hcju/isabelleps huggingface dataset:
invariant_run: fixes x :: "'a"   and xs :: "'a stream" assumes "run (x ## xs)"   and "Q x" shows "pred_stream Q (x ## xs)"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "totient n = (\<Prod>p\<in>prime_factors n. totient (p ^ multiplicity p n))"
new_pre_def: shows "new_pre \<equiv> butlast P_new"
card_of_ordIsoI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "bij_betw f A B" shows "|A| =o |B|"
ParStrict_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B ParStrict C D \<equiv> Coplanar A B C D \<and> (\<nexists>X. Col X A B \<and> Col X C D)"
image_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and r :: "'c set" shows "f ` g ` r = (f \<circ> g) ` r"
sgn_1_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(sgn a = uminus one) = less a zero"
M_def: shows "M = max C (Max {u l |l. l < i})"
sumeq: shows "(\<Sum>i\<in>p. \<bar>content i\<bar>) = sum content p"
lunit_in_hom_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.arr V (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg f) \<and> src f \<in> weak_composition.sources V H (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg f) \<and> trg f \<in> weak_composition.targets V H (prebicategory_with_homs_and_units.lunit V H \<a> \<i> trg f)"
Min_le: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq (linorder.Min less_eq A) x"
fold_congs_7: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme"   and v' :: "Literal \<Rightarrow> nat option"   and f :: "(Literal \<Rightarrow> nat option) \<Rightarrow> Literal \<Rightarrow> nat option"   and f' :: "(Literal \<Rightarrow> nat option) \<Rightarrow> Literal \<Rightarrow> nat option" assumes "r = r'"   and "getReason r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "getReason_update f r = getReason_update f' r'"
butlast_eval2: fixes x :: "'a"   and y :: "'a" shows "butlast [x, y] = [x]"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'c"   and f2 :: "'e \<Rightarrow> 'c"   and sum :: "'d + 'e" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = ((\<forall>x1. sum = Inl x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. sum = Inr x2 \<longrightarrow> P (f2 x2)))"
closed_glob_consistent_def: fixes \<R> :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "conc.closed_glob_consistent \<R> = (\<forall>c\<^sub>1 mds mem\<^sub>1 c\<^sub>2 mem\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle>\<^sub>C, \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>A. (\<forall>x. case A x of None \<Rightarrow> True | Some (v, v') \<Rightarrow> mem\<^sub>1 x \<noteq> v \<or> mem\<^sub>2 x \<noteq> v' \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma\<^sub>C mem\<^sub>1 [\<parallel>\<^sub>1 A] x \<noteq> dma\<^sub>C mem\<^sub>1 x \<longrightarrow> \<not> var_asm_not_written mds x) \<and> (\<forall>x. dma\<^sub>C mem\<^sub>1 [\<parallel>\<^sub>1 A] x = Low \<and> (x \<notin> mds AsmNoReadOrWrite \<or> x \<in> \<C>\<^sub>C) \<longrightarrow> mem\<^sub>1 [\<parallel>\<^sub>1 A] x = mem\<^sub>2 [\<parallel>\<^sub>2 A] x) \<longrightarrow> (\<langle>c\<^sub>1, mds, mem\<^sub>1 [\<parallel>\<^sub>1 A]\<rangle>\<^sub>C, \<langle>c\<^sub>2, mds, mem\<^sub>2 [\<parallel>\<^sub>2 A]\<rangle>\<^sub>C) \<in> \<R>))"
finite_insert: fixes a :: "'b"   and A :: "'b set" shows "finite (a \<triangleright> A) = finite A"
Range_Int_subset: fixes A :: "('b \<times> 'a) set"   and B :: "('b \<times> 'a) set" shows "Range (A \<inter> B) \<subseteq> Range A \<inter> Range B"
the_apartment_iso_comp: fixes A :: "'a set set"   and A' :: "'a set set"   and A'' :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "A' \<in> \<A>"   and "A'' \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> A' \<inter> A''" shows "the_apartment_iso A A'' = restrict1 (the_apartment_iso A' A'' \<circ> the_apartment_iso A A') (\<Union> A)"
ImpL_sim: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and G :: "'b formula" assumes "\<^bold>\<not> F \<triangleright> \<Gamma> \<turnstile> \<bottom>"   and "G \<triangleright> \<Gamma> \<turnstile> \<bottom>" shows "F \<^bold>\<rightarrow> G \<triangleright> \<Gamma> \<turnstile> \<bottom>"
d1: shows "v2 = Matrix.mat 8 1 (\<lambda>(i, j). [[0, 0, \<phi> $$ (1, 0), \<phi> $$ (0, 0), 0, 0, 0, 0]] ! j ! i)"
starlike_convex_subset: fixes a :: "'a"   and S :: "'a set"   and b :: "'a"   and c :: "'a" assumes "a \<in> S"   and "closed_segment b c \<subseteq> S"   and "\<And>x. x \<in> S \<Longrightarrow> closed_segment a x \<subseteq> S" shows "convex hull {a, b, c} \<subseteq> S"
basis_exists: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set" assumes "vector_space scale" obtains B :: "'b set" where "B \<subseteq> V" and "\<not> module.dependent scale B" and "V \<subseteq> module.span scale B" and "card B = vector_space.dim scale V"
snd_d_in_\<RR>: shows "snd d \<in> \<RR>"
assms: shows "\<forall>i j. f (i, j) = bot"
cos_boundaries: fixes x :: "float"   and prec :: "nat"   and n :: "nat" assumes "0 \<le> real_of_float x"   and "real_of_float x \<le> pi / 2" shows "cos (real_of_float x) \<in> {real_of_float (lb_sin_cos_aux prec (get_even n) 1 1 (x * x))..real_of_float (ub_sin_cos_aux prec (get_odd n) 1 1 (x * x))}"
sparsegrid'_def: fixes dm :: "nat" shows "sparsegrid' dm = grid (start dm) {0..<dm}"
simple_extension_as_eval_img: fixes K :: "'a set"   and x :: "'a" assumes "K \<subseteq> carrier R"   and "x \<in> carrier R" shows "simple_extension K x = (\<lambda>p. local.eval p x) ` carrier (K [X])"
prems: shows "\<langle>c'',s''\<rangle> \<rightarrow>* \<langle>Skip,s\<^sub>2\<rangle>"
dim_not_zero: shows "0 < vec.dim (solution_set A 0)"
scale_one_on: fixes x :: "'b" assumes "x \<in> U\<^sub>M" shows "(1::'a) *s x = x"
d_preserves_equation: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(d y \<cdot> x \<le> x \<cdot> d z) = (d y \<cdot> x = d y \<cdot> x \<cdot> d z)"
mem_lclss_of_state_2: fixes C :: "'a literal multiset"   and N :: "'a literal multiset set"   and P :: "'a literal multiset set"   and Q :: "'a literal multiset set" shows "((C, Processed) \<in> lclss_of_state (N, P, Q)) = (C \<in> P)"
iT_not_empty_3: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<noteq> {}"
scale_cancel_right: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'d"   and x :: "'e"   and b :: "'d" assumes "vector_space scale" shows "(scale a x = scale b x) = (a = b \<or> x = (0::'e))"
tab_simps_5: shows "cod \<rho> = r \<star> f"
initial: fixes p :: "'a"   and A :: "('b, 'c, 'a, 'd) FSM_scheme" assumes "p = initial A" shows "p \<in> nodes A"
assms_5: shows "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"
lunstream_sels_2: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lhd (lunstream g s) = (case g s of Skip s' \<Rightarrow> lhd (lunstream g s') | Yield x xa \<Rightarrow> x)"
frameImpAlphaEnt: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'d"   and A\<^sub>F' :: "name list"   and \<Psi>\<^sub>F' :: "'d"   and \<phi> :: "'e" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle> = \<langle>A\<^sub>F', \<Psi>\<^sub>F'\<rangle>"   and "A\<^sub>F \<sharp>* \<phi>"   and "A\<^sub>F' \<sharp>* \<phi>"   and "SImp \<Psi>\<^sub>F' \<phi>" shows "SImp \<Psi>\<^sub>F \<phi>"
wf_set_bits_int_simps: fixes f :: "nat \<Rightarrow> bool" shows "wf_set_bits_int f = (\<exists>n. (\<forall>n'\<ge>n. \<not> f n') \<or> (\<forall>n'\<ge>n. f n'))"
horner_sum_simps_2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b"   and x :: "'c"   and xs :: "'c list" assumes "class.comm_semiring_0 plus zero times" shows "comm_semiring_0.horner_sum plus zero times f a (x # xs) = plus (f x) (times a (comm_semiring_0.horner_sum plus zero times f a xs))"
abs_eq: fixes x :: "int" shows "uint16_of_int x = Abs_uint16 (word_of_int x)"
iFROM_inter': fixes n :: "nat"   and n' :: "nat" shows "[n\<dots>] \<inter> [n'\<dots>] = [max n n'\<dots>]"
lemma_3_10_ii_iii: fixes H :: "'a set" assumes "H \<in> filters"   and "\<forall>a. {a} ** H = H ** {a}" shows "cong_r H = cong_l H"
rel_sel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d option"   and b :: "'e option" shows "rel_option R a b = ((a = None) = (b = None) \<and> (a \<noteq> None \<longrightarrow> b \<noteq> None \<longrightarrow> R (the a) (the b)))"
p'_def: fixes t :: "real" shows "p' t = v + a * t"
Sbis_def: shows "Sbis \<equiv> gfp Sretr"
Int_Diff: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<inter> B - C = A \<inter> (B - C)"
least_eq1: shows "(LEAST n. B $ a $ n \<noteq> (0::'a)) = (LEAST n. A $ a $ n \<noteq> (0::'a))"
valid_equiv: fixes \<phi> :: "fml"   and \<psi> :: "fml" shows "valid (\<phi> \<leftrightarrow> \<psi>) = (\<forall>I. fml_sem I \<phi> = fml_sem I \<psi>)"
valid_bind_cong: fixes f :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and g :: "'e \<Rightarrow> ('d \<times> 'e) option"   and M :: "'d \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "f \<sigma> = g \<sigma>" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E f M) = (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E g M)"
filter_raw_def: fixes P :: "'e \<Rightarrow> bool"   and g :: "'f \<Rightarrow> ('e, 'f) step"   and s :: "'f" shows "filter_raw P g s = (case g s of Done \<Rightarrow> Done | Skip s' \<Rightarrow> Skip s' | Yield a s' \<Rightarrow> if P a then Yield a s' else Skip s')"
iso_hcomp: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "B.iso \<mu>"   and "B.iso \<nu>"   and "src \<nu> = trg \<mu>" shows "B.iso (\<nu> \<star> \<mu>)"
vec_real_inner_def: fixes inner_field :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inner :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a" shows "vec_real_inner inner_field inner \<equiv> inner_product_space (*) inner_field \<and> vec_real_inner_axioms inner_field inner"
axioms_2: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah" assumes "inverse_transformation A B F G \<tau>" shows "category B"
rel_interior_translation: fixes a :: "'a"   and S :: "'a set" shows "rel_interior ((+) a ` S) = (+) a ` rel_interior S"
minor: fixes t :: "'a set" assumes "t open\<^bsub>T\<^esub>"   and "s = t \<inter> carrier" shows "R"
satisfies_eqI: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and LESS :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and assigns :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'd"   and nvars :: "'c \<Rightarrow> 'b"   and Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and CONS :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and SNOC :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and Length :: "'c \<Rightarrow> nat"   and extend :: "'a \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'e"   and size :: "'e \<Rightarrow> 'b"   and zero :: "'b \<Rightarrow> 'e"   and alphabet :: "'b \<Rightarrow> 'e list"   and eval :: "'d \<Rightarrow> nat \<Rightarrow> bool"   and downshift :: "'d \<Rightarrow> 'd"   and upshift :: "'d \<Rightarrow> 'd"   and add :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and cut :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and len :: "'d \<Rightarrow> nat"   and restrict :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and Restrict :: "'a \<Rightarrow> nat \<Rightarrow> ('f, 'a) aformula"   and lformula0 :: "'f \<Rightarrow> bool"   and FV0 :: "'a \<Rightarrow> 'f \<Rightarrow> nat set"   and find0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> bool"   and wf0 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and decr0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> 'f"   and satisfies0 :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and nullable0 :: "'f \<Rightarrow> bool"   and lderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and rderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and \<AA> :: "'c"   and \<phi> :: "('f, 'a) aformula"   and \<BB> :: "'c" assumes "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 satisfies0 nullable0 lderiv0 rderiv0"   and "Formula_Operations.wf SUC wf0 (nvars \<AA>) \<phi>"   and "nvars \<AA> = nvars \<BB>"   and "\<And>m k. LESS k m (nvars \<AA>) \<Longrightarrow> assigns m \<AA> k = assigns m \<BB> k"   and "Formula_Operations.lformula lformula0 \<phi>" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ _ _. True) \<AA> \<phi> = Formula_Operations.satisfies_gen Extend Length satisfies0 (\<lambda>_ _ _. True) \<BB> \<phi>"
head_in_verts: fixes e :: "'b" assumes "e \<in> pre_digraph.arcs G" shows "head G e \<in> pre_digraph.verts G"
cont_plus2: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.plus_cpo below" shows "cont ((+) x)"
assms_1: shows "\<forall>l\<in>labels \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V p. P l p"
\<tau>move2Sync7: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and V :: "nat"   and o' :: "'addr expr1"   and e :: "'addr expr1" shows "\<tau>move2 P h stk (sync\<^bsub>V\<^esub> (o') e) (6 + length (compE2 o') + length (compE2 e)) None"
unit_factor_is_unit: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "a \<noteq> zero" shows "dvd.dvd times (unit_factor a) one"
eq: shows "add_point_to_blocks p bs rep p = image_mset (insert p) {#b \<in># \<B>. b \<in> bs#} rep p"
Model_def: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> nat"   and parOf :: "'b \<Rightarrow> nat"   and \<Phi> :: "('a, 'b) lit list set"   and D :: "'c \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> 'c list \<Rightarrow> 'c"   and intP :: "'b \<Rightarrow> 'c list \<Rightarrow> bool" shows "U.Model wtFsym wtPsym arOf parOf \<Phi> D intF intP \<equiv> U.Problem wtFsym wtPsym arOf parOf \<Phi> \<and> U.Struct wtFsym wtPsym arOf parOf D intF intP \<and> U.Model_axioms \<Phi> D intF intP"
fminusI: fixes c :: "'a"   and A :: "'a fset"   and B :: "'a fset" assumes "c |\<in>| A"   and "c |\<notin>| B" shows "c |\<in>| A |-| B"
sel_1: fixes z :: "complex" shows "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
lm122: fixes P :: "'b \<Rightarrow> bool"   and l :: "'b list" shows "set (concat (map (\<lambda>n. if P (l ! n) then [n] else []) [0..<length l])) \<subseteq> {0..<length l}"
Ap_contractE: fixes p :: "comb"   and q :: "comb"   and r :: "comb" assumes "p \<bullet> q \<rightarrow>\<^sup>1 r" obtains "p = K \<bullet> r"   | x :: "comb" and y :: "comb" where "p = S \<bullet> x \<bullet> y__" and "r = x \<bullet> q \<bullet> (y__ \<bullet> q)"   | y :: "comb" where "r = ya__ \<bullet> q" and "p \<rightarrow>\<^sup>1 ya__"   | y :: "comb" where "r = p \<bullet> y" and "q \<rightarrow>\<^sup>1 y"
2: shows "[k] \<langle>(k # ks) \<langle>f\<rangle> (l # ls) \<odot> (l # ls) \<langle>g\<rangle> (m # ms)\<rangle> [m] = [k] \<langle>f\<rangle> [l] \<odot> [l] \<langle>g\<rangle> [m] \<oplus> [k] \<langle>f\<rangle> ls \<odot> ls \<langle>g\<rangle> [m]"
maxsimpchain_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "domain.gallery (x # xs)" shows "domain.gallery xs"
inject_1: fixes x11 :: "'b"   and x12 :: "bool"   and y11 :: "'b"   and y12 :: "bool" shows "(IsInterrupted x11 x12 = IsInterrupted y11 y12) = (x11 = y11 \<and> x12 = y12)"
gen_lossless_gpv_intros: fixes p :: "('a, 'out, ('a, 'out, 'in) rpv) generat spmf" assumes "lossless_spmf p"   and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf p; input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.gen_lossless_gpv (c input)" shows "local.gen_lossless_gpv (GPV p)"
simps: fixes \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" shows "closed_glob_helper \<Gamma>' \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle> = (\<forall>x. (dma x = High \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v\<^sub>1 v\<^sub>2. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v\<^sub>1)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v\<^sub>2)\<rangle>) \<in> \<R>\<^sub>3 \<Gamma>')) \<and> (dma x = Low \<and> x \<notin> mds AsmNoWrite \<longrightarrow> (\<forall>v. (\<langle>c\<^sub>1, mds, mem\<^sub>1(x := v)\<rangle>, \<langle>c\<^sub>2, mds, mem\<^sub>2(x := v)\<rangle>) \<in> \<R>\<^sub>3 \<Gamma>')))"
simps_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" shows "power.power one times a 0 = one"
fmfilter_cong': fixes m :: "('a, 'b) fmap"   and n :: "('a, 'b) fmap"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "m = n"   and "\<And>x. x \<in> fmdom' m \<Longrightarrow> P x = Q x" shows "fmfilter P m = fmfilter Q n"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "Min (insert x A) = (if A - {x} = {} then x else min x (Min (A - {x})))"
iT_card_5: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "card [ r, mod m, c ] = (if m = 0 then Suc 0 else Suc c)"
intro: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b" assumes "mem_correct lookup update P"   and "mem_correct_empty_axioms lookup P empty" shows "mem_correct_empty lookup update P empty"
sel_3: fixes x21 :: "('d, 'c) tree"   and x22 :: "'d"   and x23 :: "'c"   and x24 :: "('d, 'c) tree" shows "nval (Node x21 x22 x23 x24) = x23"
respects_own_guarantees_def: fixes cm :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" shows "respects_own_guarantees cm \<equiv> \<forall>x. (x \<in> snd cm GuarNoReadOrWrite \<longrightarrow> doesnt_read_or_modify (fst cm) x) \<and> (x \<in> snd cm GuarNoWrite \<longrightarrow> doesnt_modify (fst cm) x)"
assms_1: fixes x :: "'a" assumes "x \<in> A" shows "carrier (f x) \<subseteq> A"
wt_cong: fixes s :: "('s, 'v) tm"   and A :: "'v pvar \<Rightarrow> zhmultiset"   and B :: "'v pvar \<Rightarrow> zhmultiset" assumes "\<forall>x\<in>vars s. A (PWt x) = B (PWt x)"   and "\<forall>x\<in>vars s. \<forall>i. A (PCoef x i) = B (PCoef x i)" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = eval_tpoly B (map_tpoly (\<lambda>x. x) zhmset_of (wt s))"
pp_total_linorder_matrix_2: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "\<forall>i. \<exists>j. f (i, j) \<noteq> bot" shows "matrix_p_algebra.dense (f \<odot> mtop)"
lunit_simps_1: fixes f :: "'a" assumes "B.ide f" shows "B.arr (B.lunit f)"
intros: fixes s :: "s"   and \<S> :: "s_context"   and i :: "nat"   and vs :: "v list"   and es :: "e list"   and ts :: "t list" assumes "store_typing s \<S>"   and "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts" shows "\<turnstile>_ i s;vs;es : ts"
simps_1: fixes P :: "'b \<Rightarrow> bool" shows "filter P [] = []"
B: fixes ys' :: "'a list"   and zs' :: "'a list" shows "xs \<simeq> {ys', zs', P} \<longrightarrow> filter Q xs \<simeq> {filter Q ys', filter Q zs', P}"
simps_5: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm" shows "dnf_modified (Or \<phi>\<^sub>1 \<phi>\<^sub>2) = dnf_modified \<phi>\<^sub>1 @ dnf_modified \<phi>\<^sub>2"
P_def: shows "P = (\<lambda>f n g r. 0 < r \<and> g holomorphic_on cball z r \<and> g z \<noteq> 0 \<and> (\<forall>w\<in>cball z r - {z}. f w = g w * (w - z) powr of_int n \<and> g w \<noteq> 0))"
Pred\<^sub>\<alpha>_eqvt: fixes p :: "perm"   and \<phi> :: "'f" shows "p \<bullet> Pred\<^sub>\<alpha> \<phi> = Pred\<^sub>\<alpha> (p \<bullet> \<phi>)"
Sum4: fixes x :: "nat \<Rightarrow> 'a" shows "sum x {0..<4} = x 0 + x 1 + x 2 + x 3"
dim_col_mat_2: fixes nr :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'b Matrix.vec" shows "dim_col (mat\<^sub>r nr nc g) = nc"
pow_if: fixes n :: "nat"   and P :: "pol" shows "Commutative_Ring.pow n P = (if n = 0 then Pc 1 else if even n then Commutative_Ring.pow (n div 2) (sqr P) else P \<langle>*\<rangle> Commutative_Ring.pow (n div 2) (sqr P))"
simps: fixes fv1 :: "'c \<Rightarrow> atom set"   and fv2 :: "'d \<Rightarrow> atom set"   and x :: "'c"   and y :: "'d" shows "prod_fv fv1 fv2 (x, y) = fv1 x \<union> fv2 y"
comparator_bool_comparator_of: shows "comparator_bool = comparator_of"
divides_mult_lI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides\<^bsub>add_monoid R\<^esub> b"   and "a \<in> carrier R"   and "c \<in> carrier R" shows "(c \<oplus> a) divides\<^bsub>add_monoid R\<^esub> (c \<oplus> b)"
CastThrow\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e' :: "nat exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and C :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
pcr_Some_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c"   and y :: "'d option" shows "pcr_Some R x y = (\<exists>z. y = Some z \<and> R x z)"