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Subtracting "a" and then "Suc 0" (which is 1) from "b" is equivalent to subtracting "Suc a" (which is "a + 1") from "b".
The product of two integers 'k' and 'l' is equal to the integer value of the product of 'k' and 'l' when they are converted to integers.
The lemma named "substNil" is defined with fixed variables "xvec" which is a list of names, and "Tvec" which is a list of type 'a. It assumes that the substitution 'sigma' is well-formed and that the list 'xvec' contains distinct elements. It shows that applying the substitution 'sigma' to the null process results in t...
If X is a bounded sequence, then the range of X is bounded above. Here, X is a function from natural numbers to real numbers.
The lemma is_UNIV_code [code] is defined for a set 'rbt' of type "'a :: {cproper_interval, finite_UNIV} set_rbt" and a set 'A' of type "'b :: card_UNIV set". It shows that the set 'A' is a universal set if and only if the cardinality of 'b' (denoted as 'a') is equal to the cardinality of 'A' (denoted as 'b'). If 'a' is...
The lemma named "chain_append_inside" assumes the following: 1. There is a long chain from function 'f' to set 'Y' between 'a₁', 'a', and 'aₙ'. 2. 'b' is not an element of set 'Y'. 3. There is a chain from 'a₁' to 'b' to 'aₙ'. 4. There is a chain from 'a₁' to 'b' to 'f k', where 'k' is less than the cardinality of set...
If a decomposition closure from M to M' exists and M' logically deduces t in the context of a composition, then M logically deduces t.
If the conjunction of A, B, and true holds, then the conjunction of B and true also holds.
The lemma named "Example1'" notes the simplification procedure "Example1_proc_simp". It shows that under the context of Gamma' and Theta', Example1' ensures that for all 'i' less than 2, the value of 'ex1_a i' equals 0, and the postcondition is false.
The set of a conditional subprobability mass function (cond_spmf) of p given A is equal to the intersection of the set of subprobability mass function of p and A.
Given the assumptions that "a" is a valid edge and the distance from the target node of "a" to "n'" is "x", there exists a unique "a'" such that the source node of "a" is equal to the source node of "a'", the distance from the target node of "a'" to "n'" is "x", "a'" is a valid edge, and the target node of "a'" is equa...
, under the assumption that both "a" and "b" belong to the cone generated by the set "hU", the difference "a - b" also belongs to the same cone. Here, "a" and "b" are elements of a commutative ring.
The mask of the successor of 0 is 1.
The scalar multiplication of the nth power of a constant 'c' and the nth power of a polynomial 'p' is equal to the nth power of the scalar multiplication of 'c' and 'p'.
If the nth power of a word 'w' is equal to the concatenation of words 'u' and 'v', and if 'w' is in the list of words 'A', then 'v' must also be in the list of words 'A'.
It is admissible that a certain value "x" belongs to the representation of the base partial order relation "r".
The nth element of the formal Laurent series obtained by shifting the series of an integer i by k positions is equal to the integer i if n is equal to -k, otherwise it is 0.
If a well-formed formula 'a' with index 'idx' exists and 'l' is less than 'idx', then the result of the function 'find0' with parameters 'k' and 'l' applied to 'a' is equivalent to 'l' being in the set of free variables 'FV0' of 'a' at level 'k'.
If a program P can transition from the expression-state pair (e, s) to the pair (e', s') through zero or more steps, then P can also transition from the expression-state pair (e.F{D}, s) to the pair (e'.F{D}, s') through the same number of steps. Here, e.F{D} and e'.F{D} represent field access expressions.
For any element 'x' in the set of 'ctermsl' of 'p', the union of 'stermsl' of 'x' is equal to 'ctermsl' of 'p'.
The lemma named "det_interchange_rows" demonstrates that the determinant of a matrix A, when rows i and j are interchanged, is equal to the determinant of A multiplied by 1 if i equals j, or by -1 if i does not equal j.
The lemma named "poly_mapping_tail_induct" with case names "0" and "tail" assumes two conditions. The first condition is that some property "P" holds for 0. The second condition is that for any polynomial "p" that is not equal to 0, if the property "P" holds for the tail of "p", then the property "P" also holds for "p"...
, assuming "t" is a binary search tree, the time taken to insert element "x" into tree "t" plus the potential of the tree after the insertion of "x" minus the potential of the tree before the insertion is less than or equal to four times the logarithm base 2 of the size of the tree plus three.
The lemma named "tank_flow" assumes that the outflow rate "c\<^sub>o" is greater than 0 and less than the inflow rate "c\<^sub>i". It shows that the invariant "I" holds for all states "s" between the minimum and maximum heights "hmin" and "hmax" respectively, under the following loop: First, the control part of the lo...
If the condition is false, then the outcome will be B2.
The lemma named "trapG1_subset" shows that "trapG1" is a subset of the closed ball with center at zero and radius 20 in the real plane.
The lemma named "sumElim" is defined with the following parameters: a fixed Psi of type 'b, P and Q of type psi ('a, 'b, 'c), and a phi of type 'c. It assumes that Psi is related to the sum of P and Q under phi, which leads to Rs. It also assumes that Psi does not entail phi. If these conditions are met, the lemma ...
If the product of the negation of p and x is less than or equal to y, and the double negation of p times x is less than or equal to z, then x is less than or equal to the supremum (least upper bound) of y and z.
The degree of the function A at indices i and j is less than or equal to the maximum degree of A.
The lemma l15 assumes that "I" is a consistent cluster, "Q" is a quorum of "p", "Q'" is a quorum of "p'", both "p" and "p'" are in "I", and the intersection of "Q", "Q'", and "W" is not empty. It shows that the intersection of "W", the truncation of "Q" at "p", and the truncation of "Q'" at "p'" is not empty.
Lemma 4.3.1: Assuming that the set of probabilities 'p' is a subset of 'W', 'w' is integrable with respect to the measure of 'p', and the set of real numbers resulting from the function 'w' applied to 'W' is bounded, it can be shown that the expected value of 'w' with respect to the probability measure 'p' is less than...
If "x" and "y" are both elements of the carrier of Zp, and neither "x" nor "y" are zero, and the sum of "x" and "y" is not zero, then the minimum of the p-adic valuation of "x" and the p-adic valuation of "y" is less than or equal to the p-adic valuation of the sum of "x" and "y".
For any substitution 'S' and type 't', the bound type instance of the scheme created from 't' under 'S' is 't' itself.
The evaluation of a sum case function, where the function f1 is applied if the input is in the left summand and the function f2 is applied if the input is in the right summand, is equivalent to binding the evaluation of the input to a sum case function that evaluates the application of f1 or f2 to the input.
If N is a normal subgroup of G, and if a is an element of the group G, and h is an element of the subgroup N, then the product of a and h, multiplied by the inverse of a, is an element of the subgroup N.
A list "xs" is distinct and not empty if and only if the list of "xs" is distinct.
The lemma named "binary_expression_leeq1" assumes that for any natural number 'n', 'a n' is in the set {0,1}. It shows that the sum of the real number conversion of 'a (n + k)' multiplied by '1/2' raised to the power of the successor of '(n + k)' is less than or equal to 1.
If set F is a subset of set G, then the composition of the relation rrstep with sets F, R, and S is also a subset of the composition of the relation rrstep with sets G, R, and S.
If 1 is in the supremum of the set {a, b}, then 1 is also in the supremum of the set {a squared, b squared}.
For any given final state, if the external reduction of the Java memory model heap (represented by "jmm_heap_read" and "jmm_heap_write") with respect to a certain thread "t", heap "h", method "M", values "vs", thread action "ta", return value "va", and new heap "h'" is valid, and if the actions of the final thread are ...
The extension of any set "h" to the universal set equals the universal set itself.
The lemma named "filtrationI" assumes that for all 't', 'F t' is a subalgebra of 'M', and for all 's' and 't', if 's' is less than or equal to 't', then 'F s' is a subalgebra of 'F t'. If these conditions are met, it shows that 'F' is a filtration of 'M'.
For a given point 'x' in a metric space, if it is assumed that for all 'x' eventually in 'X', property 'P' holds for 'x', and property 'P' also holds for 'x', then it can be shown that for all pairs '(x', y')' eventually in the product of 'X' and 'Y', property 'P' holds for 'x''.
The lemma named "update_account_mono" assumes the following conditions: 1. i is greater than or equal to 0. 2. For all n, the value of function ρ at n is less than 1. 3. For all n and m, if n is less than or equal to m, then the value of function ρ at n is less than or equal to the value of function ρ at m. 4. α is les...
The absolute value of the i-th component of a vector x in a real n-dimensional Cartesian space is less than or equal to the infinity norm of the vector x.
The cardinality of set A is less than or equal to the cardinality of set B if and only if set A is less than or equal to set B in terms of a well-ordering.
: if there is no 'DenyAll' rule in policy 'p', and the set {x, y} is in the list 'l', and all elements of policy 'p' are in the list 'l', and policy 'p' only contains single combinators, and the list 'l' is sorted with 'DenyAllFromTo x y' followed by policy 'p', and the set {x, y} is not the first list in policy 'p', a...
The lemma named "coincidence_hp" is defined with the fixed variables \<nu>, \<nu>', \<mu>, V, I, and J. It assumes the following: 1. The hybrid program (hp) \<alpha> is safe, denoted by "hpsafe (\<alpha>::('a::finite, 'b::finite, 'c::finite) hp)". 2. The interpretations I and J agree on the signature of the hybrid pro...
If a pair (x, s') is in the set defined by the execution of a generative probabilistic value (gpv) with a certain callee and initial state s, and if the invariant I holds for the initial state s, and the gpv is valid according to the interface \<I>, then the invariant I also holds for the state s'.
: If a rule set 'rs' and a packet 'p' under a match condition 'γ' transitions from a state 's' to a state 't', then the following conditions hold: 1. For any state 't', the predicate 'P' holds for an empty rule set transitioning from 't' to 't'. 2. If a match condition 'γ' matches a rule 'm' with an action 'a' on a p...
If the well-typedness of a list "is" at a given program counter "pc" and state "s" is not top, then the well-typedness of the list obtained by taking the first "pc'" elements of "is" at the same program counter "pc" and state "s" is also not top.
If x is less than y, then the successor of x in ordinal notation is less than or equal to y.
The lemma named "the_trivialI" assumes that the set of "vs" is a subset of the set containing "v" and "v'", and that "v'" is not equal to "v". It shows that the function "the_trivial" with parameters "v" and "vs" equals "None" if the set of "vs" is a subset of the set containing only "v", otherwise it equals "Some v'".
For any given destination IP address (dip), IP address (ip), routing table (rt), destination sequence number (dsn), destination sequence key (dsk), flag, hop count (hops), next hop IP address (nhip), and prefix (pre), if the IP address is not equal to the destination IP address, then the sequence number of the updated ...
Multiplying the ordinal number one on the left by any ordinal number "r" results in the same ordinal number "r". The question is whether the left side equals the right side.
The lemma named "src_some_trg" assumes that "\<mu>" is an arrow. It shows that the source of some target "\<mu>" is equal to some target "\<mu>".
For any types Ts, values vs, locations l and l', and expressions E, if the size of parameters ps is equal to the size of Ts and the size of ps is also equal to the size of vs, and if the program P with expression E evaluates the blocks of parameters, types, values, and an expression e in a certain heap and location to ...
Converting the adjugate of a matrix A to an iarray is equivalent to the adjugate of the iarray form of matrix A.
The lemma "reduced_GB_pmdl_dgrad_p_set" assumes that "d" is a Dickson grading, the set of keys "F" has finite components of terms, and "F" is a subset of the set of polynomial degrees "dgrad_p_set" with degree "d" and multiplicity "m". Under these assumptions, the lemma shows that the ideal generated by the reduced Gro...
The function "sort_leq" applied to the set "cs" and the elements "c1" and "c2" is equal to the function "exesort_leq" applied to the same elements "cs", "c1", and "c2".
The lemma named "continuous_on_blinfun_matrix" is about a function 'f' that maps from a topological space 'a to a real normed vector 'b and then to a real inner product space 'c. If 'f' is continuous on a set 'S', then the function that maps 'x' in 'S' to the inner product of 'f(x)' with 'j' and 'i' is also continuous ...
If 'a' is an element of the carrier 'G', then the group generated by 'a' is a cyclic group.
If the following conditions are met: a relation R is such that when starting from any state in set I, it always leads to a state in set J; the initial state s is in set I; and there is a transition from state s to state s' according to relation R, then the resulting state s' must be in set J.
: Assuming that: - "asi rank r cost" is true, - for all xs and ys in Y, xs equals ys or the intersection of the set of xs and the set of ys is empty, - for all xs in Y, "forward xs" is true, - the empty list is not in Y, - Y is finite, - r is an element of the set U, - U is in Y, - V is in Y, - the list formed by conc...
If "x" is an element of itself, then it can be inserted into set "H" which implies "A".
The lemma named "diamondpE" is based on two assumptions. The first assumption is that "r" forms a diamond property. The second assumption is that "r" relates "x" to "y1" and "x" to "y2". Under these assumptions, it is possible to obtain a value "z" such that "r" relates "y1" to "z" and "y2" to "z".
In the context of a tiny natural transformation, given the assumptions that alpha prime equals alpha, A prime equals A, B prime equals B, F prime equals F, and G prime equals G, it can be shown that the natural transformation N from F to G is a tiny functor from A to B in the category of tiny objects parameterized by a...
For any extended natural numbers m, n, and p, if m is greater than or equal to n, then the result of subtracting n from m and then adding p is equal to the result of adding p to m and then subtracting n.
The square root function of 'f' is differentiable at a point 'x' within a set 'S', if the function 'f' itself is differentiable at the same point within the same set, and the value of 'f' at point 'x' is not equal to zero.
The domain of the disjoint sum of two relations 'r' and 's' is equal to the disjoint union of the domains of 'r' and 's'.
If you update a function 'f' with a state 'σ' and then get the result, it is equivalent to applying the function 'f' to the result of getting the state 'σ'.
The lemma named "Gleason9_34_exists" assumes that "A" is a cut, every element "x" in "A" added to "u" is also in "A", and "z" is greater than or equal to zero. It shows that there exists an element "b" in "A" such that "b" added to "z" times "u" is also in "A".
The lemma named "wave_plus_current_minus" assumes the following: - "f" is a current in the network "Γ" - "f" is a wave in the network "Γ" - "g" is a current in the network obtained by subtracting "f" from "Γ" - "g" is a wave in the network obtained by subtracting "f" from "Γ" The lemma shows that the sum of the curren...
If there is an obligation of X towards Y, then there is an obligation of X towards the intersection of Y and X.
If matrix A is a sorted sparse matrix, then the absolute value of matrix A is also a sorted sparse matrix.
The state of a configuration (Cfg) with parameters s, d', and c' is equal to s.
If the pair (v', p') is a valid input port, then v is in the scope of (v', p') if and only if the scope' function of v', p', and v holds true.
Evaluating "r_cons" with the arguments "e" and "es" is equivalent to the function "e_cons" applied to "e" and "es".
The hyperbolic tangent of the natural logarithm of a real number x is equal to the difference of the square of x and 1 divided by the sum of the square of x and 1, given the condition that x is greater than 0.
The connection of a mapped generalized probabilistic value (gpv) with a resource is equal to the mapping of a subprobability measure function (spmf) applied to the connection of an adversary with a mapped resource.
The lemma named "parCasesB" is defined with the following parameters: P, Q, a, x, and PQ', where P, Q, and PQ' are processes in the Pi Calculus, and a and x are names. The lemma assumes the following: 1. There is a transition from the parallel composition of P and Q to PQ' with a new name x under the action a. This i...
If "a" cancels to "b" and "b" cancels to "c", then "a" cancels to "c". This is a transitive property of the "cancels_to" relation.
The lemma named "res_thm'_no_heap" assumes that the function "res_thm'" with parameters "l", "xs", "ys" has an effect from heap "h" to heap "h'" resulting in "r". It shows that the initial heap "h" is equal to the final heap "h'". This implies that the function "res_thm'" does not modify the heap.
If a sequence X converges to a positive number 'a', then the sequence formed by raising each term of X to the rational power 'q' also converges to 'a' raised to the power 'q'.
An empty list is a strict subsequence of any list 'xs'.
If the extension of v is less than or equal to the extension of v', and the ownership of v is equal to the ownership of v', and the absolute value of the length of v with respect to ts and c is greater than 0, then the length of v with respect to ts and c is less than or equal to the length of v' with respect to ts and...
The lemma named "valid_trans" assumes the following conditions: 1. Under the context of types Theta, base types B, and context Gamma, the term c0 with variable z replaced by value v is valid. 2. Under the same context but with the addition of (z, b, c0) to Gamma, the term c1 is valid. 3. The atom z is fresh in Gamma, ...
If a command 'c' is secure in the context 'X' with respect to the security lattice 'Q p G', the equivalence relation 'EQ G q', and the predicate 'Φ', and 'p' is not less or equal to 'q', then the while-loop 'While b c' is also secure in the same context, with respect to the same security lattice and equivalence relatio...
The lemma named "is_null_all" assumes that point A is not equal to point B and that the angle 'a' is null or acute. It shows that the angle 'a' is formed by points A, B, and A.
If the function "min_range_antimono" with parameters p, f, f', i, g, h returns a value of Some ((a, b), ba), then both a and ba are elements of the set of floating point numbers.
The lemma named "tcontra_halt" assumes that the Turing Machine Code (TMC) has a numerical result for "tcontra" given the code of "tcontra". If this assumption holds, it shows or results in a contradiction or falsehood.
If a write operation in phase 1 or phase 2 transitions the system from state 's' to state 's'' for a given process 'p' and disk 'd', then the set of blocks in state 's'' for a given process 'r' is a subset of the set of blocks in state 's' for the same process 'r'.
For any matrix M of type 'a (where 'a is a ring), the multiplication of M with the difference of vectors v and w is equal to the difference of the multiplication of M with v and the multiplication of M with w.
The absolute difference of x and y equals zero if and only if x equals y.
The lemma named "at_top_within_at_top_unbounded_rightI" is defined for a fixed 'a' of any linearly ordered type. It assumes that the set of all elements greater than or equal to 'a' is a subset of 's'. It then shows that the limit of 's' as it approaches positive infinity is equal to the limit as it approaches positive...
For any two non-zero natural numbers 'n' and 'm', 'n' and 'm' are coprime if and only if the intersection of their prime factors is empty.
"always, for a sequence 'xs', the first element of 'xs' is 'h' in a trace 't'" is equivalent to "the trace 't' is a constant sequence of 'h'".
The intersection of all sets in f is non-empty.
If a tree 't' is balanced and the inorder traversal of 't' is sorted, then the inorder traversal of the tree resulting from deleting 'x' from 't' is equal to the result of deleting 'x' from the inorder traversal of 't'.
If a policy "p" is a well-formed policy according to the third version of the policy formation rules, then the tail of the policy "p" (all elements of the policy except the first one) is also a well-formed policy according to the third version of the policy formation rules.