hcju/isabelleps huggingface dataset:
zor: shows "(\<sigma>, \<zeta>) \<in> oreachable (opnet np p) (otherwith (=) (net_tree_ips p) inoclosed) U"
sd_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sd_prod x y = (fst x \<oplus> fst y, snd x \<oplus> \<alpha> (fst x) (snd y))"
assms_3: shows "\<CC>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
comp_commute_below_diversity: fixes x :: "'a"   and y :: "'a" shows "irreflexive (x * y) = irreflexive (y * x)"
counterexample_def: fixes r :: "'b rexp"   and s :: "'b rexp" shows "counterexample r s = (let r' = init r; s' = init s in case closure_invariant ([([], r', s')], [], {(post r', post s')}) of None \<Rightarrow> None | Some ([], xc) \<Rightarrow> None | Some ((w, x, xa) # xb, xc) \<Rightarrow> Some (rev w))"
part_witness_Compl: fixes A :: "part_witness set" shows "- A = {part_witness.a, part_witness.b, part_witness.c} - A"
natural_transformation_axioms: shows "natural_transformation J (\<cdot>) \<chi>f.A.map D (D.cones_map f \<chi>)"
comp_to_mult: fixes f :: "nat \<Rightarrow> 'a" assumes "completely_multiplicative_function f" shows "multiplicative_function f"
Well_order_pullback: fixes r :: "('c \<times> 'c) set"   and f :: "'d \<Rightarrow> 'c" assumes "Well_order r" shows "\<exists>s. Well_order s \<and> Field s = UNIV \<and> (\<forall>x y. (f x, f y) \<in> r - Id \<longrightarrow> (x, y) \<in> s)"
bounded_Union: fixes F :: "'a set set" assumes "finite F"   and "\<forall>S\<in>F. bounded S" shows "bounded (\<Union> F)"
countable_GE_wtPsym: shows "countable (Collect GE_wtPsym)"
3: assumes "is_Abs f" shows "the_Abs f \<in> A \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+"
comm_monoid_mult_hom_1_axioms: shows "comm_monoid_mult_hom_1 hom"
containsI: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "Predicate_Compile.contains A x"
B_trusts_NS5_lemma: fixes B :: "agent"   and evs :: "event list"   and K :: "nat"   and A :: "agent"   and NA :: "msg"   and NB :: "nat" assumes "B \<notin> bad"   and "evs \<in> ns_shared"   and "Key K \<notin> analz (knows Spy evs)"   and "Says Server A (Crypt (shrK A) \<lbrace>NA, Agent B, Key K, Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>\<rbrace>) \<in> set evs"   and "Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace> \<in> parts (knows Spy evs)" shows "Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) \<in> set evs"
\<delta>: shows "cball a \<delta> \<subseteq> S"
rw: fixes k :: "nat" assumes "k < length (diag_mat A)" shows "(\<Sum>ka = 0..<length (diag_mat A). P $$ (i, ka) * A $$ (ka, k)) = P $$ (i, k) * A $$ (k, k)"
splus_right_canc: fixes t :: "'a"   and v :: "'t"   and s :: "'a" shows "(t \<oplus> v = s \<oplus> v) = (t = s)"
prod_diff_prod: fixes x :: "'a"   and y :: "'b"   and a :: "'a"   and b :: "'b" shows "x ** y - a ** b = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
R_def: fixes f :: "nat \<Rightarrow> vote"   and V :: "nat set" shows "R f V = (V \<subseteq> {0..<a + b} \<and> f \<in> extensional {1..a + b} \<and> (\<forall>i<a + b. (i \<in> V) = (f (Suc i) = A)))"
A3: shows "\<lfloor>(\<lambda>w. \<forall>x xa. (\<forall>xa. (x xa \<^bold>\<rightarrow> \<P> xa) w) \<and> (\<^bold>\<box>(\<lambda>v. \<forall>xb. xa xb v = (\<forall>xa. (x xa \<^bold>\<rightarrow> xa xb) v))) w \<longrightarrow> \<P> xa w)\<rfloor>"
1: fixes x :: "'a" assumes "\<forall>i. \<exists>n\<ge>i. P n x" shows "Max {i. P i x} = the None"
PPA_def: fixes a :: "e" shows "PPA a \<equiv> mexistsBinder (ActsOnPurpose a)"
trg_src: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "trg\<^sub>C (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
simps_2: fixes x :: "name"   and G :: "'b frame"   and F :: "'b frame" assumes "x \<sharp> G" shows "(\<lparr>\<nu>x\<rparr>F) \<otimes>\<^sub>F G = \<lparr>\<nu>x\<rparr>F \<otimes>\<^sub>F G"
integrable_add: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x + g x)"
isoI: fixes f :: "'a"   and g :: "'a" assumes "A.inverse_arrows f g" shows "A.iso f"
poly_lift_0_iff: fixes p :: "'c poly" shows "(poly_lift p = 0) = (p = 0)"
intro: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"   and "safe_distance_no_collsion_delta_axioms a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_no_collsion_delta a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>"
ltakeWhile_LCons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b llist" shows "ltakeWhile P (LCons x xs) = (if P x then LCons x (ltakeWhile P xs) else LNil)"
adds_trans: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus a b"   and "plus.adds plus b c" shows "plus.adds plus a c"
measure_pmf_UNIV: fixes p :: "'a pmf" shows "measure_pmf.prob p UNIV = 1"
not_one_le_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
add: fixes b1 :: "'a"   and b2 :: "'a" shows "f M (b1 + b2) = f M b1 + f M b2"
UNION_disjoint: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "sum g (\<Union> (A ` I)) = (\<Sum>x\<in>I. sum g (A x))"
timpl_closure'_timpls_trancl_subset': fixes c :: "('g set \<times> 'g set) set" shows "timpl_closure' {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b} \<subseteq> timpl_closure' c"
le_trans_equiv_rsp: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans t v = le_trans u v"
sup2CI: fixes B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<not> B x y \<Longrightarrow> A x y" shows "sup A B x y"
assms_8: shows "Esigma (a * u) = a * u + a * x"
relator_freeword_Q: fixes s :: "'w"   and t :: "'w" assumes "s \<in> S"   and "t \<in> S" shows "pair_relator_freeword s t \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
at_ereal: fixes r :: "real" shows "at (ereal r) = filtermap ereal (at r)"
leq_refl: fixes x :: "'c" shows "x \<le> x"
add_ac_3: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b \<squnion> (a \<squnion> c) = a \<squnion> (b \<squnion> c)"
assms: shows "m \<in> carrier (residue_ring (p ^ k))"
retract_of_def: fixes T :: "'a set"   and S :: "'a set" shows "(T retract_of S) = (\<exists>r. retraction S T r)"
runit_tensor_gen: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "\<r>[a \<otimes> b] \<otimes> c = ((a \<otimes> \<r>[b]) \<otimes> c) \<cdot> (\<a>[a, b, \<I>] \<otimes> c)"
conf_NT: fixes P :: "'a prog"   and h :: "heap"   and v :: "val" shows "(P,h \<turnstile> v :\<le> NT) = (v = Null)"
mult: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and x :: "'b"   and a :: "real" assumes "linearform V f"   and "x \<in> V" shows "f (a \<cdot> x) = a * f x"
ListSum_conv_sum: fixes xs :: "'a list"   and f :: "'a \<Rightarrow> 'b" assumes "distinct xs" shows "ListSum xs f = sum f (set xs)"
integrable_subinterval_real: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" assumes "f integrable_on {a..b}"   and "{c..d} \<subseteq> {a..b}" shows "f integrable_on {c..d}"
zero_less_abs_iff: fixes a :: "'a" shows "((0::'a) < \<bar>a\<bar>) = (a \<noteq> (0::'a))"
simps: fixes xs :: "'b list"   and ar :: "nat"   and n :: "nat"   and x :: "'b" shows "rewrite_fundef_body (Fundef xs ar) n x = Fundef (rewrite xs n x) ar"
scaleR_vector_matrix_assoc: fixes k :: "real"   and x :: "(real, 'b) vec"   and A :: "((real, 'a) vec, 'b) vec" shows "k *\<^sub>R x v* A = k *\<^sub>R (x v* A)"
pcr_poly_def: fixes T :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "pcr_poly T \<equiv> rel_fun (=) T OO cr_poly"
term_ok'_contained_tvars_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 :: "Core.term"   and idn :: "variable"   and S :: "String.literal set" assumes "term_ok' \<Sigma> t"   and "(idn, S) \<in> tvs t" shows "typ_ok_sig \<Sigma> (Tv idn S)"
mapOf_def: fixes t :: "(int \<times> 'b) Tree"   and i :: "int" shows "mapOf t i \<equiv> case tlookup fst i t of None \<Rightarrow> None | Some ia \<Rightarrow> Some (snd ia)"
PiE_mem: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
tendsto_inverse_ereal_PInf: fixes u :: "'b \<Rightarrow> ereal"   and F :: "'b filter" assumes "(u \<longlongrightarrow> \<infinity>) F" shows "((\<lambda>x. 1 / u x) \<longlongrightarrow> 0) F"
ses_inputs_are_events_def: fixes SES :: "('c, 'd) SES_rec" shows "ses_inputs_are_events SES \<equiv> I\<^bsub>SES\<^esub> \<subseteq> E\<^bsub>SES\<^esub>"
continuous_on: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
rep_eq: fixes x :: "'a \<times> 'b"   and xa :: "('a, 'b) ffun" shows "(x \<in>\<^sub>f xa) = (x \<in>\<^sub>p pfun_of xa)"
tak_m1_def: shows "tak_m1 = (\<lambda>(x, y, z). if x \<le> y then 0::'c else (1::'c))"
full_quicksort_ref_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "full_quicksort_ref R h xs \<equiv> if List.null xs then RETURN xs else quicksort_ref R h (0, length xs - 1, xs)"
normal_distributed_expectation: fixes \<sigma> :: "real"   and X :: "'a \<Rightarrow> real"   and \<mu> :: "real" assumes "0 < \<sigma>"   and "distributed M lborel X (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x))" shows "expectation X = \<mu>"
max_in_subcomplex: fixes Y :: "'a set set"   and y :: "'a set" assumes "order.greater_eq X Y \<and> SimplicialComplex Y"   and "y \<in> Y"   and "chamber y" shows "SimplicialComplex.maxsimp Y y"
good_map_bij: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "good_map (\<lambda>(x, u). h x)"
DSourcesA82_L1: shows "DSources level1 sA82 = {sA72}"
seqSubstSimps_4: fixes C :: "('a, 'b, 'c) psiCase"   and \<sigma> :: "(name list \<times> 'a list) list" shows "Case C[<\<sigma>>] = Case seqSubs'' C \<sigma>"
bij_betw_inv: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set" assumes "bij_betw f A B" shows "\<exists>g. bij_betw g B A"
trivial_group_def: fixes G :: "('e, 'f) monoid_scheme" shows "trivial_group G \<equiv> Group.group G \<and> carrier G = {\<one>\<^bsub>G\<^esub>}"
set_integral_complex_of_real: fixes M :: "'a measure"   and A :: "'a set"   and f :: "'a \<Rightarrow> real" shows "CLINT x:A|M. complex_of_real (f x) = complex_of_real (set_lebesgue_integral M A f)"
has_laplace_vector_derivative: fixes f :: "real \<Rightarrow> complex"   and L :: "complex"   and s :: "complex"   and f0 :: "complex"   and M :: "real"   and c :: "real" assumes "(f has_laplace L) s"   and "\<And>t. 0 < t \<Longrightarrow> f differentiable at t"   and "(f \<longlongrightarrow> f0) (at_right 0)"   and "exponential_order M c f"   and "c < Re s" shows "((\<lambda>t. vector_derivative f (at t)) has_laplace s * L - f0) s"
that: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)"   and "S \<subseteq> {x. a \<bullet> x = b}" shows "thesis"
closeSubst_def: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and Rel :: "('e \<times> ('d, 'e, 'f) psi \<times> ('d, 'e, 'f) psi) set" assumes "substPsi substTerm substAssert substCond" shows "substPsi.closeSubst substTerm substAssert substCond Rel \<equiv> {(\<Psi>, P, Q) |\<Psi> P Q. \<forall>\<sigma>. wellFormedSubst \<sigma> \<longrightarrow> (\<Psi>, substPsi.seqSubs substTerm substAssert substCond P \<sigma>, substPsi.seqSubs substTerm substAssert substCond Q \<sigma>) \<in> Rel}"
to_contain_is_to_divide: fixes a :: "'a list"   and b :: "'a list" assumes "a \<in> carrier (K [X])"   and "b \<in> carrier (K [X])" shows "(PIdl\<^bsub>K [X]\<^esub> b \<subseteq> PIdl\<^bsub>K [X]\<^esub> a) = a divides\<^bsub>K [X]\<^esub> b"
simps_1: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and cs' :: "com \<times> (char list \<Rightarrow> int)" shows "cs \<rightarrow>(0) cs' = (cs' = cs)"
long_pow_ex: fixes r :: "'b list"   and m :: "nat" assumes "r \<noteq> \<epsilon>" obtains n :: "nat" where "m \<le> \<^bold>|r \<^sup>@ n\<^bold>|" and "n \<noteq> 0"
simps_2: fixes x :: "nat"   and xs :: "nat list" shows "max_list (x # xs) = max x (max_list xs)"
ample_abstract_axioms: shows "ample_abstract ex en init int ind src ren"
lookup_map_entry': fixes x :: "'c"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping"   and y :: "'c" shows "Mapping.lookup (Mapping.map_entry x f m) y = (if x = y then map_option f (Mapping.lookup m y) else Mapping.lookup m y)"
nonempty_imp_ex_extreme: fixes X :: "'a set" assumes "X \<subseteq> A"   and "X \<noteq> {}" shows "\<exists>e. extreme X (\<lambda>x y. y \<sqsubseteq> x) e"
map_to_set_cong: fixes m1 :: "'e \<Rightarrow> 'f option"   and m2 :: "'e \<Rightarrow> 'f option" shows "(map_to_set m1 = map_to_set m2) = (m1 = m2)"
outside_pack: shows "pid (declclass old) \<noteq> pid (declclass new)"
divide_int_def: fixes k :: "int"   and l :: "int" shows "k div l = (if l = 0 then 0 else if sgn k = sgn l then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>) else - int (nat \<bar>k\<bar> div nat \<bar>l\<bar> + of_bool (\<not> l dvd k)))"
simps: fixes get :: "'d \<Rightarrow> nat \<Rightarrow> 'e"   and sz :: "nat"   and i :: "nat"   and l :: "'d"   and c :: "'c \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and \<sigma> :: "'c" shows "idx_iteratei_aux get sz i l c f \<sigma> = (if i = 0 \<or> \<not> c \<sigma> then \<sigma> else idx_iteratei_aux get sz (i - 1) l c f (f (get l (sz - i)) \<sigma>))"
partition_on_map: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and P :: "'c set set" assumes "inj_on f A"   and "partition_on A P" shows "partition_on (f ` A) ((`) f ` P)"
ide_in_hom_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.in_hom f f f"
intro: fixes u\<^sub>0 :: "'b list"   and u\<^sub>1 :: "'b list" assumes "u\<^sub>0 \<cdot> u\<^sub>1 \<noteq> u\<^sub>1 \<cdot> u\<^sub>0" shows "binary_code u\<^sub>0 u\<^sub>1"
scaleR_le_0_iff: fixes a :: "real"   and b :: "'a" shows "(a *\<^sub>R b \<le> (0::'a)) = (0 < a \<and> b \<le> (0::'a) \<or> a < 0 \<and> (0::'a) \<le> b \<or> a = 0)"
pq: shows "p + q = monomial (lc p) (lt p) + monomial (tc q) (tt q) + (tail p + higher q (tt q))"
bigo_def: fixes F :: "'i filter"   and g :: "'i \<Rightarrow> 'j" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
nat_power_nonzero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and nat_power :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat"   and z :: "'a" assumes "class.nat_power_field one times plus zero nat_power divide inverse minus uminus"   and "0 < n" shows "nat_power n z \<noteq> zero"
subterms_subst'': fixes M :: "('e, 'd) Term.term set"   and \<theta> :: "'d \<Rightarrow> ('e, 'd) Term.term" assumes "\<forall>x\<in>fv\<^sub>s\<^sub>e\<^sub>t M. (\<exists>f. \<theta> x = Fun f []) \<or> (\<exists>y. \<theta> x = Var y)" shows "subterms\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) = subterms\<^sub>s\<^sub>e\<^sub>t M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>"
card_UNIV_num1_def: shows "card_UNIV_class.card_UNIV = Phantom(num1) 1"
simps_19: fixes C :: "char list"   and Cs :: "char list list"   and b :: "bool"   and e :: "'a exp" shows "max_vars (INIT C (Cs,b) \<leftarrow> e) = max_vars e"
flow_undefined_1: fixes t0 :: "real"   and x0 :: "'a" assumes "t0 \<notin> T" shows "flow t0 x0 = (\<lambda>_. 0::'a)"
simps_3: fixes nat :: "nat" shows "\<infinity> \<noteq> enat nat"
szip_unfold: fixes a :: "'d"   and s1 :: "'d stream"   and b :: "'e"   and s2 :: "'e stream" shows "a ## s1 ||| b ## s2 = (a, b) ## (s1 ||| s2)"
fmrel_to_subtuple_trancl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ym :: "'e \<rightharpoonup>\<^sub>f 'd"   and xm :: "'e \<rightharpoonup>\<^sub>f 'd" assumes "reflp r"   and "(fmrel r)\<^sup>+\<^sup>+ (fmrestrict_fset (fmdom ym) xm) ym" shows "(subtuple r)\<^sup>+\<^sup>+ xm ym"
Can_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Can (Inv t)"
intro: fixes r :: "('j, 'g, 'h, 'i, 'k, 'l) semantics"   and P :: "'g \<Rightarrow> 'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "\<And>t x m ta x' m'. \<lbrakk>r t (x, m) ta (x', m'); P t x m\<rbrakk> \<Longrightarrow> P t x' m'"   and "\<And>t x m ta x' m' t'' x''. \<lbrakk>r t (x, m) ta (x', m'); P t x m; NewThread t'' x'' m' \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>\<rbrakk> \<Longrightarrow> P t'' x'' m'"   and "\<And>t x m ta x' m' t'' x''. \<lbrakk>r t (x, m) ta (x', m'); P t x m; P t'' x'' m\<rbrakk> \<Longrightarrow> P t'' x'' m'" shows "lifting_wf_axioms r P"
w_in_hom_1: shows "\<guillemotleft>cmp.chine : src (tab\<^sub>0 s \<star> p\<^sub>0) \<rightarrow> src (tab\<^sub>0 (r \<star> s))\<guillemotright>"
Ker_Im_iff: shows "(Ker = V) = (ImG = 0)"
Par_Wbis_discrL1: fixes indis :: "('a \<times> 'a) set"   and c1 :: "('b, 'c) com"   and d :: "('b, 'c) com"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis"   and "(c1, d) \<in> PL_Indis.Wbis tval aval indis"   and "PL_Indis.discr tval aval indis c2" shows "(c1 | c2, d) \<in> PL_Indis.Wbis tval aval indis"
sgn_mult_self_eq: 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 "times (sgn a) (sgn a) = zero_neq_one.of_bool one zero (a \<noteq> zero)"
distinct_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" assumes "f permutes {..<length xs}" shows "distinct (permute_list f xs) = distinct xs"
char_Pg_eq: fixes g :: "game"   and h :: "game" shows "(Abs_Pg (eq_game_rel `` {g}) = Abs_Pg (eq_game_rel `` {h})) = eq_game g h"
accpI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>y. r y x \<Longrightarrow> Wellfounded.accp r y" shows "Wellfounded.accp r x"
resTy_mthd: fixes m :: "'b \<times> 'a" shows "resTy (mthd m) = resTy m"
lexordp_antisym: fixes xs :: "'a list"   and ys :: "'a list" assumes "lexordp xs ys"   and "lexordp ys xs" shows "False"
dom_cod: fixes f :: "'a" assumes "S.arr f" shows "S.dom (S.cod f) = S.cod f"
dyn_influence_Cons_source: fixes n :: "'node"   and V :: "'var"   and n' :: "'node"   and a :: "'edge"   and as :: "'edge list" assumes "n influences V in n' via a # as" shows "sourcenode a = n"
alphaAbs_qAbs_iff_ex_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
domA_clearjunk: fixes \<Gamma> :: "('d \<times> 'e) list" shows "domA (clearjunk \<Gamma>) = domA \<Gamma>"
sr_def: fixes R :: "('c, 'd) Ring_scheme"   and S :: "'c set" shows "sr R S \<equiv> S \<subseteq> carrier R \<and> 1\<^sub>r\<^bsub>R\<^esub> \<in> S \<and> (\<forall>x\<in>S. \<forall>y\<in>S. x \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> y \<in> S \<and> x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> S)"
ReflectLAt_def: fixes M :: "'p"   and P' :: "'p"   and P :: "'p"   and A :: "'p"   and B :: "'p" shows "M ReflectLAt P' P A B \<equiv> (M Midpoint P P' \<and> Col A B M) \<and> (A B Perp P P' \<or> P = P')"
TC: fixes a :: "V" shows "TC a = a \<union>\<^sub>\<circ> \<Squnion> (TC ` elts a)"
subs: shows "restrict v2 (res ts) c \<le> lan v2"
fold_graph_insertE: fixes z :: "'a"   and x :: "'a"   and B :: "'a set"   and v :: "'a" assumes "fold_graph f z (insert x B) v"   and "x \<notin> B"   and "insert x B \<subseteq> A"   and "z \<in> A" obtains y :: "'a" where "v = f x y" and "fold_graph f z B y"
eq_fin_le_fininf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 =\<^sub>F w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>F\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
group_hom_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" shows "group_hom G H h \<equiv> Group.group G \<and> Group.group H \<and> group_hom_axioms G H h"
key_map_is_map: fixes U :: "'a set set" assumes "\<pp> \<in> U" shows "Set_Theory.map (local.key_map U) (\<O> U) pi.carrier_local_ring_at"
gen_llength_def: fixes n :: "nat"   and xs :: "'b llist" shows "gen_llength n xs = enat n + llength xs"
triangle_left: shows "(\<xi> \<star> f) \<cdot> (f \<star> \<zeta>) = f"
assms: fixes i :: "'a"   and j :: "'a" assumes "{i, j} \<subseteq> I"   and "i \<noteq> j" shows "A i \<inter> A j = {}"
preserves_reflects_arr: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.arr (Span.trg f) = Span.arr f"
eq_zero: fixes a :: "nat" assumes "\<not> coprime a n" shows "\<chi> a = 0"
swap_Send_Snapshot: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
hd_cons_def: fixes s :: "'s" shows "local.hd_cons s = hd (unstream g s)"
ereal_minus_eq_PInfty_iff: fixes x :: "ereal"   and y :: "ereal" shows "(x - y = \<infinity>) = (y = - \<infinity> \<or> x = \<infinity>)"
half_open_segment_def: fixes a :: "'b"   and b :: "'b" shows "{a--<b} = {a--b} - {b}"
C_eq_RAD: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "p \<noteq> []" shows "Cp (list2FWpolicy p) = Cp (list2FWpolicy (removeAllDuplicates p))"
i_def: shows "i \<equiv> \<lambda>x. (1 - x) *\<^sub>R nth_derivative 2 f (X + x *\<^sub>R H) H"