hcju/isabelleps huggingface dataset:
intro: fixes v :: "char list \<Longrightarrow> 'a"   and x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a" assumes "vwb_lens v"   and "v \<bowtie> x"   and "x \<bowtie> v"   and "v \<bowtie> y"   and "y \<bowtie> v"   and "v \<bowtie> a"   and "a \<bowtie> v"   and "v \<bowtie> z"   and "z \<bowtie> v" shows "myss3_axioms x y a z v"
assms: shows "st \<in> reachable (closed (pnet np (p1 \<parallel> p2))) I"
elims_3: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'j, 'k, 'l) SLin_state_scheme" assumes "\<not> P19 x" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'j, 'k, 'l) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (abortVals s2 \<noteq> {} \<longrightarrow> abortVals s1 \<noteq> {})"
subset_range_from_nat_into: fixes A :: "'c set" assumes "countable A" shows "A \<subseteq> range (from_nat_into A)"
length_of_twcmpser: fixes G :: "('e, 'f) Group_scheme"   and E :: "('g, 'h) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set" assumes "Group G"   and "Ugp E"   and "tw_cmpser G (Suc n) f" shows "length_twcmpser G (Suc n) f = Suc n - card {i. i \<le> n \<and> Qw_cmpser G f i \<cong> E}"
span_subsetI: fixes ws :: "'b vec set"   and n :: "nat"   and us :: "'b vec set" assumes "ws \<subseteq> carrier_vec n"   and "us \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) ws" shows "LinearCombinations.module.span class_ring (module_vec TYPE('b) n) us \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) ws"
mthd_is_static_simp: fixes m :: "'b \<times> 'a" shows "is_static (mthd m) = is_static m"
iMODb_singleton_subset_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and a :: "nat" shows "([ r, mod m, c ] \<subseteq> {a}) = (r = a \<and> (m = 0 \<or> c = 0))"
aform_val_msum_aforms: fixes xs :: "(real \<times> real pdevs) list"   and d :: "nat"   and e :: "nat \<Rightarrow> real"   and ys :: "(real \<times> real pdevs) list" assumes "degree_aforms xs \<le> d" shows "aform_vals e (msum_aforms d xs ys) = map2 (+) (aform_vals e xs) (aform_vals (\<lambda>i. e (i + d)) ys)"
B'_proj: fixes l :: "'lbl" shows "\<forall>t\<in>N2 (proj_unl l B). \<exists>s\<in>N2 (proj_unl l A). \<exists>\<delta>. t = s \<cdot> \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
fst_quot_of_fract_0_imp: fixes x :: "'b fract" assumes "numerator x = (0::'b)" shows "denominator x = (1::'b)"
dag: shows "\<dagger>\<^sub>D\<^sub>G\<^sub>.\<^sub>R\<^sub>e\<^sub>l \<alpha> : op_dg (dg_Rel \<alpha>) \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> dg_Rel \<alpha>"
that: fixes vs :: "'val list" assumes "liveVal' g (v # vs)" shows "thesis"
simps_2: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and P :: "'e \<Rightarrow> bool"   and U :: "'f set"   and x :: "'e"   and xs :: "'e list" shows "ipurge_tr_aux_foldr I D P U (x # xs) = ipurge_tr_aux I D U ((if P x then [x] else []) @ ipurge_tr_aux_foldr I D P (sinks_aux I D U [x]) xs)"
C_2: fixes n :: "nat" shows "C n \<subseteq> T"
affine_dependent_subset: fixes s :: "'b set"   and t :: "'b set" assumes "affine_dependent s"   and "s \<subseteq> t" shows "affine_dependent t"
simps: fixes base1 :: "'a word"   and m1 :: "nat"   and base2 :: "'a word"   and m2 :: "nat" shows "ipcidr_conjunct (base1, m1) (base2, m2) = (if ipset_from_cidr base1 m1 \<inter> ipset_from_cidr base2 m2 = {} then None else if ipset_from_cidr base1 m1 \<subseteq> ipset_from_cidr base2 m2 then Some (base1, m1) else Some (base2, m2))"
sinpoly_ub_imp_cospoly_lb: fixes x :: "real"   and k :: "nat" assumes "0 \<le> x"   and "0 < k"   and "\<And>x. 0 \<le> x \<Longrightarrow> sin x \<le> sinpoly (k - 1) x" shows "cospoly k x \<le> cos x"
absolutely_integrable_onI: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "f integrable_on S"   and "(\<lambda>x. norm (f x)) integrable_on S" shows "f absolutely_integrable_on S"
DDH1_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.DDH1 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>s. sample_uniform (order \<G>) \<bind> (\<lambda>r. let h = \<^bold>g [^] s in \<A> h (\<^bold>g [^] r) (h [^] r \<otimes> \<^bold>g)))"
exhaust_sel: fixes gpv :: "('d, 'e, 'f) gpv" obtains "gpv = GPV (the_gpv gpv)"
simps_2: fixes v :: "val" shows "eval_unop UMinus v = Intg (- the_Intg v)"
zeromaximalideal: fixes R :: "('c, 'd) ring_scheme" assumes "field R" shows "maximalideal {\<zero>\<^bsub>R\<^esub>} R"
compile_correct_step: fixes rs :: "(name \<times> pterm) fset"   and name :: "name"   and rhs :: "sterm" assumes "srewrite_step (Rewriting_Sterm.compile rs) name rhs"   and "is_fmap rs"   and "fBall rs prule" shows "(name, sterm_to_pterm rhs) |\<in>| rs"
inject: fixes x2 :: "'k"   and y2 :: "'k" shows "(Some x2 = Some y2) = (x2 = y2)"
dom_\<nu>': shows "local.dom \<nu>' = local.dom \<rho>'"
simps_1: fixes h :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" shows "Path h x [] y = (x = y)"
ideal_points_x_axis: shows "ideal_points x_axis = {of_complex (- 1), of_complex 1}"
is_natural_transformation: shows "natural_transformation VVV.comp (\<bullet>) HoHV HoVH \<alpha>.map"
derivation_times_eq_zeroI: fixes X :: "('a \<Rightarrow> real) \<Rightarrow> real"   and p :: "'a"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "X \<in> tangent_space p"   and "f \<in> diff_fun_space"   and "g \<in> diff_fun_space"   and "f p = 0"   and "g p = 0" shows "X (f * g) = 0"
smult_minus_1: fixes R :: "('a, 'b) ring_scheme"   and M :: "('a, 'c, 'd) module_scheme"   and v :: "'c" assumes "Module.module R M"   and "v \<in> carrier M" shows "\<ominus>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> v = \<ominus>\<^bsub>M\<^esub> v"
equality: fixes r :: "'a State_scheme"   and r' :: "'a State_scheme" assumes "getF r = getF r'"   and "getM r = getM r'"   and "getConflictFlag r = getConflictFlag r'"   and "getC r = getC r'"   and "more r = more r'" shows "r = r'"
ih: fixes L :: "'a literal" assumes "\<And>M. M < L \<Longrightarrow> P M" shows "P L"
sum_mset_mono: fixes K :: "'a multiset"   and f :: "'a \<Rightarrow> 'b multiset"   and g :: "'a \<Rightarrow> 'b multiset" assumes "\<And>i. i \<in># K \<Longrightarrow> f i \<subseteq># g i" shows "comm_monoid_add.sum_mset (+) {#} (image_mset f K) \<subseteq># comm_monoid_add.sum_mset (+) {#} (image_mset g K)"
flip_altsublist_adjacent_sum_list: fixes S :: "'b set"   and ss :: "'b list"   and ts :: "'b list" assumes "PreCoxeterSystem S"   and "ss \<in> lists S"   and "PreCoxeterSystem.flip_altsublist_adjacent ss ts" shows "sum_list ts = sum_list ss"
assms_2: fixes x :: "'a" assumes "x \<in> A" shows "B x \<lesssim> D x"
is_coext_def: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" shows "is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>D x\<guillemotright> \<and> f = e \<cdot>\<^sub>C F g"
not_and_lta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "\<not> (A B C LtA D E F \<and> D E F LtA A B C)"
card_set: shows "CARD('a set) = 2 ^ CARD('a)"
d_preserves_equation: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(ad (ad y) \<cdot> x \<le> x \<cdot> ad (ad z)) = (ad (ad y) \<cdot> x = ad (ad y) \<cdot> x \<cdot> ad (ad z))"
bind_returns_result_E2: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and y :: "'n" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"   and "pure f h" obtains x :: "'o" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>r y"
sup_V_def: fixes x :: "V"   and y :: "V" shows "x \<union>\<^sub>\<circ> y \<equiv> set (elts x \<union> elts y)"
upset_anti_iff: fixes y :: "'a"   and x :: "'a" shows "(\<up> y \<subseteq> \<up> x) = (x \<le> y)"
prf_oracle_def: fixes key :: "'prf_key"   and \<sigma> :: "unit"   and x :: "bool list" shows "PRF.prf_oracle key \<sigma> x = return_spmf (prf_fun key x, ())"
Cons_1: assumes "poly_inv p" shows "poly_inv (poly_mult p q)"
intro: fixes F :: "('b, 'd, 'a, 'e, 'c, 'f, 'g) Functor_scheme" assumes "\<And>f X Y. f maps\<^bsub>CatDom F\<^esub> X to Y \<Longrightarrow> F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y" shows "FunctorM_axioms F"
bind_spmf_pmf_assoc: fixes p :: "'c pmf"   and f :: "'c \<Rightarrow> 'b spmf"   and g :: "'b \<Rightarrow> 'a spmf" shows "bind_spmf (bind_pmf p f) g = bind_pmf p (\<lambda>x. bind_spmf (f x) g)"
MakeNT_dom: fixes f :: "('g, 'j, 'h, 'k, 'i, 'l) NatTrans" shows "NTCatDom f = NTCatDom (MakeNT f)"
onllD: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and P :: "('e \<times> 'd) \<times> 'f \<times> 'e \<times> 'd \<Rightarrow> bool"   and \<xi> :: "'e"   and p :: "('b, 'c, 'a, 'd) seqp"   and a :: "'f"   and \<xi>' :: "'e"   and p' :: "('b, 'c, 'a, 'd) seqp" assumes "onll \<Gamma> P ((\<xi>, p), a, \<xi>', p')" shows "\<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
iaction_def: fixes step :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<times> 'a"   and \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and \<gamma> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and \<Delta> :: "'e \<Rightarrow> 'f \<Rightarrow> 'a \<Rightarrow> 'd list \<Rightarrow> bool"   and s :: "'e"   and vl :: "'f"   and s1 :: "'a"   and vl1 :: "'d list" shows "BD_Security.iaction step \<phi> f \<gamma> \<Delta> s vl s1 vl1 \<equiv> \<exists>a1 ou1 s1' vl1'. let trn1 = Trans s1 a1 ou1 s1' in IO_Automaton.validTrans step trn1 \<and> \<phi> trn1 \<and> BD_Security.consume \<phi> f trn1 vl1 vl1' \<and> \<not> \<gamma> trn1 \<and> \<Delta> s vl s1' vl1'"
mod_hom_def: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme"   and N :: "('e, 'i, 'j) module_scheme"   and f :: "'g \<Rightarrow> 'i" shows "mod_hom R M N f \<equiv> module R M \<and> module R N \<and> mod_hom_axioms R M N f"
that: fixes r :: "real" assumes "0 < r"   and "ball z r \<subseteq> M"   and "bounded (f ` (ball z r - {z})) \<or> bounded ((inverse \<circ> f) ` (ball z r - {z}))" shows "thesis"
borel_measurable_cond_exp: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real" shows "real_cond_exp M F f \<in> borel_measurable F"
map_poly_pderiv: fixes p :: "int poly" shows "of_int_poly (pderiv p) = pderiv (of_int_poly p)"
smc_in_Arr_in_Vset: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
connectedI: fixes "open" :: "'c set \<Rightarrow> bool"   and U :: "'c set" assumes "class.topological_space open"   and "\<And>A B. \<lbrakk>open A; open B; A \<inter> U \<noteq> {}; B \<inter> U \<noteq> {}; A \<inter> B \<inter> U = {}; U \<subseteq> A \<union> B\<rbrakk> \<Longrightarrow> False" shows "topological_space.connected open U"
bijI: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f"   and "top_pres ((`) f)" shows "bij f"
eq_\<I>_converterD_WT: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv1 :: "('y, 'z, 'aa, 'ab) converter"   and conv2 :: "('y, 'z, 'aa, 'ab) converter" assumes "\<I>,\<I>' \<turnstile>\<^sub>C conv1 \<sim> conv2" shows "\<I>, \<I>' \<turnstile>\<^sub>C conv1 \<surd> = \<I>, \<I>' \<turnstile>\<^sub>C conv2 \<surd>"
distributed_variance: fixes X :: "'a \<Rightarrow> real"   and f :: "real \<Rightarrow> real" assumes "distributed M lborel X (\<lambda>x. ennreal (f x))"   and "\<And>x. 0 \<le> f x" shows "expectation (\<lambda>x. (X x - expectation X)\<^sup>2) = LBINT x. x\<^sup>2 * f (x + expectation X)"
right_spec_def: fixes xs :: "'a list"   and j :: "nat" shows "right_spec xs j = (if \<exists>i>j. xs ! i < xs ! j then Some (LEAST i. j < i \<and> xs ! i < xs ! j) else None)"
split_eqvt: fixes p :: "perm"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'c \<times> 'd" shows "p \<bullet> (case x of (x, xa) \<Rightarrow> P x xa) = (case p \<bullet> x of (x, xa) \<Rightarrow> (p \<bullet> P) x xa)"
IH: fixes out :: "'call"   and c :: "('a, 'call, 'ret) rpv"   and input :: "'ret"   and s :: "'s" assumes "IO out c \<in> set_spmf p"   and "out \<in> outs_\<I> \<I> \<Longrightarrow> input \<in> responses_\<I> \<I> out" shows "lossless_spmf (inline1 callee (c input) s)"
of_int_fls_subdegree_of_fpxs: fixes f :: "'b fpxs" shows "rat_of_int (fls_subdegree (fls_of_fpxs f)) = fpxs_val f * rat_of_nat (fpxs_root_order f)"
sorted_is_smaller: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list"   and l :: "'b set list"   and b :: "('b, 'c) Combinators" assumes "FWNormalisationCore.sorted (a # p) l"   and "in_list a l"   and "in_list b l"   and "all_in_list p l"   and "singleCombinators [a]"   and "singleCombinators p"   and "b \<in> set p" shows "smaller a b l"
std_normal_moment_odd: fixes k :: "nat" shows "has_bochner_integral lborel (\<lambda>x. std_normal_density x * x ^ (2 * k + 1)) 0"
primeness_condition_monoid_axioms_def: fixes G :: "('c, 'd) monoid_scheme" shows "primeness_condition_monoid_axioms G \<equiv> \<forall>a. a \<in> carrier G \<longrightarrow> irreducible G a \<longrightarrow> prime G a"
ord_eq_1: fixes x :: "'a" assumes "x \<in> carrier G" shows "(ord x = 1) = (x = \<one>)"
assms: shows "\<forall>m\<in>M. m dvd n"
unit_div_1_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 a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a one" shows "dvd.dvd times (divide one a) one"
assms: shows "card (keys (mapping_of p)) \<le> 1"
bit_slice_iff: fixes m :: "nat"   and w :: "'c word"   and n :: "nat" shows "bit (slice m w) n = (n < min LENGTH('b) (LENGTH('c) - m) \<and> bit w (n + LENGTH('c) - (LENGTH('c) - m)))"
lm090: fixes P :: "('c \<times> 'd) set"   and Y :: "'d set" assumes "runiq P" shows "P\<inverse> `` (Range P - Y) \<inter> P\<inverse> `` Y = {}"
sba1_double_complement: fixes x :: "'a" shows "- - - x = - x"
distrib_inf_le: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.lattice inf less_eq less sup" shows "less_eq (sup (inf x y) (inf x z)) (inf x (sup y z))"
ocircline_eq_cmat_id: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "ocircline_eq_cmat H H"
refl_on_P2R: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "refl_on (Collect P) (P2R \<phi>) = reflP P \<phi>"
all_prime_def: fixes ps :: "nat list" shows "all_prime ps = (\<forall>p\<in>set ps. prime p)"
Un_fa: shows "\<Union> (f ` a) \<in> f ` (P .\<^bold>\<le> x)"
antisym: fixes x :: "'al"   and y :: "'al" assumes "x \<in> U"   and "y \<in> U"   and "x \<le>\<^sub>o\<^sub>w y"   and "y \<le>\<^sub>o\<^sub>w x" shows "x = y"
boffa_mon_aux1: fixes x :: "('c, 'b) boffa_mon" shows "{x\<^bsub>i \<cdot> j\<^esub> |i j. i \<in> UNIV \<and> j \<in> UNIV} = {x\<^bsub>i\<^esub> |i. i \<in> UNIV}"
linear_linear: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_linear f = linear f"
set_sort: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "set (sort_key f xs) = set xs"
deriv: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "(F has_real_derivative f x) (at x within {a..b})"
eval_extensional: fixes S :: "('e, 'f) ring_scheme"   and phi :: "'a \<Rightarrow> 'e"   and p :: "'e" shows "eval R S phi p \<in> extensional (carrier P)"
add_mono: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset"   and d :: "'b multiset" assumes "a \<subseteq># b"   and "c \<subseteq># d" shows "a + c \<subseteq># b + d"
ereal_less_add: fixes a :: "ereal"   and c :: "ereal"   and b :: "ereal" assumes "\<bar>a\<bar> \<noteq> \<infinity>"   and "c < b" shows "a + c < a + b"
suffix_unsnoc: fixes xs :: "'a list"   and ys :: "'a list" assumes "suffix xs ys"   and "xs \<noteq> []" obtains x :: "'a" where "xs = butlast xs @ [x]" and "ys = butlast ys @ [x]"
abs_eq_iff: fixes x :: "'a ltl"   and y :: "'a ltl" shows "(Abs x = Abs y) = x \<equiv>\<^sub>P y"
strict_sorted_sorted_wrt: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less = sorted_wrt less"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>C.map f : C.map a \<rightarrow>\<^sub>C C.map b\<guillemotright>"
monad_altc_optionT: assumes "monad_altc return bind altc" shows "monad_altc local.return_option local.bind_option local.altc_option"
distr_const: fixes M :: "'c measure"   and c :: "'d"   and N :: "'d measure" assumes "prob_space M"   and "c \<in> space N" shows "distr M N (\<lambda>x. c) = return N c"
Par1F: fixes P :: "pi"   and \<alpha> :: "Late_Semantics.freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>l \<alpha> \<prec> P'" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<alpha> \<prec> P' \<parallel> Q"
induct: fixes P :: "('g \<Rightarrow> 'h) \<Rightarrow> 'g list \<Rightarrow> bool"   and a0 :: "'g \<Rightarrow> 'h"   and a1 :: "'g list" assumes "\<And>f x. P f [x]"   and "\<And>f x y zs. P f (y # zs) \<Longrightarrow> P f (x # y # zs)"   and "\<And>a. P a []" shows "P a0 a1"
coeff_iff_length_cond: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "length p1 = length p2" shows "(p1 = p2) = (local.coeff p1 = local.coeff p2)"
simps_17: fixes uy :: "'b"   and v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Atom uy) (Plus v va) = True"
hyps_2: shows "P,E_ \<turnstile> \<langle>e_,s\<^sub>0_\<rangle> \<rightarrow>* \<langle>throw e'_,s\<^sub>1_\<rangle>"
pr_strict'_def: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b \<times> nat"   and g :: "'b \<times> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "admissible_kbo.pr_strict' pr_strict least f g = (f \<notin> admissible_kbo.lConsts least \<and> (pr_strict f g \<or> g \<in> admissible_kbo.lConsts least))"
in_num: fixes var :: "'d set"   and trm :: "'e set"   and fmla :: "'f set"   and Var :: "'d \<Rightarrow> 'e"   and FvarsT :: "'e \<Rightarrow> 'd set"   and substT :: "'e \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'e"   and Fvars :: "'f \<Rightarrow> 'd set"   and subst :: "'f \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f"   and num :: "'e set"   and n :: "'e" assumes "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"   and "n \<in> num" shows "n \<in> trm"
sbintrunc_bintrunc': fixes n :: "nat"   and w :: "int" assumes "0 < n" shows "signed_take_bit (n - 1) (take_bit n w) = signed_take_bit (n - 1) w"
real_le_affinity: fixes m :: "'c"   and y :: "'c"   and x :: "'c"   and c :: "'c" assumes "(0::'c) < m" shows "(y \<le> m * x + c) = (inverse m * y + - (c / m) \<le> x)"
size_add_mset: fixes a :: "'k"   and A :: "'k multiset" shows "size (add_mset a A) = Suc (size A)"
parts_crypt_pwd_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and K :: "key"   and n :: "nat" assumes "s \<turnstile> s'"   and "Crypt K (Pwd n) \<in> parts (used s')"   and "Crypt K (Pwd n) \<notin> parts (used s)" shows "(\<exists>SK. K = SesK SK \<and> (Owner n, Crypt (SesK SK) (Pwd n)) \<in> s') \<or> {Pwd n, Key K} \<subseteq> spied s'"
mono: shows "\<forall>x y z. (x, y) \<in> S \<and> y \<le> z \<longrightarrow> (x, z) \<in> S"
count_empty: fixes a :: "'b" shows "count {#} a = 0"
l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_root_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_child_nodes_locs get_disconnected_nodes get_parent get_parent_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs \<equiv> (l_heap_is_wellformed type_wf known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes \<and> l_get_parent\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr type_wf get_child_nodes get_child_nodes_locs known_ptrs get_parent get_parent_locs) \<and> l_get_parent_wf type_wf known_ptr known_ptrs heap_is_wellformed parent_child_rel get_child_nodes get_parent \<and> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr known_ptrs get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_ancestors get_ancestors_locs get_root_node get_root_node_locs"
solvable_imp_trivial_derived_seq: fixes H :: "'a set" assumes "solvable_seq G H" shows "\<exists>n. (derived G ^^ n) H = {\<one>\<^bsub>G\<^esub>}"
normalize_ports_generic_preserves_normalized_not_has_disc_negated: fixes m :: "'b common_primitive match_expr"   and disc2 :: "'b common_primitive \<Rightarrow> bool"   and disc :: "'b common_primitive \<Rightarrow> bool"   and sel :: "'b common_primitive \<Rightarrow> ipt_l4_ports"   and C :: "ipt_l4_ports \<Rightarrow> 'b common_primitive"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "\<not> has_disc_negated disc2 False m"   and "wf_disc_sel (disc, sel) C"   and "\<forall>a. \<not> disc (Prot a)"   and "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"   and "m' \<in> set (normalize_ports_generic (normalize_positive_ports_step (disc, sel) C) (rewrite_negated_primitives (disc, sel) C l4_ports_negate_one) m)" shows "\<not> has_disc_negated disc2 False m'"
distinct_38: fixes x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt"   and x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "Stmt.If x51 x52 x53 \<noteq> x41 ;; x42"
fv_strand_step_subst: fixes P :: "('c, 'd) strand_step \<Rightarrow> 'd set"   and ac :: "poscheckvariant"   and x :: "('c, 'd) strand_step"   and \<delta> :: "'d \<Rightarrow> ('c, 'd) Term.term" assumes "P = fv\<^sub>s\<^sub>t\<^sub>p \<or> P = fv\<^sub>r\<^sub>c\<^sub>v \<or> P = fv\<^sub>s\<^sub>n\<^sub>d \<or> P = fv\<^sub>e\<^sub>q ac \<or> P = fv\<^sub>i\<^sub>n\<^sub>e\<^sub>q"   and "set (bvars\<^sub>s\<^sub>t\<^sub>p x) \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "fv\<^sub>s\<^sub>e\<^sub>t (\<delta> ` P x) = P (x \<cdot>\<^sub>s\<^sub>t\<^sub>p \<delta>)"