hcju/isabelleps huggingface dataset:
fmap_of_list_insort_key_fst_map: fixes f :: "'c \<Rightarrow> 'd"   and m :: "('c, 'e) fmap"   and k :: "'c"   and v :: "'e" assumes "inj f"   and "m $$ k = None" shows "fmap_of_list (insort_key fst (f k, v) (map (\<lambda>(k, v). (f k, v)) (sorted_list_of_fmap m))) = fmap_of_list (map (\<lambda>(k, v). (f k, v)) (sorted_list_of_fmap m))(f k $$:= v)"
S_def: fixes h :: "'g \<times> 'a"   and e :: "'e" shows "\<Sigma>_OR.S h e = map_spmf (\<lambda>(a, z). (a, e, z)) (S_OR h e)"
sup_top_right: fixes x :: "'a" shows "x + 1 = 1"
less_irrefl_nat: fixes n :: "nat"   and R :: "bool" assumes "n < n" shows "R"
iso_set_refl: fixes G :: "('d, 'e) monoid_scheme" shows "(\<lambda>x. x) \<in> Group.iso G G"
simps_2: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and u :: "nat"   and v :: "nat"   and w :: "nat"   and ws :: "nat list" shows "linordered_monoid.len m u v (w # ws) = dbm_add (m u w) (linordered_monoid.len m w v ws)"
ennreal2_cases: fixes x :: "ennreal"   and xa :: "ennreal" obtains r :: "real" and ra :: "real" where "0 \<le> ra__" and "x = ennreal ra__" and "0 \<le> ra" and "xa = ennreal ra"   | r :: "real" where "0 \<le> rb__" and "x = ennreal rb__" and "xa = top"   | r :: "real" where "x = top" and "0 \<le> r" and "xa = ennreal r"   | "x = top" and "xa = top"
conditionally_complete_dual: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "conditionally_complete A less_eq" shows "conditionally_complete A (\<lambda>x y. less_eq y x)"
gf_F_eq_F: fixes x :: "'s"   and y :: "'s" shows "gf_F x y 1 = F x y"
outstanding_non_volatile_writes_unshared: fixes i :: "nat"   and p :: "'p"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'p memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_writes_unshared \<S> sb"
index_smult_vec_1: fixes i :: "nat"   and v :: "'a vec"   and a :: "'a" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $ i = a * v $ i"
finsum_infinite: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "infinite A" shows "finsum V f A = \<zero>\<^bsub>V\<^esub>"
isometry_onD: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "isometry_on X f"   and "x \<in> X"   and "y \<in> X" shows "dist (f x) (f y) = dist x y"
fminus_finsert: fixes A :: "'c fset"   and a :: "'c"   and B :: "'c fset" shows "A |-| finsert a B = A |-| B |-| {|a|}"
confTs_widen: fixes P :: "'a prog"   and h :: "heap"   and loc :: "val list"   and LT :: "ty err list"   and LT' :: "ty err list" assumes "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT"   and "P \<turnstile> LT [\<le>\<^sub>\<top>] LT'" shows "P,h \<turnstile> loc [:\<le>\<^sub>\<top>] LT'"
coboundedI2: fixes b :: "'a multiset"   and c :: "'a multiset"   and a :: "'a multiset" assumes "b \<subseteq># c" shows "a \<inter># b \<subseteq># c"
tendsto_of_int_floor: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<notin> \<int>" shows "((\<lambda>x. of_int \<lfloor>f x\<rfloor>) \<longlongrightarrow> of_int \<lfloor>l\<rfloor>) F"
that: fixes S :: "'a set" assumes "finite S"   and "cbox a b = convex hull S" shows "thesis"
cblinfun_assoc_left_1: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and B :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'c"   and C :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'd" shows "A o\<^sub>C\<^sub>L (B o\<^sub>C\<^sub>L C) = A o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L C"
k_i_def: fixes i :: "nat" shows "k_i i = (SOME k. k \<le> f_aux (Suc i) \<ominus> f_i i \<and> flow (residual_network (f_i i)) k \<and> value_flow (residual_network (f_i i)) k = value_flow (residual_network (f_i i)) (f_aux (Suc i) \<ominus> f_i i) \<and> (\<forall>x. d_IN k x \<le> value_flow (residual_network (f_i i)) k))"
pullback_mono_self: fixes f :: "'a" assumes "local.mono f" shows "\<p>\<^sub>0[f, f] = \<p>\<^sub>1[f, f]"
assms: shows "\<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B a x y"
assms: shows "x \<in> carrier \<and> y \<in> carrier \<and> z \<in> carrier"
widens_refl: fixes P :: "'b prog"   and Ts :: "ty list" shows "P \<turnstile> Ts [\<le>] Ts"
inj1: shows "inj_on (\<lambda>s. (p s, s)) S"
IH: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "A = dom (thr s)"   and "mthr.\<tau>diverge s" shows "\<exists>t x. thr s t = \<lfloor>(x, no_wait_locks)\<rfloor> \<and> wset s t = None \<and> \<tau>diverge t (x, shr s)"
intro: fixes C :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and J :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ah \<Rightarrow> 'ag" assumes "category C"   and "category J"   and "functor J C D" shows "diagram J C D"
cases: fixes a1 :: "'c llist"   and a2 :: "'c llist" assumes "lprefix a1 a2" obtains xs :: "'c llist" where "a1 = LNil" and "a2 = xs__"   | xs :: "'c llist" and ys :: "'c llist" and x :: "'c" where "a1 = LCons x xs" and "a2 = LCons x ys" and "lprefix xs ys"
F_def: shows "F = (\<lambda>n. rqs_cost_exp n / (real n + 1))"
eta: fixes A :: "'a trm"   and A' :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type" assumes "A >>> A'" shows "Fn x T A >>> Fn x T A'"
closedin_closed_subtopology: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "closedin X S" shows "closedin (subtopology X S) T = (closedin X T \<and> T \<subseteq> S)"
9: shows "top * e \<squnion> top * e * (w\<^sup>T \<sqinter> - v * - v\<^sup>T \<sqinter> w\<^sup>\<star> * e\<^sup>T * top)\<^sup>\<star> * (w\<^sup>T \<sqinter> - v\<^sup>T) \<le> top * e * (w\<^sup>T \<sqinter> - v * - v\<^sup>T \<sqinter> w\<^sup>\<star> * e\<^sup>T * top)\<^sup>\<star>"
mds_consistent: shows "mds_consistent mds \<Gamma> \<S> P"
succ_to_list_def: fixes g :: "('b, 'a) graph"   and v :: "'b" shows "succ_to_list g v \<equiv> it_to_sorted_list (\<lambda>_ _. True) (succ g v)"
Limsup_le_iff: fixes F :: "'b filter"   and X :: "'b \<Rightarrow> 'a"   and C :: "'a" shows "(Limsup F X \<le> C) = (\<forall>y>C. \<forall>\<^sub>F x in F. X x < y)"
sorted_wrt_map: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"
UnionP_Mem: fixes x :: "tm"   and y :: "tm"   and z :: "tm"   and a :: "tm" shows "{UnionP x y z, a IN z} \<turnstile> a IN x OR a IN y"
parts_insert_knows_A: fixes X :: "msg"   and A :: "agent"   and evs :: "event list" shows "parts (insert X (knows A evs)) = parts {X} \<union> parts (knows A evs)"
nrows_eq_card_rows: fixes A :: "(('d, 'e) vec, 'f) vec" shows "nrows_iarray (matrix_to_iarray A) = CARD('f)"
le_mult_floor: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "\<lfloor>a\<rfloor> * \<lfloor>b\<rfloor> \<le> \<lfloor>a * b\<rfloor>"
productF_path_inclusion: fixes w :: "('d \<times> 'e) list"   and r1 :: "'f list"   and r2 :: "'g list"   and A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and p1 :: "'f"   and p2 :: "'g" assumes "length w = length r1"   and "length r1 = length r2"   and "productF A B FAIL AB"   and "well_formed A"   and "well_formed B"   and "path A (w || r1) p1 \<and> path B (w || r2) p2"   and "p1 \<in> nodes A"   and "p2 \<in> nodes B" shows "path AB (w || r1 || r2) (p1, p2)"
count_inverse: fixes x :: "'k multiset" shows "Abs_multiset (count x) = x"
rPar1: fixes \<Psi>\<^sub>Q :: "'b"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* C" shows "Prop (P' \<parallel> Q)"
length_filter_less: fixes x :: "'d"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool" assumes "x \<in> set xs"   and "\<not> P x" shows "length (filter P xs) < length xs"
consistent_swapFourSyms: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "consistent t" shows "consistent (swapFourSyms t a b c d)"
compl_inf: fixes x :: "'a"   and y :: "'a" shows "- inf_class.inf x y = sup_class.sup (- x) (- y)"
CLS_REP: fixes F :: "('a, 'a set) Maps.arr" assumes "Maps.arr F" shows "\<lbrakk>\<lbrakk>Maps.REP F\<rbrakk>\<rbrakk> = F"
liouville_lambda_def: fixes n :: "nat" shows "liouville_lambda n = (if n = 0 then 0::'a else (- (1::'a)) ^ size (prime_factorization n))"
const_name_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.const_name unconst (const name) = name"
quotient_of_denom_pos': fixes r :: "rat" shows "0 < snd (quotient_of r)"
new_tv_typ_le: fixes n :: "nat"   and m :: "nat"   and t :: "typ" assumes "n \<le> m"   and "new_tv n t" shows "new_tv m t"
assms: shows "Var x \<lhd> \<sigma> \<noteq> Var x \<lhd> \<eta>"
akra_bazzi_params_def: fixes k :: "nat"   and as :: "real list"   and bs :: "real list" shows "akra_bazzi_params k as bs \<equiv> (length as = k \<and> length bs = k) \<and> k \<noteq> 0 \<and> (\<forall>a. a \<in> set as \<longrightarrow> 0 \<le> a) \<and> (\<forall>b. b \<in> set bs \<longrightarrow> b \<in> {0<..<1})"
CH_domain: fixes A :: "'a set"   and h :: "'c" shows "CH A \<inter> {x. Xh x = h} = Ch h (A \<inter> {x. Xh x = h})"
distinct_disc_2: fixes process :: "'a process" assumes "isChoice process" shows "\<not> isAction process"
l3: fixes x :: "'a"   and y :: "'a" assumes "- x \<le> - y" shows "- x \<sqinter> y = bot"
sum_divisors_moebius_mu_times_multiplicative_revisited: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and N :: "nat" assumes "multiplicative_function f"   and "0 < n"   and "0 < N" shows "(\<Sum>d | d dvd n \<and> coprime N d. moebius_mu d * f d) = (\<Prod>p\<in>{p. p \<in># prime_factorization n \<and> \<not> p dvd N}. (1::'a) - f p)"
assoc: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" shows "a \<inter># b \<inter># c = a \<inter># (b \<inter># c)"
ccspan_singleton_scaleC: fixes a :: "complex"   and \<psi> :: "'b" assumes "a \<noteq> 0" shows "ccspan {a *\<^sub>C \<psi>} = ccspan {\<psi>}"
that: assumes "nop_onto A (1\<^sub>\<nat>) f" shows "thesis"
extended_Gromov_product_add_ge: fixes x :: "'b"   and y :: "'b"   and xi :: "'b Gromov_completion" shows "ereal (dist x y - deltaG TYPE('b)) \<le> extended_Gromov_product_at x xi (to_Gromov_completion y) + extended_Gromov_product_at y xi (to_Gromov_completion x)"
i_in2: shows "\<iota> \<in> Inf_from (active_subset (Liminf_llist (lmap snd Ns)))"
unique_keys_set_def: fixes S :: "('b \<times> 'c) set" shows "unique_keys_set S = (\<forall>i x y. (i, x) \<in> S \<longrightarrow> (i, y) \<in> S \<longrightarrow> x = y)"
ran_upd_mem: fixes v :: "'a"   and m :: "'b \<Rightarrow> 'a option"   and k :: "'b" shows "v \<in> ran (m(k \<mapsto> v))"
disj_zero_right: fixes conj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and disj :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and compl :: "'b \<Rightarrow> 'b"   and zero :: "'b"   and one :: "'b"   and x :: "'b" assumes "boolean_algebra conj disj compl zero one" shows "disj x zero = x"
elims: fixes x :: "'b alt"   and y :: "'b" assumes "tree x = y" obtains t :: "'b" where "x = Same ta__" and "y = ta__"   | t :: "'b" where "x = Diff t" and "y = t"
x: shows "swaps (drop (length (paid_A ! n) - m) (paid_A ! n)) (s_A n) = s'_A n m"
assms_4: shows "g \<in> A \<rightarrow> A"
hyps: fixes p :: "real poly" assumes "degree p < degree pa__"   and "p \<noteq> 0" shows "sgnx (poly p) (at_left a) = (if even (order a p) then sgnx (poly p) (at_right a) else - sgnx (poly p) (at_right a))"
tranclp_into_rtranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "r\<^sup>+\<^sup>+ a b" shows "r\<^sup>*\<^sup>* a b"
tNDComm: fixes a :: "'a set set"   and b :: "'b set set"   and c :: "'a set set"   and d :: "'b set set" assumes "twoNetsDistinct a b c d" shows "twoNetsDistinct c d a b"
atlasE: fixes x :: "'a" assumes "x \<in> carrier" obtains c :: "('a, 'b) chart" where "c \<in> atlas" and "x \<in> domain c"
eventually_frequently_const_simps_4: fixes C :: "bool"   and P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "(\<forall>\<^sub>F x in F. C \<or> P x) = (C \<or> eventually P F)"
left_diff_distrib: fixes x :: "(real, 3) Finite_Cartesian_Product.vec"   and y :: "(real, 3) Finite_Cartesian_Product.vec"   and z :: "(real, 3) Finite_Cartesian_Product.vec" shows "cross3 (x - y) z = cross3 x z - cross3 y z"
assms_2: shows "fmdom' (snd a) \<noteq> {}"
UP_integral: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "UP_domain R"   and "p \<otimes>\<^bsub>UP R\<^esub> q = \<zero>\<^bsub>UP R\<^esub>"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "p = \<zero>\<^bsub>UP R\<^esub> \<or> q = \<zero>\<^bsub>UP R\<^esub>"
ces_append_one: fixes v1 :: "'c \<times> nat"   and es :: "('c \<times> nat) edge list"   and e :: "('c \<times> nat) edge"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.ces v1 (es @ [e]) v2 subs = (SubRel.ces v1 es (src e) subs \<and> SubRel.ces (src e) [e] v2 subs)"
coinduct: fixes X :: "'a list set \<Rightarrow> 'a list set \<Rightarrow> bool"   and x1 :: "'a list set"   and x2 :: "'a list set" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>K L. x1 = K \<and> x2 = L \<and> ([] \<in> K) = ([] \<in> L) \<and> (\<forall>x. X (Deriv x K) (Deriv x L) \<or> bisimilar (Deriv x K) (Deriv x L))" shows "bisimilar x1 x2"
is_module: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme"   and W :: "'i set" assumes "vectorspace K V"   and "subspace K W V" shows "LinearCombinations.submodule K W V"
real_distribution_distr: fixes X :: "real \<Rightarrow> real" assumes "\<mu>.random_variable borel X" shows "real_distribution (distr \<mu> borel X)"
inverse_permutation_of_list_unique: fixes xs :: "('b \<times> 'b) list"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "list_permutes xs A"   and "(x, y) \<in> set xs" shows "inverse_permutation_of_list xs y = x"
unit_simps_3: fixes a :: "'a" assumes "C.obj a" shows "trg\<^sub>D (G.unit a) = G.map\<^sub>0 a"
nstut_imp: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "NSTUTINV F"   and "NSTUTINV G" shows "NSTUTINV (F \<longrightarrow> G)"
wf_iff_acyclic_if_finite: fixes r :: "('g \<times> 'g) set" assumes "finite r" shows "wf r = acyclic r"
of_int_round_abs_le: fixes x :: "'a" shows "\<bar>of_int (round x) - x\<bar> \<le> (1::'a) / (2::'a)"
intro: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
jkbpC_traces_of_length: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview" shows "{t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview. tLength t = n} = PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n"
DERIV_fun_powr: fixes g :: "real \<Rightarrow> real"   and m :: "real"   and x :: "real"   and r :: "real" assumes "(g has_real_derivative m) (at x)"   and "0 < g x" shows "((\<lambda>x. g x powr r) has_real_derivative r * g x powr (r - real 1) * m) (at x)"
a: shows "\<theta>_n,\<theta>_c<M> = Ax x a"
of_nat_eq_0_iff: fixes m :: "nat" shows "(of_nat m = (0::'b)) = (m = 0)"
Ln_series: fixes z :: "complex" assumes "cmod z < 1" shows "(\<lambda>n. (- 1) ^ Suc n / of_nat n * z ^ n) sums Ln (1 + z)"
dom_inv: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C"   and "category.iso C f" shows "partial_magma.dom C (category.inv C f) = partial_magma.cod C f"
elims_3: fixes h :: "nat"   and n_ops :: "nat"   and x :: "sat_plan_variable" assumes "\<not> cnf_to_dimacs.valid_state_var h n_ops x" obtains t :: "nat" and k :: "nat" where "x = Operator t__ k__" and "\<not> (t__ < h \<and> k__ < n_ops)"   | t :: "nat" and k :: "nat" where "x = State t k" and "\<not> t < h"
cfun_map_map: fixes f1 :: "'b \<rightarrow> 'd"   and g1 :: "'e \<rightarrow> 'c"   and f2 :: "'d \<rightarrow> 'f"   and g2 :: "'g \<rightarrow> 'e"   and p :: "'f \<rightarrow> 'g" shows "cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) = cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p"
clear_locals_def: fixes \<xi> :: "state" shows "clear_locals \<xi> = \<xi>\<lparr>msg := SOME x. True, data := SOME x. True, dests := SOME x. True, dip := SOME x. True, oip := SOME x. True, hops := SOME x. True, dsn := SOME x. True, dsk := SOME x. True, osn := SOME x. True, sip := SOME x. x \<noteq> ip \<xi>, handled := SOME x. True\<rparr>"
suffix_state_prop: fixes P :: "'b \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "\<lceil>P\<rceil> (\<sigma> |\<^sub>s i) = P (\<sigma> i)"
wf_type_iff_typ_ok_sig: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and T :: "typ" shows "wf_type \<Sigma> T = typ_ok_sig \<Sigma> T"
X_def: shows "X = codomain c \<inter> inv_chart c -` (src.carrier \<inter> f -` UNIV)"
rewrite_B2_to_cases: fixes s :: "'a option"   and t :: "'a option"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "B2 s t f = (case s of None \<Rightarrow> True | Some s1 \<Rightarrow> case t of None \<Rightarrow> True | Some t1 \<Rightarrow> f s1 t1)"
p_empty: shows "P_of_state (Liminf_state Sts) = {}"
det_addrow_mat: fixes k :: "nat"   and l :: "nat"   and n :: "nat"   and a :: "'b" assumes "k \<noteq> l" shows "Determinant.det (addrow_mat n a k l) = (1::'b)"
circline_type_zero_card_eq1: fixes H :: "circline" assumes "circline_type H = 0" shows "\<exists>z. circline_set H = {z}"
Suc_le_mono: fixes n :: "nat"   and m :: "nat" shows "(Suc n \<le> Suc m) = (n \<le> m)"
E_sup_SUP: fixes f :: "nat \<Rightarrow> 's stream \<Rightarrow> ennreal"   and s :: "'s" assumes "\<And>i. f i \<in> borel_measurable St"   and "incseq f" shows "E_sup s (\<lambda>x. \<Squnion>i. f i x) = (\<Squnion>i. E_sup s (f i))"
return_pmf_inj: fixes x :: "'a"   and y :: "'a" shows "(return_pmf x = return_pmf y) = (x = y)"
TESL_interp_stepwise_tagrel_coind_unfold: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'b tag_const \<times> 'b tag_const \<Rightarrow> bool"   and n :: "nat" shows "\<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
insert_Diff: fixes a :: "'c"   and A :: "'c set" assumes "a \<in> A" shows "insert a (A - {a}) = A"
sp_fundefs_compile_env: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'fun \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and F1 :: "'a"   and F2 :: "'f" assumes "compile_env \<A> \<O> F1 = Ok F2"   and "\<And>x. rel_option (\<lambda>ar fd. arity fd = ar) (\<A> x) (Finca_get F1 x)" shows "sp_fundefs (Fubx_get F2)"
unit_in_vhom: fixes a :: "'a" assumes "B.obj a" shows "\<guillemotleft>\<i>\<^sub>B[a] : a \<star>\<^sub>B a \<Rightarrow>\<^sub>B a\<guillemotright>"
prime_mod_4_cases: fixes p :: "nat" assumes "prime p" shows "p = 2 \<or> [p = 1] (mod 4) \<or> [p = 3] (mod 4)"
inj_on_proj: shows "inj_on proj (S - rel_interior S)"
leftmost_unique: fixes i :: "nat"   and a :: "symbol list"   and j :: "nat" assumes "leftmost i a"   and "leftmost j a" shows "i = j"
map_sel: fixes f1 :: "'u \<Rightarrow> 'r"   and f2 :: "'v \<Rightarrow> 's"   and a :: "('u, 'v, 't) gpv" shows "the_gpv (map_gpv f1 f2 a) = map_spmf (map_generat f1 f2 ((\<circ>) (map_gpv f1 f2))) (the_gpv a)"
Cauchy_integral_circlepath_simple: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real"   and w :: "complex" assumes "f holomorphic_on cball z r"   and "cmod (w - z) < r" shows "((\<lambda>u. f u / (u - w)) has_contour_integral 2 * complex_of_real pi * \<i> * f w) (circlepath z r)"
hyps_2: assumes "getCn state' \<noteq> 1"   and "InvariantUniq (getM (applyExplain (getCl state') state'))"   and "InvariantCFalse (getConflictFlag (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state'))"   and "InvariantCEntailed (getConflictFlag (applyExplain (getCl state') state')) F0 (getC (applyExplain (getCl state') state'))"   and "InvariantClCharacterization (getCl (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantCnCharacterization (getCn (applyExplain (getCl state') state')) (getC (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantClCurrentLevel (getCl (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state'))"   and "InvariantGetReasonIsReason (getReason (applyExplain (getCl state') state')) (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) (set (getQ (applyExplain (getCl state') state')))"   and "InvariantEquivalentZL (getF (applyExplain (getCl state') state')) (getM (applyExplain (getCl state') state')) F0"   and "getConflictFlag (applyExplain (getCl state') state')"   and "0 < currentLevel (getM (applyExplain (getCl state') state'))" shows "let state' = applyExplainUIP (applyExplain (getCl state') state') in isUIP (opposite (getCl state')) (getC state') (getM state')"
abs_eq: fixes xa :: "real_alg_3"   and x :: "real_alg_3" shows "Real_Alg_Quotient xa * Real_Alg_Quotient x = Real_Alg_Quotient (mult_3 xa x)"
bind_assoc: fixes x :: "'m"   and f :: "'a fset \<Rightarrow> 'm"   and g :: "'a fset \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
id_univalent: fixes x :: "'a set" shows "univalent (Id_on x)"
prems_2: fixes i :: "nat" assumes "i < length (G # Gs)" shows "gs ! i \<in> carrier ((G # Gs) ! i)"
minimum_spanning_forest_def: fixes F :: "('c, 'd) graph"   and G :: "('c, 'd) graph" shows "minimum_spanning_forest F G \<equiv> spanning_forest F G \<and> optimal_forest F G"