hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "exp0 list"   and a5 :: "'b state \<times> (v list, v) result" assumes "evaluate_list a1 a2 a3 a4 a5" obtains ck :: "bool" and env :: "v sem_env" and s :: "'b state" where "a1 = ckb__" and "a2 = envb__" and "a3 = sb__" and "a4 = []" and "a5 = (sb__, Rval [])"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and vs :: "v list" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = ckc__" and "a2 = envc__" and "a3 = s1__" and "a4 = eb__ # esa__" and "a5 = (s3__, Rval (v1a__ # vs))" and "evaluate ckc__ envc__ s1__ eb__ (s2__, Rval v1a__) \<and> evaluate_list ckc__ envc__ s2__ esa__ (s3__, Rval vs)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and err :: "v error_result" and s :: "'b state" and s' :: "'b state" where "a1 = ckd__" and "a2 = envd__" and "a3 = s" and "a4 = ec__ # esb__" and "a5 = (s', Rerr err__)" and "evaluate ckd__ envd__ s ec__ (s', Rerr err__)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and err :: "v error_result" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = ck" and "a2 = env" and "a3 = s1" and "a4 = e # es" and "a5 = (s3, Rerr err)" and "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rerr err)"
arr_Rel_ArrVal_in_Vset: shows "T\<lparr>ArrVal\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
A: fixes a :: "'a" shows "((1::'a) r\<rightarrow> a) r\<rightarrow> (1::'a) = (1::'a)"
zero_right_mult_decreasing: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and x :: "'c" assumes "class.idempotent_left_semiring one times sup less_eq less bot" shows "less_eq (times x bot) x"
nbt_subset_imp_card_leq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier"   and "y \<in> carrier" shows "(no_better_than x carrier relation \<subseteq> no_better_than y carrier relation) = (card (no_better_than x carrier relation) \<le> card (no_better_than y carrier relation))"
arr_cod_iff_arr: fixes f :: "'c" shows "Cop.arr (Cop.cod f) = Cop.arr f"
invariantsHoldInValidRuns: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "(stateA, stateB) \<in> transitionRelation F0 decisionVars" shows "invariantsHoldInState stateB F0 decisionVars"
previous_segments_of_aformE: fixes y :: "real \<times> real"   and z :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(y, z) \<in> set (segments_of_aform X)" obtains x :: "real \<times> real" where "(x, y) \<in> set (segments_of_aform X)"
inj_Some: fixes A :: "'c set" shows "inj_on Some A"
p_prod_assoc: fixes X :: "'a list set"   and Y :: "'a list set"   and Z :: "'a list set" shows "p_prod (p_prod X Y) Z = p_prod X (p_prod Y Z)"
rev_path_append: fixes xs :: "('a \<times> 'b \<times> 'a) list"   and ys :: "('a \<times> 'b \<times> 'a) list" shows "rev_path (xs @ ys) = rev_path ys @ rev_path xs"
carrier_is_submodule: shows "submodule (carrier (UP Zp)) Zp (UP Zp)"
residues_prime_cyclic': shows "\<exists>x\<in>{0<..<p}. units.ord x = p - 1"
equalmodstrick1: fixes a :: "nat"   and b :: "nat"   and p :: "nat" assumes "pdifference a b mod p = 0" shows "a mod p = b mod p"
rev_line_integral: shows "line_integral F basis (reversepath \<gamma>) = - i"
S1: shows "Struct wtFsym wtPsym arOf resOf intT intF1 intP"
hfunE: fixes B :: "hf"   and C :: "hf"   and F :: "hf" assumes "hfun B C F" obtains "\<And>Y. Y \<^bold>\<in> B \<Longrightarrow> (\<exists>!Z. \<langle>Y, Z\<rangle> \<^bold>\<in> F) \<and> (\<forall>Z. \<langle>Y, Z\<rangle> \<^bold>\<in> F \<longrightarrow> Z \<^bold>\<in> C)"
swap: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and B :: "'e set"   and A :: "'d set" shows "sup_monoid.sum (\<lambda>i. sup_monoid.sum (g i) B) A = sup_monoid.sum (\<lambda>j. sup_monoid.sum (\<lambda>i. g i j) A) B"
induct: fixes x :: "'a tree"   and P :: "'a tree \<Rightarrow> bool" assumes "tfinite x"   and "\<And>t. \<lbrakk>\<And>t'. t' |\<in>| cont t \<Longrightarrow> tfinite t'; \<And>t'. t' |\<in>| cont t \<Longrightarrow> P t'\<rbrakk> \<Longrightarrow> P t" shows "P x"
prj_simps_8: shows "C.cod Prj\<^sub>0\<^sub>1 = \<nu>.apex"
to_hma\<^sub>m_def: fixes a :: "'d mat" shows "Mod_Type_Connect.to_hma\<^sub>m a = (\<chi>i j. a $$ (mod_type_class.to_nat i, mod_type_class.to_nat j))"
distinct_6: fixes x31 :: "'b expr'"   and x32 :: "'b expr'"   and x2 :: "int" shows "Add' x31 x32 \<noteq> Lit' x2"
simps_2: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and x1 :: "'d"   and x2 :: "'e" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
boundedI: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> a \<le> x" shows "Max A \<le> x"
shead_bottom_iff: fixes xs :: "[:'r:]" shows "(shead\<cdot>xs = \<bottom>) = (xs = \<bottom> \<or> xs = [::])"
iMin_ge_iff: fixes I :: "'a set"   and n :: "'a" assumes "I \<noteq> {}" shows "(n \<le> iMin I) = (\<forall>a\<in>I. n \<le> a)"
sum_telescope'': fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "m \<le> n" shows "(\<Sum>k = Suc m..n. f k - f (k - 1)) = f n - f m"
surjective: fixes r :: "('a, 'b, 'c) Propagate.configuration_scheme" shows "r = \<lparr>c_work = c_work r, c_pts = c_pts r, c_imp = c_imp r, \<dots> = Propagate.configuration.more r\<rparr>"
easy_direction: shows "consistent_sign_vectors_R qs (set S) \<subseteq> consistent_sign_vectors_R qs UNIV"
matches_add_match_simp_helper: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>add_match m rs, Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t"
two_D_integral_to_one_D: shows "integral UNIV f = integral UNIV (\<lambda>x. integral UNIV (\<lambda>y. f (y, x)))"
part_to_full': fixes C :: "com"   and ps :: "char list \<Rightarrow> int option"   and m :: "nat"   and ps' :: "char list \<Rightarrow> int option"   and s :: "char list \<Rightarrow> int" assumes "(C, ps) \<Rightarrow>\<^sub>A m \<Down> ps'" shows "(C, Partial_Evaluation.emb ps s) \<Rightarrow> m \<Down> Partial_Evaluation.emb ps' s"
map_charpair_map_pairtrue_R: fixes s :: "'w"   and t :: "'w" assumes "s \<in> S"   and "t \<in> S" shows "map (charpair S) (pair_relator_list s t) = map pairtrue (pair_relator_list s t)"
eq_ivl_def: fixes p1 :: "int extended \<times> int extended"   and p2 :: "int extended \<times> int extended" shows "eq_ivl p1 p2 = (\<gamma>_rep p1 = \<gamma>_rep p2)"
sorted_wrt01: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "length xs \<le> 1" shows "sorted_wrt P xs"
assms_3: shows "y \<in> circline_set l \<inter> circline_set y_axis"
stable_ordIso1: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "stable r"   and "r' =o r" shows "stable r'"
nres_order_simps_1: fixes M :: "'b nres" shows "SUCCEED \<le> M"
solve_equation: fixes H :: "'a set"   and x :: "'a"   and y :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>G\<lparr>carrier := adjoin G H a\<rparr>\<^esub> x"
index_non_zero_implies_no_group: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "int (\<B> index ps) = \<Lambda>" shows "size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 0"
bind_alt1: fixes m :: "'m"   and m' :: "'m"   and f :: "'a \<times> 's \<Rightarrow> 'm" shows "bind (alt m m') f = alt (bind m f) (bind m' f)"
DERIV_const: fixes k :: "'b"   and F :: "'b filter" shows "((\<lambda>x. k) has_field_derivative (0::'b)) F"
adopt_node_wf_is_l_adopt_node_wf: shows "l_adopt_node_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes get_disconnected_nodes ShadowRootClass.known_ptrs Shadow_DOM.adopt_node"
mklr_inverse: fixes y :: "'a set" assumes "y \<in> {x. x \<in> admS}" shows "unlr (mklr y) = y"
replicate_zmset_def: fixes n :: "nat"   and x :: "'b" shows "replicate_zmset n x = (add_zmset x ^^ n) {#}\<^sub>z"
rewrite_X_enat_atoms: fixes \<phi> :: "'a ltln" shows "atoms_ltln (fst (rewrite_X_enat \<phi>)) \<subseteq> atoms_ltln \<phi>"
factors_exist: fixes a :: "'a" assumes "a \<in> carrier G"   and "a \<notin> Units G" shows "\<exists>fs. set fs \<subseteq> carrier G \<and> factors G fs a"
N_syms: shows "\<forall>i>N. num_syms (f i) = num_syms (f (Suc i))"
lea_obtuse_obtuse: fixes D :: "'p"   and E :: "'p"   and F :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Obtuse D E F"   and "D E F LeA A B C" shows "Obtuse A B C"
ortho_c_scalprod0: fixes z1 :: "complex"   and z2 :: "complex" assumes "z1 \<noteq> 0"   and "z2 \<noteq> 0" shows "(\<angle>c z1 z2 = pi / 2) = (scalprod z1 z2 = 0)"
sup_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<sqinter> y = y"
emeasure_density: fixes f :: "'a \<Rightarrow> ennreal"   and M :: "'a measure"   and A :: "'a set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = \<integral>\<^sup>+ x. f x * indicator A x \<partial>M"
cblinfun_apply_cblinfun_compose: fixes a :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'b"   and b :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'd"   and c :: "'c" shows "(a o\<^sub>C\<^sub>L b) *\<^sub>V c = a *\<^sub>V b *\<^sub>V c"
weak_sharing_consis_axioms: shows "weak_sharing_consis (t # ts)"
mat_of_rows_def: fixes n :: "nat"   and rs :: "'a Matrix.vec list" shows "mat_of_rows n rs = Matrix.mat (length rs) n (\<lambda>(i, j). rs ! i $v j)"
unit_counit_F: fixes y :: "'d" assumes "D.ide y" shows "F y = \<epsilon>o (F y) \<cdot>\<^sub>C F (\<eta>o y)"
rel_eq: shows "rel_generat (=) (=) (=) = (=)"
assms_4: shows "card (reachable_s PROB s) - 1 < length as"
char_root': fixes a :: "'a" assumes "a \<in> carrier G" shows "\<chi> a ^ order G = 1"
norm_of_real_add1: fixes x :: "real" shows "norm (of_real x + (1::'b)) = \<bar>x + 1\<bar>"
Collect_inj: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "p2s P = p2s Q" shows "P = Q"
1: shows "real (size1 l) \<le> (1 - \<alpha>) * real (size1 \<langle>l, a, r\<rangle>)"
closed_env_cons: fixes v :: "exp"   and \<rho>'' :: "(nat \<times> exp) list"   and a :: "nat" assumes "FV v = {}"   and "closed_env \<rho>''" shows "closed_env ((a, v) # \<rho>'')"
join_self: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.cpo below" shows "cpo.join below x x = x"
cases: fixes a :: "('fenv, 'henv, ('fun, 'dyn) frame) state" assumes "final a" obtains F :: "'fenv" and f :: "'fun" and fd :: "('dyn, 'var, 'fun, 'op) instr fundef" and pc :: "nat" and H :: "'henv" and \<Sigma> :: "'dyn list" where "a = State F H [Frame f pc \<Sigma>]" and "F_get F f = Some fd" and "pc = length (body fd)"
my_wf_induct_triple'': fixes r1 :: "('a \<times> 'a) set"   and r2 :: "('b \<times> 'b) set"   and r3 :: "('c \<times> 'c) set"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c" assumes "wf (r1 <*lex*> r2 <*lex*> r3)"   and "\<And>x1 x2 x3. (\<And>y1 y2 y3. ((y1, y2, y3), x1, x2, x3) \<in> r1 <*lex*> r2 <*lex*> r3 \<longrightarrow> P y1 y2 y3) \<Longrightarrow> P x1 x2 x3" shows "P x1 x2 x3"
integral_density: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. 0 \<le> g x" shows "integral\<^sup>L (density M (\<lambda>x. ennreal (g x))) f = LINT x|M. g x *\<^sub>R f x"
a_rcos_self: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "abelian_subgroup H G"   and "x \<in> carrier G" shows "x \<in> H +>\<^bsub>G\<^esub> x"
M: shows "if_dens \<delta> f b \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>)"
independent_vnm_def: fixes C :: "'b pmf set"   and P :: "('b pmf \<times> 'b pmf) set" shows "independent_vnm C P = (\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r)"
mHom_func: fixes R :: "('e, 'f) Ring_scheme"   and f :: "'g \<Rightarrow> 'h"   and M :: "('g, 'e, 'i) Module_scheme"   and N :: "('h, 'e, 'j) Module_scheme" assumes "Ring R"   and "f \<in> mHom R M N" shows "f \<in> carrier M \<rightarrow> carrier N"
induced_hom_Abs_freelist_conv_sum_list: fixes S :: "'a set"   and P :: "('a \<times> bool) list set"   and f :: "'a \<Rightarrow> 'b"   and ss :: "'a list" assumes "GroupByPresentationInducedFun S P f"   and "ss \<in> lists S" shows "GroupByPresentationInducedFun.induced_hom S P f (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` P\<rangle>) (FreeGroup S) `` {Abs_freelist ss})) = sum_list (map f ss)"
se'_3: shows "snd se' \<notin> set (awalk_verts u p)"
inputInduct: fixes P :: "pi"   and a :: "name"   and u :: "name"   and P' :: "pi"   and F :: "'b \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'b" assumes "P \<longmapsto> a<u> \<prec> P'"   and "\<And>a x P u C. \<lbrakk>x \<sharp> C; x \<noteq> u; x \<noteq> a\<rbrakk> \<Longrightarrow> F C (a<x>.P) a u (P[x::=u])"   and "\<And>P a u P' b C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a u P'"   and "\<And>P a u P' b c C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a u P'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u P'"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto> a<u> \<prec> Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a u Q'"   and "\<And>P a u P' Q C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P' \<parallel> Q)"   and "\<And>Q a u Q' P C. \<lbrakk>Q \<longmapsto> a<u> \<prec> Q'; \<And>C. F C Q a u Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a u (P \<parallel> Q')"   and "\<And>P a u P' x C. \<lbrakk>P \<longmapsto> a<u> \<prec> P'; x \<noteq> a; x \<noteq> u; x \<sharp> C; \<And>C. F C P a u P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) a u (<\<nu>x>P')"   and "\<And>P a u P' C. \<lbrakk>P \<parallel> !P \<longmapsto> a<u> \<prec> P'; \<And>C. F C (P \<parallel> !P) a u P'\<rbrakk> \<Longrightarrow> F C (!P) a u P'" shows "F C P a u P'"
ldropn_llist_of: fixes n :: "nat"   and xs :: "'c list" shows "ldropn n (llist_of xs) = llist_of (drop n xs)"
compact_imp_complete: fixes s :: "'b set" assumes "compact s" shows "complete s"
tickFree_implies_front_tickFree: fixes s :: "'a event list" assumes "tickFree s" shows "front_tickFree s"
sel_2: fixes x31 :: "'c"   and x32 :: "'d" shows "un_Yield1 (Yield x31 x32) = x31"
inverse_arrows_def: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" shows "VoV.inverse_arrows f g \<equiv> VoV.ide (VoV.comp g f) \<and> VoV.ide (VoV.comp f g)"
FAssThrow1: fixes E :: "char list \<Rightarrow> ty option"   and r :: "nat \<times> char list list"   and F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "P,E \<turnstile> \<langle>Throw r\<bullet>F{Cs} := e\<^sub>2,s\<rangle> \<rightarrow> \<langle>Throw r,s\<rangle>"
FOREACH_def: fixes S :: "'a set" shows "FOREACH S \<equiv> FOREACH\<^sub>C S (\<lambda>_. True)"
ex_a_pos: shows "\<exists>a\<in>set as. 0 < a"
ltake_lappend1: fixes n :: "enat"   and xs :: "'b llist"   and ys :: "'b llist" assumes "n \<le> llength xs" shows "ltake n (xs $ ys) = ltake n xs"
splus_left_canc: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and v :: "'c"   and u :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "(term_powerprod.splus pair_of_term term_of_pair t v = term_powerprod.splus pair_of_term term_of_pair t u) = (v = u)"
landau_function_family_chain_primfuns: fixes fs :: "(primfun \<times> 'a) list" assumes "sorted (map fst fs)"   and "distinct (map fst fs)" shows "landau_function_family_chain at_top fs (eval_primfun' \<circ> fst)"
emeasure_subprob_space_less_top: fixes M :: "'c measure"   and A :: "'c set" assumes "subprob_space M" shows "emeasure M A \<noteq> top"
mult_strict_mono': fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less c d"   and "less_eq zero a"   and "less_eq zero c" shows "less (times a c) (times b d)"
gen_fib_fib: fixes n :: "nat"   and m :: "nat" shows "gen_fib (fib n) (fib (Suc n)) m = fib (n + m)"
extract_var_single: fixes v :: "nat"   and n :: "nat"   and a :: "'b" shows "extract_var (monom (Poly_Mapping.single v n) a) v = monom 0 (monom (Poly_Mapping.single v n) a)"
pre: shows "check_precond_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta>"
parallel_oracle_conv_plus_oracle: fixes oracle1 :: "'h \<Rightarrow> 'j \<Rightarrow> ('l \<times> 'h) spmf"   and oracle2 :: "'i \<Rightarrow> 'k \<Rightarrow> ('m \<times> 'i) spmf" shows "oracle1 \<ddagger>\<^sub>O oracle2 = oracle1\<dagger> \<oplus>\<^sub>O \<dagger>oracle2"
icheck_curr_init': fixes e :: "'c exp"   and e' :: "'c exp"   and P :: "'d prog"   and C :: "char list" assumes "ss_exp e = e'"   and "icheck P C e'" shows "curr_init P e = \<lfloor>C\<rfloor>"
mset_eq: shows "image_mset f (mset_set A) = image_mset f' (mset_set A)"
admissible_dom_prog: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" shows "dom_prog.admissible (\<lambda>f. \<forall>x h h' r. h \<turnstile> f x \<rightarrow>\<^sub>r r \<longrightarrow> h \<turnstile> f x \<rightarrow>\<^sub>h h' \<longrightarrow> P x h h' r)"
measure_mono_fmeasurable: fixes A :: "'a set"   and B :: "'a set"   and M :: "'a measure" assumes "A \<subseteq> B"   and "A \<in> sets M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
dom_sep: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp (list2FWpolicy p))" shows "x \<in> dom (Cp (list2FWpolicy (separate p)))"
fpsmodrel_def: fixes p :: "'b" shows "fpsmodrel p \<equiv> {(f, g). \<forall>n. f $ n mod p = g $ n mod p}"
elims_2: fixes x :: "rat le_constraint"   and xa :: "QDelta le_constraint" assumes "compatible_cs x xa" obtains p :: "linear_poly" and c :: "rat" and q :: "linear_poly" and d :: "QDelta" where "x = Le_Constraint Leq_Rel p__ ca__" and "xa = Le_Constraint Leq_Rel q__ d__" and "q__ = p__ \<and> d__ = QDelta ca__ 0"   | p :: "linear_poly" and c :: "rat" and q :: "linear_poly" and d :: "QDelta" where "x = Le_Constraint Lt_Rel p c" and "xa = Le_Constraint Leq_Rel q d" and "q = p \<and> qdfst d = c"
no_trailing_Cons: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list" shows "no_trailing P (x # xs) = (no_trailing P xs \<and> (xs = [] \<longrightarrow> \<not> P x))"
restrict_complement_singleton_eq: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a" shows "f |` (- {x}) = f(x := None)"
a_Hcarr: fixes h :: "'a" assumes "h \<in> I" shows "h \<in> carrier R"
ends_in_segment_1: fixes a :: "'b"   and b :: "'b" shows "a \<in> closed_segment a b"
S\<phi>: shows "seq_sem I \<phi> = UNIV"
pgwt_is_empty_synth: fixes t :: "('fun, 'var) Term.term" shows "public_ground_wf_term t = ({} \<turnstile>\<^sub>c t)"
intros: fixes M :: "hmultiset"   and e :: "nat" assumes "\<forall>m\<in>#hmsetmset M. local.encode 0 e \<le> m" shows "local.aligned\<^sub>e e M"
greaterThan_def: fixes l :: "'a" shows "{l<..} \<equiv> {x. l < x}"
strip_comb_induct: fixes P :: "'a \<Rightarrow> bool"   and t :: "'a" assumes "\<And>x y. P x \<Longrightarrow> P (app x y)"   and "\<And>t. unapp t = None \<Longrightarrow> P t" shows "P t"
assms_8: fixes p :: "perm" assumes "supp p \<sharp>* c" shows "p \<bullet> f bs y c = f (p \<bullet> bs) (p \<bullet> y) c"
app: fixes A :: "'a trm"   and x :: "'a"   and T :: "PreSimplyTyped.type"   and A' :: "'a trm"   and B :: "'a trm" assumes "A \<noteq> Fn x T A'"   and "NF A"   and "NF B" shows "NF (App A B)"
key_gen_def: shows "key_gen = sample_uniform q \<bind> (\<lambda>a. sample_uniform q \<bind> (\<lambda>b. sample_uniform q \<bind> (\<lambda>x1. let y1 = (a * x1 + b) mod q in return_spmf ((a, b), x1, y1))))"
W'_5: fixes x :: "'a" assumes "x \<in> A" shows "f x = f' x"
ref_WHILEIT_invarI: fixes I :: "'c \<Rightarrow> bool"   and s :: "'c"   and M :: "'d nres"   and R :: "('d \<times> 'c) set"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" assumes "I s \<Longrightarrow> M \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I\<^esup> b f s)" shows "M \<le> \<Down> R (WHILE\<^sub>T\<^bsup>I\<^esup> b f s)"
undefined: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m" assumes "rha a \<noteq> rha b"   and "a \<in> A" shows "ma a b = None"
fmeasurable_UN_bound: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and M :: "'d measure"   and B :: "real" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M"   and "\<And>I'. \<lbrakk>I' \<subseteq> I; finite I'\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure M (\<Union> (A ` I')) \<le> B" shows "\<Union> (A ` I) \<in> fmeasurable M"
pdevs_of_list_map_scaleR: fixes r :: "real"   and xs :: "'b list" shows "pdevs_of_list (map ((*\<^sub>R) r) xs) = scaleR_pdevs r (pdevs_of_list xs)"
encR: fixes S :: "'a"   and Enc :: "'a \<Rightarrow> 'b"   and SRel :: "('a \<times> 'a) set"   and TRel :: "('b \<times> 'b) set" shows "(SourceTerm S, TargetTerm (Enc S)) \<in> encoding.indRelSTEQ Enc SRel TRel"
a: shows " M \<longrightarrow>\<^sub>1 M1"
affine_diffs_subspace_subtract: fixes S :: "'a set"   and a :: "'a" assumes "affine S"   and "a \<in> S" shows "subspace ((\<lambda>x. x - a) ` S)"
G_relpow_matpow_pos: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and i :: "'b"   and j :: "'b"   and n :: "nat" assumes "pf_nonneg_mat A"   and "(i, j) \<in> fixed_mat.G A ^^ n" shows "(0::'a) < matpow A n $h i $h j"
ik\<^sub>s\<^sub>t_induct: fixes P :: "('c, 'd) strand_step list \<Rightarrow> bool"   and a0 :: "('c, 'd) strand_step list" assumes "P []"   and "\<And>t S. P S \<Longrightarrow> P (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v S. P S \<Longrightarrow> P (send\<langle>v\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va vb S. P S \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t # S)"   and "\<And>v va S. P S \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va\<rangle>\<^sub>s\<^sub>t # S)" shows "P a0"