hcju/isabelleps huggingface dataset:
le_oLimitI: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal"   and n :: "nat" assumes "x \<le> f n" shows "x \<le> oLimit f"
Triangle: fixes t :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes " t \<longrightarrow>\<^sub>d t1"   and " t \<longrightarrow>\<^sub>1 t2" shows " t2 \<longrightarrow>\<^sub>1 t1"
simps_3: fixes hosts :: "'a set"   and flows_fix :: "('a \<times> 'a) set"   and flows_state :: "('a \<times> 'a) set"   and more :: "'b" shows "flows_state \<lparr>hosts = hosts, flows_fix = flows_fix, flows_state = flows_state, \<dots> = more\<rparr> = flows_state"
zero_pdivides: fixes p :: "'a list list" shows "[] pdivides\<^bsub>K [X]\<^esub> p = (p = [])"
arrivemsg_def: fixes i :: "nat"   and P :: "'b \<Rightarrow> bool"   and a :: "'b node_action" shows "arrivemsg i P a \<equiv> case a of ii\<not>ni:arrive(m) \<Rightarrow> ii = {i} \<longrightarrow> P m | _ \<Rightarrow> True"
last_non_root_interval: fixes p :: "real poly"   and ub :: "real" assumes "p \<noteq> 0" obtains lb :: "real" where "lb < ub" and "\<forall>z. lb \<le> z \<and> z < ub \<longrightarrow> poly p z \<noteq> 0"
bfs2_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and succ :: "nat \<Rightarrow> nat list nres"   and src :: "nat"   and dst :: "nat" shows "Graph.bfs2 c succ src dst \<equiv> if src = dst then return (Some []) else Graph.pre_bfs2 c succ src dst \<bind> (\<lambda>br. case br of None \<Rightarrow> return None | Some (d, PRED) \<Rightarrow> Graph.extract_rpath c src dst PRED \<bind> (\<lambda>p. return (Some p)))"
mat_of_row_index: fixes x :: "nat"   and y :: "'b Matrix.vec" assumes "x < dim_vec y" shows "mat_of_row y $$ (0, x) = y $v x"
fls_base_factor_to_fps_to_fls: fixes f :: "'a fps" shows "fls_base_factor_to_fps (fps_to_fls f) = unit_factor f"
pderiv_eq_0_iff: fixes p :: "'b poly" shows "(pderiv p = 0) = (degree p = 0)"
antisym_conv1: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
eta_preserves_term_ok': 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 r :: "Core.term"   and s :: "Core.term" assumes "term_ok' \<Sigma> r"   and "r \<rightarrow>\<^sub>\<eta> s" shows "term_ok' \<Sigma> s"
exhaust: fixes y :: "'a Predicate.pred" obtains x :: "'a \<Rightarrow> bool" where "y = pred.Pred x"
exprBinopTypeable: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and e1 :: "expr"   and bop :: "bop"   and e2 :: "expr"   and T :: "secLevel" assumes "\<Gamma> \<turnstile> e1 \<guillemotleft>bop\<guillemotright> e2 : T" shows "(\<exists>T1. \<Gamma> \<turnstile> e1 : T1) \<and> (\<exists>T2. \<Gamma> \<turnstile> e2 : T2)"
less_list_def': fixes xs :: "'a list"   and ys :: "'a list" shows "(xs < ys) = strict_prefix xs ys"
not_eta_reducible_decr: fixes t :: "Core.term"   and lev :: "nat" assumes "\<not> eta_reducible t"   and "\<not> loose_bvar1 t lev" shows "\<not> eta_reducible (decr lev t)"
all_rev_wrap_def: fixes P :: "'b list \<Rightarrow> prop" shows "all_rev_wrap P \<equiv> (\<And>x. PROP P (rev x))"
order_iff_strict: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a < b \<or> a = b)"
nb\<^sub>1: shows "\<forall>op\<in>set (a # ops). is_operator_applicable_in s op"
compat_different: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and cms\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and cms\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list"   and i :: "nat"   and x :: "'b" assumes "sifum_security stop eval some_val some_val'"   and "sifum_security.makes_compatible dma eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i \<sqsubset> length cms\<^sub>1"   and "x \<in> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "mem\<^sub>1 x \<noteq> mem\<^sub>2 x \<and> dma x = Low"
cycle_free_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and i :: "nat" assumes "linordered_monoid.cycle_free m n"   and "i \<le> n" shows "Le (0::'c) \<preceq> m i i"
induct: fixes P :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a acom \<Rightarrow> 'a acom \<Rightarrow> bool"   and a0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a1 :: "'a acom"   and a2 :: "'a acom" assumes "\<And>f a1 a2. P f (SKIP {a1}) (SKIP {a2})"   and "\<And>f x e a1 x' e' a2. P f (x ::= e {a1}) (x' ::= e' {a2})"   and "\<And>f c1 c2 c1' c2'. \<lbrakk>P f c1 c1'; P f c2 c2'\<rbrakk> \<Longrightarrow> P f (c1;; c2) (c1';; c2')"   and "\<And>f b c1 c2 a1 b' c1' c2' a2. \<lbrakk>P f c1 c1'; P f c2 c2'\<rbrakk> \<Longrightarrow> P f (IF b THEN c1 ELSE c2 {a1}) (IF b' THEN c1' ELSE c2' {a2})"   and "\<And>f a1 b c a2 a3 b' c' a4. P f c c' \<Longrightarrow> P f ({a1} WHILE b DO c {a2}) ({a3} WHILE b' DO c' {a4})"   and "\<And>a v va vb vc. P a (v ::= va {vb}) (SKIP {vc})"   and "\<And>a v va vb vc vd. P a (v ::= va {vb}) (vc;; vd)"   and "\<And>a v va vb vc vd ve vf. P a (v ::= va {vb}) (IF vc THEN vd ELSE ve {vf})"   and "\<And>a v va vb vc vd ve vf. P a (v ::= va {vb}) ({vc} WHILE vd DO ve {vf})"   and "\<And>a v va vb. P a (v;; va) (SKIP {vb})"   and "\<And>a v va vb vc vd. P a (v;; va) (vb ::= vc {vd})"   and "\<And>a v va vb vc vd ve. P a (v;; va) (IF vb THEN vc ELSE vd {ve})"   and "\<And>a v va vb vc vd ve. P a (v;; va) ({vb} WHILE vc DO vd {ve})"   and "\<And>a v va vb vc vd. P a (IF v THEN va ELSE vb {vc}) (SKIP {vd})"   and "\<And>a v va vb vc vd ve vf. P a (IF v THEN va ELSE vb {vc}) (vd ::= ve {vf})"   and "\<And>a v va vb vc vd ve. P a (IF v THEN va ELSE vb {vc}) (vd;; ve)"   and "\<And>a v va vb vc vd ve vf vg. P a (IF v THEN va ELSE vb {vc}) ({vd} WHILE ve DO vf {vg})"   and "\<And>a v va vb vc vd. P a ({v} WHILE va DO vb {vc}) (SKIP {vd})"   and "\<And>a v va vb vc vd ve vf. P a ({v} WHILE va DO vb {vc}) (vd ::= ve {vf})"   and "\<And>a v va vb vc vd ve. P a ({v} WHILE va DO vb {vc}) (vd;; ve)"   and "\<And>a v va vb vc vd ve vf vg. P a ({v} WHILE va DO vb {vc}) (IF vd THEN ve ELSE vf {vg})"   and "\<And>a vc v va vb. P a (SKIP {vc}) (v ::= va {vb})"   and "\<And>a vc vd v va vb. P a (vc;; vd) (v ::= va {vb})"   and "\<And>a vc vd ve vf v va vb. P a (IF vc THEN vd ELSE ve {vf}) (v ::= va {vb})"   and "\<And>a vc vd ve vf v va vb. P a ({vc} WHILE vd DO ve {vf}) (v ::= va {vb})"   and "\<And>a vb v va. P a (SKIP {vb}) (v;; va)"   and "\<And>a vb vc vd v va. P a (vb ::= vc {vd}) (v;; va)"   and "\<And>a vb vc vd ve v va. P a (IF vb THEN vc ELSE vd {ve}) (v;; va)"   and "\<And>a vb vc vd ve v va. P a ({vb} WHILE vc DO vd {ve}) (v;; va)"   and "\<And>a vd v va vb vc. P a (SKIP {vd}) (IF v THEN va ELSE vb {vc})"   and "\<And>a vd ve vf v va vb vc. P a (vd ::= ve {vf}) (IF v THEN va ELSE vb {vc})"   and "\<And>a vd ve v va vb vc. P a (vd;; ve) (IF v THEN va ELSE vb {vc})"   and "\<And>a vd ve vf vg v va vb vc. P a ({vd} WHILE ve DO vf {vg}) (IF v THEN va ELSE vb {vc})"   and "\<And>a vd v va vb vc. P a (SKIP {vd}) ({v} WHILE va DO vb {vc})"   and "\<And>a vd ve vf v va vb vc. P a (vd ::= ve {vf}) ({v} WHILE va DO vb {vc})"   and "\<And>a vd ve v va vb vc. P a (vd;; ve) ({v} WHILE va DO vb {vc})"   and "\<And>a vd ve vf vg v va vb vc. P a (IF vd THEN ve ELSE vf {vg}) ({v} WHILE va DO vb {vc})" shows "P a0 a1 a2"
keysFor_parts_insert: fixes K :: "nat"   and X :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "K \<in> keysFor (parts (insert X G))"   and "X \<in> synth (analz H)" shows "K \<in> keysFor (parts (G \<union> H)) \<or> Key (invKey K) \<in> parts H"
hm_new_rule: shows "<emp> hm_new <is_hashmap Map.empty>"
G_def: shows "G = (\<lambda>x. g x * psi (real_cond_exp M F X x))"
lfp_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "mono f" shows "lfp f = f (lfp f)"
I: fixes n :: "nat"   and s0 :: "state"   and L :: "lenv"   and accC :: "qtname"   and T :: "tys"   and C :: "assigned"   and v :: "vals"   and s1 :: "state"   and Y :: "vals"   and Z :: "'a" assumes "\<forall>t\<in>A. G\<Turnstile>n\<Colon>t"   and "s0\<Colon>\<preceq>(G, L)"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "normal s0 \<Longrightarrow> \<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> C"   and "G\<turnstile>s0 \<midarrow>t\<succ>\<midarrow>n\<rightarrow> (v, s1)"   and "P Y s0 Z" shows "Q v s1 Z \<and> s1\<Colon>\<preceq>(G, L)"
nlists: fixes xs :: "'c list"   and A :: "'c set"   and n :: "nat" assumes "set xs \<subseteq> A"   and "length xs = n" shows "xs \<in> nlists A n"
run_pause_writer: fixes pause :: "'d \<Rightarrow> ('e \<Rightarrow> 'a) \<Rightarrow> 'a"   and out :: "'d"   and c :: "'e \<Rightarrow> ('b, 'c, 'a) writerT" shows "run_writer (pause_writer pause out c) = pause out (\<lambda>input. run_writer (c input))"
closed_open: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "topological_space.closed open S = open (- S)"
x: shows "x \<in> {real_of_float l1..real_of_float u1}"
DirProds_Sigma: fixes G :: "'c \<Rightarrow> 'd \<Rightarrow> ('e, 'f) monoid_scheme"   and J :: "'c \<Rightarrow> 'd set"   and I :: "'c set" shows "DirProds (\<lambda>i. DirProds (G i) (J i)) I \<cong> DirProds (\<lambda>(i, j). G i j) (Sigma I J)"
derivation_ge_shift_simp: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and i :: "nat"   and l :: "nat"   and r :: "nat" assumes "derivation_ge D i"   and "l \<le> i"   and "l \<le> r" shows "derivation_shift D l r = derivation_shift D 0 (r - l)"
keys_cmul_iff: fixes i :: "'b"   and c :: "int"   and x :: "'b \<Rightarrow>\<^sub>0 int" shows "(i \<in> keys (frag_cmul c x)) = (i \<in> keys x \<and> c \<noteq> 0)"
deliver_insert_exists: fixes es :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and xs :: "('a \<times> 'b \<times> bool) list"   and a :: "'a" assumes "es prefix of j"   and "apply_operations es = Some xs"   and "a \<in> element_ids xs" shows "\<exists>i v f n. Deliver (i, Insert (a, v, f) n) \<in> set es"
colx: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p"   and C :: "'p" assumes "A \<noteq> B"   and "Col X Y A"   and "Col X Y B"   and "Col A B C" shows "Col X Y C"
cospan: shows "trg (tab\<^sub>1 h) = trg (tab\<^sub>0 g)"
zip_map1: fixes f :: "'d \<Rightarrow> 'b"   and xs :: "'d list"   and ys :: "'c list" shows "zip (map f xs) ys = map2 (\<lambda>x. Pair (f x)) xs ys"
three_div_two: shows "3 div Suc (Suc 0) = 1"
enum_all_finite_3_def: fixes P :: "finite_3 \<Rightarrow> bool" shows "enum_all P = (P a\<^sub>1 \<and> P a\<^sub>2 \<and> P a\<^sub>3)"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_S: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>s\<^sub>s\<^sub>t S \<union> pair ` setops\<^sub>s\<^sub>s\<^sub>t S)"
tls1: shows "tls1 = lmap (fst \<circ> snd) stls1"
bool_tests_1: fixes b :: "'a bool_expr" shows "bool_taut_test b = (\<forall>s. val_bool_expr b s)"
sprod_map_spair': fixes f :: "'f \<rightarrow> 'e"   and g :: "'h \<rightarrow> 'g"   and x :: "'f"   and y :: "'h" assumes "f\<cdot>\<bottom> = \<bottom>"   and "g\<cdot>\<bottom> = \<bottom>" shows "sprod_map\<cdot>f\<cdot>g\<cdot>(:x, y:) = (:f\<cdot>x, g\<cdot>y:)"
intro: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R"   and "Idomain_axioms R" shows "Idomain R"
A6: shows "\<forall>a b. B a b a \<longrightarrow> a = b"
omax_def: fixes R :: "('b \<times> 'b) set set" shows "omax R \<equiv> SOME r. isOmax R r"
id_arrow: fixes A :: "'o" assumes "A \<in> Ob" shows "Id A \<in> Ar"
assms_2: fixes x :: "'a ltln" assumes "x \<in> X" shows "sf x \<subseteq> X"
const_charn: fixes X :: "'a \<Rightarrow> 'b"   and \<tau> :: "'a"   and \<tau>' :: "'a" assumes "const X" shows "X \<tau> = X \<tau>'"
linear: fixes a :: "real \<times> real"   and X :: "'a \<times> (real \<times> real) pdevs"   and b :: "real \<times> real" assumes "a \<in> set (inl (snd X))"   and "b \<in> set (inl (snd X))" shows "ccw 0 a b \<or> ccw 0 b a"
main_step: fixes qs :: "rat poly list" shows "set (find_consistent_signs qs) = consistent_sign_vectors qs UNIV"
mat_append_col_access_in_mat: fixes A :: "'a mat"   and b :: "'a vec"   and i :: "nat"   and j :: "nat" assumes "dim_row A = dim_vec b"   and "i < dim_row A"   and "j < dim_col A" shows "row (mat_append_col A b) i $ j = row A i $ j"
D_p_1: shows "D \<in> N_of_state (Sup_state Sts) \<union> P_of_state (Sup_state Sts) \<union> Q_of_state (Sup_state Sts)"
find_fund_sol_step_def: fixes D :: "nat" shows "find_fund_sol_step D = (\<lambda>x. case x of Inl (y, y') \<Rightarrow> case get_nat_sqrt y' of None \<Rightarrow> Inl (y + 1, y' + D * (2 * y + 1)) | Some x \<Rightarrow> Inr (x, y))"
assms: shows "(a, b) \<in> rtrancl_on F r"
sum_set_simps_1: fixes x :: "'c" shows "setl (Inl x) = {x}"
axioms_21: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and \<phi>C :: "'c \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'e"   and \<phi>D :: "'d \<times> 'd \<Rightarrow> 'd \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<Phi> :: "'d \<times> 'c \<Rightarrow> 'e"   and \<Psi> :: "'d \<times> 'c \<Rightarrow> 'e" assumes "hom_adjunction C D S \<phi>C \<phi>D F G \<Phi> \<Psi>" shows "set_valued_transformation (product_category.comp (dual_category.comp D) C) S (hom_functor.map D S \<phi>D \<circ> product_functor.map (dual_category.comp D) C (dual_category.comp D) D (identity_functor.map (dual_category.comp D)) G) (hom_functor.map C S \<phi>C \<circ> product_functor.map (dual_category.comp D) C (dual_category.comp C) C (dual_functor.map F) (identity_functor.map C)) \<Psi>"
cases: fixes a1 :: "exp"   and a2 :: "exp" assumes "a1 \<in> isubexprs(a2)" obtains e :: "exp" and fi :: "nat" where "a1 = e__" and "a2 = FieldProj e__ fi"   | e :: "exp" and m :: "nat" and es :: "exp list" where "a1 = ea__" and "a2 = MethodInvk ea__ m__ es__"   | ei :: "exp" and es :: "exp list" and e :: "exp" and m :: "nat" where "a1 = ei__" and "a2 = MethodInvk eb__ m esa__" and "ei__ \<in> set esa__"   | ei :: "exp" and es :: "exp list" and C :: "nat" where "a1 = ei" and "a2 = New C__ es" and "ei \<in> set es"   | e :: "exp" and C :: "nat" where "a1 = e" and "a2 = Cast C e"
\<tau>_map_\<Gamma>: fixes f :: "'c set \<Rightarrow> 'b set"   and s :: "'c trace"   and i :: "nat" shows "\<tau> (map_\<Gamma> f s) i = \<tau> s i"
crel_vs_iterate_state: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and g :: "'f \<Rightarrow> 'g"   and f :: "'f \<Rightarrow> ('b, 'h) state"   and xs :: "'f list" assumes "((=) ===>\<^sub>T R) g f" shows "crel_vs (=) () (iterate_state f xs)"
lexordp_def: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" shows "List.lexordp r xs ys = ((xs, ys) \<in> lexord {(x, y). r x y})"
ideD_3: fixes a :: "'a" assumes "S.ide a" shows "S.cod a = a"
plus_minus: fixes y :: "i"   and x :: "i" assumes "PP y x" shows "y \<oplus> (x \<ominus> y) = x"
bal_tab_correct: shows "\<forall>i<length bal_tab. is_floor (bal_tab ! i) i"
subst_atms_empty: fixes \<sigma> :: "'s" shows "{} \<cdot>as \<sigma> = {}"
interpret_floatariths_append: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and vs :: "real list" shows "interpret_floatariths (xs @ ys) vs = interpret_floatariths xs vs @ interpret_floatariths ys vs"
get_via_put: fixes x :: "'d \<Longrightarrow> 'e"   and s :: "'e" assumes "vwb_lens x" shows "get\<^bsub>x\<^esub> s = (THE v. put\<^bsub>x\<^esub> s v = s)"
e2_def: shows "e2 = (\<lambda>N. (\<Prod>ia = 0..<ji. (of_nat N - of_nat ia) / of_nat (ji - ia)) * la ^ (N - ji) * (complex_of_real c / of_nat N ^ (m - 1)))"
map_project_empty: fixes A :: "('b \<times> 'c list) list set" shows "map \<pi> ` A - {[]} = map \<pi> ` (A - {[]})"
FactRing_iso: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "ring_hom_ring R S h"   and "h ` carrier R = carrier S" shows "R Quot a_kernel R S h \<simeq> S"
mat_of_cblinfun_cblinfun_apply: fixes F :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and u :: "'c" shows "vec_of_basis_enum (F *\<^sub>V u) = mat_of_cblinfun F *\<^sub>v vec_of_basis_enum u"
StepActEvent_def: fixes TS :: "('e \<times> (('e, 'd) expr \<times> ('f data \<Rightarrow> bool) \<times> 'd set \<times> 'f pupdate) \<times> 'e) set" shows "StepActEvent TS \<equiv> \<Union> (Actevent (Label TS))"
cone_\<chi>: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b" assumes "limit_cone J C D a \<chi>" shows "\<chi> \<in> {\<chi>. Limit.cone J C D a \<chi>}"
wt_elim_cases_9: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and unop :: "unop"   and e :: "expr"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (UnOp unop e)\<Colon>T" obtains Te :: "ty" where "T = Inl (PrimT (unop_type unop))" and "E,dt\<Turnstile>e\<Colon>-Te" and "wt_unop unop Te"
dvdI: fixes a :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and k :: "'c" assumes "a = times b k" shows "dvd.dvd times b a"
cond_spmf_fst_def: fixes p :: "('d \<times> 'c) spmf"   and a :: "'d" shows "cond_spmf_fst p a = map_spmf snd (cond_spmf p ({a} \<times> UNIV))"
OPT2_C: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list"   and u :: "'a list"   and v :: "'a list" assumes "x \<noteq> y"   and "qs = u @ v"   and "u = [] \<or> u = [x]"   and "v \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = length v div 2"
inbetween_inset: fixes x :: "'c"   and xs :: "'c list"   and a :: "'c"   and b :: "'c" assumes "x \<in> set (between xs a b)" shows "x \<in> set xs"
trancl_into_trancl: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> r\<^sup>+"   and "(b, c) \<in> r" shows "(a, c) \<in> r\<^sup>+"
zero_le_dist: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "0 \<le> dist x y"
ief_lens_axioms_def: fixes x :: "'b \<Longrightarrow> 'c" shows "ief_lens_axioms x \<equiv> \<forall>\<sigma> v. put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
mono_mprefix_ref: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and Q :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq> Q x" shows "Mprefix A P \<sqsubseteq> Mprefix A Q"
d1_def: shows "st1.d1 = prod_list st1.dims1"
wf_dual_lderiv_dual: fixes n :: "nat"   and r :: "'b rexp_dual"   and w :: "'a" assumes "wf_dual n r" shows "wf_dual n (lderiv_dual w r)"
contI2: fixes f :: "'a \<Rightarrow> 'b" assumes "monofun f"   and "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk> \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))" shows "cont f"
connected_diff_ball: fixes S :: "'a set"   and a :: "'a"   and r :: "real" assumes "connected S"   and "cball a r \<subseteq> S"   and "2 \<le> DIM('a)" shows "connected (S - ball a r)"
fusion2_aux2: fixes X :: "('a \<times> 'a) set"   and Q :: "('a \<times> 'a) set" shows "rdom (rel_nabla_bin X Q) \<subseteq> rdom (rel_nabla_bin X Q \<inter> rel_ad (relfdia (X\<^sup>*) Q) \<union> relfdia (X\<^sup>*) Q)"
fps_deriv_const: fixes c :: "'a" shows "fps_deriv (fps_const c) = 0"
intro: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd" assumes "functor A B F"   and "functor B C G" shows "composite_functor A B C F G"
acc_wfI: fixes r :: "('k \<times> 'k) set" assumes "\<forall>x. x \<in> Wellfounded.acc r" shows "wf r"
wrap_iso_arcs_iso_arcs: fixes G :: "('e, 'f) pre_digraph"   and hom :: "('e, 'f, 'g, 'h) digraph_isomorphism"   and x :: "'f"   and f :: "'f \<Rightarrow> 'f" assumes "pre_digraph.digraph_isomorphism G hom"   and "x \<in> arcs G" shows "pre_digraph.wrap_iso_arcs G hom f (iso_arcs hom x) = iso_arcs hom (f x)"
ord_of_nat_vempty: shows "[]\<^sub>\<circ> = 0\<^sub>\<nat>"
states_Reverse_nfa: fixes MS :: "'b dfa" shows "nfa.states (Reverse_nfa MS) = dfa.states MS"
red_sat_eq_sat: fixes N :: "'f set" shows "reduc_saturated N = saturated (N - Red_F N)"
irreducible_mult: fixes a :: "'a"   and b :: "'a" shows "irreducible (a * b) = (a dvd (1::'a) \<and> irreducible b \<or> b dvd (1::'a) \<and> irreducible a)"
nonzero_inverse_mult_distrib: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b"   and b :: "'b" assumes "field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>"   and "b \<noteq> \<zero>\<^bsub>R\<^esub>" shows "inv\<^bsub>R\<^esub> (a \<otimes>\<^bsub>R\<^esub> b) = inv\<^bsub>R\<^esub> b \<otimes>\<^bsub>R\<^esub> inv\<^bsub>R\<^esub> a"
var_not_in_linear_fm: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and F :: "atom fm" assumes "var \<notin> vars b"   and "var \<notin> vars c" shows "freeIn var (linear_substitution_fm var b c F)"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'c list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
hyps: fixes x2a :: "('a, 'c) Term.term" assumes "x2a \<in> set ts" shows "scf_term scf (x2a \<cdot> \<sigma>) = scf_term scf x2a \<cdot> (\<lambda>x. scf_term scf (\<sigma> x))"
preserves_reflects_arr: fixes f :: "'j" shows "C.arr (D f) = J.arr f"
cardSuc_finite: fixes r :: "('b \<times> 'b) set" assumes "Card_order r" shows "finite (Field (cardSuc r)) = finite (Field r)"
product_integral_prod: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> integrable ((count_space \<circ> B') i) (f i)" shows "LINT x|Pi\<^sub>M I (count_space \<circ> B'). (\<Prod>i\<in>I. f i (x i)) = (\<Prod>i\<in>I. integral\<^sup>L ((count_space \<circ> B') i) (f i))"
poly_strict_mono_def: fixes p :: "('b monom \<times> 'a) list"   and v :: "'b" shows "poly_strict_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<alpha> v \<succ> \<beta> v \<longrightarrow> eval_poly \<alpha> p \<succ> eval_poly \<beta> p"
Parallel: fixes xs :: "('c com \<times> 'c set \<times> ('c \<times> 'c) set \<times> ('c \<times> 'c) set \<times> 'c set) list"   and rely :: "('c \<times> 'c) set"   and guar :: "('c \<times> 'c) set"   and pre :: "'c set"   and post :: "'c set" assumes "\<forall>i<length xs. rely \<union> (\<Union>j\<in>\<lbrace>\<acute>(<) (length xs) \<and> \<acute>(\<noteq>) i\<rbrace>. Guar (xs ! j)) \<subseteq> Rely (xs ! i)"   and "(\<Union>j\<in>\<lbrace>\<acute>(<) (length xs)\<rbrace>. Guar (xs ! j)) \<subseteq> guar"   and "pre \<subseteq> (\<Inter>i\<in>\<lbrace>\<acute>(<) (length xs)\<rbrace>. Pre (xs ! i))"   and "(\<Inter>i\<in>\<lbrace>\<acute>(<) (length xs)\<rbrace>. Post (xs ! i)) \<subseteq> post"   and "\<forall>i<length xs. \<turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]" shows "\<turnstile> xs SAT [pre, rely, guar, post]"
not_less: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(\<not> ko.lt ko x y) = ko.le ko y x"
sccs_verts_da: shows "H.sccs_verts = G.sccs_verts"
abs_eq: fixes xa :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a"   and x :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "MPoly xa - MPoly x = MPoly (xa - x)"