x \<squnion> y)"
irred_terms_and_reduced_subst: fixes f :: "'a trm \<Rightarrow> 'a trm"   and S :: "'a eclause set"   and \<eta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list"   and E :: "'a trm set"   and I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and C :: "'a eclause" assumes "f = (\<lambda>t. trm_rep t S)"   and "\<eta> = map_subst f \<sigma>"   and "all_trms_irreducible (subst_set E \<sigma>) f"   and "I = int_clset S"   and "equivalent_on \<sigma> \<eta> (vars_of_cl (cl_ecl C)) I"   and "lower_on \<eta> \<sigma> (vars_of_cl (cl_ecl C))"   and "E = trms_ecl C"   and "\<forall>x\<in>S. \<forall>y. y \<in> trms_ecl x \<longrightarrow> dom_trm y (cl_ecl x)"   and "C \<in> S"   and "fo_interpretation I" shows "all_trms_irreducible (subst_set E \<eta>) f"
insert_queue_invar: fixes q :: "('c, 'd) BinomialTree list"   and e :: "'c"   and a :: "'d" assumes "queue_invar q" shows "queue_invar (local.insert e a q)"
1: shows "(x \<otimes>\<^bsub>F\<^esub> (y \<otimes>\<^bsub>F\<^esub> z)) a = x a \<otimes> (y a \<otimes> z a)"
tt_in_keys: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0" shows "tpp p \<in> keys p"
finprod_one: fixes A :: "'d set" shows "(\<Otimes>i\<in>A. \<one>) = \<one>"
in_set_replicate: fixes x :: "'k"   and n :: "nat"   and y :: "'k" shows "(x \<in> set (replicate n y)) = (x = y \<and> n \<noteq> 0)"
extremum_unique: fixes a :: "'a" shows "(top_class.top \<le> a) = (a = top_class.top)"
ASSIGN: fixes e :: "Expr"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and x :: "Var" assumes "\<forall>s ss.  s \<approx> ss  \<longrightarrow> evalE e s = evalE e ss" shows "G \<rhd> Assign x e : Sec (\<lambda>(s, t).  s \<approx> update t x (evalE e t) )"
downset_setp_upset_setp_dual: shows "upclosed_set \<circ> (`) \<partial> = downclosed_set"
isomorphic_trg: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (trg\<^sub>D (F a)) (F a)"
simps_1: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "inf\<^sub>- (And \<phi>\<^sub>1 \<phi>\<^sub>2) = and (inf\<^sub>- \<phi>\<^sub>1) (inf\<^sub>- \<phi>\<^sub>2)"
row_op: fixes A :: "((real, 'n) vec, 'n) vec"   and m :: "'n"   and n :: "'n"   and c :: "real" assumes "P A"   and "m \<noteq> n" shows "P (\<chi>i. if i = m then row m A + c *\<^sub>R row n A else row i A)"
Entry_loop: fixes g :: "'g"   and ns :: "'node list" assumes "invar g"   and "g \<turnstile> Entry g-ns\<rightarrow>Entry g" shows "ns = [Entry g]"
assms_1: shows "P \<leadsto>\<guillemotleft>weakBisim\<guillemotright> Q"
A: shows "g_size m \<equiv> iterate_size (iteratei m)"
llp: fixes y :: "'a"   and x :: "'a" assumes "test y" shows "(d x \<le> y) = (x \<le> y \<sqinter> x)"
Liminf_const: fixes F :: "'b filter"   and c :: "'c" assumes "F \<noteq> \<bottom>" shows "Liminf F (\<lambda>x. c) = c"
ARR_mkarr: fixes t :: "'c local.term" assumes "Arr t" shows "ARR (mkarr t)"
splus_def: fixes t :: "'b"   and v :: "'b \<times> 'c" shows "pprod.splus t v = (t + pprod.pp_of_term v, pprod.component_of_term v)"
SINK_plus_current: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and g :: "'k \<times> 'k \<Rightarrow> ennreal" shows "MFMC_Network.SINK (plus_current f g) = MFMC_Network.SINK f \<inter> MFMC_Network.SINK g"
assms: shows "(c, ictxt, fctxt) \<in> decompose_com c0"
natural_transformation_axioms: shows "natural_transformation A B F K \<rho>_\<sigma>\<tau>.map"
uncurry_def: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "uncurry f \<equiv> \<lambda>(a, b). f a b"
variety_of_antimono: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and G :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" assumes "F \<subseteq> G" shows "\<V> G \<subseteq> \<V> F"
c_def: shows "c = y \<bullet>\<^sub>C x / (x \<bullet>\<^sub>C x)"
F2: fixes e :: "real" assumes "0 < e"   and "ball a e \<subseteq> S" shows "(\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) field_differentiable at a"
distinct_1: fixes x2 :: "'g" shows "Bottom \<noteq> sum_bot.Left x2"
Rep_atoms: fixes x :: "'a atoms" shows "Rep_atoms x \<in> range atom_map"
inf_continuous_mono: fixes F :: "'a \<Rightarrow> 'b" assumes "inf_continuous F" shows "mono F"
card_partition: fixes C :: "'a set set"   and k :: "nat" assumes "finite C"   and "finite (\<Union> C)"   and "\<And>c. c \<in> C \<Longrightarrow> card c = k"   and "\<And>c1 c2. \<lbrakk>c1 \<in> C; c2 \<in> C; c1 \<noteq> c2\<rbrakk> \<Longrightarrow> c1 \<inter> c2 = {}" shows "k * card C = card (\<Union> C)"
disjuptoR0: shows "disj_dlHPP_Bisimulation_Up_To_R' d PP R R0"
vec_le_iff_diff_le_0: fixes a :: "'b vec"   and b :: "'b vec" shows "(a \<le> b) = (a - b \<le> 0\<^sub>v (dim_vec a))"
a: shows "mod_type_class.to_nat a = mod_type_class.to_nat b \<and> mod_type_class.to_nat a + 1 < Suc k \<and> mod_type_class.to_nat b + 1 < Suc k"
dvd_mod_imp_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times c (modulo a b)"   and "dvd.dvd times c b" shows "dvd.dvd times c a"
reduction_eval: fixes t :: "db"   and t' :: "db"   and v :: "db" assumes "t \<rightarrow>\<^sub>v t'"   and "t' \<Down> v" shows "t \<Down> v"
runit'_in_hom_2: fixes f :: "('a S.E.term, 'a) S.arr" assumes "S.ide f" shows "\<guillemotleft>S.runit' f : f \<Rightarrow>\<^sub>S f \<star>\<^sub>S S.src f\<guillemotright>"
fls_regpart_plus: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_regpart (f + g) = fls_regpart f + fls_regpart g"
in_Span_obtain_same_length_coeffs: fixes n :: "'v"   and ms :: "'v list" assumes "n \<in> RSpan ms" shows "\<exists>rs. set rs \<subseteq> UNIV \<and> length rs = length ms \<and> n = rs \<bullet>\<cdot> ms"
sum_le_singleton: fixes A :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "A \<subseteq> {x}" shows "sum f A = (if x \<in> A then f x else (0::'c))"
length_partition_by_median_4: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length r \<le> length l"
wls_vsubst_trans: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y2 :: "'var"   and y :: "'var" assumes "wls s X"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[y2 // y1]_ys = X #[y2 // y]_ys"
in_segment_1: fixes x :: "'b"   and a :: "'b"   and b :: "'b" shows "(x \<in> {a--b}) = (\<exists>u\<ge>0. u \<le> 1 \<and> x = (1 - u) *\<^sub>R a + u *\<^sub>R b)"
chambercomplex_image: shows "ChamberComplex (g \<turnstile> X)"
mem_cone: fixes S :: "'a set"   and x :: "'a"   and c :: "real" assumes "cone S"   and "x \<in> S"   and "0 \<le> c" shows "c *\<^sub>R x \<in> S"
finite_Collect_conjI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "finite {x. P x} \<or> finite {x. Q x}" shows "finite {x. P x \<and> Q x}"
add_nonneg_eq_0_iff: fixes x :: "'a"   and y :: "'a" assumes "(0::'a) \<le> x"   and "(0::'a) \<le> y" shows "(x + y = (0::'a)) = (x = (0::'a) \<and> y = (0::'a))"
iso_class_eq: fixes F :: "'a set"   and G :: "'a set" assumes "is_iso_class F"   and "is_iso_class G"   and "F \<inter> G \<noteq> {}" shows "F = G"
simps: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "inv_check_left_moving (as, lm) (s, l, r) ires = (inv_check_left_moving_on_leftmost (as, lm) (s, l, r) ires \<or> inv_check_left_moving_in_middle (as, lm) (s, l, r) ires)"
inference_closed_def: fixes S :: "'a eclause set" shows "inference_closed S = (\<forall>P C' D \<theta>. derivable D P S \<theta> FirstOrder C' \<longrightarrow> D \<in> S)"
unsigned_of_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_nat n) = semiring_1.of_nat one plus zero (take_bit LENGTH('c) n)"
advantage_def: fixes \<G> :: "'a cyclic_group"   and \<A> :: "'a \<Rightarrow> nat spmf" assumes "dis_log \<G>" shows "dis_log.advantage \<G> \<A> \<equiv> spmf (dis_log.dis_log \<G> \<A>) True"
mem_val_w32_mod_state: fixes asi :: "word8"   and a :: "word36"   and s :: "'b sparc_state"   and new_undef :: "bool"   and new_traps :: "Trap set"   and new_state_var :: "sparc_state_var"   and new_dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and new_user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and new_cpu_reg :: "CPU_register \<Rightarrow> word32" shows "mem_val_w32 asi a s = mem_val_w32 asi a (s\<lparr>cpu_reg := new_cpu_reg, user_reg := new_user_reg, dwrite := new_dwrite, state_var := new_state_var, traps := new_traps, undef := new_undef\<rparr>)"
thz: shows "det (\<chi>k. if k = i then z else row k A) = (0::'a)"
llist_le_anti_sym: fixes s :: "'a llist"   and t :: "'a llist" assumes "s \<le> t"   and "t \<le> s" shows "s = t"
Quotient_alt_def5: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "Quotient R Abs Rep T = (T \<le> Grp UNIV Abs \<and> Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>)"
prefixes_converg_le: fixes s :: "nat \<Rightarrow> nat option"   and t :: "nat"   and j :: "nat"   and t' :: "nat" assumes "total_cons s"   and "eval (total_cons.r_prefixes s) [t, j] \<down>"   and "t' \<le> t" shows "eval (total_cons.r_prefixes s) [t', j] \<down>"
closures_universal: fixes \<phi> :: "'a \<Rightarrow> \<o>" assumes "\<And>x. Axioms.axiom (\<phi> x)" shows "Axioms.axiom (\<^bold>\<forall>x. \<phi> x)"
split_int_less_real': fixes a :: "int"   and b :: "real" shows "(real_of_int a + b < 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> < 0 \<or> real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0)"
cases: fixes x :: "('form, 'rule) graph_node" obtains r :: "'rule" where "x = Rule r__"   | r :: "'form" where "x = Assumption ra__"   | r :: "'form" where "x = Conclusion r"   | "x = Helper"
into_WW_lenlex_iff: fixes k :: "nat"   and ms :: "nat list"   and ns :: "nat list" shows "((into_WW k ms, into_WW k ns) \<in> lenlex less_than) = ((ms, ns) \<in> lenlex less_than)"
arr_inv: fixes f :: "'a2" assumes "C2.iso f" shows "C2.arr (C2.inv f)"
cong2: fixes xf :: "nat \<Rightarrow> real" shows "(\<lambda>f. evaluate_net (insert_weights s m2 (\<lambda>i. f (i + (a + (if s then 0 else count_weights s m1))))) (drop (length (input_sizes (insert_weights s m1 (\<lambda>i. xf (i + a))))) inputs) $ j) = (\<lambda>f. evaluate_net (insert_weights s m2 (if s then \<lambda>i. f (i + a) else (\<lambda>i. f (i + count_weights s m1 + a)))) (drop (length (input_sizes (insert_weights s m1 (\<lambda>i. f (i + a))))) inputs) $ j)"
sign_simps_51: fixes ak :: "'b"   and bk :: "'b" shows "(ak * bk < (0::'b)) = ((0::'b) < ak \<and> bk < (0::'b) \<or> ak < (0::'b) \<and> (0::'b) < bk)"
inf_assoc: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf (inf x y) z = inf x (inf y z)"
fresh_subst'': fixes z :: "name"   and t\<^sub>2 :: "trm"   and t\<^sub>1 :: "trm" assumes "z \<sharp> t\<^sub>2" shows "z \<sharp> t\<^sub>1[z::=t\<^sub>2]"
rh_def: fixes l :: "'b tree"   and r :: "'b tree" shows "rh l r = (if size l < size r then 1 else 0)"
numeral_add_unfold_funpow: fixes k :: "num"   and a :: "'a" shows "numeral k + a = ((+) (1::'a) ^^ numeral k) a"
poly_mod_prime_type_def: fixes m :: "int" shows "poly_mod_prime_type TYPE('a) m \<equiv> poly_mod_type TYPE('a) m"
extractable_def: fixes K :: "msg set"   and I :: "msg set" shows "extractable K I \<equiv> {implInsec A B M |A B M. implInsec A B M \<in> I} \<union> {implAuth A B M |A B M. implAuth A B M \<in> I} \<union> {implConfid A B M |A B M. implConfid A B M \<in> I \<and> (A, B) \<in> broken K} \<union> {implSecure A B M |A B M. implSecure A B M \<in> I \<and> (A, B) \<in> broken K}"
bounded_linear_intro: fixes f :: "'a \<Rightarrow> 'b"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_linear f"
Conc_not_MT: fixes p :: "'a list"   and x :: "'a"   and xs :: "'a list" assumes "p = x # xs" shows "p \<noteq> []"
distinct_80: fixes x7 :: "logic_type"   and x5 :: "load_store_type" shows "logic_type x7 \<noteq> load_store_type x5"
negligible_sphere: fixes a :: "'c"   and e :: "real" shows "negligible (sphere a e)"
4: shows "\<forall>t\<in>{A..}. ((\<lambda>k. indicat_real {A..k} t *\<^sub>R local.laplace_integrand s t) \<longlongrightarrow> local.laplace_integrand s t) at_top"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Delete x2 = Delete y2) = (x2 = y2)"
integer_shiftr_def: fixes x :: "integer"   and n :: "integer" shows "integer_shiftr x n = (if n < 0 then undefined x n else drop_bit (nat_of_integer n) x)"
clsrel: shows "G\<turnstile>C\<prec>\<^sub>C D"
coll_trans: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "coll 0 a b"   and "coll 0 a c"   and "a \<noteq> 0" shows "coll 0 b c"
construct_witness_def: fixes Q :: "'b \<Rightarrow> 'a tree option"   and r :: "('b, 'a) ta_rule" shows "construct_witness Q r \<equiv> NODE (rhsl r) (map (\<lambda>q. the (Q q)) (rhsq r))"
MVT'_gr: fixes t :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real \<Rightarrow> real"   and s :: "real" assumes "\<And>s. s \<in> {0..t} \<Longrightarrow> (f has_derivative f' s) (at s within {0..t})"   and "\<And>s. s \<in> {0..t} \<Longrightarrow> (g has_derivative g' s) (at s within {0..t})"   and "\<And>x. x \<in> {0..t} \<Longrightarrow> g' x s \<le> f' x s"   and "g 0 < f 0"   and "0 < s \<and> s \<le> t"   and "0 < t" shows "g s < f s"
measurable_count_space_const: fixes c :: "'d"   and M :: "'c measure" shows "(\<lambda>x. c) \<in> M \<rightarrow>\<^sub>M count_space UNIV"
adc: shows "a \<bullet> (d - c) \<noteq> 0"
simps_1: fixes carrier :: "'d set"   and eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and more :: "'e" shows "(.=\<^bsub>\<lparr>carrier = carrier, eq = eq, \<dots> = more\<rparr>\<^esub>) = eq"
NormalizeI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P Language.normalize c Q,A"
min_satisfying_Some: fixes P :: "'a \<Rightarrow> bool"   and l :: "'a list"   and x :: "'a" shows "min_satisfying P l = Some x \<longrightarrow> x \<in> set l \<and> P x \<and> (\<forall>x'\<in>set l. x' < x \<longrightarrow> \<not> P x')"
min_less_iff_conj: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and z :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "less z (ord.min less_eq x y) = (less z x \<and> less z y)"
is_element_ptr\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) element_ptr" shows "is_element_ptr ptr = (case ptr of Ref x \<Rightarrow> True | element_ptr.Ext element_ptr \<Rightarrow> False)"
univalent_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i j k. f (i, j) \<noteq> bot \<and> f (i, k) \<noteq> bot \<longrightarrow> j = k" shows "matrix_stone_relation_algebra.univalent f"
case_map_generat: fixes io :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and a :: "'e \<Rightarrow> 'b"   and b :: "'f \<Rightarrow> 'c"   and d :: "'g \<Rightarrow> 'd"   and r :: "('e, 'f, 'g) generat" shows "(case map_generat a b d r of Pure x \<Rightarrow> pure x | IO x xa \<Rightarrow> io x xa) = (case r of Pure x \<Rightarrow> (pure \<circ> a) x | IO out x \<Rightarrow> (io (b out) \<circ> d) x)"
is_glb_above_iff: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'c set"   and x :: "'c"   and u :: "'c" assumes "class.po below"   and "po.is_glb below S x" shows "below u x = po.is_lb below S u"
ex_post_efficient: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "set_pmf (sds R) \<inter> pareto_losers R = {}"
plus_int_code_1: fixes k :: "int" shows "k + 0 = k"
continuous_on_cnj: fixes s :: "'a set"   and g :: "'a \<Rightarrow> complex" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. cnj (g x))"
yun_factorization_def: fixes Gcd :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> 'c poly"   and p :: "'c poly" shows "yun_factorization Gcd p = (if p = 0 then (0::'c, []) else let c = lead_coeff p; q = smult (inverse c) p in (c, yun_gcd.yun_monic_factorization Gcd q))"
cr_envT_stateT_run: fixes s :: "'f"   and M :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "(cr_envT_stateT s M ===> eq_onp ((=) s) ===> M) run_env run_state"
simps_4: fixes f :: "'d"   and args :: "'e \<Rightarrow> ('d, 'e) trm"   and \<sigma> :: "('d, 'f, 'e) subst" shows "Tsubst ($f f args) \<sigma> = (case SFunctions \<sigma> f of None \<Rightarrow> $f f | Some f' \<Rightarrow> TsubstFO f') (\<lambda>i. Tsubst (args i) \<sigma>)"
LFP_closed: fixes f :: "'a \<Rightarrow> 'a" shows "LFP\<^bsub>L\<^esub> f \<in> carrier L"
rep_eq: fixes x :: "'d \<Rightarrow>\<^sub>L 'c"   and xa :: "'b \<Rightarrow>\<^sub>L 'd" shows "blinfun_apply (x o\<^sub>L xa) = blinfun_apply x \<circ> blinfun_apply xa"
Trans: shows "<\<nu>y>P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> yP'"
P: fixes S :: "'a set"   and T :: "'a set" assumes "P S \<and> P T" shows "P (S \<inter> T)"
lex_fun_antisym: fixes s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" assumes "rlex_fun t s"   and "rlex_fun s t" shows "s = t"
ratfps_dvd_code: fixes x :: "'b ratfps"   and y :: "'b ratfps" shows "(x dvd y) = (y = 0 \<or> x \<noteq> 0 \<and> ratfps_subdegree x \<le> ratfps_subdegree y)"
deduct1: fixes ps :: "('a, 'b) fm list"   and p :: "('a, 'b) fm"   and q :: "('a, 'b) fm" assumes "ps \<turnstile> p \<^bold>\<longrightarrow> q" shows "p # ps \<turnstile> q"
map_poly_comm_semiring_hom_axioms: shows "map_poly_comm_semiring_hom of_int"
phi_empty2: fixes n :: "nat"   and c :: "'a list"   and b :: "bool list"   and i :: "'a list" assumes "length init = 0" shows "\<phi> n (c, b, i) = 0"
word_of_int_minus: fixes i :: "int" shows "word_of_int (2 ^ LENGTH('a) - i) = word_of_int (- i)"
PowD: fixes A :: "'b set"   and B :: "'b set" assumes "A \<in> Pow B" shows "order.greater_eq B A"
fps_is_left_unit_iff_zeroth_is_left_unit: fixes f :: "'b fps" shows "(\<exists>g. 1 = f * g) = (\<exists>k. (1::'b) = f $ 0 * k)"
surjective: fixes r :: "('a, 'b, 'c) sas_plus_problem_scheme" shows "r = \<lparr>variables_of = r\<^sub>\<V>\<^sub>+, operators_of = r\<^sub>\<O>\<^sub>+, initial_of = r\<^sub>I\<^sub>+, goal_of = r\<^sub>G\<^sub>+, range_of = range_of r, \<dots> = sas_plus_problem.more r\<rparr>"
mdc_tuple: fixes C :: "char list"   and Cs :: "char list list" shows "mdc (C, Cs) = C"
mat_zero_def: shows "\<delta> \<equiv> \<lambda>j i. zero_class.zero::'d"
not_conj_deMorgans: fixes P :: "'a upred"   and Q :: "'a upred" shows "(\<not> (P \<and> Q)) = (\<not> P \<or> \<not> Q)"
comp_fun_idem_axioms: shows "comp_fun_idem MaxR_f"
hyps_6: shows "P \<turnstile> C sees M: Ts\<rightarrow>T = (pns, body) in D"
cong_add_trans_1: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and n :: "'a"   and y :: "'a" assumes "[a = b + x] (mod n)"   and "[x = y] (mod n)" shows "[a = b + y] (mod n)"
rep_eq: fixes x :: "'c"   and xa :: "'c set_dlist" shows "list_of_dlist (remove x xa) = equal_base.list_remove1 ceq' x (list_of_dlist xa)"
fun_upd_def: fixes f :: "'f \<Rightarrow> 'g"   and a :: "'f"   and b :: "'g" shows "f(a := b) = (\<lambda>x. if x = a then b else f x)"
general_U_def: fixes p :: "'a pmf" shows "local.general_U p = (if local.some_best \<approx>[\<R>] local.some_worst then 1 else local.my_U p)"
attractor_step_bounded_by_V: fixes W :: "'a set"   and S :: "'a set"   and p :: "Player" assumes "W \<subseteq> V"   and "S \<subseteq> V" shows "W \<union> S \<union> directly_attracted p S \<subseteq> V"
insert_ident: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "x \<notin> A"   and "x \<notin> B" shows "(insert x A = insert x B) = (A = B)"
ide_char: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "category C" shows "partial_magma.ide C a = (partial_magma.arr C a \<and> partial_magma.dom C a = a \<and> partial_magma.cod C a = a)"
wlsBinp_iff: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "wlsBinp delta binp = (wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 wlsAbs (barOf delta) binp)"
mismatchId: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<noteq>b]P \<simeq> P"
Rep_fs_set_inject: fixes x :: "'b fs_set"   and y :: "'b fs_set" shows "(Rep_fs_set x = Rep_fs_set y) = (x = y)"
Span_in_carrier: fixes K :: "'a set"   and Us :: "'a list" assumes "K \<subseteq> carrier R"   and "set Us \<subseteq> carrier R" shows "R.Span K Us \<subseteq> carrier R"
nonzero_power_divide: fixes a :: "'a"   and b :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "b \<noteq> \<zero>" shows "(a \<oslash> b) [^] n = a [^] n \<oslash> b [^] n"
smulextp_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool"   and m :: "'a multiset"   and n :: "'a multiset" shows "smulextp f m n = ((m, n) \<in> s_mul_ext {(x, y). snd (f x y)} {(x, y). fst (f x y)})"
bound: fixes I' :: "'a set" assumes "I' \<subseteq> I"   and "finite I'" shows "Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B"
assms_2: shows "X \<subseteq> {0..<n + 1}"
run_return_state: fixes return :: "'c \<times> 'b \<Rightarrow> 'a"   and x :: "'c"   and s :: "'b" shows "run_state (return_state return x) s = return (x, s)"
fib_add: fixes n :: "nat"   and k :: "nat" shows "fib (Suc (n + k)) = fib (Suc k) * fib (Suc n) + fib k * fib n"
negative_part_def: fixes C :: "'at Literal set" shows "negative_part C = {L. (\<exists>A. L = Neg A) \<and> L \<in> C}"
path_weight_path: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b"   and loc1 :: "'a"   and loc2 :: "'a" assumes "graph weights"   and "s \<in>\<^sub>A graph.path_weight weights loc1 loc2" obtains xs :: "('a \<times> 'b \<times> 'a) list" where "graph.path weights loc1 loc2 xs \<and> (\<forall>ys. graph.path weights loc1 loc2 ys \<longrightarrow> \<not> foldr (+) (map (\<lambda>(s, l, t). l) ys) (0::'b) < foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b))" and "distinct xs" and "foldr (+) (map (\<lambda>(s, l, t). l) xs) (0::'b) = s"
card1: fixes A :: "'b set" assumes "card A = 1" shows "\<exists>a. A = {a}"
monomial_0I: fixes c :: "'a"   and t :: "'b" assumes "c = (0::'a)" shows "monomial c t = 0"
lattice_ow_axioms: shows "lattice_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2 sup\<^sub>2"
scale_zero_left: fixes x :: "'b poly" shows "smult (0::'b) x = 0"
reaches1_reaches_iff1: fixes x :: "'a"   and y :: "'a" shows "A.reaches1 x y = (\<exists>z. A x z \<and> A.reaches z y)"